Mathematical Problems in Engineeringdownloads.hindawi.com/journals/specialissues/760807.pdf · Mathematical Problems in Engineering Theory, Methods, and Applications Guest Editors:

MathematicalProblems inEngineeringTheory, Methods, and Applications

Guest Editors: Cristian Toma, Carlo Cattani, Ezzat G. Bakhoum, and Ming Li

Special IssuePropagation Phenomena and Transitions in Complex Systems 2012

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Propagation Phenomena and Transitionsin Complex Systems 2012

Mathematical Problems in Engineering

Propagation Phenomena and Transitionsin Complex Systems 2012

Guest Editors: Cristian Toma, Carlo Cattani,

Ezzat G. Bakhoum, and Ming Li

Copyright q 2012 Hindawi Publishing Corporation. All rights reserved.

This is a special issue published in “Mathematical Problems in Engineering.” All articles are open access articles distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

Editorial BoardEihab Abdel-Rahman, Canada

Rashid K. Abu Al-Rub, USA

Salvatore Alfonzetti, Italy

Igor Andrianov, Germany

Sebastian Anita, Romania

W. Assawinchaichote, Thailand

Erwei Bai, USA

Ezzat G. Bakhoum, USA

Jose Manoel Balthazar, Brazil

Rasajit K. Bera, India

Jonathan N. Blakely, USA

Stefano Boccaletti, Spain

Daniela Boso, Italy

M. Boutayeb, France

Michael J. Brennan, UK

John Burns, USA

Salvatore Caddemi, Italy

Piermarco Cannarsa, Italy

Jose E. Capilla, Spain

Carlo Cattani, Italy

Marcelo M. Cavalcanti, Brazil

Diego J. Celentano, Chile

Mohammed Chadli, France

Arindam Chakraborty, USA

Yong-Kui Chang, China

Michael J. Chappell, UK

Xinkai Chen, Japan

Kui Fu Chen, China

Kue-Hong Chen, Taiwan

Jyh Horng Chou, Taiwan

Slim Choura, Tunisia

Swagatam Das, India

Filippo de Monte, Italy

M. do R. de Pinho, Portugal

Antonio Desimone, Italy

Yannis Dimakopoulos, Greece

Baocang Ding, China

Joao B. R. Do Val, Brazil

Daoyi Dong, Australia

Balram Dubey, India

Horst Ecker, Austria

M. Onder Efe, Turkey

Elmetwally Elabbasy, Egypt

Alex Elias-Zuniga, Mexico

Anders Eriksson, Sweden

Vedat Suat Erturk, Turkey

Qi Fan, USA

Moez Feki, Tunisia

Ricardo Femat, Mexico

Rolf Findeisen, Germany

R. A. Fontes Valente, Portugal

C. R. Fuerte-Esquivel, Mexico

Zoran Gajic, USA

Ugo Galvanetto, Italy

Xin-Lin Gao, USA

Furong Gao, Hong Kong

Behrouz Gatmiri, Iran

Oleg V. Gendelman, Israel

Didier Georges, France

Paulo Batista Goncalves, Brazil

Oded Gottlieb, Israel

Quang Phuc Ha, Australia

Muhammad R. Hajj, USA

Thomas Hanne, Switzerland

Katica R. Hedrih, Serbia

C. Cruz Hernandez, Mexico

M.I. Herreros, Spain

Wei-Chiang Hong, Taiwan

J. Horacek, Czech Republic

Chuangxia Huang, China

Gordon Huang, Canada

Yi Feng Hung, Taiwan

Hai-Feng Huo, China

Asier Ibeas, Spain

Anuar Ishak, Malaysia

Reza Jazar, Australia

Zhijian Ji, China

J. Jiang, China

J. J. Judice, Portugal

Tadeusz Kaczorek, Poland

Tamas Kalmar-Nagy, USA

Tomasz Kapitaniak, Poland

Hamid Reza Karimi, Norway

Metin O. Kaya, Turkey

Nikolaos Kazantzis, USA

Farzad Khani, Iran

Kristian Krabbenhoft, Australia

Jurgen Kurths, Germany

Claude Lamarque, France

F. Lamnabhi-Lagarrigue, France

Marek Lefik, Poland

Stefano Lenci, Italy

Jian Li, China

Shihua Li, China

Shanling Li, Canada

Tao Li, China

Ming Li, China

Teh-Lu Liao, Taiwan

P. Liatsis, UK

Shueei M. Lin, Taiwan

Jui-Sheng Lin, Taiwan

Yuji Liu, China

Wanquan Liu, Australia

Bin Liu, Australia

Fernando Lobo Pereira, Portugal

Paolo Lonetti, Italy

Vassilios C. Loukopoulos, Greece

Junguo Lu, China

Chien-Yu Lu, Taiwan

Alexei Mailybaev, Brazil

Manoranjan Maiti, India

Oluwole D. Makinde, South Africa

Rafael Martinez-Guerra, Mexico

Bohdan Maslowski, Czech Republic

Driss Mehdi, France

Roderick Melnik, Canada

Xinzhu Meng, China

Yuri Vladimirovich Mikhlin, Ukraine

Gradimir V. Milovanovic, Serbia

Ebrahim Momoniat, South Africa

Trung Nguyen Thoi, Vietnam

Hung Nguyen-Xuan, Vietnam

Ben T. Nohara, Japan

Anthony Nouy, France

Sotiris K. Ntouyas, Greece

Gerard Olivar, Colombia

Claudio Padra, Argentina

Francesco Pellicano, Italy

Vu Ngoc Phat, Vietnam

A. Pogromsky, The Netherlands

Seppo Pohjolainen, Finland

Stanislav Potapenko, Canada

Sergio Preidikman, USA

Carsten Proppe, Germany

Hector Puebla, Mexico

Justo Puerto, Spain

Dane Quinn, USA

Ruben R. Garcıa, Spain

K. R. Rajagopal, USA

Gianluca Ranzi, Australia

Sivaguru Ravindran, USA

G. Rega, Italy

Pedro Ribeiro, Portugal

J. Rodellar, Spain

R. Rodriguez-Lopez, Spain

A. J. Rodriguez-Luis, Spain

Ignacio Romero, Spain

Hamid Reza Ronagh, Australia

Carla Roque, Portugal

Mohammad Salehi, Iran

Miguel A. F. Sanjuan, Spain

Ilmar Ferreira Santos, Denmark

Nickolas S. Sapidis, Greece

Bozidar Sarler, Slovenia

Andrey V. Savkin, Australia

Massimo Scalia, Italy

Mohamed A. Seddeek, Egypt

Alexander P. Seyranian, Russia

Leonid Shaikhet, Ukraine

Cheng Shao, China

Daichao Sheng, Australia

Tony Sheu, Taiwan

Zhan Shu, UK

Dan Simon, USA

Luciano Simoni, Italy

Grigori M. Sisoev, UK

Christos H. Skiadas, Greece

Davide Spinello, Canada

Sri Sridharan, USA

Rolf Stenberg, Finland

Changyin Sun, China

Jitao Sun, China

Xi-Ming Sun, China

Andrzej Swierniak, Poland

Allen Tannenbaum, USA

Cristian Toma, Romania

Irina N. Trendafilova, UK

Alberto Trevisani, Italy

Jung-Fa Tsai, Taiwan

Kuppalapalle Vajravelu, USA

Victoria Vampa, Argentina

Josep Vehi, Spain

Stefano Vidoli, Italy

Xiaojun Wang, China

Dan Wang, China

Youqing Wang, China

Yongqi Wang, Germany

Moran Wang, USA

Yijing Wang, China

Cheng C. Wang, Taiwan

Gerhard-Wilhelm Weber, Turkey

Jeroen A.S. Witteveen, USA

Kwok-Wo Wong, Hong Kong

Zheng-Guang Wu, China

Ligang Wu, China

Wang Xing-yuan, China

X. Frank XU, USA

Xuping Xu, USA

Jun-Juh Yan, Taiwan

Xing-Gang Yan, UK

Suh-Yuh Yang, Taiwan

Mahmoud T. Yassen, Egypt

Mohammad I. Younis, USA

Huang Yuan, Germany

S. P. Yung, Hong Kong

Ion Zaballa, Spain

Arturo Zavala-Rio, Mexico

Ashraf M. Zenkour, Saudi Arabia

Yingwei Zhang, USA

Xu Zhang, China

Liancun Zheng, China

Jian Guo Zhou, UK

Zexuan Zhu, China

Mustapha Zidi, France

Contents

Propagation Phenomena and Transitions in Complex Systems 2012, Cristian Toma,Carlo Cattani, Ezzat G. Bakhoum, and Ming LiVolume 2012, Article ID 251791, 3 pages

Fast Detection of Weak Singularities in a Chaotic Signal Using Lorenz System andthe Bisection Algorithm, Tiezheng Song and Carlo CattaniVolume 2012, Article ID 102848, 10 pages

Fractional Calculus and Shannon Wavelet, Carlo CattaniVolume 2012, Article ID 502812, 26 pages

Parallel Motion Simulation of Large-Scale Real-Time Crowd in a Hierarchical EnvironmentalModel, Xin Wang, Jianhua Zhang, and Massimo ScaliaVolume 2012, Article ID 918497, 15 pages

Optimization of Resource Control for Transitions in Complex Systems, Florin PopVolume 2012, Article ID 625861, 12 pages

Mathematical Models of Dissipative Systems in Quantum Engineering, Andreea Sterian andPaul SterianVolume 2012, Article ID 347674, 12 pages

Power-Law Properties of Human View and Reply Behavior in Online Society, Ye Wu, Qihui Ye,Lixiang Li, and Jinghua XiaoVolume 2012, Article ID 969087, 7 pages

Sinogram Restoration for Low-Dosed X-Ray Computed Tomography Using Fractional-OrderPerona-Malik Diffusion, Shaoxiang Hu, Zhiwu Liao, and Wufan ChenVolume 2012, Article ID 391050, 13 pages

Difference-Equation-Based Digital Frequency Synthesizer, Lu-Ting Ko, Jwu-E. Chen,Yaw-Shih Shieh, Hsi-Chin Hsin, and Tze-Yun SungVolume 2012, Article ID 784270, 12 pages

Kernel Optimization for Blind Motion Deblurring with Image Edge Prior, Jing Wang, Ke Lu,Qian Wang, and Jie JiaVolume 2012, Article ID 639824, 10 pages

Dual-EKF-Based Real-Time Celestial Navigation for Lunar Rover, Li Xie, Peng Yang,Thomas Yang, and Ming LiVolume 2012, Article ID 578719, 16 pages

Hidden-Markov-Models-Based Dynamic Hand Gesture Recognition, Xiaoyan Wang, Ming Xia,Huiwen Cai, Yong Gao, and Carlo CattaniVolume 2012, Article ID 986134, 11 pages

Stable One-Dimensional Periodic Wave in Kerr-Type and Quadratic Nonlinear Media,Roxana Savastru, Simona Dontu, Dan Savastru, Marina Tautan, and Vasile BabinVolume 2012, Article ID 532610, 6 pages

Cutting Affine Moment Invariants, Jianwei Yang, Ming Li, Zirun Chen, and Yunjie ChenVolume 2012, Article ID 928161, 12 pages

Homotopy Perturbation Method and Variational Iteration Method for Harmonic WavesPropagation in Nonlinear Magneto-Thermoelasticity with Rotation, Khaled A. Gepreel,S. M. Abo-Dahab, and T. A. NofalVolume 2012, Article ID 827901, 30 pages

Simplicial Approach to Fractal Structures, Carlo Cattani, Ettore Laserra, and Ivana BochicchioVolume 2012, Article ID 958101, 21 pages

Gaussian Curvature in Propagation Problems in Physics and Engineering, Ezzat G. BakhoumVolume 2012, Article ID 371890, 10 pages

Solving Linear Coupled Fractional Differential Equations by Direct Operational Method andSome Applications, S. C. Lim, Chai Hok Eab, K. H. Mak, Ming Li, and S. Y. ChenVolume 2012, Article ID 653939, 28 pages

Study of the Fractal and Multifractal Scaling Intervening in the Description of FractureExperimental Data Reported by the Classical Work: Nature 308, 721722(1984),Liliana Violeta Constantin and Dan Alexandru IordacheVolume 2012, Article ID 706326, 10 pages

Multidimensional Wave Field Signal Theory: Transfer Function Relationships,Natalie BaddourVolume 2012, Article ID 478295, 27 pages

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 251791, 3 pagesdoi:10.1155/2012/251791

EditorialPropagation Phenomena and Transitions inComplex Systems 2012

Cristian Toma,1 Carlo Cattani,2 Ezzat G. Bakhoum,3 and Ming Li4

1 Faculty of Applied Sciences, University Politehnica of Bucharest, 70709 Bucharest, Romania2 Department of Mathematics, University of Salerno, 84084 Fisciano, Italy3 Department of Electrical and Computer Engineering, University of West Florida,Pensacola, FL 32514, USA

4 School of Information Science and Technology, East China Normal University,No. 500 Dong-Chuan Road, Shanghai 2002411, China

Correspondence should be addressed to Cristian Toma, [email protected]

Received 14 June 2012; Accepted 14 June 2012

Copyright q 2012 Cristian Toma et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

An increasing challenge in advanced engineering applications based on efficient math-

ematical models for propagation and transition phenomena can be noticed nowadays.

Fractal theory and special mathematical functions are used in modeling very small-scale

material properties (energy levels and induced transitions) for the design of nanostructures.

Differential geometry is adapted for solving nonlinear partial differential equations with very

great number of variables for modeling dynamics and transitions in complex optoelectronics

systems. Propagation aspects implying commutative and/or additive consequences of

quantum physics are used extensively in the design of long-range transmission systems. Time

series with extremely high-transmission rates are used for multiplexed transmission systems

for large communities, such as traffic in computer networks or transportations, financial time

series, and time series of fractional order in general. All these advanced engineering subjects

require efficient mathematical models in the development of classical tools for complex

systems. The objective in such applications is to take into consideration efficiency aspects

of mathematical and physical models required by basic phenomena of propagation and

transitions in complex systems, when specific limitations are involved (very long distance

propagation phenomena, fractal aspects and transitions in nanostructures, and complex

systems with great number of variables and infinite spatiotemporal extension of material

media). Using advanced mathematical tools for modeling propagation and transition

phenomena, this special issue presents high qualitative and innovative developments for

eficient mathematical approaches of propagation phenomena and transitions in complex

2 Mathematical Problems in Engineering

systems. Significant results were obtained in the research fields of low-scale physical

structures, propagation of waves in advanced materials, dynamics of complex systems,

and efficient signal and image analysis based on fundamental mathematical and physical

laws.

This special issue involves 19 original papers, selected by the editors so as to present

the most significant results in the previously mentioned topics. These papers are organised

as follows:

(a) Three papers on specific fractal approach for oscillation, propagation and diffusion

properties of low-scale structures: “Fractional calculus and Shannon wavelets” by C.

Cattani, “Simplicial approach to fractal structure” by I. Bochicchio, C. Cattani, and

E. Laserra, and “Sinogram restoration for low-dosed x-ray computed tomography usingfractional-order perona-malik diffusion” by S. Hu, Z. Liao, and W. Chen.

(b) Four papers on specific methods for analysis of complex movements: “Cutting afinemoment invariants” by J. Yang, M. Li, Z. Chen, and Y. Chen, “Dual-ekf based real-timecelestial navigation for lunar rover” by L. Xie, P. Yang, T. T. Yang, and M. Li, “Parallelmotion simulation of large-scale real-time crowd in a hierarchical environmental model” by

X. Wang, J. Zhang, and M. Scalia, and “Hidden Markov models based dynamic handgesture recognition” by X. Wang, M. Xia, H. Cai, Y. Gao, and C. Cattani.

(c) Five papers on accurate and efficient mathematical models for propagation

phenomena: “Gaussian curvature in propagation problems in physics and engineering”

by E. G. Bakhoum, “Multidimensional wave field signal theory: transfer functionrelationships” by N. Baddour, “Homotopy perturbation method and variational iterationmethod for harmonic waves propagation in nonlinear magneto-thermoelasticity withrotation” by S. M. Abo-Dahab, K. A. Gepreel, and T. A. Nofal, “Mathematical modelsof dissipative systems in quantum engineering” by A. Sterian and P. Sterian, and

“Stable one-dimensional periodic wave in kerr-type and quadratic nonlinear media” by R.

Savastru, S. Dontu, D. Savastru, M. Tautan, and V. Babin.

(d) Three papers on mathematical tools for analyzing the dynamics of complex

systems: “Solving linear coupled fractional differential equations by direct operationalmethod and some applications” by S. C. Lim, M. Li, C. H. Eab, K. H. Mak, and S. y.

Chen, “Difference-equation-based digital frequency synthesizer” by L. T. Ko, J. E. Chen,

Y. S. Shieh, H. C. Hsin, and T. Y. Sung, and ”Fast detection of weak singularities ina chaotic signal using Lorenz system and the bisection algorithm” by T. Song and C.

Cattani.

(e) Two papers on efficient image analysis based on fundamental mathematical and

physical laws: “Kernel optimization for blind motion deblurring with image edge prior”by J. Wang, K. Lu, Q. Wang, and J. Jia, and “Power-law properties of human view andreply behavior in online society” by Y. Wu, Q. Ye, J. Xiao, and L. Li.

(f) Two papers on scaling and optimization aspects: “Kernel optimization for blind motiondeblurring with image edge prior” by F. Pop, and “Study of the fractal and multifractalscaling intervening in the description of fracture experimental data reported by the classicalwork” by C. L. Violeta and D. Iordache.

Mathematical Problems in Engineering 3

Acknowledgment

Guest Editor Ming Li would like to acknowledge the supports in part by the 973 plan under

the project number 2011CB302802, and the National Natural Science Foundation of China

under the project Grant numbers 61070214 and 60873264.

Cristian TomaCarlo Cattani

Ezzat G. BakhoumMing Li


Research ArticleFast Detection of Weak Singularities ina Chaotic Signal Using Lorenz System andthe Bisection Algorithm

Tiezheng Song1 and Carlo Cattani2

1 School of Electrical Engineering and Automation, Hefei University of Technology, Anhui Province,Hefei City 230009, China

2 Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy

Correspondence should be addressed to Tiezheng Song, [email protected]

Received 1 March 2012; Accepted 1 May 2012

Academic Editor: Cristian Toma

Copyright q 2012 T. Song and C. Cattani. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

Signals with weak singularities are important for condition monitoring, fault forecasting, andmedicine diagnosis. However, the weak singularity in a signal is usually hidden by strong noise. Anovel fast method is proposed for detecting a weak singularity in a noised signal by determininga critical threshold towards chaos for the Lorenz system. First, a rough critical threshold value iscalculated by local Lyapunov exponents with a step size 0.1. Second, the exact threshold valueis calculated by the bisection algorithm. The advantage of the method will not only reduce thecomputation costs, but also show the weak singular signal which can be accurately identifiedfrom strong noise. When the variance of an external signal method embeds into a Lorenz system,according to the parametric equivalent relation between the Lorenz system and the originalsystem, the critical threshold value of the parameter in a Lorenz system is determined.

1. Introduction

In engineering, most weak singular information often is submerged into strong signals, such

as the peaks, the discontinuities, and so forth. Moreover, when the some weak singular points

are magnified slowly with time, at the moment when the fault occurs, the output signals

usually contain jump points that are often singular points. Therefore, weak singular detection

has played an important role in condition monitoring, fault forecast and medicine diagnosis

[1, 2]. For example, some weak singular vibration signals in machine processes are important

for fault forecasting.

The weak-signal detection is a central problem in the general field of signal processing

and the use of chaos theory in weak-signal detection, and it is also a topic of interest in chaos


0

10

20

30

40

50

y(t)

0

10

20

30

40

50

y(t)

x(t) t

−20 −10 0 10 20 0 5 10 15 20

Figure 1: r = 24, phase plane of periodic state (sampling time 20 s).

control. At present, however, this research is mainly theory and simulation with MATLAB

in terms of the Duffing-Holmes oscillator [3–6]. Whether other chaos system had better to

characteristic than the Duffing-Holmes oscillator for detecting weak a singular signal. In this

paper, a weak singular signal embedded in the strong signal is detected by Lorenz system.

In 1963, an atmospheric scientist named E.N. Lorenz of M.I.T. proposed a simple model for

thermally induced fluid convection in the atmosphere [7]. In Lorenz’s mathematical model

of convection, three state variables are used (x, y, z). The variable x is proportional to the

amplitude of the fluid velocity circulation in the fluid ring, while y and z measure the

distribution of temperature around the ring. The so-called Lorenz systems may be derived

formally from the Navier-Stokes partial differential equations of fluid mechanics. The Lorenz

model reads in standard notation as follows:

x = a(x − y

),

y = −xz + rx − y,

z = xy − bz.

(1.1)

For a = 10 and b = 8/3 (a favorite set of parameters for experts in the field, integrated with

fourth-order Runge-Kutta method with a fixed step size, t = 0.01 s, there is an attractor for

r = 24 for which the origin is of periodic state (Figure 1). The r = 24.5 gives the phase plane

of chaotic state in which the other two attractors on the x-y plane become unstable spirals

which is called a strange attractor (sometimes called the “butterfly attractor”) and y(t) take

on a complex chaotic trajectory as shown in Figure 2.

Many researchers [8–10] analyze the Lorenz characteristic using r as a control variable.

Upwards, in terms of r = 24 and r = 25, a Lorenz system has proved that there is a huge

difference in the phase space trajectories between the chaotic state and the periodic state,

and this difference can be used for the detection of weak singular signals in strong noise.

Meanwhile, if Lyapunov exponents are adopted as the threshold value evaluated roughly


0

10

20

30

40

50

y(t)

0

10

20

30

40

50

y(t)

x(t) t

−20 −10 0 10 20 0 10 20 30 40 50

Figure 2: r = 24.5, phase plane of chaotic state (sampling time 50 s).

for a chaotic critical state, the bisection algorithm can fast approach any accurate threshold

value. Thus, when an external signal is embedded into parameter r, one chaotic threshold is

determined conveniently which can detect a weak singular signal in strong noise.

2. The Chaotic Behavior of the Detecting Lorenz byLyapunov Exponent

The Lyapunov exponent (LE) is frequently computed measure for characterizing of chaotic

dynamics. It describes a method for diagnosing whether or not a system is chaotic. For a

discrete mapping x(t+ 1) = f[x(t)], we calculate the local expansion of a flow by considering

the difference of two trajectories as follows:

x(t + 1) − y(t + 1) = f(x(t)) − f(y(t)

) ∂f

∂x[x(t)] ·

[x(t) − y(t)

]. (2.1)

If this grows like

∣∣x(t + 1) − y(t + 1)∣∣ eλ

∣∣x(t) − y(t)∣∣, (2.2)

then the exponent λ is called the Lyapunov exponent. If it is positive, bounded flows will

generally be chaotic. We can solve for this exponent, asymptotically,

λ ln

(∣∣xn+1 − yn+1

∣∣∣∣xn − yn

∣∣). (2.3)


Since the Lorenz system is in three dimensions, it has three Lyapunov exponents. How effi-

cient and reliable can algorithms to compute Lyapunov exponents be? For three-dimensional

mapping as Lorenz system

xn+1 = f1

(xn,yn, zn

)= a

(xn − yn

),

yn+1 = f2

(xn,yn, zn

)= −xnzn + rxn − yn,

zn+1 = f3

(xn,yn, zn

)= xnyn + bzn.

(2.4)

We get a Jacobian matrix for Lorenz flow

J(xn,yn, zn

)=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂f1

∂xn,∂f1

∂yn,∂f1

∂zn

∂f2

∂xn,∂f2

∂yn,∂f2

∂zn

∂f3

∂xn,∂f3

∂yn,∂f3

∂zn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦=

⎡⎣ a −a 0

r − zn −1 −xn

yn xn −b

⎤⎦. (2.5)

Algorithms to compute eigenvalues of matrices are remarkably efficient: supposing the point

successive mapping from the initial point P0(x0,y0, z0) to P1(x1,y1, z1), P2(x2,y2, z2),. . .,Pn(xn,yn, zn), the Jackobian matrix of the previous (n − 1)th point is J0 = J(x0,y0, z0),J1 = J(x1,y1, z1), J2 = J(x2,y2, z2), . . ., and Jn−1 = J(xn−1,yn−1, zn−1). Defining J(n) =Jn−1Jn−2 ···J1J0, the module of eigenvalue for J(n) is J1

(n), J2(n), and J3

(n), and J1(n) > J2

(n) >

J(n)3 , the Lyapunov exponents are defined as follows:

λ1 = limn

n

√J(n)1 , λ2 = lim

n

n

√J(n)2 , λ3 = lim

n

n

√J(n)3 . (2.6)

When (1.1) is in the chaotic state, at least one of the three Lyapunov exponents in (2.6) is

positive. The value is called maximum Lyapunov exponent. The chaotic behavior of the

detection (1.1) is established on the basis of maximum Lyapunov exponents. If the system

is not a point attractor, then the largest exponent cannot be negative. The Lyapunov exponent

links with self-similarity of fractal dimension [11, 12].

3. Threshold Calculated Based on Lyapunov Exponents

To confirm the existence of the weak singular signal, we need to define a proper index

for denoting the change in the states of Lorenz system. The index should be sensitive to a

weak singular signal, but insensitive to the random noise from the viewpoint of statistical

characteristics. Thus, the dynamic properties of Lorenz system are reflected statistically by

Lyapunov exponents which are described in the following as [13–15]:Let initial condition: [0.00001, 0.00001, 0.00001], with about typically 30 points in the

region r = [20, 30] chosen to calculate the Lyapunov exponents (LE), the computation’s

precision of r is two digits after the decimal point, shown in Table 1. The LE curve is plotted

in Figure 3. Obviously, when r = 24.05, (1.1) takes on the chaotic state, and when r = 24.10,

(1.1) takes on the periodic state.


Table 1: Lyapunov exponents in Lorenz.

No. r Max LE No. r Max LE

1 20 −0.15 16 24.4 0.782

2 20.4 −0.141 17 24.8 0.82

3 20.8 −0.127 18 25.2 0.833

4 21.2 −0.114 19 25.6 0.836

5 21.6 −0.099 20 26 0.844

6 22 −0.087 21 26.4 0.857

7 22.4 −0.074 22 26.8 0.873

8 22.8 −0.06 23 27.2 0.881

9 23.2 −0.047 24 27.6 0.892

10 23.6 −0.033 25 28 0.907

11 23.8 −0.027 26 28.4 0.909

12 24 −0.018 27 28.8 0.922

13 24.05 −0.014 28 29.2 0.923

14 24.1 0.736 29 29.6 0.939

15 24.2 0.758 30 30 0.945

1.2

1

0.8

0.6

0.4

0.2

0

−0.220 21 22 23 24 25 26 27 28 29 30

Ly

ap

un

ov

ex

po

nen

ts

Threshold r

Figure 3: The relational curve between max. LE and r.

The LE changes from positive to negative corresponding to the region r = [24.05, 24.1],and denotes the chaotic system’s extreme sensitivity to the changed parameters. If the

threshold r is equal to 24.05, and computation precision of r is only three effective digits after

decimal point, as the critical threshold between a chaotic and periodic state, the sensitivity

property is not precise enough.

4. Quickly Approaching Critical Threshold withthe Bisection Algorithm

First, the rough region of the system threshold r is estimated by Lyapunov exponents

with computation precision to be one digit after decimal point. Whatever the region of

r = [24.05, 24.1] is always sensitivity region changed from chaotic state to large periodic

state. Since the bisection algorithm can converge to an optimizing solution quickly [16],


Table 2: Threshold r based on the bisection algorithm in the region r = [24.05, 24.1].

Step Periodic, r1 Phase plane Chaos, r2 Phase plane (r1 + r2)/2 Phase plane

1 24.05 24.1 24.075

2 24.078 24.079 24.0785

3 24.0782 24.0783 24.07825

4 24.07820 24.07821

the threshold value is determined by the bisection algorithm in the region r = [24.05, 24.1].For the initial condition [0.00001, 0.00001, 0.00001], in order to improve the sensitivity of (1.1),the computation precision of r has risen from five digits after decimal point. The steps are as

follows:

(1) Because 24.1 corresponds to the chaotic state and 24.05 corresponds to the periodic

state, r = 24.075 is the midpoint value between 24.05 (chaotic) and 24.1 (periodic).

(2) Because r = 24.075 corresponds to periodic states, the region of r is [24.075, 24.1].Then r is accumulated from 24.075 to 24.1 with the step 0.001 up to 24.079 which

corresponds to the chaotic state and 24.078 which corresponds to the periodic state.

The 24.0785 is the middle value between 24.078 (periodic) and 24.079 (chaotic).

(3) Because r = 24.0785 corresponds to chaotic state, the region of r is taken [24.078,

24.0785]. Then r is accumulated from 24.078 to 24.0785 with the step 0.0001 up to

r = 24.0783 which corresponds to chaotic state and r = 24.0782 which corresponds

to periodic state. The 24.07825 is the middle value between 24.0782 (periodic) and

24.0783 (chaotic).

(4) Because r = 24.07825 corresponds to the chaotic state, the interval of r is [24.0782,

24.07825]. Then r is accumulated from 24.0782 to 24.07825 with the step 0.00001 up

to 24.07821 which corresponds to chaotic state.

(5) Finally, the threshold value calculated is 24.07820. When a weak noisy signal is

merged into (1.1), it takes on the large-scale chaotic state. The calculating process is

shown Table 2.

4.1. How Is an External Signal Merged into Lorenz System

The Lorenz system of differential equations contains the item of the power two, where a, b

are constants, parameter r can be motivated by exterior stimulations to generate a chaotic

trajectory or periodic trajectory. We can adjust the amplitude r of the reference signal to the

special value as in the chaotic critical state. The value is called the threshold value. How will

the external signal be embedded into the control variable r in (1.1)?The variance of a random signal is a measure of its statistical dispersion, indicating

how far from the expected value its values typically are. The variance of a real-valued random


0 200 400 600 800 1000

0

200

400

600

800

−200

−400

−600

−800

f(t)

t

Figure 4: Random signal with variance 7.59 × 104 (r = 24.0782, sampling time 20 s).

signal is its second central moment, and the variance is simply the square of the standard

deviation and also happens to be its second cumulant.

Let the time sequence of a random signal f(t) be x1,x2,x3, . . . ,xN.

The mean value is x = 1/N∑N

i=1 xi, and the variance of the time sequence is

σ =1

N − 1

N∑i=1

(xi − x)2, (4.1)

Since variance determines within what range values concentrated in a series fluctuate

around the series mean and provides a quantitative measure of these fluctuations [17], the

variance of an external signal f(t) is merged into r, that is as follows:

r = r0 + var(f(t)

). (4.2)

Var(f(t)) is variance function in MATLAB.

So long as the threshold is adjusted appropriately, the behavior of the Lorenz system

will be changed dramatically from chaotic states to periodic states. Then Lorenz equation

becomes

x = a(x − y

),

y = −xz +[r0 + var

(f(t)

)]x − y,

z = xy − bz.

(4.3)


0

10

20

30

40

50

y(t)

0

10

20

30

40

50

y(t)

x(t) t

−20 −10 0 10 20 0 5 10 15 20

Figure 5: x-y plane of f(t) merged into (4.3) (r = 24.0782, sampling time 20 s).

0 200 400 600 800 10000

50

100

150

200

250

300

350

400

s(t)

t

Figure 6: One-pulse signal (sampling time 50 s).

In (4.3), random signal f(t) is shown as Figure 4, its variance is var(f(t)) = 7.59 × 104,

then r0 = r − var(f(t)) = 24.07820 − 7.59 = 16.4882, (4.3) takes on periodic state (Figure 5).When one weak noisy signal s(t) (Figure 6) is merged into input f(t) (Figure 7), that is

f1(t) = S(m) + f(t), the variance of f1(t) is Var(f1(t)) = 8.04 × 104 = 8.04, r = r0 + var(S(m)) =16.4882 + 8.04 = 24.5282, then (4.3) takes on chaotic state (Figure 8).

5. Conclusion

Since the bisection algorithm can quickly converge to the critical threshold whose precision

can be changed freely, searching any precision grade of the critical threshold of a Lorenz

system will spent less time. If the variance of a random signal has a constant or a limited

range band, in case weak-singularities signal happens and arouses the variance of the random

signal to change infinitely small, the weak singularities signal can be detected.


0 200 400 600 800 1000

0

200

400

600

800

−200

−400

−600

−800

f1(t)

t

Figure 7: f1(t) = s(t) + f(t) (sampling time 50 s).

0

10

20

30

40

50

y(t)

x(t)

−20 −10 0 10 20

Figure 8: x-y plane of r = 24.5282 (sampling time 50 s).

Since the Runge-Kutta method of fourth-order is one kind of approximate solution

method for dynamic equations, a difference time step size will impact the computation’s pre-

cision for the threshold value.

References

[1] C. Toma, “Advanced signal processing and command synthesis for memory-limited complexsystems,” Mathematical Problems in Engineering, Article ID 927821, 13 pages, 2012.

[2] C. Cattani, “Wavelet based approach to fractals and fractal signal denoising,” Transactions onComputational Science VI, vol. 5730, pp. 143–162, 2009.

[3] N.-Q. Hu, X.-S. Wen, and M. Chen, “application of the Duffing chaotic oscillator for early faultdiagnosis-I. Basic theory,” International Journal of Plant and Management, vol. 7, no. 2, pp. 67–75, 2006.


[4] Y. Li and B. Yang, “Chaotic system for the detection of periodic signals under the background ofstrong noise,” Chinese Science Bulletin, vol. 48, no. 5, pp. 508–510, 2003.

[5] D. Liu, H. Ren, L. Song, and H. Li, “Weak signal detection based on chaotic oscillator,” in Proceedingsof the IEEE Industry Applications Conference, 40th IAS Annual Meeting, pp. 2054–2058, October 2005.

[6] B. Le, Z. Liu, and T. Gu, “Chaotic oscillator and other techniques for detection of weak signals,” IEICETransactions on Fundamentals of Electronics, Communications and Computer Sciences, vol. E88-A, no. 10,pp. 2699–2701, 2005.

[7] F. C. Moon, Chaotic and Fractal Dynamics, A Wiley-Interscience Publication, John Wiley & Sons, NewYork, NY, USA, 1992.

[8] C. K. Chen, J. J. Yan, T. L. Liao, and M. L. Hung, “Chaos suppression of generalized lorenz system:adaptive fuzzy sliding mode control approach,” in Proceedings of the IEEE Conference on Soft Computingon Industrial Applications (SMCia’08), pp. 318–321, June 2008.

[9] I. Pehlivan and Y. Uyaroglu, “A new chaotic attractor from general Lorenz system family andits electronic experimental implementation,” Turkish Journal of Electrical Engineering and ComputerSciences, vol. 18, no. 2, pp. 171–184, 2010.

[10] D. S. Lehrman, “A critique of Konrad Lorenz’s theory of instinctive behavior,” The Quarterly Reviewof Biology, vol. 28, no. 4, pp. 337–363, 1953.

[11] M. Li and W. Zhao, “Visiting power laws in cyber-physical networking systems,” MathematicalProblems in Engineering, vol. 2012, Article ID 302786, 13 pages, 2012.

[12] M. Li, C. Cattani, and S. Y. Chen, “Viewing sea level by a one-dimensional random function with longmemory,” Mathematical Problems in Engineering, vol. 2011, Article ID 654284, 13 pages, 2011.

[13] Q. H. Alsafasfeh and M. S. Al-Arni, “New chaotic behavior from lorenz and rossler systems and itselectronic circuit implementation,” Circuits and Systems, vol. 2, pp. 101–105, 2011.

[14] A. M. Al-Roumy, “The study of a new lorenz-like model,” Journal of Basrah Researches, vol. 37, no. 3 A,2011.

[15] M. Moghtadaei and M. R. H. Golpayegani, “Complex dynamic behaviors of the complex Lorenzsystem,” Scientia Iranica, vol. 19, no. 3, pp. 733–738, 2012.

[16] A. K. Kaw, E. E. Kalu, and D. Ngyen, Numerical Methods with Applications, 1st edition, 2008, http://numericalmethods.eng.usf.edu/topics/textbook index.html.

[17] R. G. Lyons, Understanding Digital Signal Processing, Prentice Hall PTR, 2004.


Research ArticleFractional Calculus and Shannon Wavelet

Carlo Cattani

Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy

Correspondence should be addressed to Carlo Cattani, [email protected]

Received 18 February 2012; Accepted 13 May 2012


Copyright q 2012 Carlo Cattani. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

An explicit analytical formula for the any order fractional derivative of Shannon wavelet is givenas wavelet series based on connection coefficients. So that for any L2(R) function, reconstructedby Shannon wavelets, we can easily define its fractional derivative. The approximation error isexplicitly computed, and the wavelet series is compared with Grunwald fractional derivative byfocusing on the many advantages of the wavelet method, in terms of rate of convergence.

1. Introduction

Shannon wavelet theory [1, 2] is based on a family of orthogonal functions having many

interesting properties. They enjoy the many advantages of wavelets [3, 4]; moreover, being

analytical functions they are infinitely differentiable. Thus, enabling us to define the so-

called connection coefficients [5–7] for any order derivative. Connection coefficients are an

expedient tool for the projection of differential operators, useful for computing the wavelet

solution of integrodifferential equations [8–13].Wavelets are localized functions, in time and/or frequency, which are the basis for

energy-bounded functions and in particular for L2(R)-functions. So that localized pulse

problems [14, 15] can be easily approached and analyzed. Moreover, wavelet allows the

multiscale decomposition of problems, thus emphasizing the contribution of each scale. By

defining a suitable inner product on the orthogonal family of scaling/wavelet functions, any

L2(R)-function can be approximated at a fixed scale, by a truncated series having, as basis, the

scaling functions and the wavelet functions. The wavelet coefficients of these series represent

the contribution of each scale.

Shannon wavelets are related to the harmonic wavelets [3, 5, 8], being the real part

thereof, and to the well-known sinc function, which is the basic function in signal analysis.

It should be also noticed that, as compared with other wavelet families, the main advantage


of Shannon wavelets is that they are analytical functions, thus being infinitely differentiable.

Moreover, they are sharply bounded in the frequency domain, so that, by taking into account

the Parseval identity, any computation can be easily performed by their Fourier transforms.

The theory of connection coefficients was initially given [10, 13] for the compactly

supported wavelet families, such as the Daubechies wavelets [4]. The computation of these

connection coefficients was based on the recursive equations of the wavelet theory and the

explicit forms of these coefficients were given only up to the second order derivatives. The

connection coefficients are the wavelet coefficients of the derivatives of the wavelet basis.

These coefficients are a fundamental tool for the approximation of differential operators, with

respect to the wavelet basis.

In some recent papers, the connection coefficients for Shannon wavelets have been

explicitly computed up to any order derivative with a finite analytical form. This is due to

the analytical form of Shannon wavelets and the discovery by Cattani of a suitable series

expansion for the connection coefficients [2, 6, 7].In the following, we will define the wavelet representation of fractional derivative,

so that the fractional derivative of an L2(R)-function can be easily computed by knowing

the connection coefficients. The fractional derivatives of the Shannon scaling/wavelet basis

are defined and the error of the approximation will be explicitly computed. Moreover, a

comparison with the classical definition of Grunwald formula [16, 17] is given, by showing

the major performance of wavelets, in terms of rate of convergence.

In particular, Section 2 gives some preliminary remarks, definitions, and properties

about Shannon wavelets. Their corresponding connection coefficients are discussed in

Section 3. This Section deals with some properties of connection coefficients, functional

equalities, and error of approximation. Fractional derivatives of the Shannon scaling function

and wavelets are given in Section 4. In this section, it is also shown that the fractional

derivative is a semigroup. The error of the approximation is explicitly computed and

compared with classical definitions of the fractional derivative, and in particular with the

Grunwald formula.

2. Preliminary Remarks

In this section, some remarks on Shannon wavelets and connection coefficients are given (see

also [7]).Shannon wavelet theory (see e.g. [1, 2, 6, 7, 9]) is based on the scaling function ϕ(x),

also known as sinc function, and the wavelet function ψ(x), respectively, defined as

ϕ(x) = sincxdef=

sin πx

πx=

eπix − e−πix

2πix,

ψ(x) =sin 2π(x − (1/2)) − sin π(x − (1/2))

π(x − (1/2))

=e−2 i π x

(−i + ei π x + e3 i π x + i e4 i π x)

2π(x − (1/2)).

(2.1)


The corresponding families of translated and dilated instances wavelet [1, 2, 6, 7, 9], on which

is based the multiscale analysis [4], are

ϕnk(x) = 2n/2ϕ(2nx − k) = 2n/2 sin π(2nx − k)

π(2nx − k)

= 2n/2 eπi(2nx−k) − e−πi(2

nx−k)

2πi(2nx − k),

ψnk (x) = 2n/2 sin 2π(2nx − k − (1/2)) − sin π(2nx − k − (1/2))

π(2nx − k − (1/2))

=2n/2

2π(2nx − k − (1/2))

2∑s=1

i1+sesπi(2nx−k) − i1−se−sπi(2

nx−k) ,

(2.2)

being, in particular,

ϕ00(x) = ϕ(x), ψ0

0(x) = ψ(x), ϕ0k(x) = ϕk(x) = ϕ(x − k),

ψ0k(x) = ψk(x) = ψ(x − k).

(2.3)

Let

f(ω) = f(x) def=1

2π

∫∞

−∞f(x)e−iωxdx, f(x) =

∫∞

−∞f(ω)eiωxdω (2.4)

be the Fourier transform of the function f(x) ∈ L2(R), and its inverse transform, respectively.

The Fourier transform of (2.1) give us [2]

ϕ(ω) =1

2πχ(ω + 3π) =

⎧⎨⎩1

2π, −π ≤ ω < π

0, elsewhere

ψ(ω) =1

2πeiω/2

[χ(2ω) + χ(−2ω)

],

(2.5)

with

χ(ω) =

{1, 2π ≤ ω < 4π

0, elsewhere.(2.6)

Analogously for the dilated and translated instances of scaling/wavelet function, in

the frequency domain, it is

ϕnk(ω) =

2−n/2

2πeiωk/2nχ

(ω

2n+ 3π

)ψnk (ω) =

2−n/2

2πeiω(k+1/2)/2n

[χ

(ω

2n−1

)+ χ

(− ω

2n−1

)].

(2.7)


Both families of Shannon scaling and wavelet are L2(R)-functions therefore, for each

f(x) ∈ L2(R) and g(x) ∈ L2(R), the inner product is defined as

⟨f, g

⟩ def=∫∞

−∞f(x)g(x)dx = 2π

∫∞

−∞f(ω)g(ω)dω = 2π

⟨f , g

⟩, (2.8)

where the bar stands for the complex conjugate.

Shannon wavelets fulfill the following orthogonality properties (for the proof see e.g.,

[2, 7]):

⟨ψnk (x), ψ

mh (x)

⟩= δnmδhk,

⟨ϕ0k(x), ϕ

0h(x)

⟩= δkh ,

⟨ϕ0k(x), ψ

mh (x)

⟩= 0, m ≥ 0, (2.9)

δnm, δhk being the Kronecker symbols.

2.1. Properties of the Shannon Wavelet

According to (2.2), Shannon wavelets can be easily computed at some special points, being

in particular

ϕk(h) = ϕh(k) = ϕ(h − k) = ϕ(k − h) = δkh, (h, k ∈ Z), (2.10)

so that

ϕk(x) =

{0, x = h/= k, (h, k ∈ Z)1, x = h = k, (h, k ∈ Z).

(2.11)

It is also [7]

ψnk (h) = (−1)2nh−k 21+n/2(

2n+1h − 2k − 1)π,

(2n+1h − 2k − 1/= 0

)

ψnk (x) = 0, x = 2−n

(k +

1

2± 1

3

), (n ∈ N, k ∈ Z)

limx→ 2−n(h+(1/2))

ψnk (x) = −2n/2δhk.

(2.12)

In the following, we will be interested on the maximum values of these functions

which can be easily computed. The maximum value of the scaling function ϕk(x) can be

found at the integers x = k

max[ϕk(xM)

]= 1, xM = k, (2.13)


and the max values of ψnk(x) are

max[ψnk (xM)

]= 2n/2 3

√3

π, xM =

⎧⎪⎪⎨⎪⎪⎩−2−n

(k +

1

6

)2−n−1

3(18k + 7).

(2.14)

Both families of scaling and wavelet functions belong to L2(R), thus having a bounded

range and (slow) decay to zero

limx→±∞

ϕnk(x) = 0, lim

x→±∞ψnk (x) = 0. (2.15)

Let B ⊂ L2(R) the set of functions f(x) in L2(R) such that the integrals

αkdef=⟨f(x), ϕk(x)

⟩ (2.8)=

∫∞

−∞f(x)ϕ0

k(x)dx

βnkdef=⟨f(x), ψn

k (x)⟩ (2.8)

=∫∞

−∞f(x)ψn

k(x)dx

(2.16)

exist with finite values, then it can be shown [2–4, 7] that the series

f(x) =∞∑

h=−∞αh ϕh(x) +

∞∑n=0

∞∑k=−∞

βnkψnk (x) (2.17)

converges to f(x).According to (2.8), the coefficients can be also computed in the Fourier domain [7] so

that

αk =∫π

−πf(ω) eiωkdω,

βnk = 2−n/2

[∫2n+1π

2nπ

f(ω)eiω(k+1/2)/2ndω +∫−2nπ

−2n+1π

f(ω)eiω(k+1/2)/2ndω

].

(2.18)

In the frequency domain, (2.17) gives [7]

f(ω) =1

2πχ(ω + 3π)

∞∑h=−∞

αhei ωh

+1

2πχ

(ω

2n−1

) ∞∑n=0

∞∑k=−∞

2−n/2βnkeiω(k+1/2)/2n

+1

2πχ

(− ω

2n−1

) ∞∑n=0

∞∑k=−∞

2−n/2βnkeiω(k+1/2)/2n .

(2.19)


When the upper bound for the series of (2.17) is finite, then we have the approximation

f(x) ∼=K∑

h=−Kαhϕh(x) +

N∑n=0

S∑k=−S

βnkψnk (x). (2.20)

The error of the approximation has been estimated in [7].

2.2. Reconstruction of the Derivatives

In order to represent the differential operators in wavelet bases, we have to compute the

wavelet decomposition of the derivatives. It can be shown [2, 7] that the derivatives of the

Shannon wavelets are orthogonal functions:

d

dx ϕh(x) =

∞∑k=−∞

λ( )hk

ϕk(x),

d

dx ψmh (x) =

∞∑n=0

∞∑k=−∞

γ ( )mnhk ψn

k (x),

(2.21)

being

λ( )kh

def=

⟨d

dx ϕ0k(x), ϕ

0h(x)

⟩, γ ( )

mnkh

def=

⟨d

dx ψnk (x), ψ

mh (x)

⟩, (2.22)

the connection coefficients [2, 5, 6, 8–13].The computation of connection coefficients can be easily performed in the Fourier

domain, thanks to the equality (2.8)

λ( )kh

= 2π

⟨ d

dx ϕk(x), ϕh(x)

⟩, γ ( )

mnkh = 2π

⟨ d

dx ψnk(x), ψm

h(x)

⟩. (2.23)

In fact, in the Fourier domain, the -order derivative of the (scaling) wavelet functions

are simply

d

dx ϕnk(x) = (iω) ϕn

k(ω),d

dx ψnk(x) = (iω) ψn

k (ω), (2.24)

and, according to (2.7),

d

dx ϕnk(x) = (iω)

2−n/2

2πeiωk/2nχ

(ω

2n+ 3π

),

d

dx ψnk(x) = (iω)

2−n/2

2πeiω(k+(1/2))/2n

[χ

(ω

2n−1

)+ χ

(− ω

2n−1

)].

(2.25)


It has been shown [2, 6, 7] that the any order connection coefficients (2.22)1 of the

Shannon scaling functions ϕk(x) are

λ( )kh

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩(−1)k−h

i

2π

∑s=1

!πs

s![i(k − h)] −s+1

[(−1)s − 1

], k /=h

i π +1

2π( + 1)

[1 + (−1)

], k = h,

(2.26)

or, by defining

μ(m) = sign(m) =

⎧⎪⎪⎨⎪⎪⎩1, m > 0

−1, m < 0

0, m = 0,

(2.27)

shortly as,

λ( )kh

=i π

2( + 1)

[1 + (−1)

](1 − ∣∣μ(k − h)

∣∣)+ (−1)k−h

∣∣μ(k − h)∣∣ i

2π

∑s=1

!πs

s![i(k − h)] −s+1

[(−1)s − 1

],

(2.28)

when ≥ 1, and for = 0,

λ(0)kh

= δkh. (2.29)

For the proof see [2].Analogously for the connection coefficients (2.22)2 we have that the any order

connection coefficients of the Shannon scaling wavelets ψnk(x) are

γ ( )nmkh = μ(h − k)δnm

{ +1∑s=1

(−1)[1+μ(h−k)](2 −s+1)/2 !i −s π −s

( − s + 1)!|h − k|s (−1)−s−2(h+k)2n −s−1

×{

2 +1[(−1)4h+s + (−1)4k+

]− 2s

[(−1)3k+h+ + (−1)3h+k+s

]}}, k /=h

γ ( )nmkh = δnm

[i π 2n −1

+ 1

(2 +1 − 1

)1 +

((−1)

)], k = h,

(2.30)


or, shortly

γ ( )nmkh = δnm

{i (1 − ∣∣μ(h − k)

∣∣)π 2n −1

+ 1

(2 +1 − 1

)(1 + (−1)

)

+ μ(h − k) +1∑s=1

(−1)[1+μ(h−k)](2 −s+1)/2 !i −s π −s

( − s + 1)! |h − k|s (−1)−s−2(h+k)2n −s−1

×{

2 +1[(−1)4h+s + (−1)4k+

]− 2s

[(−1)3k+h+ + (−1)3h+k+s

]}},

(2.31)

for ≥ 1, and

γ (0)nmkh = δkhδ

nm, (2.32)

= 0, respectively.

For the proof see [2].

3. Remarks on Connection Coefficients

3.1. Recursiveness

The connection coefficients fulfill some recursive formula as follows.

Theorem 3.1. The connection coefficients (2.26) are recursively given by

λ( +1)kh

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩ + 1

k − hλ( )kh

− (−1)k−hi π +1

k − h

[(−1) + 1

], k /=h

iπ + 1

+ 2λ( )kh

+(−i) +1π +1

+ 2, k = h,

(3.1)

Proof. Let us show first when k = h. From the definition (2.26), it is

λ( +1)kk

=i +1π +2

2π( + 2)

[1 + (−1) +1

]= iπ

( + 1)( + 2)

i π +1

2π( + 1)

[1 + (−1) +1 + (−1) − (−1)

]= iπ

( + 1)( + 2)

i π +1

2π( + 1)

[1 + (−1) + 2(−1) +1

],

(3.2)

from where (3.1)2 follows. Analogously with simple computation we obtain (3.1)1.


Shorty and with some caution, (3.1) can be written as

λ( +1)kh

= (1 − δkh)

[ + 1

k − hλ( )kh

− (−1)k−hi π +1

k − h

[(−1) + 1

]]

+ δkh

[iπ

+ 1

+ 2λ( )kh

+(−i) +1π +1

+ 2

],

(3.3)

that is,

λ( +1)kh

=[(1 − δkh)

+ 1

k − h+ δkhiπ

+ 1

+ 2

]λ( )kh

− (1 − δkh)(−1)k−hi π +1

k − h

[(−1) + 1

]+ δkh

(−i) +1π +1

+ 2.

(3.4)

It is not so easy to find out a similar property also for the γ-coefficients as a function of

however, there is a simple rule for the recursiveness of the scale (upper) indexes, as follows.

Theorem 3.2. The connection coefficients (2.30) are recursively given by the matrix at the lowestscale level:

γ ( )nn

kh = 2 (n−1)γ ( )11kh . (3.5)

Proof. As can be seen from (2.30) parameter n appears only in the factor

2n −1, (3.6)

so that (3.5) follows from the identity

2n −1 = 2 (n−1)2 −1. (3.7)

Moreover, it can be shown also that

γ (2 +1)nnkh = −γ (2 +1)nnhk , γ (2 )nnhk = γ (2 )

nnhk . (3.8)

3.2. Taylor Series

By using the connection coefficients, it is easy to show the following theorem.


Theorem 3.3. If f(x) ∈ Bψ ⊂ L2(R) and f(x) ∈ CS the Taylor series of f(x) in x0 is

f(x) = f(x0)

+S∑r=1

[ ∞∑h,k=−∞

αh λ(r)hkϕk(x0) +

∞∑n=0

∞∑k,s=−∞

2r(n−1)βnkγ(r)11

skψns (x0)

](x − x0)r

r!+ RS(x,x0),

(3.9)

being αh and βnkgiven by (2.16), (2.18) and RS(x,x0) the error.

Proof. From (2.17), the -order derivative ( ≤ S) is

f ( )(x) =∞∑

h=−∞αh

d

dx ϕh(x) +

∞∑n=0

∞∑k=−∞

βnkd

dx ψnk (x),

(2.21)=

∞∑h=−∞

αh

∞∑k=−∞

λ( )hkϕk(x) +

∞∑n=0

∞∑k=−∞

βnk

∞∑m=−∞

∞∑s=−∞

γ ( )mnsk ψm

s (x),

=∞∑

h,k=−∞αh λ

( )hkϕk(x) +

∞∑n,m=0

∞∑k,s=−∞

βnkγ( )mn

sk ψms (x),

(3.10)

so that by taking into account (3.5) the proof follows.

In particular, by a suitable choice of the initial point x0, (3.9) can be simplified. For

instance, at the integers, x0 = h, (h ∈ Z), according to (2.10), (2.12) and (3.5), it is

f(x) ∼= f(h) +S∑r=1

[ ∞∑h=−∞

αh λ(r)hh

+∞∑n=0

∞∑k,s=−∞

(−1)2nh−s 2r(n−1)+1+n/2(2n+1h − 2s − 1

)πβnkγ

(r)11sk

](x − h)r

r!.

(3.11)

3.3. Functional Equations

The connection coefficients fulfill some identities as follows.

Theorem 3.4. For any k ∈ Z and ∈ N, it is

(iω) e−iωk =∞∑

h=−∞λ( )khe−iωh, −π ≤ ω ≤ π, (3.12)

or

(iω) =∞∑

h=−∞λ( )khe−iω(h−k), −π ≤ ω ≤ π, ∀k ∈ Z. (3.13)


Proof. From (2.21), by a Fourier transform of both sides and taking into account (2.24), we get

(iω) ϕk(ω) =∞∑

h=−∞λ( )khϕh(ω)

(iω) e−iωkχ(ω + 3π)(2.7)=

∞∑h=−∞

λ( )khe−iωhχ(ω + 3π),

(3.14)

from where the identity (3.12) follows.

In particular, by assuming, without restrictions, k = 0, we have the following (see

Figure 1).

Corollary 3.5. For any ∈ N it is

(iω) =∞∑

h=−∞λ( )0he−iωh, −π ≤ ω ≤ π, (3.15)

so that λ( )0h

are the Fourier coefficients of the power (iω) .

Analogously, from (2.21)2, we have the following.

Theorem 3.6. For any k ∈ Z and , n ∈ N it is

(iω) e−iω(k+1/2)/2n =∞∑

h=−∞γ ( )

nnkhe−iω(h+1/2)/2n , ω ∈

[−2n+1π,−2nπ

]∪[2nπ , 2n+1π

], (3.16)

or

(iω) =∞∑

h=−∞γ ( )

nnkhe−iω(h−k)/2n , ω ∈

[−2n+1π ,−2nπ

]∪[2nπ , 2n+1π

]. (3.17)

In particular, with k = 0, and taking into account (3.5), we have the following.

Corollary 3.7. For any , n ∈ N it is

(iω) = 2 (n−1)∞∑

h=−∞γ ( )

11

0he−iωh/2n , ω ∈

[−2n+1π ,−2nπ

]∪[2nπ , 2n+1π

]. (3.18)

As a consequence of the previous theorems we have the following.

Theorem 3.8. For any , n ∈ N it is

(iω) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩∞∑

h=−∞λ( )0he−iωh, −π ≤ ω ≤ π

2 (n−1)∞∑

h=−∞γ ( )

110he−iωh/2n , ω ∈ [−2n+1π,−2nπ

] ∪ [2nπ, 2n+1π].

(3.19)


−−1

1 x

(a)

−−1

1 x

(b)

−−1

x

(c)

−−1

x

(d)

Figure 1: Approximation of (iω) (plain) by the r.h.s of (3.15) at different scale: (a) �[(iω)3], k =5, |hmax| = 5; (b) �[(iω)3], k = 5, |hmax| = 10; (c) [(iω)2], k = 7, |hmax| = 5; (d) [(iω)2], k = 7, |hmax| =8.

There we have the following.

Corollary 3.9. The Fourier transform of the derivatives of a function is

d

dx f(x) =f(ω) ×

⎧⎪⎪⎪⎨⎪⎪⎪⎩∞∑

h=−∞λ( )0he−iωh, −π ≤ ω ≤ π

2 (n−1)∞∑

h=−∞γ ( )

11khe−iωh/2n , ω ∈ [−2n+1π,−2nπ

] ∪ [2nπ, 2n+1π].

(3.20)


If we express eiω as a Taylor series we have

eiω =∞∑ =0

(iω)

!, (3.21)

so that eiω with −π ≤ ω ≤ π is the solution of the functional equation

X =∞∑ =0

∞∑h=−∞

1

!λ( )0hX−h. (3.22)

Moreover, the theorem of moments

∫R

x f(x)dx = i df(ω)dω

(3.23)

can be written as

∫R

x f(x)dx = i f(ω) ×

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

∞∑h=−∞

λ( )khe−iωh, −π ≤ ω ≤ π

∞∑h=−∞

γ ( )nnkhe−iω(h−k)/2n , ω ∈ [−2n+1π,−2nπ

] ∪ [2nπ, 2n+1π].

(3.24)

3.4. Error of the Approximation by Connection Coefficients

For a fixed scale of approximation in (2.21), it is possible to estimate the error as follows. It

should be noticed that the approximation depends on a the upper bound of the limits in the

sums.

Theorem 3.10 (error of the approximation of scaling functions derivatives). The error of theapproximation in (2.21)1 is given by

∣∣∣∣∣ d

dx ϕh(x) −

N∑k=−N

λ( )hk

ϕk(x)

∣∣∣∣∣ ≤ ∣∣∣λ( )h(−N−1) + λ( )h(N+1)

∣∣∣. (3.25)

Proof. The error of the approximation (2.21)1 is defined as

d

dx ϕh(x) −

N∑k=−N

λ( )hkϕk(x) =

−N−1∑k=−∞

λ( )hkϕk(x) +

∞∑k=N+1

λ( )hkϕk(x). (3.26)


Concerning the r.h.s, and according to (2.13), it is

−N−1∑k=−∞

λ( )hk

ϕk(x) +∞∑

k=N+1

λ( )hk

ϕk(x)

≤ maxx∈R

[−N−1∑k=−∞

λ( )hk

ϕk(x) +∞∑

k=N+1

λ( )hk

ϕk(x)

]

= λ( )h(−N−1) ϕ−N−1(x) + λ

( )h(N+1) ϕN+1(x) ≤ λ

( )h(−N−1) + λ

( )h(N+1).

(3.27)

Theorem 3.11 (error of the approximation of wavelet functions derivatives). The error of theapproximation in (2.21)2 is given by

∣∣∣∣∣ d

dx ψmh (x) −

N∑n=0

S∑k=−S

γ ( )mnhk ψn

k (x)

∣∣∣∣∣ ≤∣∣∣∣∣2 (m−1)+m/2 3

√3

π

[γ ( )

11h(−S−1) + γ ( )

11h(S+1)

]∣∣∣∣∣. (3.28)

Proof. The error of the approximation is

d

dx ψmh (x) −

N∑n=0

S∑k=−S

γ ( )mnhk ψn

k (x) =∞∑

n=N+1

[ −S−1∑k=−∞

γ ( )mnhk ψn

k (x) +∞∑

k=S+1

γ ( )mnhk ψn

k (x)

]. (3.29)

If m < N, the r.h.s. according to (2.30) is zero; therefore, we assume that m > N so that the

last equation becomes

d

dx ψmh (x) −

N∑n=0

S∑k=−S

γ ( )mnhk ψn

k (x) =

[ −S−1∑k=−∞

γ ( )mmhk ψn

k (x) +∞∑

k=S+1

γ ( )mmhk ψn

k (x)

]

(3.5)= 2 (m−1)

[ −S−1∑k=−∞

γ ( )11hk +

∞∑k=S+1

γ ( )11hk

]ψmk (x)

≤ 2 (m−1) max

{[ −S−1∑k=−∞

γ ( )11hk +

∞∑k=S+1

γ ( )11hk

]ψmk (x)

}

= 2 (m−1)[γ ( )

11h(−S−1)ψm

(−S−1)(x) + γ ( )11h(S+1)ψm

(S+1)(x)]

≤ 2 (m−1)[γ ( )

11h(−S−1) maxψm

(−S−1)(x) + γ ( )11h(S+1) maxψm

(S+1)(x)]

(2.14)= 2 (m−1)2m/2 3

√3

π

[γ ( )

11h(−S−1) + γ ( )

11h(S+1)

].

(3.30)


4. Fractional Derivatives of the Wavelet Basis

The simplest way to define the fractional derivative is based on the assumption that the

noninteger derivative of the exponential function formally coincides with the derivative with

integer order so that

dν

dxνeax = aνeax ν ∈ Q. (4.1)

For negative values of ν, this formula still holds true and it represents the integration.

It is known that the fractional derivative cannot be analytically computed except for

some special functions, such as (see e.g., [16–18]) the following:

dν

dxνeax = aνeax ,

dν

dxνcosax = aν cos

(ax +

π

2ν),

dν

dxνsinax = aν sin

(ax +

π

2ν).

(4.2)

From these, classical examples, we can see that the fractional derivative can be also

interpreted as an interpolating function between derivatives with integer order, so that

dν

dxνf(x) = (1 − ν)f(x) + ν

d

dxf(x), 0 ≤ ν ≤ 1. (4.3)

More in general, let f(x) be a single-valued real function, then the Riemann-Liouville

fractional order derivative is defined as [16]

dν

dxνf(x) def=

1

Γ(1 − ν)d

dx

∫x

0

f(ξ)(x − ξ)ν

dξ, (0 < ν < 1, x > 0), (4.4)

Γ(ν) being the gamma function.

Other equivalent representations were given by Caputo (for a differentiable function)

dν

dxνf(x) def=

1

Γ(1 − ν)

∫x

0

f ′(ξ)(x − ξ)ν

dξ, 0 < ν < 1, (4.5)

and by Grunwald (see e.g., [17, 18])

dν

dxνf(x) = lim

N→∞1

Γ(−ν)( x

N

)−νN−1∑k=0

Γ(k − ν)Γ(k + 1)

f

[(1 − k

N

)x

], (0 < ν < 1, x > 0). (4.6)

However, a drawback in the Grunwald definition, as well as in the Riemann-Liouville, is that

it cannot be computed for negative values of the variable (x < 0).


4.1. Fractional Derivative of the Shannon Scaling Function

Let us assume that the fractional order derivative is defined by a linear interpolation of the

integer order derivatives, so that the fractional derivative of the scaling-wavelet basis

d +ν

dx +νϕh(x),

d +ν

dx +νψmh (x). (4.7)

with

0 ≤ ν ≤ 1, (4.8)

can be defined as

d +ν

dx +νϕh(x)

def= (1 − ν)d

dx ϕh(x) + ν

d +1

dx +1ϕh(x),

d +ν

dx +νψmh (x)

def= (1 − ν)d

dx ψmh (x) + ν

d +1

dx +1ψmh (x).

(4.9)

Let us show the following.

Theorem 4.1. The fractional derivative of the Shannon scaling functions is

d +ν

dx +νϕh(x)

def=∞∑

k=−∞λ( +ν)hk

ϕk(x) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩∞∑

k=−∞

[(1 − ν)λ( )

hk+ νλ

( +1)hk

]ϕk(x), > 0

∞∑k=−∞

[(1 − ν)δhk + νλ

(1)hk

]ϕk(x), = 0.

(4.10)

Proof. From (4.9), by taking into account (2.21), it is

d +ν

dx +νϕh(x)

def= (1 − ν)∞∑

k=−∞λ( )hkϕk(x) + ν

∞∑k=−∞

λ( +1)hk

ϕk(x)

(3.1)=

∞∑k=−∞

[(1 − ν)λ( )

hk+ νλ

( +1)hk

]ϕk(x),

(4.11)

and, when = 0,

dν

dxνϕh(x) =

∞∑k=−∞

[(1 − ν)δhk + νλ

(1)hk

]ϕk(x). (4.12)

With this definition, the fractional order derivative of the scaling functions is a

commutative operator according to the following.


Theorem 4.2. The operator (4.10) is a semigroup, so that

dμ

dxμ

dν

dxνϕh(x) =

dν

dxν

dμ

dxμϕh(x) =

dμ+ν

dxμ+ν ϕh(x). (4.13)

Proof. Without loss of generality, let us show that

dμ

dxμ

dν

dxνϕ(x) =

dν

dxν

dμ

dxμϕ(x). (4.14)

According to (4.10)2, it is

dν

dxνϕ0(x) =

∞∑k=−∞

[(1 − ν)δ0k + νλ

(1)0k

]ϕk(x), (4.15)

that is

dν

dxνϕ(x) = (1 − ν)ϕ(x) + ν

⎡⎢⎢⎣λ(1)00 ϕ(x) +∞∑

k /= 0k=−∞

λ(1)0kϕk(x)

⎤⎥⎥⎦, (4.16)

and, taking into account (2.26), by explicit computation we have

dν

dxνϕ(x) = (1 − ν)ϕ(x) + ν

∞∑k /= 0k=−∞

(−1)k

kϕk(x). (4.17)

By deriving, with respect to μ, we have

dμ

dxμ

dν

dxνϕ(x) = (1 − ν)

dμ

dxμϕ(x) + ν

∞∑k /= 0k=−∞

(−1)k

k

dμ

dxμϕk(x)

(4.17)= (1 − ν)

⎡⎢⎢⎣(1 − μ)ϕ(x) + μ

∞∑k /= 0k=−∞

(−1)k

kϕk(x)

⎤⎥⎥⎦+ ν

∞∑k /= 0k=−∞

(−1)k

k

dμ

dxμϕk(x),

(4.18)


that is, according to (2.26),

dμ

dxμ

dν

dxνϕ(x) = (1 − ν)

⎡⎢⎢⎣(1 − μ)ϕ(x) + μ

∞∑k /= 0k=−∞

(−1)k

kϕk(x)

⎤⎥⎥⎦+ ν

∞∑k /= 0k=−∞

(−1)k

k

∞∑s=−∞

[(1 − μ

)δsk + μλ

(1)sk

]ϕs(x)

= (1 − ν)

⎡⎢⎢⎣(1 − μ)ϕ(x) + μ

∞∑k /= 0k=−∞

(−1)k

kϕk(x)

⎤⎥⎥⎦+ ν

(1 − μ

) ∞∑k /= 0k=−∞

(−1)k

kϕk(x) + νμ

∞∑k /= 0k=−∞

(−1)k

k

∞∑s=−∞

λ(1)skϕs(x).

(4.19)

From where,

dμ

dxμ

dν

dxνϕ(x) = (1 − ν)

(1 − μ

)ϕ(x) +

[(1 − ν)μ + ν

(1 − μ

)] ∞∑k /= 0k=−∞

(−1)k

kϕk(x)

+ νμ∞∑

k /= 0k=−∞

(−1)k

k

∞∑s=−∞

λ(1)skϕs(x),

(4.20)

the proof follows due to the symmetry of the change μ → ν.

It can be easily seen that together with (4.17) also the following equations hold:

dν

dxνϕ1(x) = (1 − ν)ϕ1(x) + ν

∞∑k /= 0k=−∞

(−1)k

k − 1ϕk(x)

dν

dxνϕ−1(x) = (1 − ν)ϕ−1(x) + ν

∞∑k /= 0k=−∞

(−1)k

1 + kϕk(x),

(4.21)

and, in general,

dν

dxνϕh(x) = (1 − ν)ϕh(x) + ν

∞∑k /= 0k=−∞

(−1)k

k − hϕk(x). (4.22)

Moreover, when μ + ν = 1, then we can see that the definition (2.26) reduces to the

ordinary derivative, according to the following.


x

1

−1

= 0

= 1

A

A

Figure 2: Fractional derivative of the scaling functions (dν/dxν)ϕ(x) with upper limit N = 4 at differentvalues of ν = 0, 1/5, 2/5, 3/5, 4/5, 1.

Theorem 4.3. When μ + ν = 1, then

dμ

dxμ

dν

dxνϕh(x) =

dμ+ν

dxμ+ν ϕh(x) =d

dxϕh(x). (4.23)

Proof. If we restrict to ϕ(x), according to the definition (2.26), it is

dμ

dxμ

dν

dxνϕ(x) =

∞∑k=−∞

[(1 − (μ + ν

))δ0k +

(μ + ν

)λ(1)0k

]ϕk(x), (4.24)

and since (μ + ν) = 1 we have

dμ

dxμ

dν

dxνϕ(x) =

d

dxϕ(x) =

∞∑k=−∞

λ(1)0kϕk(x), (4.25)

According to the definition (4.10), the fractional derivative is an interpolation between

integer order derivative (see Figure 2).

4.2. Error of the Approximation of (4.10)

In the definition (4.10), the fractional derivative depends on a fixed bound N of the infinite

series. In this section, it will be shown that the rate of convergence of the series, on the r.h.s of

(4.10), is quite fast; already with low values of N, the approximation is quite good (Figure 3).


1 < N < 10

−x

(a)

10 < N < 50

−x

(b)

Figure 3: Fractional derivative of the scaling functions (d3/10/dx3/10)ϕ(x) with upper limit N = 1, . . . , 10(a) and N = 10, . . . , 50 (b).

4.2.1. Rate of Convergence

If we compare the fractional derivative (dν/dxν)ϕh(x) given by (4.10) with the Grunwald

definition (4.6), we can see that the approximation by connection coefficients is good (see

Figure 4), with a lower number of terms. Moreover, the definition based on connection

coefficients can be extended also to negative values of the variable.

Since we have defined the fractional derivative on an infinite series N → ∞, as well

as the Grunwald formula, we can explicitly compute the error of the approximation as the

difference between the approximated value at N + 1 and the corresponding value of the

infinite series at N. For instance, with respect to (4.10), it is

ενN = maxx∈R

∣∣∣∣∣∣N+1∑

k=−(N+1)

λ( +ν)hk

ϕk(x) −N∑

k=−Nλ( +ν)hk

ϕk(x)

∣∣∣∣∣∣, (4.26)

while for the Grunwald formula (4.6) we have

ενN = maxx>0

∣∣∣∣∣ 1

Γ(−ν)( x

N + 1

)−ν N∑k=0

Γ(k − ν)Γ(k + 1)

f

[(1 − k

N + 1

)x

]

− 1

Γ(−ν)( x

N

)−νN−1∑k=0

Γ(k − ν)Γ(k + 1)

f

[(1 − k

N

)x

]∣∣∣∣∣,(4.27)

Let us show the following.


1

−1

2x

(a)

1

−1

2x

(b)

1

−1

2x

(c)

1

−1

2x

(d)

Figure 4: Fractional derivative of the scaling functions (dν/dxν)ϕh(x) by Grunwald approximation (4.6)(shaded) and connection coefficients interpolation (4.10)2 (plain): (a) ν = 1/10, h = 0 with upper limitN = 1 (connection coefficients) and N = 4 (Grunwald); (b) ν = 1/10, h = 1 with upper limit N = 1(connection coefficients) and N = 1 (Grunwald); (c) ν = 1/20, h = 1 with upper limit N = 2 (connectioncoefficients) and N = 8 (Grunwald); (d) ν = 9/10, h = 1 with upper limit N = 10 (connection coefficients)and N = 50 (Grunwald).

Theorem 4.4. For = 0, the approximation error of (4.10)2 is given by

ενN = 2ν

∣∣∣∣∣ (−1)N+1h

(N + 1)2 − h2

∣∣∣∣∣. (4.28)


Proof. By taking into account (4.22), it is

N+1∑k=−(N+1)

λ( +ν)hk

ϕk(x) −N∑

k=−Nλ( +ν)hk

ϕk(x) = ν

[(−1)N+1

−(N + 1) − hϕ−(N+1)(x)

+(−1)N+1

(N + 1) − hϕ(N+1)(x)

]

(2.13)< ν

[(−1)N+1

−(N + 1) − h+

(−1)N+1

(N + 1) − h

]

=2ν(−1)N+1h

(N + 1)2 − h2.

(4.29)

Analogously, the following can be shown.

Theorem 4.5. For x > 0, the approximation error of (4.6)2 is given by

ενN =Nν

Γ(−ν)Γ(N − ν)Γ(N + 1)

. (4.30)

Proof. At the integer x = 1, it is

1

Γ(−ν)(

1

N + 1

)−ν N∑k=0

Γ(k − ν)Γ(k + 1)

f

[(1 − k

N + 1

)]− 1

Γ(−ν)(

1

N

)−νN−1∑k=0

Γ(k − ν)Γ(k + 1)

f

[(1 − k

N

)]

<1

Γ(−ν)Nν

N∑k=0

Γ(k − ν)Γ(k + 1)

f

[(1 − k

N + 1

)]− 1

Γ(−ν)NνN−1∑k=0

Γ(k − ν)Γ(k + 1)

f

[(1 − k

N

)](2.13)<

Nν

Γ(−ν)Γ(N − ν)Γ(N + 1)

.

(4.31)

4.3. Fractional Derivative of the Shannon Wavelet

Analogously to (4.10), the following can be proved.

Theorem 4.6. The fractional derivative of the Shannon wavelet functions is

d +ν

dx +νψmh (x)

def=∞∑

k=−∞γ ( +ν)

mmhk ψm

k (x)

=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩2 (m−1)

[ ∞∑k=−∞

(1 − ν)γ ( )11hk + ν 2m−1γ ( +1)11

hk

]ψmk(x), > 0[ ∞∑

k=−∞(1 − ν)δhk + ν2m−1γ (1)

11hk

]ψmk(x), = 0.

(4.32)


Proof. From (4.9), by taking into account (2.21)

d +ν

dx +νψmh (x) = (1 − ν)

∞∑n=0

∞∑k=−∞

γ ( )mnhk ψn

k (x) + ν∞∑n=0

∞∑k=−∞

γ ( +1)mnhk ψn

k (x)

(3.5)=

[(1 − ν)

∞∑n=0

∞∑k=−∞

δmn2 (n−1)γ ( )11hk + ν

∞∑n=0

∞∑k=−∞

δmn2( +1)(n−1)γ ( +1)11hk

]ψnk (x)

=

[(1 − ν)

∞∑k=−∞

2 (m−1)γ ( )11hk + ν

∞∑k=−∞

2( +1)(m−1)γ ( +1)11hk

]ψmk (x)

= 2 (m−1)

[ ∞∑k=−∞

(1 − ν)γ ( )11hk + ν 2m−1γ ( +1)11

hk

]ψmk (x).

(4.33)

Analogously to the fractional derivative of the scaling function, also for the wavelet

function, the fractional order derivatives are enveloped by the integer order derivatives

(Figure 5).

4.4. Fractional Derivative of an L2(R) Function

Let f(x) ∈ B ⊂ L2(R) be a function such that (2.17) holds, then its fractional derivative can be

computed as

dν

dxνf(x) =

∞∑h=−∞

αhdν

dxνϕh(x) +

∞∑n=0

∞∑k=−∞

βnkdν

dxνψnk (x), (4.34)

where the fractional derivatives of the scaling functions ϕh(x) and wavelets ψnk(x) are given

by (4.10) and (4.32), respectively.

For instance, a good approximation of y = e−x2

is (Figure 6)

e−x2 ∼= 0.97ϕ(x) + 0.39

[ϕ−1(x) + ϕ1(x)

]. (4.35)

The fractional derivative is

dν

dxνe−x

2 ∼= 0.97dν

dxνϕ(x) + 0.39

dν

dxν

[ϕ−1(x) + ϕ1(x)

], (4.36)


= 1

= 0

x

−1

1

A

A

Figure 5: Fractional derivative of the wavelet functions (dν/dxν)ψ00(x) with upper limit N = 4 at different

values of ν = 0, 1/5, 2/5, 3/5, 4/5, 1.

= 1

= 0

x3−3

−1

1A

A

Figure 6: Fractional derivative of the function y = e−x2

with upper limit N = 4 at different values ofν = 0, 1/5, 2/5, 3/5, 4/5, 1.


so that by using (4.17) and (4.21) we have

dν

dxνe−x

2 ∼= 0.97

⎡⎢⎢⎣(1 − ν)ϕ(x) + ν∞∑

k /= 0k=−∞

(−1)k

kϕk(x)

⎤⎥⎥⎦+ 0.39(1 − ν)

[ϕ−1(x) + ϕ1(x)

]

+ 0.39ν

⎡⎢⎢⎣ ∞∑k /= 0k=−∞

(−1)k+1

kϕk(x) +

∞∑k /= 0k=−∞

(−1)k

k − 1ϕk(x)

⎤⎥⎥⎦,(4.37)

5. Conclusion

In this paper, fractional calculus has been revised by using Shannon wavelets. Fractional

derivatives of the Shannon scaling/wavelet functions, based on connection coefficients,

are explicitly computed and the approximation error is estimated. In the comparison with

the classical Grunwald formula of fractional derivative, Shannon wavelets and connection

coefficients make a better approximation and rate of convergence.

References

[1] C. Cattani, “Shannon wavelet analysis,” in Proceedings of the International Conference on ComputationalScience (ICCS ’07), Y. Shi, G. D. van Albada, J. Dongarra, and P. M. A. Sloot, Eds., Lecture Notes inComputer Science, LNCS 4488, Part II, pp. 982–989, Springer, Beijing, China, May 2007.

[2] C. Cattani, “Shannon wavelets theory,” Mathematical Problems in Engineering, vol. 2008, Article ID164808, 24 pages, 2008.

[3] C. Cattani and J. Rushchitsky, Wavelet and Wave Analysis as applied to Materials with Micro orNanostructure, vol. 74 of Series on Advances in Mathematics for Applied Sciences, World ScientificPublishing, Singapore, 2007.

[4] I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in AppliedMathematics, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1992.

[5] C. Cattani, “Harmonic wavelet solutions of the Schrodinger equation,” International Journal of FluidMechanics Research, vol. 30, no. 5, pp. 463–472, 2003.

[6] C. Cattani, “Connection coefficients of Shannon wavelets,” Mathematical Modelling and Analysis, vol.11, no. 2, pp. 117–132, 2006.

[7] C. Cattani, “Shannon wavelets for the solution of integrodifferential equations,” Mathematical Problemsin Engineering, vol. 2010, Article ID 408418, 22 pages, 2010.

[8] C. Cattani, “Harmonic wavelets towards the solution of nonlinear PDE,” Computers & Mathematicswith Applications, vol. 50, no. 8-9, pp. 1191–1210, 2005.

[9] E. Deriaz, “Shannon wavelet approximation of linear differential operators,” Institute of Mathematicsof the Polish Academy of Sciences, no. 676, 2007.

[10] A. Latto, H. L. Resnikoff, and E. Tenenbaum, “The evaluation of connection coefficients of compactlysupported wavelets,” in Proceedings of the French-USA Workshop on Wavelets and Turbulence, Y. Maday,Ed., pp. 76–89, Springer, 1992.

[11] E. B. Lin and X. Zhou, “Connection coefficients on an interval and wavelet solutions of Burgersequation,” Journal of Computational and Applied Mathematics, vol. 135, no. 1, pp. 63–78, 2001.

[12] J. M. Restrepo and G. K. Leaf, “Wavelet-Galerkin discretization of hyperbolic equations,” Journal ofComputational Physics, vol. 122, no. 1, pp. 118–128, 1995.

[13] C. H. Romine and B. W. Peyton, “Computing connection coefficients of compactly supported waveletson bounded intervals,” Tech. Rep. ORNL/TM-13413, Computer Science and Mathematics Division,Mathematical Sciences Section, Oak Ridge National Laboratory, Oak Ridge, Tenn, USA, 1997.


[14] G. Toma, “Specific differential equations for generating pulse sequences,” Mathematical Problems inEngineering, vol. 2010, Article ID 324818, 11 pages, 2010.

[15] C. Toma, “Advanced signal processing and command synthesis for memory-limited complexsystems,” Mathematical Problems in Engineering, vol. 2012, Article ID 927821, 13 pages, 2012.

[16] K. B. Oldham and J. Spanier, The Fractional Calculus., Academic Press, London, UK, 1970.[17] B. Ross, A Brief History and Exposition of the Fundamental Theory of Fractional Calculus, Fractional Calculus

and Applications, vol. 457 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1975.[18] L. B. Eldred, W. P. Baker, and A. N. Palazotto, “Numerical application of fractional derivative model

constitutive relations for viscoelastic materials,” Computers and Structures, vol. 60, no. 6, pp. 875–882,1996.


Research ArticleParallel Motion Simulation ofLarge-Scale Real-Time Crowd in a HierarchicalEnvironmental Model

Xin Wang,1 Jianhua Zhang,2 and Massimo Scalia3

1 College of Computer Science and Technology, Zhejiang University of Technology, Hangzhou 310023, China2 TAMS Group, Department of Informatics, University of Hamburg, Vogt-Koelln-Straße 30,22527 Hamburg, Germany

3 Department of Mathematics, Sapienza University of Rome, Piazzale Aldo Moro 2, 00185 Rome, Italy

Correspondence should be addressed to Xin Wang, [email protected]

Received 17 February 2012; Accepted 28 March 2012

Academic Editor: Carlo Cattani

Copyright q 2012 Xin Wang et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

This paper presents a parallel real-time crowd simulation method based on a hierarchicalenvironmental model. A dynamical model of the complex environment should be constructedto simulate the state transition and propagation of individual motions. By modeling of a virtualenvironment where virtual crowds reside, we employ different parallel methods on a topologicallayer, a path layer and a perceptual layer. We propose a parallel motion path matching methodbased on the path layer and a parallel crowd simulation method based on the perceptual layer.The large-scale real-time crowd simulation becomes possible with these methods. Numericalexperiments are carried out to demonstrate the methods and results.

1. Introduction

Real-time crowd simulation is one of the important research directions [1, 2] in computer

games, movies, and virtual reality. In the last decade, Shao et al. [3] proposed a

multilevel model for the virtual crowd simulation in the face of visual effects, perception,

routing, interactive and other issues directly and efficient support. Virtual environment

modeling consists of two parts, that is, geometric environment modeling and nongeometric

environment modeling [4]. Geometric environment model corresponds to the geometric layer

of the hierarchical environmental model, and nongeometric environment model corresponds


to the topology layer, the path layer, and the perception layer. There have existed some

parallel methods [5], which can fast compute a scene path map based on the topology layer.

However, they lacked parallel methods to accelerate real-time crowd simulation based on the

path layer and the perception layer.

Given a motion path, matching human movements with it is a common problem in the

field of character animation. Usually artist manually tuned the motion data. However, when

lots of human movements need to be matched with lots of paths, only using manual method

will be impractical and cannot meet the real-time requirements. Based on our analysis, we

find that a parallel matching algorithm is suitable for motion path matching. Because motion

matching between paths is independent of each other, the path segments among every two

points within the calculation of the discrete sampling are independent. Based on the above

investigations, this paper employs a parallel motion path matching algorithm based on the

path layer.

Researchers often use the agent-based simulation model in crowd simulation. We

found that single agent status updates only need to consider it within a limited range state

of the world. Based on this observation, we designed a parallel crowd simulation method,

employing a parallel update strategy and dividing the scene into bins (each bin is a square

area; agents are distributed among all bins); we then determined the scope of each agent. For

the nonadjacent bins, agents can employ parallel update strategy, which is divided into four

batches of parallel updates. Experimental results show that this strategy can greatly increase

update efficiency.

In the real world, each person makes appropriate movement decisions according to

the state of the environment around him [6, 7]. For example, when a car approaches, people

will maintain their status or change their walking direction according to the car’s driving

direction, speed, and distance. The agent model proposed by Reynolds [8] can simulate these

situations. The current research employed a similar but relatively simple agent model. Our

agent is controlled by three forces: [9] avoiding the force of obstacles; avoiding collision with

other agent forces; following path force [10, 11]. The three forces are in order of decreasing

priority.

This paper presents parallel real-time crowd simulation algorithms based on a

hierarchy environmental model and fully taps the multilevel environmental model in the

parallelism, making the simulation of large-scale real-time crowds possible.

2. Related Work

Given a motion path, matching human movements with it is a common problem in the

field of character animation. In the game industry, a relatively simple and efficient way is

to loop the motion clip while the character moves along a motion path. This method makes

the movements of characters appear mechanical, monotonous, and unrealistic in terms of

imitating body movements. To achieve realistic effects, some scholars have proposed complex

motion synthesis methods [12–14]. The general approach first uses motion data, which are

captured to construct a graph-like data structure. In searching this data structure for the

right movement sequences to match input path, these sequences meet specific constraints

that stitch up human posture, finally forming the results. When matching motion data with

motion path, there exist some difficulties [15] as follows: (1) in the construction of the motion

graph, if the graph is large and complex, determining what motion can be synthesized from

the graph is difficult, and the range of new movements is unknown [16]; (2) the structure


of the motion graph is complex, which takes a lot of time to search on the graph each

time; hence, this structure cannot improve system performance [17]; (3) for the presence

of path constraints, there are some low-level connection relationships in the motion graph,

so there will be some unnecessary local shaking when synthesizing motion [12]; (4) some

constraints need to be considered (e.g., obstacles); however, such methods are difficult to

estimate.

Lau and Kuffner [18] described a similar method for this paper. The method

organizes motion data into a finite state machine (FSM); each state represents a high-

level semantical movement, such as walking and running. If two states can be connected

together, there will be one connection in the state machine. Motion state machine is

used to find the character’s motion data when synthesizing motion, which can overcome

the difficulties mentioned above. For multiple paths, this paper divided long path into

short path segments. The length of these short path segments is almost equal to the

motion segments in a motion state machine. Thus, when performing matching work,

the efficiency of matching can be improved. This paper also used parallel computing

strategy to greatly reduce the time required for each planning step. In addition, when

using motion state machine to match, a benefit arises; that is, while the current matching

segment crosses paths with other motion path segments, a collision situation is possible;

if such a situation happens, there will be no displacement motion data (idle) for the

role to effectively avoid the collision. On the other hand, when more human movements

are required to match more paths, solely using such methods cannot meet the real-time

requirements.

The agent-based crowd simulation model is accorded the characteristics of crowd

movement in the real world and has a very flexible control strategy (e.g., controlling each

parameter of each agent). Therefore, agent-based crowd simulation model is widely used.

However, for large-scale real-time crowd simulation, each agent needs to update its own

status according to its surrounding environment, and each agent is a dynamic obstacle to

other agents. Computing cost increases rapidly as the number of agent increases [19] in the

face of time complexity O(n2). Thus, when n is large, the time required for each frame will

significantly increase, making the real-time status update of each agent impossible. Hence,

we need to find a new method to do large-scale crowd simulation.

3. Overview

The system consists of two parts: (1) hierarchy environment model and (2) multiple parallel

algorithms based on the hierarchy (see Figure 1).The hierarchy environmental model [20], as shown in Figure 1 (right panel), consists

of two parts: geometric and non-geometric environment model. The non-geometric environ-

ment model includes the topology layer, path layer, and perception layer.

Based on the multi-level environment, the topology layer, path layer, and perception

layer employ different parallel algorithms to speed up real-time crowd simulation, respec-

tively, as shown in Figure 1 (left panel). We employed the existing Voronoi diagram algorithm

[4] based on topology layer to segment the scene quickly and get the path layer map. We also

used parallel motion path matching based on path layer to quickly match human motion for

path, as well as parallel real-time crowd simulation based on perception layer to fully tap the

crowd behavior in parallelism.


Voronoi diagram

scene segmentation

Parallel motion

path matching

Parallel real-time

crowd simulation

Topology

layer

Path

layer

Perceptual

layer Room 5

Room 3

Room 4Room 3

Room 1

Room 5

Room 2

Room4Room3

Room1

Room5

Room2

Nongeometric environment model

A hierarchy environmental model

13

4

2

Geometric

layer

Geometric environment

model

Figure 1: System overview.

4. Parallel Motion Path Matching Based on Path Layer

This section explains parallel motion path matching based on path layer. The section contains

three parts: (1) motion state machine, (2) motion path parameterization, and (3) extraction of

motion segments.

4.1. Motion State Machine

The main differences between the motion state machine proposed by Kovar et al. [12] and

the motion state machine constructed in this paper are as follows: (1) trajectory arc length

of the node of motion segments used by each state is fixed and equal in this paper’s motion

state machine; (2) the length of most of the motion segments in this paper is longer than

that in Kovar et al. [12]. For example, there are 100 frame movements about walking used

in this paper, but the length of the movements in Da Silva et al. [21] is only 25 frames;

(3) the motion state machine in this paper employed group-based hierarchical status node,

which can ensure that “walk left” and “run left” are on the same level, and all directional

run and walk movements comprise a position move node on the up level. Three facts exist

in the motion synthesis process, which lead to the above differences: (1) this paper used

sectional matching; (2) the movement range of the motion path to be matched is large; (3) the

connection relationships among states are more complex.

The motion state machine (MSM) used in this paper is shown in Figure 2. The MSM

has a total of 12 base states, which have their own names and include 36 motion segments

(except start and end states). The motion segments’ arc length of “Crawl,” “Squat move,”

and “Stride over” is indeterminate, because these motion segments are mainly used to deal

with the path segments about obstacles. These are marked directly by users to explain which


Start

End

Crawl

Squat

move

Stride

over

Idle

Run

forward

Run leftRun

right

L

F

R

Position move

Walk

forward

Walk

left

Walk

right

Figure 2: Finite motion state machine.

motion segments are used, so that the system does not need to match in the rear. The function

of the “Idle” state’s motion segments is to make the virtual characters wait, so that the system

can avoid the collision between virtual characters. The states, which include “run forward,”

“run left,” “run right,” “walk forward,” “walk left,” and “walk right,” describe the ways in

which virtual characters can match according to the specific path situation. Furthermore, the

motion path’s arc length of motion segments is equal, which makes the matching of motion

path segments easy.

There are some directed edges among states in MSM. These directed edges express

whether two motion segments can be connected. For example, there is a directed edge

between “run right” and “walk right,” which means that the motion segments of “run right”

can stitch with the motion segments of “walk right.” Similarly, the motion segments of

“walk left” can stitch with the motion segments of “run left.” The motion machine in this

paper adopts a group-based hierarchical status node. For example, “walk forward” and “run

forward” constitute F, “run left” and “walk left” constitute L, and “run right” and “walk

right” constitute R. Finally, L, F, and R constitute locomotion. The child states can inherit

the father states’ connection. For example, the connection between F and R explains that

“walk forward” and “run right” can connect with each other, and the connection between the

“crawl” state and l locomotion can lead to the connection between “crawl” and “run left.”

This paper also prepares some transitional motion segments. These motion data are

used to connect the transitional fragments among motion segments. There are eight frames.

The transitional fragments exist at the beginning and the end of motion segments.


4.2. Motion Path Parameterization

In this paper, there is a path planner responsible for generating a parameterized original

movement path called “T” and decides the grid size of path planning pace. If the planning

space is planed in a lattice structure, which is composed of m ∗m square, the space between

2d point sequence is m, but in the specific application, the space of sample points cannot

keep the same with the sample degree of the original path. For example, in this research,

the human root node movement size between two frames is s in the MSM. However, the

motion clips in the MSM were prepared before motion synthesis and cannot be resampled.

The system initially needs to use some fitting methods to parameterize the original path

and then resamples the parametric path based on the desired sampling size. The resulting

path can be used to match the crowd animation better in the following phase. In detail,

our method constructs the cubic polynomial curve p(u) with C1 continuous based on

the original data point. If there is a curve formula called p(u), the system begins with a

starting point p0 and records those points p1, p2, . . . , pn, whose arc length spacing is s and

corresponding parameters u1, u2, . . . , un. The characteristics of cubic polynomial spline curve

with C1 continuous are simple and easy to control, and human motion path is generally not

so smooth that curve with C1 continuous is enough to make the motion path smooth through

fitting.

In addition, the algorithm needs the corresponding tangent vector of each sample

point. The system can calculate p(u) curve’s tangent vector in p0, p1, . . . , pn through

parameters u0, u1, . . . , un. Then, it can represent the motion path after normalizing with the

coordinates of parameters point and tangent vector, called T1:

T1 ={pi, �pi | i = 0, 1, . . . , n

}. (4.1)

The arc length in our motion segments in the MSM is almost equal. The arc length

of motion state i is represented by li. If li is n times as long as s, motion path T1 used in

experiment general is n times longer than s. Hence, it can cut up the whole path into many

small segments whose length is n ∗ s. For example, suppose there are n + 1 points from pj to

pj+n. The algorithm can find motion segments called motionk in the MSM, which is similar

to the path curve shape. If there are some obstacles in the path, direct matching cannot be

performed. The path is marked by specific segments of motion data through interaction.

Then, it can stitch directly in the matching phrase.

There are some crossings of motion paths and obstacles in T1. Thus, matching directly

to T1 is not enough. An artificial mark on this cross-path, such as “crawl” state, is needed.

In the matching, when meeting this path, motion data are directly found under the “crawl”

state and are associated with motion path T2:

T2 ={pi, �pi,

(motion

(j, t)= motionk

)r| i = 0, 1, . . . , n; 0 ≤ j < n; 0 < t ≤ n − j; r = 0, 1 . . . , u

}.

(4.2)

The path segments composed of continuous t points from pj have been associated with

motion data called motionk; r means that this mark is the r-section in all u+ 1 marks. Figure 3

shows the normalized conversion process from curve T to curve T2.


SS

S

Motionk

T

T

1

T1

−p0

−p1

−p2

−p3

−p4

p0

p1

p2

p3

−p4

p i−1 p i p i+1

p i−1 p i p i+1 −pn 1

−pn

−pn 1

pn

pi−1pi pi+1

−

−

p1

p0

p2

p3

p4

p0

p1

p2

p3

p4

pi−1 pi+1pipn

pn−1

pn−1

pn

m

m

Figure 3: Normalized conversion process from curve T to curve T2.

T2

di

−p0

−p1

−p2

−p3

p4

−p5

p5

−pn 1

p0

p1

p2

p3

p4

p5

p6

p5

p7

pi−

−−

1pi pi+1

pi+t

pn 1

pn

p5+k

d

Figure 4: Schematic diagram of T2 segmentation.

4.3. Extraction of Motion Segments

T1 changes into T2 through parametric resampling and marking the specific segment. Setting

the path consists of n + 1 points, and total arc length is n ∗ s. The basic motion segment of

the root node’s path curves arc length is certain, set as d, such as “walk” and “run.” d is also

integer times as s, set as k. Thus, T2 needs to be cut into path segments, whose arc length is d.

The set of this path segments is D. Consider

D = {di | i = 0, 1, . . . , Dsize − 1}, (4.3)

where di means the ith motion path segment. Dsize means the total number of segments after

cutting:

di ={pi | i = b + 0, b + 1, . . . , b + k, b = di starting coordinates

}. (4.4)

Figure 4 shows the T2 segmentation process.


Block 0 Block 1 Block 2 Block 3 Block m

Grid (m , 1)

······

Figure 5: Schematic diagram of Grid (m, 1).

The next work that needs to be done is using the distance formula D to find proper

motion segment motionk for each di.

This paper designs a one-dimensional Grid (motionNum, 1) for single curve.

motionNum means the number of motion segments in the database, and one Grid has

motionNum blocks. Each block is responsible for calculating its representative motion

segments. For example, block0 is only responsible for calculating the matching degree

between the current line and the 0th motion segment. Each block is one-dimensional. For

example, in block (PointNum, 1), PointNum means the number of discrete points currently

calculating the curve segment matching. Each thread in the block means the distance between

a discrete point in the curve and the corresponding point in the motion segment. For example,

thread0 is responsible for calculating the distance of 0th discrete point. The formula is

distancei =√

CurvePointi − MotionPointi. (4.5)

Each thread only needs to calculate its own data, and it does not need to interact with

other threads. After the final computation, thread0 accumulates matching value, calculating

by each thread, and acquires the total matching value between the curve and the motion

segment. The computation formula is

MatchDegree =n∑i=0

distancei. (4.6)

After calculating the matching value with all motion segments, the minimum

matching value is obtained through comparison. The final result is

Motion ={j | MatchDegreej = min

i∈(0,n)(MatchDegreei

)}. (4.7)

The matching effect of single curve is shown in Figure 6, and its internal thread

organization way is depicted in Figure 5. Before calculating the distance between a discrete

point in curve and the corresponding point in motion segment, every point in a motion

segment needs to do a rotating translation operation. This step is necessary because motion

segment only converts to a curve’s local coordinate. The matching value can be obtained

accurately. Now, the system selects thread0 to calculate rotation and offset. Meanwhile, the

other threads wait for the calculating result of thread. Then, they share the calculating result

of thread0.

We need to design a two-dimensional Grid (motionNum, CurveNum) to extend to

multiple curves, as shown in Figure 7. motionNum means the number of motion segments.


Figure 6: Matching effect of single curve.

Block(0,0)

Block(1,0)

Block(2,0)

Block(3,0)

Block(n,0)

Block(0,1)

Block(1,1)

Block(2,1)

Block(3,1)

Block(n,1)

Block(0,2)

Block(1,2)

Block(2,2)

Block(3,2)

Block(n,2)

Block(0,3)

Block(1,3)

Block(2,3)

Block(3,3)

Block(n,3)

Block(0,m )

Block(1,m )

Block(2,m )

Block(3,m )

Block(n,m )······

······

······

······

······

.

.

....

.

.

....

Grid (m ,n)

Figure 7: Schematic diagram of Grid (m,n).

CurveNum means the number of curves. Others are similar to the single curve. The matching

effect of multiple curves is illustrated in Figure 8.

This method can achieve better expansibility. If there are more motion segments that

need to match, then the dimension of Grid only increases.

5. Parallel Real-Time Crowd Simulation Based on Perception Layer

This section explains how to parallel real-time crowd simulation based on perception layer.

This section contains three parts: (1) scene segmentation; (2) nearest neighbor query; (3)parallel real-time crowd simulation.

5.1. Scene Segmentation

In agent-based crowd simulation, each agent has its own perceived range. In each update,

each agent must query the scope of its perception to obtain the range of obstacles or

information of other agents. Then, based on the information to calculate the forces acting

on the agents, their speed and direction can be adjusted, and the location can be updated.

Neighbor queries generally employ serial query that is executed when the current agent is

updated. Then, the next agent begins to update. In this algorithm, using parallel neighbor


Figure 8: Matching effect of multiple curves.

Agent

120 3

4 5

67

8

9 1011

12

13 14 15

16

17 18 19

2021 22 23

2425 26

Figure 9: Scene segmentation and agent distribution (circle represents the agent).

query, the main query is the bin, rather than each agent. Through these neighbor queries,

the bin can be obtained, in which agents need to consider all the potential neighbor agents.

Before the implementation of parallel neighbors’ query, the entire scene needs to be evenly

segmented to get all the information about the bin.

The segmentation needs to be completed in the initialization process. The size of the

entire scene is 1200 × 1200; setting the side length of each bin is 4 for the square, so that

the whole scene is to be divided into 300 × 300 bins. The square side length is 4, because the

sensing range of agent sets a circular area that is a query radius of 4. The formula is as follows:

queryRadius = predTime ∗ maxVelocity ∗ 2. (5.1)

The predTime is the forward predictive time, set as 1 s. maxVelocity is the agent’s

maximum move speed, set as 2 m/s. In the query process, only 8 bins need to be considered,

which are around the center bin.

As shown in Figure 9, the scene is evenly divided into a number of bins, the scene of

the agents located in each bin. There are multiple agents in a bin; however, a bin can also have

no agent.


Observation range ofgreen bin

0 1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20 21

22 23 24 25 26 27 28 29 30 31 32

33 34 35 36 37 38 39 40 41 42 43

44 45 46 47 48 49 50 51 52 53 54

55 56 57 58 59 60 61 62 63 64 65

66 67 68 69 70 71 72 73 74 75 76

77 78 79 80 81 82 83 84 85 86 87

88 89 90 91 92 93 94 95 96 97 98

99 100 101 102 103 104 105 106 107 108 109

Figure 10: Observation range of bin (dashed circle).

5.2. Nearest Neighbor Query

After scene segmentation and statistical distribution among the agents in each bin, the nearest

neighbor query is employed. In the scene segmentation step, each Agent’s observation range

is a circular area of radius 4, and the bin side is also 4. Thus, the algorithm only needs to check

the bin and approximately 8 bins. Figure 10 shows that the number of green bins is 49 and the

observation range is the dashed circle. Hence, the bin numbers that need to be checked are

37, 38, 39, 48, 49, 50, 59, 60, and 61. As the neighbor queries between the bins only read data

with no write operation, this paper chooses the parallel query; the query results are saved in

the corresponding data structures.

5.3. Parallel Real-Time Crowd Simulation

After completing the nearest neighbor querying, crowd velocity, and position updating, the

next Agent begins to update. At the step of parallel nearest neighbor query, the algorithm

has queried all the neighbors of the bin. However, as the Agents of the adjacent bin would

interact, this step of the update operation cannot be completely parallel. For example, for bin

48 and bin 49, if the two parallels update, there would be a read/write conflict. The reason

is that when the Agents of bin 48 are updating, the Agents of bin 49 should be considered;

hence, the reading data may be outdated Agent data of bin 49, and vice versa. Therefore, this

paper chooses partial parallel update strategy; steps are described below in detail.

Referring to the behavior rules of pedestrians in the real world and traditional Agent

model, the Agent model used in this paper is acted upon by three kinds of forces, namely, (1)avoid the force of obstacles; (2) avoid collision with other Agent forces; (3) follow the path of

force. Figure 11 illustrates each kind of force.

Three kinds of forces decline in priority order. Avoiding the behavior of obstacles is the

highest priority; that is, when Agent k detects it will crash into obstacle d, it also may have

collided with Agent g, and then it would only consider avoiding obstacle d. Considering that

the obstacle is stationary, if the Agent does not adjust its speed, it will be hit and move. Even

if Agent k does not make an adjustment, Agent g itself can adjust the speed for Agent k.

Avoiding force is calculated as follows. Agent forward predictive distance with the observed

radius builds up a rectangular area; in the region of collision, we can find the nearest obstacle.

As shown in Figure 11 for obstacle d, avoiding force (Force k) for the current Agent is pull


Force j

Agent jP’

Pj’ i’

Agent i

PathObstacle e

Force h

Agenth

Obstacle dAgent g

Agent k

Force k

Figure 11: Schematic diagram of agent’s three kinds of forces.

force and the obstacle repulsive force direction. Another situation is when Agent j moves to

prediction j’ position, and Agent i will reach position i at the same time. When the distance

between position j and position i is less than the distance between the center of the radius of

their observations, a collision will be assumed; thus, they will not be influenced by the force

of path-following, but by the Agent force between the impact of avoidance, the avoidance of

force as a lateral tension (Force j), to deviate the current running direction. In Figure 11, there

is a lane in the path (dotted line parallel with the path and the distance between the path);the current updated Agent will forecast forward for some distance to predict the location of

P projected onto the path P’ point. If the distance between P’ and P is greater than the lane,

then the Agent’s travel direction deviates from the path; otherwise, there is no deviation. As

shown in Figure 11, Agent h would be acted upon by the force of path following (Force h), for

the projection point P’ of the direction of the Agent center of the connection to slowly move

closer to the path direction.

This paper makes parallel updates on the Agent rates four times. As shown in

Figure 10, the bin has a total of four colors. Each time parallel updates the bin that has the

same color. When all colors have been updated, a general update is completed. Agents within

the same bin have an impact on each other, but they cannot be updated in parallel; hence, the

serial update can only be used. In the implementation process, each block represents a bin,

following the Agent model above; meanwhile, according to the priority of various types of

forces, the research also carries out the appropriate adjustments.

6. Experimental Results

In general, the experiment platform uses a computer of 4 core (Q9950 CPU at 2.83 GHz

and 4 GB of RAM), equipped with a NVIDIA GTX 280 graphics card, which has 30

multiprocessors. Each multiprocessor has 8 cores; the total number of cores is 240.

The CPU-based algorithm achieves serial algorithms, and the GPU-based algorithm achieves

the proposed parallel algorithms in this paper.


0 20 40 60 80 100 120 140 200 400 600 800 1000

CurveNum

120000

100000

80000

60000

40000

20000

0

Parallel (GPU) time (ms)

Serial (CPU) time (ms)

Ex

ecu

tio

n t

ime(m

s)

Figure 12: Comparison between CPU-based and GPU-based algorithms.

Table 1: Comparison between CPU and GPU execution time.

10 100 200 500 1000

CPU execution time (ms) 1214.9124 11999.31 25312.8457 56183.2070 112101.531

GPU execution time (ms) 70.2647 549.388 1049.8966 2572.0297 5188.6655

Figure 12 clearly shows that in the path matching based on path layer, with the increase

in the number of curves that are required for matching, the CPU’s execution time increases

significantly, whereas the GPU’s execution time barely increases.

Table 1 lists the matching time based on the path matching on the path layer with the

CPU matching algorithm and GPU matching algorithm under some curve lines. The data in

the table indicate that GPU parallel algorithm would increase by about 20 times than the CPU

algorithm in execution speed.

Figure 13 shows the frame rate comparison between GPU parallel algorithm and CPU

serial algorithm in real-time crowd simulation based on the perceived layer.

As can be seen from Figure 13 GPU-based parallel computing speed is significantly

higher than the CPU-based parallel computing speed in crowd simulation. With the increase

in the number of Agents, CPU-based FPS drops very quickly. When the number reaches

more than 5,000, real-time simulation results are not achieved. Although the GPU program

number is above 10000 of the Agent, FPS can reach 24 or more, fully meeting the real-time

requirement. When the number is 1000, the FPS of the GPU is 60.

In addition, as the curves indicate, the FPS of GPU-based algorithms is significantly

higher than that of the CPU-based algorithms, but it still has not reached 10 or more times.

The main reason is the speed of Agent updating; there are a lot of conditional executions,

such as statements reducing the parallel computing efficiency of the GPU-based algorithm.

7. Discussion

Through the proposed parallel algorithms in this paper, we achieve a completely parallel

real-time crowd simulation based on a hierarchy environmental model, making topology


1000 3000 5000 7000 9000 11000

60

55

50

45

40

35

30

25

20

15

10

5

0

Number of agents

FPS

FP

S

GPU’

CPU’ FPS

Figure 13: Performance comparison between CPU and GPU in crowd simulation.

layer, path layer, and perception layer have corresponding parallel algorithms to speed up

the calculation.

This paper describes the detailed process with the path layer-based motion path

matching, through the structure of motion state machine, setting up the transfer relationship

among movement segments. Specifying the sequence of key points, this paper chooses cubic

spine curve fitting to get a continuous path and then samples the sequence of points needed.

We use the distance function to calculate the matching degree between the motion segment

and the path, then accumulate the value of discrete points, and get the total matching degree

of the corresponding segment. Moreover, this paper explains the use of parallel computing

algorithms to accelerate and verify the algorithm through data analysis.

This paper introduces parallel computing algorithm based on the perception layer to

achieve real-time simulation of large crowds. Using scene segmentation evenly, according to

his own location, each Agent is assigned to the appropriate bin where the original calculations

must be serialized into parallel computing. Each bin is a separate update unit.

Experimental results suggest that the proposed method in this paper is consistent with

the serial method in effect, but efficiency has been greatly improved. Given that the Agent

model used in this paper is relatively simple, the focus of future work is how to use more

complex models to achieve a more realistic real-time crowd simulation, reduce the occurrence

times of logic-based computing in the parallel algorithm framework, and further improve the

algorithm’s parallelism.

Acknowledgments

This work was supported by Natural Science Foundation of Zhejiang Province (Y1110882,

Y1110688, R1110679), Department of Education of Zhejiang Province (Y200907765,

Y201122434), and Doctoral Fund of Ministry of Education of China (20113317110001).


References

[1] N. Farenc, S. R. Musse, E. Schweiss et al., “A paradigm for controlling virtual humans in urbanenvironment simulations,” Applied Artificial Intelligence, vol. 14, no. 1, pp. 69–91, 2000.

[2] F. Tecchia, C. Loscos, R. Conroy et al., “Agent Behavior Simulator (ABS): a platform for urbanbehavior development,” in Proceedings of Games Technology Conference, 2001.

[3] M. Shao, X. Wang, and Y. Hou, “Crowd evacuation simulation based on a hierarchy environmentalmodel,” in Proceedings of the IEEE 10th International Conference on Computer-Aided Industrial Designand Conceptual Design: E-Business, Creative Design, Manufacturing (CAID & CD ’09), pp. 1075–1078,November 2009.

[4] S. Chen, Y. Wang, and C. Cattani, “Key issues in modeling of complex 3D structures from videosequences,” Mathematical Problems in Engineering, vol. 2012, Article ID 856523, 17 pages, 2012.

[5] M. Denny, “Solving geometric optimization problems using graphics hardware,” Computer GraphicsForum, vol. 22, no. 3, pp. 441–451, 2003.

[6] S. Chen and Z. Wang, “Acceleration strategies in generalized belief propagation,” IEEE Transactionson Industrial Informatics, vol. 8, no. 1, pp. 41–48, 2012.

[7] S. Y. Chen, H. Tong, Z. Wang, S. Liu, M. Li, and B. Zhang, “Improved generalized belief propagationfor vision processing,” Mathematical Problems in Engineering, vol. 2011, Article ID 416963, 12 pages,2011.

[8] C. W. Reynolds, “A distributed behavioral model,” in Proceedings of the ACM Computer Graphics(SIGGRAPH ’87), M. C. Stone, Ed., pp. 25–34, 1987.

[9] C. W. Reynolds, Steering Behaviors for Autonomous Characters, Sony Computer Entertainment America,Boulevard Foster City, Calif, USA, 1999.


[11] M. Li and W. Zhao, “Representation of a stochastic traffic bound,” IEEE Transactions on Parallel andDistributed Systems, vol. 21, no. 9, pp. 1368–1372, 2010.

[12] L. Kovar, M. Gleicher, and F. Pighin, “Motion graphs,” in Proceedings of the ACM Transactions onGraphics (ACM SIGGRAPH ’02), pp. 473–482, July 2002.

[13] O. Arikan and D. A. Forsyth, “Interactive motion generation from examples,” in Proceedings of theACM Transactions on Graphics (ACM SIGGRAPH ’02), pp. 483–490, July 2002.

[14] J. Lee, J. Chai, P. S. A. Reitsma, J. K. Hodgins, and N. S. Pollard, “Interactive control of avatarsanimated with human motion data,” in Proceedings of the ACM Transactions on Graphics (ACMSIGGRAPH ’02), pp. 491–500, July 2002.

[15] S. Y. Chen, H. Tong, and C. Cattani, “Markov models for image labeling,” Mathematical Problems inEngineering, vol. 2012, Article ID 814356, 18 pages, 2012.

[16] P. S. A. Reitsma and N. S. Pollard, “Evaluating motion graphs for character navigation,” in Proceedingsof the ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pp. 89–98, Grenoble, France,2004.

[17] J. Lee and K. H. Lee, “Precomputing avatar behavior from human motion data,” Graphical Models, vol.68, no. 2, pp. 158–174, 2006.

[18] M. Lau and J. J. Kuffner, “Behavior planning for character animation,” in Proceedings of the 5thEurographics Symposium on Computer Animation (ACM SIGGRAPH ’05), pp. 271–280, Los Angeles,Calif, USA, July 2005.

[19] S. M. Lavalle, Planning Algorithms, Cambridge University Press, Cambridge, Mass, USA, 2006.

[20] S. Chen, J. Zhang, Y. Li, and J. Zhang, “A hierarchical model incorporating segmented regions andpixel descriptors for video background subtraction,” IEEE Transactions on Industrial Informatics, vol. 8,no. 1, pp. 118–127, 2012.

[21] A. R. Da Silva, W. S. Lages, and L. Chaimowicz, “Improving boids algorithm in GPU using estimatedself occlusion,” in Proceedings of SBGames’08: Computing Track, Computers in Entertainment (CIE), pp.41–46, 2008.


Research ArticleOptimization of Resource Control forTransitions in Complex Systems

Florin Pop

Faculty of Automatic Control and Computers, University Politehnica of Bucharest,Splaiul Independentei 313, 060042 Bucharest, Romania

Correspondence should be addressed to Florin Pop, [email protected]

Received 7 March 2012; Accepted 2 April 2012


Copyright q 2012 Florin Pop. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

In complex systems like Large-Scale Distributed Systems (LSDSs) the optimization of resource con-trol is an open issue. The large number of resources and multicriteria optimization requirementsmake the optimization problem a complex one. The importance of resource control increases withthe need of use for industrial process and manufacturing, being a key solution for QoS assuring.This paper presents different solutions for multiobjective decentralized control models for tasksassignment in LSDS. The transaction in real-time complex system is modeled in simulation bytasks which will be scheduled and executed in a distributed system, so a set of specificationsand requirements are known. The paper presents a critical analysis of existing solutions andfocuses on a genetic-based algorithm for optimization. The contribution of the algorithm is thefitness function that includes multiobjective criteria for optimization in different way. Severalexperimental scenarios, modeled using simulation, were considered to offer a support for analysisof near-optimal solution for resource selection.

1. Introduction

The resources control in complex systems requires information about resources and tasks. A

near-optimal assignment could be made based on some criterion function, such as minimum

execution time or load balancing. There has been a steadily increasing interest, supported

by advanced technological and economic developments, into dealing with very complex

(dynamical) systems describing natural phenomena or manufacturing processes [1]. The

particular interest in the study of complex systems is tipping points where one observes a

sudden change in the dynamics, sometimes referred to as critical transitions, modeled as

tasks to be submitted for execution on physical resources. For instance, medical conditions

such as asthma attacks and epileptic seizures can change quickly from regular to irregular


behavior, the financial markets are known to suddenly break trends in a crisis, and climate

conditions and ecological environments can change rather abruptly. The understanding of

the dynamical behavior near tipping points would enable human interaction to attenuate or

control the consequences of critical transitions [2].The scheduling process is considered to be the core of resources control in complex

systems. Due to the NP-complete nature of scheduling algorithms, current research directions

are focused on finding suboptimal (near-optimal) solutions, which can be further divided

into the following two general categories: approximate and heuristic algorithms. At global

level, two-phase scheduling solution comprised of a set of heuristic subalgorithms to achieve

optimized scheduling performance over the scope of overall resources is a new research

subject in the present [3, 4].Today, engineers face an increasing challenge in advanced applications with different

requirements and constrains. Innovative developments for efficient mathematical approaches

focused on approximate algorithms, heuristics-based methods, and bio-inspired models. The

approximate algorithms use formal computational models, but instead of searching the entire

solution space for an optimal solution, they are satisfied when a solution that is sufficiently

good is found. In the case where a metric is available for evaluating a solution, this technique

can be used to decrease the time taken to find an acceptable schedule. The factors which

determine whether this approach is worthy of pursuit include [5, 6] availability of a function

to evaluate a solution, the time required to evaluate a solution, the ability to judge the

value of an optimal solution according to some metric, and availability of a mechanism for

intelligently pruning the solution space. The paper proposes a mathematical approach for

resources control based on a multicriteria optimization genetic algorithm. One of the well-

known problems of genetic algorithms is that, for large solution space, the convergence time

is high.

The rest of the paper is structured as follows. Section 2 describes the optimization

methods. Section 3 approaches the multidimensional optimization methods for real-time

system. The proposed genetic algorithm is described in Section 4. The tests conducted on the

proposed algorithm (Section 5) highlight the improvement provided by this new approach

not only in terms of convergence time, but also in terms of solution quality.

2. Optimization Methods for Decentralized Control

Optimization methods for resource control use heuristic (multiobjective) approaches. The

allocation problem considers a set of n tasks, T = {T1, T2, . . . , Tn}, for some finite integer n,

that models a set o transitions that will use a multiple processor system (e.g., cluster system)in which each transition can be characterized by multiple parameters: Ti = {ai, ti, Ci, ri, ωi . . .},

where ai is the arrival time (the time when the transition is produce), ti is the execution time(it can be estimated or calculated), Ci is the completion time (with the following condition:

ai + ti ≤ Ci), ri is a rate 0 < ri ≤ 1 (can be a normalized priority with∑n

i=1 ri = 1), 0 ≤ ωi ≤ 1 is

a weight (with the special normalization condition∑n

i=1 ωi = 1), and we can have some other

parameters which characterize the task.

In this model, a complex system has a number of m resources R = {R1, R2, . . . , Rm}.

Each resource Rj has specific characteristics like capacity, latency, memory type and space,

CPU processing characteristics, and storage limitation. The most important characteristic

used in the proposed model is the utilization rate (uj) that measures the processing capacity.

The workj done by Rj in order to process a task Ti is defined as its running time multiplied


by the resource utilization rate, workj(Ti) = tiuj ≤ (Ci − ai)uj . A valid schedule is defined

as Sched = {(Ti, Rj) | Ti ∈ T, Rj ∈ R}. Similarly, the work of set of scheduled tasks is

work(T, R) =∑

(Ti,Rj )∈ Sched workj(Ti). Usually it is assumed that in the case of malleable tasks

the work of a task cannot be decreased by spending more processors on it (preservation of

work). Similarly the work of a task cannot be decreased by using virtualization.

To evaluate the efficiency o resource utilization a resource is considered to be Active(working) or Idle (waiting for new tasks). Efficiency E(t) at time t is E(t) = Resource

Active(t)/(Resource Active(t) + Resource Idle(t)). In general we are looking for a feasible

solution to scheduling problem. This is a schedule which meets all the requirements and

constrains.

For optimization, there are bottleneck objectives and sum objectives. The scheduling

problem considers the following objective for optimization: maximum completion time (Cmax =maxi{Ci}), weighted completion time (Cw =

∑ni=1 wiCi), or maximum lateness (Lmax = maxi{|(Ci−

ai) − ti|}).Another important aspect of scheduling optimization considers real-time systems.

These type of systems are defined as those systems in which the correctness of the system

depends not only on the logical result of computation, but also on the time at which the results

are produced. If the timing constraints of the system are not met, system failure is said to have

occurred. Hence, it is essential that the timing constraints of the system are guaranteed to be

met.

2.1. Heuristics for Resources Control

Opportunistic Load Balancing (OLB)

The Opportunistic Load Balancing heuristic selects the task Ti arbitrarily from the group of

tasks and assigns it to the next resource that is expected to be available [7, 8]. It does not

consider the workj(Ti), which may lead to very high Cmax. If all tasks are scheduled with

respect to condition a1 < a2 < · · · < an the heuristic is called First Come First Served (FCFS) [9].

Minimum Execution Time (MET)

The heuristic assigns each task selected arbitrarily to the machine with the least expected

execution time for that task [10].

Minimum Completion Time (MCT)

The heuristic assigns each task selected in arbitrary order to the machine with the minimum

expected completion time for that task [10]. The MCT combines the benefits of OLB and MET

and tries to avoid the circumstances in which OLB and MET perform poorly.

Min-Min

The heuristic begins with the set T of all task to be unscheduled. Then, the set C of minimum

possible completion times of all tasks on any of the machines is computed: Cmin = mini{Ci}.

The task with the Cmin is then assigned on a processor with minimum expected work.


Max-Min

The heuristic is very similar to min-min, but considers Cmax and then assigns the task on a

processor with minimum expected work [10, 11].

Duplex

The heuristic is a combination of the min-min and max-min heuristics. The heuristic performs

both of the min-min and max-min heuristics and used the better solution [8, 10, 11].

Genetic Algorithms (GAs)

They are used for searching large solution spaces with multiple possible schedule of tasks.

Each possible schedule is modeled by a chromosome that has a fitness value, which is the

result of an objective function designed in accordance with the performance criteria of the

problem (Cmax or Lmax) [12].

Simulated Annealing (SA)

It is an iterative technique that considers only one possible solution (mapping) for each task at

a time [13]. This solution is modeled like in a GA. SA uses a procedure that probabilistically

allows poorer solutions to be accepted to attempt to obtain a better search of the solution

space.

A∗

Heuristic is a search technique that has been applied in various task allocation problems. The

A∗ heuristic begins at a root node that is a null solution. As the tree grows, nodes represent

partial schedule. A pruning process is performed to limit the maximum number of active

resources at any one time. A cost function f(Sched) is associated with a partial solution (e.g.,

f(Sched) = Cmax).Guaranteeing timing behavior requires that the system could be predicted. Predictabil-

ity means that when a task is activated it should be possible to determine its execution time

with certainty. It is also desirable that the system attains a high degree of utilization while

satisfying the timing constraints of the system [14–16].

2.2. Resource Control in Real-Time Complex System

A complex system is said to be real-time if there exists at least one task Ti ∈ T , which falls into

one of the following categories.

(1) Task Ti is a hard real-time task. The execution of the task Ti should be completed by

a given deadline, ai + ti ≤ Ci.

(2) Task Ti is a soft real-time task. If a task Ti finishes the work after a given deadline Ci, thepenalty is pays. A penalty function P(Ti) is defined for the task. If ai + ti ≤ Ci, the

penalty function P(Ti) = 0. Otherwise P(Ti) = (ai + ti) − Ci > 0.


(3) Task Ti is a firm real-time task. If a task Ti finishes the work before a given deadline

Ci, the more rewards it gains. A reward function R(Ti) is defined for the task. If

ai+ ti ≥ Ci, the reward function R(Ti) = 0 is zero. Otherwise R(Ti) = Ci− (ai+ ti) > 0.

The set of real-time tasks T can be a combination of hard, firm, and soft real-time

tasks. Let TS be the set of all soft real-time tasks in T . The penalty function of the system is

P(T) =∑|TS|

i=1 P(TS,i). Let TF be the set of all soft real-time tasks in T . Similarly, the reward

function of the system is R(T) =∑|TF |

i=1 R(TF,i).The following goals should be considered in scheduling a real-time system: (i) meeting

the timing constraints of the system; (ii) preventing simultaneous access to shared resources

and devices; (iii) attaining a high degree of utilization while satisfying the timing constraints

of the system; (iv) reducing the cost of context switches caused by preemption; (v) reducing

the communication cost in real-time distributed systems. In addition, the following criteria

are considered in advanced real-time systems: (vi) considering a combination of hard, firm,

and soft real-time activities, which implies the possibility of applying dynamic scheduling

policies that respect the optimality criteria; (vii) task scheduling for a real-time system whose

behavior is dynamically adaptive, reconfigurable, reflexive, and intelligent; (viii) covering

reliability, security, and safety. Basically, the scheduling problem is to determine a schedule

for the execution of the task so that they are all completed before the overall deadline [14, 15].

3. Multidimensional Optimization Methods: Applications

Multidimensional optimization methods are useful when the search space is likely to have

many local optima, making it hard to locate the global optimum. In low-dimensional or

constrained problems it may be enough to apply a local optimizer starting at a set of possible

start points, generated either randomly or systematically (for instance, at systems locations),and choose the best result. However this approach is less likely to locate the true optimum as

the ratio of volume of the search region to number of starting points increases. The application

of different multidimensional optimization method proves that finding the global optimum

is a hard problem.

Application of Simplex Method

Scheduling of vehicles in the container terminal is often studied as a static problem in the

literature, where all information, including the number of task, their arrival time, and so

forth, is known beforehand. The objective is generally minimizing the total traveling and/or

waiting times of the vehicles. When the situation changes, for example, new jobs arrive or a

section of the terminal is blocked, new solutions are generated from scratch.

Application of Simulated Annealing Method

A parallel approach of a modular simulated annealing (MSA) algorithm, a shortened

SA algorithm, applied to classical job-shop scheduling (JSS) problems is presented. The

JSS problems tackled are very well-known difficult benchmarks, which are considered to

measure the quality of such systems.


Step 1. User requests

Step 3. Task preparation

for scheduling

Step 4. Monitoring thesystem receive theavailable resources

Step 6. Migrate to the

best solution

Step 7. Get a localoptimum and end the GA

Step 8. Aggregate the

results for a near-globaloptimum

Step 9. Keep the schedule

in a dedicated repository

Step 5. Run thescheduling algorithm(GA)

Step 2. Create the “batch

of tasks“

Figure 1: Main actions of proposed algorithm.

Application of Genetic Algorithms

GAs are developed for solving the machine-component grouping problem required for

example a cellular manufacturing systems. GA provides a collection of satisfactory solutions

for a two-objective environment (minimizing cell load variation and minimizing volume of

inter cell movement), allowing the decision maker to then select the best alternative.

4. Genetic Algorithm for Resources Control in LDSD

In [17] Iordache et al. present a genetic algorithm for decentralized scheduling. The

description of the scheduling algorithm in a logical flow of activities is described in the

following steps. The important contribution of this algorithm is the fitness function that

considers multiobjective criteria for optimization (see Figure 1).

Step 1. A user requests that one or more tasks are scheduled.

Step 2. The input is processed as a “batch of tasks” (group of tasks). The batch of tasks is

broadcast to all the resources in the cluster.

Step 3. The resources receive the group of tasks to be scheduled. The tasks are inserted-sorted

in a queue according to a sorting criteria like arriving time (ai) or scheduling priority (ri).If the number of tasks in the queue is less than a predefined length of the chromosome,

they wait for τ units of time before starting the genetic algorithm. If the chromosome is still

not complete at the end of the waiting period, a noninfluential padding is added. On the

contrary, if the length of an arriving group of tasks exceeds the predefined dimension of the

chromosome, some tasks are saved in the waiting queue and will be scheduled at the next

time.


Step 4. On each resource, a tool keeps an up-to-date status of the computers in the LSDS on

which tasks are sent for execution, by constantly interrogating a monitoring system.

Step 5. The resources in the cluster run the GA. Each resource starts with a different, specific

initialization of the genetic algorithm. The subsequent steps of the GA are similar for all the

nodes in the cluster, and so is the fitness formula. The clients will compute different optimum

from which the best one will be chosen.

Step 6. The migration of the best current solutions is performed after each step of the GA,

thus ensuring that the population finds a better optimum. The resources exchange the fittest

individuals and insert them into the next generation.

Step 7. The reproduction process stops after a finite, predefined number of steps. Each

resource in the cluster computes its optimal individual.

Step 8. Each resource sends its optimum to all the other nodes in the cluster and the final

optimal individual is decided.

Step 9. The scheduling obtained is saved in a history file on each resource in the cluster of

resources.

The fitness function is an essential element of proposed GA. It gives an appreciation

of the quality of a potential solution according to the problem’s specification. For the

scheduling problem, the goal is to obtain task assignments that ensure minimum execution

time, maximum processor utilization, a well-balanced load across all machines, and last

but not least to ensure that the precedence of the task’s is not violated. According to the

chromosome encoding and genetic operators presented previously all individuals respect

the task dependencies, so the focus should be on the other goals of the problem. The fitness

function has the following representations: (1) F =∑

i cifi or (2) F =∏

ifi, where fi encode

a criterion in fitness function and ci is a weight for a criterion (∑

i ci = 1). In both cases, if

0 ≤ fi ≤ 1, then 0 ≤ F ≤ 1. For the proposed genetic scheduling algorithm three criteria are

considered: load balancing over the resources, f1 = tmin/tmax = mini{ti}/maxi{ti}, average idletime of the resources, f2 = (1/n)

∑ni=1 ti/tmax, and the schedule penalty, f3 = |Sched|/|T |, where

|Sched| represents the length of a schedule (the number of tasks that respect deadlines and

resource restrictions) and |T | is the total number of tasks.

5. Experimental Methodology and Results

5.1. Simulation Environment

Due to the complexity of the LSDS, involving many resources and many jobs being

concurrently executed in heterogeneous environments, there are not many simulation tools

to address the general problem of LSDS computing. The simulation instruments tend to

narrow the range of simulation scenarios to specific subjects, such as scheduling or data

replication. The simulation model provided by MONARC is more generic than others, as

demonstrated in [18]. It is able to describe various actual distributed system technologies

and provides the mechanisms to describe concurrent network traffic, to evaluate different

strategies in data replication, and to analyze job scheduling procedures. In order to provide a

realistic simulation, all the components of the system and their interactions were abstracted.


DB serverDB server

CPU CPU CPU

Task scheduler

Activity Task Transition

Regional centerRegional

center

Regionalcenter

LAN

WAN

Figure 2: MONARC simulation tool: the Regional center model for LSDS control.

The chosen model is equivalent to the simulated system in all the important aspects. A first

set of components was created for describing the physical resources of the distributed system

under simulation. The largest one is the regional center (see Figure 2), which contains a site

of processing nodes (CPU units), database servers and mass storage units, as well as one or

more local and wide area networks.

The maturity of the simulation model was demonstrated in previous work. For

example, a number of data replications experiments were conducted in [17], presenting

important results for the future LHC experiments, which will produce more than 1 PB of data

per experiment and year, data that needs to be then processed. In [19] the simulation model

was used to conduct a series of simulation experiments to compare a number of different

scheduling algorithms.

5.2. Evaluation Criteria for LSDS Control

It is quite difficult to make a comparison among different control systems for LSDS,

since each of them is suitable for different situations. For different control systems, the

class of targeted applications and LSDS resource configurations may differ significantly.

The adequate evaluation criteria for LSDS control systems are as follows. (i) ApplicationPerformance Promotion involves reviewing how well the applications can benefit from the

deployment of the control system (ii) System Performance Promotion concerns how well the

whole system can benefit (iii) Control and Efficient Allocation it is desired so that the LSDS

control system can always produce good allocation. However, it is also required that the

scheduling system should introduce additional overhead as low as possible. (iv) Reliability—

a reliable LSDS control system should provide some level of fault tolerance. An LSDS


80

75

70

65

60

55

50

45

40

35

30

GA convergence

0 10 20 30 40 50 60 70 80 90 100

Generation number

Mak

esp

an(C

max)

Figure 3: Makespan comparison for the scheduling of 38 tasks on 8 processors.

is a large collection of loosely coupled resources, and therefore it is inevitable that some

of the resources may fail due to diverse reasons. The control system should handle such

frequent resource failures. For example, in case of resource failure, the control system should

guarantee an applications completion. (v) Scalability—since an LSDS environment is in nature

heterogeneous and dynamic, a scalable scheduling infrastructure should maintain good

performance with not only increasing number of applications, but also increasing number

of participating resources with diverse heterogeneity.

When designing the control infrastructure for LSDSs, these criteria are expected to

receive careful consideration. Emphasis may be laid on different concerns among these

evaluation criteria according to practical needs in real situations. The performance of

scheduling algorithms for LSDS control is usually estimated using a certain number of

standard parameters, like total time or schedule length. In the tests performed we used the

following evaluation parameters [20, 21]:

(i) total schedule length (SL)—Cmax;

(ii) convergences time—the number of generations needed to obtain performances

better then a certain threshold;

(iii) load balancing (where umed, denotes the average utilization for all processors in the

system):

L = 1 − 1

umed

√√√√ 1

n

n∑i=1

(ui − umed)2, 0 < L ≤ 1. (5.1)

The load balancing of system, for a given schedule, converges to 1 when all resources

have approximately the same utilization rate, equal to makespan. In these conditions the

square deviation Δ → 0.

5.3. Experimental Results

The test case considered task dependencies containing 38 tasks. The processors’ topology

contained 8 processors connected in a full mesh. The results presented in Figure 3 show that


Mak

esp

an(s)

1000900800700600500400300200100

0

200

805

211

816

219

896

203

810

GA Duplex SA Min-min

90 tasks + 8 resources

506 tasks 38 resources−

Figure 4: Makespan comparison for the following scenarios: 90 tasks + 8 resources, 506 tasks − 38 resources.

Min-min

SA

Duplex

GA

Proc4

Proc3

Proc2

Proc1

0 50 100 150 200 250 300 350 400 450

Execution time (s)

Figure 5: Processor utilization overview for 90 tasks.

the genetic algorithm has a very good convergence (after 50 generations there is no significant

improvement, so the algorithm could be stopped).In order to analyze the schedule length of different dependent task scheduling

algorithms, it has been used a processor topology containing 8 processors connected in a

full mesh. Two tests have been run for DAGs containing 90 and 506 tasks (see Figure 4).For the first test (90 tasks—left side of the figure), the best result, 263 time units, was

provided by the proposed GA. On the second place came the GA without the initialization

phase with the value of makespan equal to 266. From the classic scheduling algorithm,

Duplex provided the best solution equal to 281, while the result of SA was the worst equal

to 292. The test containing 506 tasks was an extreme test. The best result, 1073 time units,

was offered by GA, proving once more the importance of the proposed algorithm. The worst

solution was given by SA. The other compared algorithm is min-min [22] (see Figure 5).Memory is another important factor since it is the characteristic that controls most

of the allocation algorithms and also since it cannot be oversubscribed. As can be seen, the

memory allocator gets to the maximum memory value slower and thus allows for better


UsedNo Share

Total

Low (<50%)

High ( 50%)

9080706050403020100

Time (s)

0 250 500 750 1000 1250

Memory type: Load:

Mem

ory

(GB)

Figure 6: Memory usage for Best Fit allocation.

performance. Also this allocator is the first to leave the maximum value barrier when the

load is decreased (see Figure 6).

6. Conclusions

We present in this paper an algorithm for controling the resources allocation for special tasks

type (transitions) in LSDS (considered to be a complex one). The novelty of the proposed

algorithm is represented by the multicriteria optimization fitness function for special tasks

with specific requirements and constrains. The process was modulated using a genetic

scheduling algorithm. The paper analyzed the existing methods for control optimization

in LSDS. The multidimensional optimization criteria were considered with the real-time

behavior introducing two measures for evaluation: penalty and reward. In accordance

with this behavior, the convergence of proposed control method is very good in terms of

convergence, solution cost, and memory usage.

The most important contribution of this paper is the innovative method for the

optimization of dependent task scheduling control in LSDS. Inspired from the natural

models, this algorithm evolves an initial population of chromosomes in order to achieve

a good average fitness for the population. The experimental results have proven that the

proposed algorithm offers the best solutions in most cases. For comparison were used several

classical algorithms such as SA, Duplex, and min-min.

Acknowledgments

The research presented in this paper is supported by the Romanian Project: SORMSYS-Resource Management Optimization in Self-Organizing Large-Scale Distributes Systems (Contract

no. 5/28.07.2010, Project CNCSIS-PN-II-RU-PD ID: 201). The work has been cofunded

by the Sectorial Operational Program Human Resources Development 2007–2013 of the

Romanian Ministry of Labor, Family and Social Protection through the Financial Agreement

POSDRU/89/1.5/S/62557.


References

[1] C. Toma, E. G. Bakhoum, and M. Li, “Propagation phenomena and transitions in complex systems:Efficient mathematical models,” Mathematical Problems in Engineering, vol. 2012, Article ID 429129, 3pages, 2012.

[2] E. G. Bakhoum and C. Toma, “Specific mathematical aspects of dynamics generated by coherencefunctions,” Mathematical Problems in Engineering, vol. 2011, Article ID 436198, 10 pages, 2011.

[3] J. Kołodziej and F. Xhafa, “Integration of task abortion and security requirements in GA-based meta-heuristics for independent batch Grid scheduling,” Computers and Mathematics with Applications, vol.63, no. 2, pp. 350–364, 2012.

[4] Y. Huang, N. Bessis, P. Norrington, P. Kuonen, and B. Hirsbrunner, “Exploring decentralized dynamicscheduling for grids and clouds using the community-aware scheduling algorithm,” Future GenerationComputer Systems. In press.

[5] H. Abdu, H. Lutfiyya, and M. A. Bauer, “Optimizing management functions in distributed systems,”Journal of Network and Systems Management, vol. 10, no. 4, pp. 505–530, 2002.

[6] T. L. Casavant and J. G. Kuhl, “A taxonomy of scheduling in general-purpose distributed computingsystems,” IEEE Transactions on Software Engineering, vol. 14, no. 2, pp. 141–154, 1988.

[7] R. Armstrong, D. Hensgen, and T. Kidd, “The relative performance of various mapping algorithms isindependent of sizable variances in run-time predictions,” in Proceedings of the Seventh Hetero-geneousComputing Workshop (HCW ’98), pp. 79–87, IEEE Computer Society, Washington, DC, USA, 1998.

[8] J. Wu, X. Xu, P. Zhang, and C. Liu, “A novel multi-agent reinforcement learning approach for jobscheduling in Grid computing,” Future Generation Computer Systems, vol. 27, no. 5, pp. 430–439, 2011.

[9] T. Wang, X.-S. Zhou, Q.-R. Liu, Z.-Y. Yang, and Y.-L. Wang, “An adaptive resource scheduling algo-rithm for computational grid,” in Proceedings of the IEEE Asia-Pacific Conference on Services Computing(APSCC ’06), pp. 447–450, IEEE Computer Society, Washington, DC, USA, 2006.

[10] B. Sotomayor, R. S. Montero, I. M. Llorente, and I. Foster, “Virtual infrastructure management inprivate and hybrid clouds,” IEEE Internet Computing, vol. 13, no. 5, pp. 14–22, 2009.

[11] W. Zhao, K. Ramamritham, and J. A. Stankovic, “Preemptive scheduling under time and resourceconstraints,” IEEE Transactions on Computers, vol. 36, no. 8, pp. 949–960, 1987.

[12] J. Koodziej and F. Xhafa, “Enhancing the genetic-based scheduling in computational grids by astructured hierarchical population,” Future Generation Computer Systems, vol. 27, no. 8, pp. 1035–1046,2011.

[13] T. Ma, Q. Yan, W. Liu, and C. Mengmeng, “A survey on grid task scheduling,” International Journal ofComputer Applications in Technology, vol. 41, no. 3-4, pp. 303–309, 2011.

[14] G. Logothetis, Specification, Modelling, Verification and Runtime Analysis of Real Time Systems, IOS Press,2004.

[15] E. Fersman and W. Yi, “A generic approach to schedulability analysis of real-time tasks,” NordicJournal of Computing, vol. 11, no. 2, pp. 129–147, 2004.

[16] G. C. Buttazzo, Hard Real-Time Computing Systems: Predictable Scheduling Algorithms and Applications,Springer, 3rd edition, 2011.

[17] G. V. Iordache, M. S. Boboila, F. Pop, C. Stratan, and V. Cristea, “A decentralized strategy for geneticscheduling in heterogeneous environments,” Multiagent and Grid Systems, vol. 3, no. 4, pp. 355–367,2007.

[18] C. Dobre, C. Stratan, and V. Cristea, “Realistic simulation of large scale distributed systems usingmonitoring,” in Proceedings of the 7th International Symposium on Parallel and Distributed Computing(ISPDC ’08), pp. 434–438, IEEE Computer Society, Washington, DC, USA, July 2008.

[19] F. Pop, C. Dobre, G. Godza, and V. Cristea, “A simulation model for grid scheduling analysisand optimization,” in Proceedings of the International Symposium on Parallel Computing in ElectricalEngineering (PARELEC ’06), pp. 133–138, IEEE Computer Society, Washington, DC, USA, 2006.

[20] S. Venugopal and R. Buyya, “An SCP-based heuristic approach for scheduling distributed data-intensive applications on global grids,” Journal of Parallel and Distributed Computing, vol. 68, no. 4,pp. 471–487, 2008.

[21] F. Pop, C. Dobre, and V. Cristea, “Performance analysis of grid DAG scheduling algorithms usingMONARC simulation tool,” in Proceedings of the 7th International Symposium on Parallel and DistributedComputing (ISPDC ’08), pp. 131–138, IEEE Computer Society, Washington, DC, USA, July 2008.

[22] F. Xhafa and A. Abraham, “Computational models and heuristic methods for Grid schedulingproblems,” Future Generation Computer Systems, vol. 26, no. 4, pp. 608–621, 2010.


Review ArticleMathematical Models of Dissipative Systems inQuantum Engineering

Andreea Sterian and Paul Sterian

Academic Center of Optical Engineering and Photonics, Polytechnic University of Bucharest,313 Spl. Independentei, 060042 Bucharest, Romania

Correspondence should be addressed to Paul Sterian, [email protected]


Academic Editor: Ezzat G. Bakhoum

Copyright q 2012 A. Sterian and P. Sterian. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The paper shows the results of theoretical research concerning the modeling and characterizationof the dissipative structures generally, the dissipation being an essential property of the systemwith self-organization which include the laser-type systems also. The most important resultspresented are new formulae which relate the coupling parameters ain from Lindblad equation withenvironment operators Γi; microscopic quantitative expressions for the dissipative coefficients ofthe master equations; explicit expressions which describe the changes of the environment densityoperator during the system evolution for fermion systems coupled with free electromagnetic field;the generalized Bloch-Feynman equations for N-level systems with microscopic coefficients inagreement with generally accepted physical interpretations. Based on Maxwell-Bloch equationswith consideration of the interactions between nearing atomic dipoles, for the dense optical mediawe have shown that in the presence of the short optical pulses, the population inversion oscillatesbetween two extreme values, depending on the strength of the interaction and the optical pulseenergy.

1. Introduction

An essential problem of the quantum information systems is the controllability and observ-

ability of the quantum systems. In this context, Fermi systems are essential for several

important physical effects in quantum engineering as the dynamics of semiconductor na-

nostructures and high temperature superconductivity, nuclear resonances, fusion-fission

reactions, and analysis of optical quantum systems. These effects are essentially determined

by the dissipative coupling of the system.

Dissipation in quantum systems is a complex phenomenon which raises important

theoretical investigations. A dissipative system is a system of interest, coupled with another

system usually considered as being of much larger-environment. Fundamental and difficult


problem of dissipative quantum theory is to design the total system (system of interest +environment) on the space system of interest. In this way obtain a quantum master equation

describing the evolution of the system using two terms: (1) a hamiltonian term for processes

with energy conservation and (2) a nonhamiltonian term with coefficients that depend on the

dissipative coupling. A master equation is based on approximations that consist in mediating

rapid oscillations of reduced density matrix describing the interaction.

Such an approximation is the assumption that the evolution operators of a dissipative

system forms a semigroup, not a group like for isolated systems. In this framework was

derived a quantum master equation with dissipative terms which is consistent with all

principles of quantum mechanics. Considering two operators, coordinate q and momentum

p, master equation was used to describe the harmonic oscillator. In this theoretical framework,

dissipation is described by the friction and diffusion coefficients that satisfy certain conditions

called basic restrictions and Heisenberg’s uncertainty relations are observed during the whole

evolution of the system.

A rigorous method for deducting the master equation with microscopic expressions of

the dissipative coefficients is developed in the literature.

For a weak dissipative coupling one obtains a master equation of Lindblad form [1],but with the microscopic expressions of the dissipative coefficients.

In the development of quantum theory of dissipative systems an important step was

the connection between Lindblad’s generator and the previous phenomenological descrip-

tions, realized by Sandulescu and Scutaru [2]. Besides, we must mention Isar et al.’s con-

tributions [3]. This school developed by the above-mentioned researchers in the field are

well recognized in the scientific world [4–8].Firstly, in the paper general expressions which relate the coupling parameters ain in

Lindblad equation with environment operators Γi have been established [9–11]. In this way,

became possible deeper causality understanding of processes of friction and diffusion and of

related quantum effects: broadening and shift of spectral lines, tunneling rates, bifurcations

and instability [12, 13].Secondly, for a system of fermions, coupled with a dissipative environment quanti-

tative microscopic expressions for the coefficients of the dissipative master equation depend-

ing on the potential matrix elements, the densities of states of the environment and the occu-

pation probabilities of these states are presented [14–19].The study continue with the systems of fermions coupled by electric dipole inter-

actions of free electromagnetic field for which has established general explicit expressions

which describe the changes of the environment density operator during the system evolution.

This description is not restricted to the Born approximation, taking into account the envi-

ronment time evolution as a function of the system evolution. The results of the dissipative

dynamics of the system of fermions in the presence of laser field are applicable to the dis-

sipative structures [14, 20–29].Next, generalized Bloch-Feynman equations for N-level systems with microscopic

coefficients in agreement with generally accepted physical interpretations are presented.

In the last part, we study the dynamics of dense media under the action of ultrafast

optical pulses using Maxwell-Bloch formalism to include interaction between close atomic

dipoles [30–34]. It is shown that, in a system initially without inversion, in the presence

of optical pulses, the final population has two extreme values, results which contribute to

understanding the specific mechanisms of switching for applications, with specific examples

concerning the coherent radiation generation and amplification [35–43]. A computational

specific software, to verify the experimental and numerical existing models and in the same


time to discover new important situations for operative systems design and implementation,

was developed [44–48].

2. Relationship between Coupling Coefficients in Lindblad MasterEquation and Environment Observables

Research on dissipative processes has led to evidence for the first time concerning the

relationship between coupling coefficients ain in the Lindblad equation:

ρ ≡ − i

ħ

[H,ρ

]+

1

2ħ

∑n

{[Xnρ,X

+n

]+[Xn, ρX

+n

]}(2.1)

depending on the system Hamiltonian H and the operators of opening Xn:

Xn ≡∑i

ainsi, (2.2)

where si are system operators, ain are complex coupling coefficients or amplitudes and Γioperators of environment defined using the interaction Hamiltonian as

HSE = ħ∑i

siΓi. (2.3)

These relationships have been established under the form [10]

∑n

aina∗jn = 2ħ

⟨ΓiΓj

⟩, (2.4)

and allow an understanding of the physical causes of quantum processes of friction and

diffusion, with their known effects: broadening and shift of spectral lines [11], increased rates

of tunneling, nonlinear characteristics, leading to bifurcation, instability, and chaos.

3. Microscopic Quantitative Expressions ofthe Dissipative Coefficients in Master Equations

A general quantum master equation for a many-level many-particle system, with microscopic

coefficients, that preserves the quantum-mechanical properties of the density matrix was

obtained [12]:

d

dtρ(t) = − i

ħ

[H,ρ(t)

]+∑i,j

λij{[

c+i cjρ(t), c+j ci]+[c+i cj , ρ(t)c

+j ci]}

(3.1)

with dissipative coefficients:

λij = λFij + λBij (3.2)


including a component λFij for a dissipative environment of fermions and a component λBij for

a dissipative environment of bosons.

Equation (3.1) is of Lindblad’s form, with dissipative operators depending on the

transition/population operators c+i cj . For a system with N levels, the total number of these

operators is N2 − 1 the number of the independent operators defined.

If we denote by V F and VB the interaction dissipative potentials of the environment

containing YF fermions and YB bosons, respectively, it is possible to write the expressions

of the coefficients λFij si λBij for the resonant transition |j〉 → |i〉 of the system coupled with

|β〉 → |α〉 environmental transition, with fermionic state having densitiesaly gFα , g

Fβ

and

populations fFα (εα), f

Fβ(εβ), and bosonic states with densities gB

α , gBβ

and population fBα (εα) si

fBβ(εβ). The probability that the final state |α〉 of the environment to be free is 1 − f(εα) while

the probability the initial state of the environment to be occupied is f(εβ).General expressions of the dissipative coefficients are written for this type of

interaction in the form:

λFij =π

ħYF

∫ ∣∣∣〈αi|V F∣∣βj⟩∣∣∣2[

1 − fFα (εα)

]fFβ

(εβ)gFα (εα)g

Fβ

(εβ)dεβ, εα − εβ = εj − εi,

λBij =π

ħYB

∫ ∣∣〈αi|VB∣∣βj⟩∣∣2[

1 + fBα (εα)

]fBβ

(εβ)gBα (εα)g

Bβ

(εβ)dεβ, εα − εβ = εj − εi.

(3.3)

4. The Environment Dynamics Correlated with that ofa Fermion Systems Coupled with Free Electromagnetic Field

We consider a system of Z charged fermions with the coordinates rn and momenta pn (n =1, 2, . . . , Z) in a single-particle potential U(1)(rn), while U(2)(rn, rm) represents the two-

particle residual potential. This system is coupled to the modes ν of the free electromagnetic

field. In order to describe the dynamics of this system, for simplicity, we neglect the particle

spin and its dimensions with respect to the electromagnetic field wavelength (the electric

dipole approximation). In this case, the total hamiltonian is of the form [14]

HT =Z∑n=1

(pn − eA

B)2

2m+

Z∑n=1

U(1)(rn) +1

2

Z∑n,m=1

U(2)(rn, rm) +HB. (4.1)

In the total hamiltonian (4.1),

V = − e

m

Z∑n=1

pnAB

(4.2)

is the system-field interaction potential, while

HS =Z∑n=1

p2n

2m+

Z∑n=1

U(1)(rn) +1

2

Z∑n,m=1

U(2)(rn, rm) (4.3)


is the fermion system hamiltonian, and

HB =∑ν

H(ν) (4.4)

is the field hamiltonian, where

H(ν) = ħων

(a+νaν +

1

2

)(4.5)

is the field mode ν hamiltonian.

Let us take the density operator χ(t) of the total system with hamitonian (4.1) and the

reduced density matrix

ρ(t) = TrB{χ(t)

}(4.6)

over the environment states.

The total density operator χ(t) satisfies the equation of motion:

dχ

dt= − i

ħ

[εV R(t) + εV (t), χ(t)

], (4.7)

where the sign above χ designs operators within the framework of interaction picture of the

system and environment

χ(t) = e(i/ħ)(HB+HS

0 )tχ(t)e−(i/ħ)(HS0 +H

B)t, (4.8)

while ε is an intensity parameter used to show the orders of the series expansion of this

density. Considering the radiation field of the black body in the initial state R, the total density

operator of the system can be taken under the form:

χ(t) = R ⊗ ρ(t) + εχ(1)(t) + ε2χ(2)(t) + · · · , (4.9)

where χ(1)(t), χ(2)(t) represent modifications of the field during the system evolution. The first

term of this expression corresponds to the Born approximation when the environment state

is a constant state R, while the higher-order terms, which satisfy the normalization relations

TrB{χ(1)(t)

}= TrB

{χ(2)(t)

}= · · · = 0, (4.10)

describe the environment dynamics that is correlated to the system dynamics. For an equation

of motion of the form

dρ

dt= εB(1)[ρ(t), t] + ε2B(2)[ρ(t), t] (4.11)


From (4.7), (4.9), and (4.11) we get a system of coupled equations:

R ⊗ B(1)[ρ(t), t] + dχ(1)

dt= − i

ħ

[V R(t) + V (t), R ⊗ ρ(t)

],

R ⊗ B(2)[ρ(t), t] + dχ(2)

dt= − i

ħ

[V R(t) + V (t), χ(1)(t)

].

(4.12)

By calculating the partial traces over the environment states and using the normalization

conditions (4.10), from these equations we get successively the terms of the equation of

motion (4.11):

B(1)[ρ(t), t] = − i

ħTrB

[V R(t) + V (t), R ⊗ ρ(t)

],

B(2)[ρ(t), t] = − i

ħTrB

[V R(t) + V (t), χ(1)(t)

],

(4.13)

while, integrating by time, we get ”excitation” terms of the total density operator (4.9):

χ(1)(t) =∫ t

0

{− i

ħ

[V R

(t′)+ V

(t′), R ⊗ ρ

(t′)] − R ⊗ B(1)[ρ(t′), t′]}dt′,

χ(2)(t) =∫ t

0

{− i

ħ

[V R

(t′)+ V

(t′), χ(1)(t′)] − R ⊗ B(2)[ρ(t′), t′]}dt′.

(4.14)

The first-order equation (4.13) represents the system evolution when the environment is

considered as being in a constant state R, while for the higher-order term (28), we take

into consideration some changes of the environment matrix (4.14). Further on, we will show

that the first-order terms (4.13) describe the hamiltonian dynamics of the system, while the

second-order term (28) describes system one-particle transitions related to environment.

5. The Generalized Bloch-Feynman Equations

An alternative description of dissipative system dynamics is given by Bloch-Feynman

equations for systems of fermions obtained by defining the pseudo-spin operators [14].In particular, for a system with two-level known Bloch-Feynman, equations are

obtained, where, Q12 is the field operator, P12 is the polarization operator, and N2 is

population operator:

d

dt〈Q12〉 = −γ⊥〈Q12〉 +ω21〈P12〉,

d

dt〈P12〉 = −ω21〈Q12〉 − γ⊥〈P12〉,

d

dt〈N2〉 = −γ||

[〈N2〉 −N

(0)2

],

(5.1)


with microscopic coefficients γ⊥ si γ|| expressed by dissipative coefficients λij of the master

equation:

γ⊥ = λ12 + λ21 + λ11 + λ22, (5.2)

γ|| = 2(λ12 + λ21), (5.3)

N(0)2 =

λ21

λ12 + λ21. (5.4)

The condition 2γ⊥ ≥ γ|| is a confirmation of master equation (4.7) which led to the estab-

lishment of Bloch equations-Feynman, because this condition is verified experimentally.

6. Dynamics of Dense Media under the Action of Short Optical Pulses

Maxwell-Bloch equations of a two-level atomic medium generalized to include interactions

between the dipoles approach [15, 16, 32] have been used to describe the system dynamics

under the action of ultrafast optical pulses. These equations, for systems with homogeneous

broadening of spectral lines in about semiclassical treating, were established using the density

matrix formalism as

dw

dt= −γL(w + 1) +

μ

ħ

(E∗Rab + ER∗

ab

), (6.1)

dRab

dt= −[γT + i(Δ + εw)

]Rab −

μ

2ħEw. (6.2)

In the above equations, w is the inversion of population, Rab nondiagonal elements of density

matrix slow variable, indices a and b refer to lower and higher energy states, with the gap

ħω0, EL is slowly varying local field Δ = ω0 − ω is the frequency deviation in relation to the

center frequency of the field resonance frequency, μ is the transition matrix element of the

electric dipole, and γ||, γ⊥ are longitudinal and transverse relaxation rates.

Contributions of the dipole-dipole interactions occur in (6.2) by term iεwRab, where

ε = nμ2/3ħεd � ω0 is the strength parameter of dipole-dipole interactions having

a dimension of a frequency. Equations (6.1) and (6.2) for the atomic variables and for

field variables realise the description Maxwell-Bloch of optically dense environment. These

equations were generalized and used to study intrinsic optical bistability, propagation effects

in nonlinear media, and so forth.

For numerical simulation, we considered the case resonant (Δ = 0), a characteristic

distance between dipoles much smaller than the wavelength of the central field (propagation

effects are negligible) and ultrafast pulses (pulses much shorter than γ−1|| ; this enables us to


−1

−0.5

0

0.5

1

w(t

)

0.8 0.9 1 1.1 1.2 1.3

Ω0/

20

30

Figure 1: Final state of population inversion, depending on the hyperbolic secant pulse maximum value

E(t) = E0sech(t/τp) (solid line), ετp = 20 (continuous line) and ετp = 30.

neglect dissipation processes). In these conditions, the matrix element Rab is decomposed

into its real and imaginary parts Rab = 1/2(ν + iu), resulting in system

du

dt′= −(ετp)νw,

dν

dt′=(ετp

)(u +

Ωε

)w,

dw

dt′= −(ετp)(Ω

ε

)ν

(6.3)

whose outcome is possible only numerically.

In the above equations t′ = t/τp is the normalized time, τp is the measured width pulse,

Ω(t) = μE(t)/h is the instantaneous Rabi frequency, and E(t) is the intensty of electrical pulse.

In Figure 1, we present the final population inversion function of maximum Rabi

frequency for hyperbolic secant pulses E(t) = E0sech(t/τp). As long as the Rabi frequency

has a value so that Ω0/ε < 1, the final population inversion is w = −1. In the region Ω0/ε > 1,

the final population inversion has an oscillatory behavior, almost rectangular wave. As the

parameter ετp value is greater, the oscillation period decreases, the transitions become abrupt,

and the first half cycle of the rectangular wave becomes more centered to Ω0/ε = 1.

In Figure 2, temporal evolution of the system is presented for a hyperbolic secant pulse with

a peak higher than one (when t → ∞, the population inversion performs a number of

oscillations before reaching a value 1; under certain conditions when t → ∞, after a number

of oscillations, the system remains in the ground state).


−1

0

0.5

1.5

1

−0.5

w,Ω/

w

Ω/

−8 −6 −4 −2 0 2 4 6 8

t

Figure 2: Temporal evolution of the system for a hyperbolic secant pulse, E(t) = E0sech(t/τp).

7. Conclusions

General expressions which relate the coupling parameters ain in Lindblad equation with envi-

ronment operators Γi have been established. These expressions allow deeper understanding

of causal processes of friction- and diffusion-related quantum effects: broadening and shift of

spectral lines, tunneling rates, bifurcations, and instability.

For a system of fermions coupled with a dissipative environment quantitative micro-

scopic expressions for the coefficients of the dissipative master equation are presented.

These coefficients depend on the potential matrix elements, the densities of states of

the environment, and the occupation probabilities of these states.

Expressions of the dependence of the particle distributions on temperature are taken

into account. It can be shown that a system of fermions located in a dissipative environment

of bosons tends to a Bose-Einstein distribution.

Studying the systems of fermions coupled by electric dipole interactions of free elec-

tromagnetic field, has established general explicit expressions which describe the changes of

the environment density operator during the system evolution for fermion systems coupled

with free electromagnetic field. This description is not restricted to the Born approximation,

taking into account the environment time evolution as a function of the system evolution. The

study can be continued with the calculation of the higher-order term of the reduced matrix

equation in order to describe the correlated transition of the system particles. The results of

the dissipative dynamics of the system of fermions in the presence of laser field are applicable

to the dissipative structures.

Generalized Bloch-Feynman equations for N-level systems with microscopic coef-

ficients in agreement with generally accepted physical interpretations are presented. On

this basis, the problem of a quantum system control is explicitly formulated in terms of

microscopic quantities: matrix elements of the dissipative two-body potential, densities of

the environment states, and occupation probabilities of these states.


Studying the dynamics of dense media under the action of ultrafast optical pulses

using Maxwell-Bloch formalism to include interaction between close atomic dipoles showed

that, in a system initially without inversion, in the presence of optical pulses, the final

population has two extreme values, the ratio of Rabi frequency and the parameter that des-

cribes the interactions between close dipoles, which contribute to understanding the specific

mechanisms of switching.

References

[1] G. Lindblad, “On the generators of quantum dynamical semigroups,” Communications inMathematicalPhysics, vol. 48, no. 2, pp. 119–130, 1976.

[2] A. Sandulescu and H. Scutaru, “Open quantum systems and the damping of collective modes in deepinelastic collisions,” Annals of Physics, vol. 173, no. 2, pp. 277–317, 1987.

[3] A. Isar, A. Sandulescu, H. Scutaru, E. Stefanescu, and W. Scheid, “Open quantum systems,” Inter-national Journal of Modern Physics E, vol. 3, no. 2, pp. 635–714, 1994.

[4] E. Stefanescu and A. Sandulescu, “Microscopic coefficients for the quantum master equation of aFermi system,” International Journal of Modern Physics E, vol. 11, no. 2, pp. 119–130, 2002.

[5] A. Sandulescu and E. Stefanescu, “New optical equations for the interaction of a two-level atom witha single mode of the electromagnetic field,” Physica A, vol. 161, no. 3, pp. 525–538, 1989.

[6] E. Stefanescu, R. J. Liotta, and A. Sandulescu, “Giant resonances as collective states with dissipativecoupling,” Physical Review C, vol. 57, no. 2, pp. 798–805, 1998.

[7] G. W. Ford, J. T. Lewis, and R. F. Connell, “Master equation for an oscillator coupled to the electro-magnetic field,” Annals of Physics, vol. 252, no. 2, pp. 362–385, 1996.

[8] E. Stefanescu, A. Sandulescu, and W. Scheid, “The collisional decay of a fermi system interacting witha many-mode electromagnetic field,” International Journal of Modern Physics E, vol. 9, no. 1, pp. 17–50,2000.

[9] E. N. Stefanescu, P. E. Sterian, and A. R. Sterian, “Fundamental interactions in dissipative quantumsystems,” The Hyperion Scientific Journal, vol. 1, no. 1, pp. 87–92, 2000.

[10] E. Stefanescu and P. Sterian, “Exact quantum master equations for Markoffian systems,” Optical Engi-neering, vol. 35, no. 6, pp. 1573–1575, 1996.

[11] E. Stefanescu, A. Sandulescu, and P. Sterian, “Open systems of fermions interacting with the electro-magnetic field,” Romanian Journal of Optoelectronics, vol. 9, no. 2, pp. 35–54, 2001.

[12] E. Stefanescu, P. Sterian, and A. Gearba, “Spectral line broadening in the two level systems withdissipative coupling,” Romanian Journal of Optoelectronics, vol. 6, no. 2, pp. 17–22, 1998.

[13] E. Stefanescu, P. Sterian, and A. Gearba, “Absorption spectra in dissipative two level system media,”in Proceedings of the EPS 10 Trends in Physics 10th General Conference of the European Physical Society,September 1996.

[14] E. N. Stefanescu, A. R. Sterian, and P. E. Sterian, “Study of the fermion system coupled by electricdipol interaction with the free electromagnetic field,” in Proceedings of the Advanced Laser Tehnologies,A. Giardini, V. I. Konov, and V. I. Pustavoy, Eds., vol. 5850 of Proceeding of the SPIE, pp. 160–165, 2005.

[15] E. Stefanescu and P. Sterian, “Dynamics of a Fermi system in a complex dissipative environment,”in Proceedings of the International Conference on Advanced Laser Technologies (ALT ’02), vol. 4762 ofProceeding of SPIE, pp. 247–259, Constanta, Romania, 2002.

[16] E. Stefanescu and P. Sterian, “Non-Markovian effects in dissipative systems,” in Proceedings of the 5thConference on Optics (Romopto ’97), vol. 3405 of Proceeding of SPIE, pp. 877–882, Bucharest, Romania,1998.

[17] E. Stefanescu, A. Sandulescu, and P. Sterian, “Quantum tunneling through a dissipative barrier,” inProceedings of the 4th Conference In Optics (Romopto ’94), vol. 2461, pp. 218–225, 1995.

[18] E. Stefanescu, P. Sterian, and A. R. Sterian, “The lindblad dynamics of a Fermi system in a particledissipative environment,” in Proceedings of the International Conference on Advanced Laser Technologies(ALT ’03), vol. 5147 of Proceeding of SPIE, pp. 160–168, Adelboden, Switzerland, 2003.

[19] E. Stefanescu, P. Sterian, and A. Sandulescu, “Quantum tunneling in open systems,” in Proceedings ofthe 1st General Conference of the Balkan Physical Union, K. M. Paraskevopoulos, Ed., Hellenic PhysicalSociety, 1991.

[20] E. Stefanescu and A. Sandulescu, “Dynamics of a superradiant dissipative system of tunneling elec-trons,” Romanian Journal of Physics, vol. 49, pp. 199–207, 2004.


[21] E. Stefanescu, “Quantum master equation for a system of charged fermions interacting with the elec-tromagnetic field,” Romanian Journal of Physics, vol. 48, pp. 763–770, 2003.

[22] E. Stefanescu and A. Sandulescu, “The decay of a Fermi system through particle-particle dissipation,”Romanian Journal of Physics, vol. 47, pp. 199–219, 2002.

[23] E. Stefanescu and A. Sandulescu, “Dissipative processes through Lindblad’s master equation,” Ro-manian Journal of Optoelectronics, vol. 7, p. 59, 2000.

[24] E. Stefanescu and P. Sterian, “Open systems of fermions interacting with the electromagnetic field,”Romanian Journal of Optics, vol. 9, pp. 35–55, 2001.

[25] E. Stefanescu, P. Sterian, I. M. Popescu et al., “Time-dependent problem algorithm of the optical bi-stability,” Revue Roumaine de Physique, vol. 31, pp. 345–350, 1986.

[26] . E. Stefanescu and P. Sterian, “Optical equations for an open system of fermions,” in Proceedings of the11th General Conference of the European Physical Society, Trends in Physics, London, UK, Septmber 1999.

[27] E. Stefanescu, I. M. Popescu, and P. Sterian, “The Semi-classical Approach of the Optical Bistability,”Revue Roumaine de Physique, vol. 29, no. 2, pp. 183–188, 1984.

[28] E. Stefanescu and P. Sterian, “Dynamics of a fermi system in a complex dissipative environment,” inProceedings of the Advanced Laser Technologies, vol. 4762 of Proceeding of SPIE, pp. 247–259, Constanta,Romania, Septmber 2001.

[29] E. Stefanescu and A. Sandulescu, “Dynamics of a superradiant dissipative system of electrons tunnel-ing in a micro-cavity,” Romanian Journal of Physics, vol. 50, pp. 629–638, 2005.

[30] C. M. Bowden and J. P. Dowling, “Near-dipole-dipole effects in dense media: generalized Maxwell-Bloch equations,” Physical Review A, vol. 47, no. 2, pp. 1247–1251, 1993.

[31] V. Ninulescu and A. R. Sterian, “Dynamics of a two-level medium under the action of short opticalpulses,” in Proceedings of the Computational Science and Its Applications (ICCSA ’05), O. Gervasi, M. L.Gavrilova, V. Kumar et al., Eds., vol. 3482 of Lecture Notes in Computer Science-LNCS, pp. 635–642,Springer, 2005.

[32] C. Vasile, A. R. Sterian, and V. Ninulescu, “Optically dense media described by help of Maxwell-Bloch equations,” Romanian Journal of Optoelectronics, vol. 10, no. 4, pp. 81–86, 2002.

[33] M. E. Crenshaw, M. Scalora, and C. M. Bowden, “Ultrafast intrinsic optical switching in a dense med-ium of two-level atoms,” Physical Review Letters, vol. 68, no. 7, pp. 911–914, 1992.

[34] V. Ninulescu, I. Sterian, and A. R. Sterian, “Switching effect in dense medie,” in Proceedings of the Inter-national Conference, Advanced Laser Technologies (ALT ’01), vol. 4762 of Proceeding of SPIE, pp. 227–234,2002.

[35] A. R. Sterian, “Coherent radiation generation and amplification in erbium doped systems,” in Ad-vances in Optical Amplifiers, P. Urquhart, Ed., InTech, Vienna, Austria, 2011.

[36] A. R. Sterian and V. Ninulescu, “Nonlinear phenomena in erbium doped lasers,” in Proceedings of theComputational Science and Its Applications (ICCSA ’05), O. Gervasi et al., Ed., vol. 3482 of Lecture Notesin Computer Science, LNCS, pp. 643–650, Springer, 2005.

[37] A. R. Sterian, Laserii ın Ingineria Electrica, Printech Publishing House, Bucharest, Romania, 2003.

[38] A. R. Sterian and V. Ninulescu, “The nonlinear dynamics of erbium-doped fiber laser,” in Proceedingsof the 13th International School on Quantum Electronics: Laser Physics and Applications, Burgas, Bulgaria,September 2004.

[39] M. Ghelmez, C. Toma, M. Piscureanu, and A. R. Sterian, “Laser signals’ nonlinear change in fattyacids,” Chaos, Solitons and Fractals, vol. 17, no. 2-3, pp. 405–409, 2003.

[40] A. R. Sterian, Amplificatoare Optice, Printech Publishing House, Bucharest, Romania, 2006.

[41] C. Rosu, D. Manaila-Maximean, D. Donescu, S. Frunza, and A. R. Sterian, “Influence of polarizingelectric fields on the electrical and optical properties of polymer-clay composite system,” ModernPhysics Letters B, vol. 24, no. 1, pp. 65–73, 2010.


[43] E. G. Bakhoum and C. Toma, “Dynamical aspects of macroscopic and quantum transitions due tocoherence function and time series events,” Mathematical Problems in Engineering, vol. 2010, Article ID428903, 13 pages, 2010.

[44] F. C. Maciuc, C. I. Stere, A. R. Sterian et al., “Time evolution and multiple parameters variations in atime dependent numerical model applied for Er+3 laser system,” in Proceedings of the 1th InternationalSchool on Quantum Electronics: Laser Physics and Applications, P. A. Atanasov and S. Cartaleva, Eds.,vol. 4397 of Proceeding of SPIE, pp. 84–89, 2001.


[45] A. R. Sterian and F. C. Maciuc, “Numerical model of an EDFA based on rate equations,” in Proceedingsof the 12th International School an Quantum Electronics, Laser Physics and Application, vol. 5226 of Pro-ceeding of SPIE, pp. 74–78, 2003.

[46] C. Cattani and I. Bochicchio, “Wavelet analysis of bifurcation in a competition model,” in Proceedingsof the Computational Science (ICCS ’07), vol. 4488, part 2 of Lecture Notes in Computer Science, pp. 990–996, 2007.

[47] F. C. Maciuc, C. I. Stere, and A. R. Sterian, “Rate equations for an Erbium laser system, a numericalapproach,” in Proceedings of the 6th Conference on Optics (ROMOPTO ’01), vol. 4430 of Proceeding ofSPIE, pp. 136–146, 2001.

[48] D. A. Iordache, P. Sterian, F. Pop, and A. R. Sterian, “Complex computer simulations, numerical arti-facts, and numerical phenomena,” International Journal of Computers, Communications and Control, vol.5, no. 5, pp. 744–754, 2010.


Research ArticlePower-Law Properties of Human View andReply Behavior in Online Society

Ye Wu,1, 2 Qihui Ye,2 Lixiang Li,3 and Jinghua Xiao1, 2

1 State Key Lab of Information Photonics and Optical Communications,Beijing University of Posts and Telecommunications, Beijing 100876, China

2 School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China3 Information Security Center, Beijing University of Posts and Telecommunications, Beijing 100876, China

Correspondence should be addressed to Ye Wu, [email protected]


Academic Editor: Ming Li

Copyright q 2012 Ye Wu et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Statistical properties of the human comment behavior are studied using data from “Tianya” and“Tieba” which are very popular online social systems (or forums) in China. We find that both thereply number R and the view number V of a thread in a subforum obey the power-law distributionsP(R) = Rα and P(V ) ∝ V β, respectively, which indicates that there exists a kind of highly populartopics. These topics should be specially paid much attention, because they play an important role inthe public opinion formation and the public opinion control. In addition, the relationship betweenR and V also obeys the power-law function R ∝ V γ . Based on the human comment habit, a modelis introduced to explain the human view and reply behaviors in the forum. Numerical simulationsof the model fit well with the empirical results. Our findings are helpful for discovering collectivepatterns of human behaviors and the evolution of public opinions on the virtual society as well asthe real one.

1. Introduction

Statistical properties and models of human behaviors have received much attention in dif-

ferent scientific fields, such as sociology, psychology, and economics. However, most of the

existing findings are only qualitative analyses for the lack of real data about the complexity

of human behaviors. Usually, it is assumed that the human behavior is a Poisson progress

[1, 2] which is a kind of the Markov progress. However, some researchers found that the in-

terevent time distribution of some human behaviors is power-law which means that it is a

non-Markov [3–14] one. More and more researchers are interested in it for its importance in

the theory and potential applications.

As an important part of modern life and human dynamics, the human behavior on the

Internet also attracts more and more attention. Chmiel et al. investigated the flows of visitors


migrating between different portal subpages. A model of portal surfing was developed where

a browsing process corresponds to a self-attracting walk on weighted networks with a short

memory [15]. Grabowski found that the distribution of human activity has the form of a

power-law [16] distribution. Based on the data from “Tianya”, Wu et al. found that the

dynamics of human comment in the online society is non-Markov. Further, they proposed

a model to explain it [17]. All these researches indicated that some kinds of human behavior

in on-line systems were non-Markov. They have some common statistic properties. More and

more researchers considered the forum as a virtual society to study the property and the

evolution of complex friendship networks [18, 19].A forum is very important for the information and the spreading of public opinions.

Many public opinions were also formatted and then spread in the forum. Analyzing the

user behavior in the forum is not only helpful for understanding the human behavior

and enhancing the information spreading, but also for designing a better website which is

important for the information spreading. Recently in China, the news about controlling public

opinions on purpose by news have attracted more and more attention. There was a report

that at least a half of public opinions in the Internet were proposed by some companies on

purpose. So it is very important to study the human comment behavior in the forum. Yu et

al. analyzed the view and reply data in the forum which was the beginning of researches on

the human comment behavior in the forums. They found that the view and reply numbers

of a thread in the sub-forum were power-law. However, they mainly considered statistic

properties of the behavior and did not present a model to explain the basic mechanism [20].In this paper, we consider the data collected from “Tianya” and “Tieba” which are very

popular on-line social system in China and different from those in [17]. We show that both

the view number (V ) and the reply number (R) of a thread in the sub-forum obey power-

law distributions which confirmed Yu et al.’s finding [20]. The relationship between V and

R is also power-law. These present that a lot of topics are important in the formation and

evolution of public opinions. Furthermore, based on the human habit, a model is proposed to

explain these phenomena. Numerical simulations are given to explain the human comment

behavior in the forum. We hope it is useful for understanding complex human behaviors in

the forums.

This paper is organized as follows: in Section 2, the origin of the data is introduced.

The statistical results are presented in Section 3. The model and numerical simulations are

presented in Section 4. Finally, our conclusion is given in Section 5.

2. Description of the Original Data

Our data are obtained from “Tianya” (http://www.tianya.cn) and “Tieba” (http://tieba.bai-

du.com), which are two most popular on-line social systems in China. Our data are collected

from the sub-forums of “Tianya” and “Tieba.” Each user is assigned a different identity name

(ID) in the forums. A topic in the sub-forum is called a thread. A thread is a minimal unit, and

it can be divided into a root thread and the reply threads. A root thread is a new topic, and

the reply threads are related to a root one. The users discuss the public opinion in both the

root and reply threads. Until 2010/02/11, there were 33,296,350 IDs in “Tianya,” and about

200,000 IDs on average were on-line at the same time. The topics and the public opinion in

“Tianya” and “Tieba” reflect part of the public opinions of the real society in China. Our data

sets are collected from the threads in four sub-forums. The types of these topics are different

from public news to personal stories which indicate that our results are general for different


Table 1: Detailed format of a subforum.

Topic Last update ID Reply/View Last update time

A ID 1 123/23124 2009/02/12, 12:11:05

B ID 3 3243/323532 2009/02/12, 12:10:05

C ID 32 323/42421 2009/02/12, 12:03:05

· · · · · ·Y ID 31 43/232 2009/02/12, 12:01:05

Table 2: Detailed informations about four randomly selected sub-forums.

Subforum A B C D

Total threads 19,492 19,479 27,359 5,302

Total clicks 1.05 × 109 1.62 × 109 2.33 × 108 5.19 × 106

Total replies 1.11 × 108 1.43 × 108 7.89 × 105 1.77 × 105

Duration (day) 247 2569 3 15

contents. The format of the data is shown in Table 1, where the first column is the title of a

thread, the second one gives the author’s name of a root thread, the third one shows R and

V , and the last one is the last update time of a thread.

3. Statistical Results

In the forum, the view and reply times of a thread reflect the influencing ability of a topic.

Further, more reply times mean more discussions and more communications. These two

parameters play an important role in the public opinion formation and the web design.

Hence, we study statistical properties of V and R in the thread of each sub-forum. Four sub-

forums are randomly selected as our data sets. The topics and some prosperities are listed in

Table 2.

The distributions of V and R in each sub-forum are shown in Figure 1 from which

we can clearly see that all the distributions are power-law, although the threads differ in

their contents. Their exponents vary with different sub-forums. These results show that the

process of human comments is non-Markov which is the same as the human dynamics of

the letter and e-mail communications, the web browsing, online movie watching, and broker

trades. The heavy tail of the distribution allows for much more numbers of threads which

have larger amounts of V and R than the Poisson progress. The thread which has more V and

R has much more influences on the public opinion. The number of these kind of threads is

so large that they cannot be ignored. A large population will read the thread by which their

opinions may be influenced. So we must pay much attention to them.

As is known to all, the more the view, the more the reply. However, the quantity

relationship between V and R is not very easy to know and it is the basic property of a

thread. Hence, next we mainly focus on the relationship between the human’s view and reply

behaviors in Figure 2. We found that it can be illustrated as a straight line in a log-log plot,

which means R ∝ V γ . It is easy to understand that the more the view, the more the reply.

Moreover, the nonlinear relationship here also means that the reply number increases slower

than the view one when the view number is large enough. It also indicates that human’s

interest in reply decreases as the increment of V .


100

10−2

10−4

102 104 106

P(R

,V)

R,V

(a)

100

10−2

10−4

102 104 106100

P(R

,V)

R,V

(b)

100

10−2

10−4

100 105 101010−6

R,V

P(R

,V)

(c)

100

10−5

100

R,V

P(R

,V)

102 104 106

(d)

Figure 1: Distributions of V (∗) and R (•) in each sub-forum. The solid line and the dashed one showthe slopes of fitting function for distributions of V and R, respectively, where (a) sub-forum A, the slopesα = 1.40 ± 0.01, β = 1.44 ± 0.02, (b) sub-forum B, the slopes α = 1.35 ± 0.02, β = 1.12 ± 0.01, (c) sub-forum C,the slopes α = 1.51 ± 0.01, β = 1.68 ± 0.01, (d) sub-forum D, the slopes α = 1.12 ± 0.02, β = 1.76 ± 0.02.

4. The Model and the Simulations

In order to get a better understanding of our empirical observations in Section 3, we propose

a model based on our intuitive experience about the human comment habit. We see that the

view number of each sub-forum increases more quickly as the time evolves. There are many

threads on each sub-forum. Each thread will be viewed based on its content and its previous

view time. Hence, our model is defined by the following scheme.

Step 1 (growing). At time t = 0, there are a few threads on the sub-forum, and each thread

has a random small V and R. At each step, a new thread is created, and there are c ∗ tθ views

on the old thread. All the old threads have the probabilities to be viewed.

Step 2 (view habit). The probability that an old thread is viewed at each step is based on

its attraction Π(i) = Ai(t)/ΣAi(t), where Ai(t) is the attraction of a thread i at time t and it is

reflected by the previous view number Vi(t), that is, Ai(t) = A(0)+Vi(t). Here A(0) represents

the initial attraction which is different due to different topics.

Step 3 (reply habit). At each step, when the user views a thread, he has a probability P(i) =L ∗ (R(i)/V (i))η to reply the thread.


104

100

108

R

V

100 104

(a)

104

100

108

R

V

100 104

(b)

104

100

108

R

V

100 104

(c)

104

100

R

V

100 104

(d)

Figure 2: The power-law relationship between V and R, where (a) sub-forum A, the slope γ = 0.77, (b) sub-forum B, the slope γ = 0.89, (c) sub-forum C, the slope γ = 0.85, (d) sub-forum D, the slope γ = 0.90.

Mathematically, the model is similar to the growing networks in [21]. Based on the

analysis of the growing network in this paper, we obtain that the distribution of Vi is a power-

law one, that is, P(Vi) ∝ (Vi)−α at a large enough time t where the exponent α is 1+ 1/(1+ θ) .

To compare our model with empirical observation results, let us take the sub-forum C

in our data sets as an example. Here we use the parameters θ = 0.9, L = 0.1, η = 0.5 in the

simulation. The results are shown in Figure 3. Figure 3(a) presents that the distribution of the

view is indeed a power-law one with a similar exponent as that from the data. Figure 3(b)shows that the reply number also obeys a power-law distribution. The nonlinear relationship

between the view and reply times is shown in Figure 3(c) which is the same as that from

the data. In Figure 3(d), we further study the relationship between the parameter η and the

slope γ . We see that γ decreases as η increases. From the analyses above, we can see that

the proposed model can well-describe most important features in the human view and reply

behaviors in online social systems.

5. Conclusion

In this paper, we analyze the statistical properties of the view and reply behaviors in on-line

social systems. We find that they are different types of interactive human dynamics which

are non-Markov. The view and the reply behaviors follow power-law distributions, and the

relationship between them also follows a power-law one. A model based on the personal


100

10−2

10−4

102 104

V

100

P(V

)

(a)

100

10−2

10−4

R

100

P(R

)

102 104 106

(b)

100

102 104100

102

R

V

106

(c)

1

0.8

0.6

0.2 0.4 0.6

γ

η

(d)

Figure 3: Simulation results of the model whose parameters are selected as θ = 0.9, L = 0.1, η = 0.5, wherethe dashed line shows the slope of the fitting function. (a) The distribution of V of a thread. The slopeof fitting function is α = 1.51 ± 0.02. (b) The distribution of R of a thread. The slope of fitting function isα = 1.41± 0.04. (c) The relationship between V and R. The slope of fitting function is γ = 0.9± 0.03. (d) Therelationship between the slope γ and the parameter η.

attraction is introduced to explain the human complex behavior. Numerical simulations of the

model fit well with empirical results. Our work is useful to understand the human complex

behavior in realistic society, for example, the human discussion behavior in a meeting

or group communications in trunked mobile telephony [22]. We expect that quantitative

understanding of human view and reply behaviors, when combined with additional content

analyses, will open a new perspective on distinguishing fraud public opinions from realistic

opinions.

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (Grants nos.

61104152, 60804046), the Fundamental Research Funds for the Central Universities (Grant

no. 2011R01), the Foundation for the Author of National Excellent Doctoral Dissertation of

China (Grant no. 200951), and the Asia Foresight Program under NSFC Grant (Grant no.

61161140320).

References

[1] F. A. Haight, Handbook of the Poisson Distribution, John Wiley & Sons, New York, NY, USA, 1967.[2] P. Reynolds, Call Center Staffing, The call Center School Press, Lebanon, Tenn, USA, 2003.


[3] A. L. Barabasi, “The origin of bursts and heavy tails in human dynamics,” Nature, vol. 435, no. 7039,pp. 207–211, 2005.

[4] Y. Wu, C. Zhou, J. Xiao, J. Kurths, and H. J. Schellnhuber, “Evidence for a bimodal distribution inhuman communication,” Proceedings of the National Academy of Sciences of the United States of America,vol. 107, no. 44, pp. 18803–18808, 2010.

[5] A. Vazquez, J. G. Oliveira, Z. Dezso, K. I. Goh, I. Kondor, and A. L. Barabasi, “Modeling bursts andheavy tails in human dynamics,” Physical Review E, vol. 73, no. 3, Article ID 036127, pp. 1–19, 2006.

[6] U. Harder and M. Paczuski, “Correlated dynamics in human printing behavior,” Physica A, vol. 361,no. 1, pp. 329–336, 2006.

[7] T. Zhou, H. A. T. Kiet, B. J. Kim, B. H. Wang, and P. Holme, “Role of activity in human dynamics,”EuroPhysics Letters, vol. 82, no. 2, Article ID 28002, 2008.

[8] M. Li and W. Zhao, “Visiting power laws in cyber-physical networking systems,” Mathematical Prob-lems in Engineering, vol. 2012, Article ID 302786, 13 pages, 2012.


[10] M. Li, “Fractal time series-a tutorial review,” Mathematical Problems in Engineering, vol. 2010, ArticleID 157264, 26 pages, 2010.

[11] Z. Jiao, Y. Q. Chen, and I. Podlubny, Distributed-Order Dynamic Systems, Springer, New York, NY, USA,2011.

[12] H. Sheng, H. Sun, Y. Chen, and T. Qiu, “Synthesis of multifractional Gaussian noises based on varia-ble-order fractional operators,” Signal Processing, vol. 91, no. 7, pp. 1636–1644, 2011.

[13] H. Sheng, Y. Q. Chen, and T. Qiu, “Heavy-tailed distribution and local long memory in time series ofmolecular motion on the cell membrane,” Fluctuation and Noise Letters, vol. 10, no. 1, pp. 93–119, 2011.

[14] H. Sheng, Y. Q. Chen, and T. S. Qiu, Fractional Processes and Fractional Order Signal Processing, Springer,New York, NY, USA, 2012.

[15] A. Chmiel, K. Kowalska, and J. A. Holyst, “Scaling of human behavior during portal browsing,” Phys-ical Review E, vol. 80, no. 6, Article ID 066122, 7 pages, 2009.

[16] A. Grabowski, “Human behavior in online social systems,” European Physical Journal B, vol. 69, no. 4,pp. 605–611, 2009.

[17] Y. Wu, C. Zhou, M. Chen, J. Xiao, and J. Kurths, “Human comment dynamics in on-line social sys-tems,” Physica A, vol. 389, no. 24, pp. 5832–5837, 2010.

[18] A. Grabowski, N. Kruszewska, and R. A. Kosinski, “Properties of on-line social systems,” EuropeanPhysical Journal B, vol. 66, no. 1, pp. 107–113, 2008.

[19] G. Csanyi and B. Szendroi, “Structure of a large social network,” Physical Review E, vol. 69, no. 3,Article ID 036131, 5 pages, 2004.

[20] J. Yu, Y. Hu, M. Yu, and Z. Di, “Analyzing netizens’ view and reply behaviors on the forum,” PhysicaA, vol. 389, no. 16, pp. 3267–3273, 2010.

[21] S. N. Dorogovtsev and J. F. F. Mendes, “Effect of the accelerating growth of communications networkson their structure,” Physical Review E, vol. 63, no. 2, Article ID 025101, pp. 1–4, 2001.

[22] D. G. Xenikos, “Modeling human dialogue-the case of group communications in trunked mobiletelephony,” Physica A, vol. 388, no. 23, pp. 4910–4918, 2009.


Research ArticleSinogram Restoration for Low-Dosed X-RayComputed Tomography Using Fractional-OrderPerona-Malik Diffusion

Shaoxiang Hu,1 Zhiwu Liao,2 and Wufan Chen3

1 School of Automation Engineering, University of Electronic Science and Technology of China,Chengdu 611731, China

2 School of Computer Science, Sichuan Normal University, Chengdu 610101, China3 Institute of Medical Information and Technology, School of Biomedical Engineering,Southern Medical University, Guangzhou 510515, China

Correspondence should be addressed to Zhiwu Liao, [email protected]

Received 18 January 2012; Accepted 16 March 2012


Copyright q 2012 Shaoxiang Hu et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Existing integer-order Nonlinear Anisotropic Diffusion (NAD) used in noise suppressing willproduce undesirable staircase effect or speckle effect. In this paper, we propose a new scheme,named Fractal-order Perona-Malik Diffusion (FPMD), which replaces the integer-order derivativeof the Perona-Malik (PM) Diffusion with the fractional-order derivative using G-L fractionalderivative. FPMD, which is a interpolation between integer-order Nonlinear Anisotropic Diffusion(NAD) and fourth-order partial differential equations, provides a more flexible way to balancethe noise reducing and anatomical details preserving. Smoothing results for phantoms and realsinograms show that FPMD with suitable parameters can suppress the staircase effects and speckleeffects efficiently. In addition, FPMD also has a good performance in visual quality and root meansquare errors (RMSE).

1. Introduction

Radiation exposure and associated risk of cancer for patients receiving CT examination have

been an increasing concern in recent years. Thus minimizing the radiation exposure to

patients has been one of the major efforts in modern clinical X-ray CT radiology [1–8].A simple and cost-effective means to achieve low-dose CT applications is to lower X-

ray tube current (mA) as low as achievable [6, 7]. However, the presentation of strong noise

degrades the quality of low-dose CT images dramatically and decreases the accuracy of the

diagnosis dose.Filtering noise from clinical scans is a challenging task, since these scans contain arti-

facts and consist of many structures with different shape, size, and contrast, which should be


preserved for making correct diagnosis. Many strategies have been proposed to reduce

the noise, for example, by nonlinear noise filters [8–20] and statistics-based iterative image

reconstructions (SIIRs) [21–29].The SIIRs utilize the statistical information of the measured data to obtain good denois-

ing results but are limited for their excessive computational demands for the large CT image

size. Although the nonlinear filters show effectiveness in reducing noise both in sinogram

space and image space, they cannot handle the noise-induced streak artifacts. Some nonlinear

filters, such as Nonlinear Anisotropic Diffusion (NAD), even produce new artifacts in L-CT

denoising [30–37].To eliminate the undesirable staircase effect, high-order PDEs (typically fourth-order

PDEs) for image restoration have been introduced in [38–43]. Though these methods can

eliminate the staircase effect efficiently, they often lead to a speckle effect [44].Recently, fractional-order PDEs have been studied in many fields [38–49]. The frac-

tional derivative can be seen as the generalization of the integer-order derivative. It has been

studied by many mathematicians (e.g., Euler, Hardy, Littlewood, and Liouville) [47]. Not

until Mandelbrot found fractals and applied the G-L fractional derivative to the Brownian

motion did the fractional derivative cause great attention. There are many methods that can

define the fractional derivative. The usual definitions among them involve G-L fractional

derivative, Cauchy-integral fractional derivative, frequency-domain (Fourier-domain) frac-

tional derivative.

Li and Zhao investigate relation between the data of cyber-physical networking systems

and power laws and then suggest that power-law-type data may be governed by stochasti-

cally differential equations of fractional order [45]. They also propose that one-dimensional

random functions with long-range dependence (LRD) based on a specific class of processes

called the Cauchy-class (CC) process maybe a possible model of sea level data [46].You and Kaveh develop a class of fractional-order multiscale variational model using

G-L definition of fractional-order derivative and propose an efficient condition of the conver-

gence for the model [38]. The experiments show that the model can improve the peak signal-

to-noise ratio, preserve texture, and eliminate the stair effect efficiently.

Bai and Feng proposed a class of fractional-order anisotropic diffusion equations based

on PM equation for image denoising using Fourier-domain fractional derivative in [49].The numerical results showed that both of the staircase effect and the speckle effect can be

eliminated effectively by using the fractional-order derivative.

Inspired from previous works and in order to eliminate the staircase effects and pre-

serve anatomical details, we propose to replace the first-order and the second-order deriva-

tive of the PM Diffusion with the fractional-order derivative using G-L fractional derivative.

It should be indicated that the method proposed in this paper, which is carried on the

sinogram space directly, is different to the method proposed in [49], which is carried on the

Fourier space.

The arrangement of this paper is as follows. In Section 2, the noise model of Low-dosed

CT (L-CT) is introduced; and then the PM diffusion is given in Section 3, new fractional-order

PM method is developed using G-L fractional definition in Section 4; the experiment results

are shown and discussed in Section 5; the final part is the conclusions and acknowledgement.

2. Noise ModelsBased on repeated phantom experiments, low-mA (or low-dose) CT-calibrated projection

data after logarithm transform were found to follow approximately a Gaussian distribution


with an analytical formula between the sample mean and sample variance, that is, the noise

is a signal-dependent Gaussian distribution [20].In this section, we will introduce signal-independent Gaussian noise (SIGN), Poisson

noise, and signal-dependent Gaussian noise.

2.1. Signal-Independent Gaussian Noise (SIGN)

SIGN is a common noise for imaging system. Let the original projection data be {xi}, i = 1,

. . . , m, where i is the index of the ith bin. The signal has been corrupted by additive noise

{ni}, i = 1, . . . , m and one noisy observation

yi = xi + ni, (2.1)

where yi, xi, ni are observations for the random variables Yi, Xi, and Ni where the uppercase

letters denote the random variables and the lower-case letters denote the observations for

respective variables. Xi is normal N(0, σ2X); Ni is normal N(0, σ2

N) and independent to the

Gaussian random variable Xi. Thus Yi is normal N(0, σ2X + σ2

N).

2.2. Poisson Model and Signal-Dependent Gaussian Model

The photon noise is due to the limited number of photons collected by the detector [36]. For

a given attenuating path in the imaged subject, N0(i, α) and N(i, α) denote the incident and

the penetrated photon numbers, respectively. Here, i denote the index of detector channel or

bin, and α is the index of projection angle. In the presence of noises, the sinogram should be

considered as a random process and the attenuating path is given by

ri = − ln

[N(i, α)N0(i, α)

], (2.2)

where N0(i, α) is a constant and N(i, α) is Poisson distribution with mean N.

Thus we have

N(i, α) = N0(i, α) exp(−ri). (2.3)

Both its mean value and variance are N.

Gaussian distributions of ploy-energetic systems were assumed based on limited

theorem for high-flux levels and followed many repeated experiments in [20]. We have

σ2i

(μi

)= fi exp

(μi

γ

), (2.4)

where μi is the mean and σ2i is the variance of the projection data at detector channel or bin i,

γ is a scaling parameter, and fi is a parameter adaptive to different detector bins.

The most common conclusion for the relation between Poisson distribution and

Gaussian distribution is that the photon count will obey Gaussian distribution for the case


with large incident intensity and Poisson distribution with feeble intensity [20]. In addition,

in [36], the authors deduce the equivalency between Poisson model and Gaussian model.

Therefore, both theories indicate that these two noises have similar statistical properties and

can be unified into a whole framework.

3. Perona-Malik Diffusion

In image smoothing, Nonlinear Anisotropic Diffusion (NAD), also called Perona-Malik dif-

fusion (PMD), is a technique aiming at reducing image details without removing significant

parts of the image contents, typically edges, lines, or textures, which are important for the

image [50].With a constant diffusion coefficient, the anisotropic diffusion equations reduce to the

heat equation, which is equivalent to Gaussian blurring. This is ideal for smoothing details

but also blurs edges. When the diffusion coefficient is chosen as an edge seeking function, the

resulting equations encourage diffusion (hence smoothing) within regions and stop it near

strong edges. Hence the edges can be preserved while smoothing from the image [50].Formally, NAD is defined as

∂u(x,y, t

)∂t

= div(g(x,y, t

)∇u(x,y, t

)), (3.1)

where u(x,y, 0) is the initial gray scale image, u(x,y, t) is the smooth gray scale image at time

t, ∇ denotes the gradient, div(·) is the divergence operator, and g(x,y, t) is the diffusion

coefficient. g(x,y, t) controls the rate of diffusion and is usually chosen as a monotonically

decreasing function of the module of the image gradient. Two functions proposed in [50] are

g(∥∥∇u

(x,y, t

)∥∥) = e−(‖∇u(x,y,t)‖/σ)2

, (3.2)

g(∥∥∇u

(x,y, t

)∥∥) = 1

1 +(∥∥∇u

(x,y, t

)∥∥/σ)2, (3.3)

where ‖ · ‖ is the module of the vector and the constant σ controls the sensitivity to edges.

Perona and Malik propose a simple method to approach the modules of gradients,

which is called PM diffusion [50]. Its discretization for Laplacian operator is

u(i, j, t + 1

)= u

(i, j, t

)+

1

4

[cN · ∇2

Nu(i, j, t

)+cS · ∇2

Su(i, j, t

)+ cE · ∇2

Eu(i, j, t

)+ cW · ∇2

Wu(i, j, t

)],

(3.4)

where

∇2Nu

(i, j, t

)= u

(i − 1, j, t

) − u(i, j, t

),

∇2Su(i, j, t

)= u

(i + 1, j, t

) − u(i, j, t

),


∇2Eu(i, j, t

)= u

(i, j + 1, t

) − u(i, j, t

),

∇2Wu

(i, j, t

)= u

(i, j − 1, t

) − u(i, j, t

). (3.5)

According to (3.2)-(3.3), the diffusion coefficient is defined as a function of module of the

gradient. However, computing a gradient accurately in discrete data is very complex and the

module of the gradient is simplified as the absolute values of four directions and diffusion

coefficients are

cN(i, j, t

)= g

(∣∣∣∇2Nu

(i, j, t

)∣∣∣),cS(i, j, t

)= g

(∣∣∣∇2Su(i, j, t

)∣∣∣),cE(i, j, t

)= g

(∣∣∣∇2Eu(i, j, t

)∣∣∣),cW(i, j, t

)= g

(∣∣∣∇2Wu

(i, j, t

)∣∣∣),(3.6)

where | · | is the absolute value of the number and g(·) is defined in (3.2) or (3.3).The main default for PM diffusion is that it will lead to staircase effect or sometimes

details oversmoothing. In order to eliminate the staircase effects and preserve anatomical

details, we propose to replace the first-order and the second-order derivative of the PM Dif-

fusion with the fractional-order derivative using G-L fractional derivative. The new diffusion

model will be introduced in the next section.

4. The Fractional-Order PM Diffusion (FPMD)

The FPMD is developed using G-L fractional-order derivative, which is defined as [38]

Dαg(x) = limh→ 0+

∑k≥0 (−1)kCα

kg(x − kh)

hα, α > 0, (4.1)

where g(x) is a real function, α > 0 is a real number, Cαk= Γ(α + 1)/[Γ(k + 1)Γ(α − k + 1)] is

the generalized binomial coefficient and Γ(·) denotes the Gamma function. If h = 1, the finite

fractional difference is

�αg(x) =K−1∑k=0

(−1)kCαkg(x − k). (4.2)

An image U will be a 2-dimensional matrix of size N × N and its discrete fractional-

order gradient ∇αu is an 8-dimensional vector:

∇αu(i, j)

=(∇α

0u(i, j),∇α

1u(i, j),∇α

2u(i, j),∇α

3u(i, j),∇α

4u(i, j),∇α

5u(i, j),∇α

6u(i, j),∇α

7u(i, j))T

,(4.3)

where T represents the transpose of the vector and ∇αuk(i, j), k = 0, . . . , 7 are defined as

∇α0u(i, j)=

K−1∑k=0

(−1)kCαku(i, j + k

), ∇α

1u(i, j)=

K−1∑k=0

(−1)kCαku(i − k, j + k

),


∇α2u(i, j)=

K−1∑k=0

(−1)kCαku(i − k, j

), ∇α

3u(i, j)=

K−1∑k=0

(−1)kCαku(i − k, j − k

),

∇α4u(i, j)=

K−1∑k=0

(−1)kCαku(i, j − k

), ∇α

5u(i, j)=

K−1∑k=0

(−1)kCαku(i + k, j − k

),

∇α6u(i, j)=

K−1∑k=0

(−1)kCαku(i + k, j

), ∇α

7u(i, j)=

K−1∑k=0

(−1)kCαku(i + k, j + k

).

(4.4)

Thus

∇2αu(i, j)

=(∇2α

0 u(i, j),∇2α

1 u(i, j),∇2α

2 u(i, j),∇2α

3 u(i, j),∇2α

4 u(i, j),∇2α

5 u(i, j),∇2α

6 u(i, j),∇2α

7 u(i, j))T

,

(4.5)

where T represents the transpose of the vector. From (4.3), we have

∇2α0 u

(i, j)=

K−1∑k=0

(−1)kCαk∇α

0u(i, j + k

), ∇2α

1 u(i, j)=

K−1∑k=0

(−1)kCαk∇α

1u(i − k, j + k

),

∇2α2 u

(i, j)=

K−1∑k=0

(−1)kCαk∇α

2u(i − k, j

), ∇2α

3 u(i, j)=

K−1∑k=0

(−1)kCαk∇α

3u(i − k, j − k

),

∇2α4 u

(i, j)=

K−1∑k=0

(−1)kCαk∇α

4u(i, j − k

), ∇2α

5 u(i, j)=

K−1∑k=0

(−1)kCαk∇α

5u(i + k, j − k

),

∇2α6 u

(i, j)=

K−1∑k=0

(−1)kCαk∇α

6u(i + k, j

), ∇2α

7 u(i, j)=

K−1∑k=0

(−1)kCαk∇α

7u(i + k, j + k

).

(4.6)

Let

g =(g0, g1, g2, g3, g4, g5, g6, g7

)T,

(4.7)

where T represents the transpose of the vector and gk, k = 0, . . . , 7 is defined as

gk =g(∣∣∇α

ku(i, j)∣∣)∑7

n=0 g(∣∣∇α

nu(i, j)∣∣) , k = 0, 1, . . . , 7, (4.8)

where ∇αku(i, j), k = 0, . . . , 7, defined in (4.3) are the components of vector ∇αu(i, j) and∑7

n=0 g(|∇αnu(i, j)|) is the normalized constant, g is the decreasing function of absolute value


of ∇αku(i, j), k = 0, . . . , 7. Following (2.2) and (2.3), g(|∇uα

k(x,y, t)|) can be defined as

g(∣∣∇uα

k

(x,y, t

)∣∣) = e−(|∇uαk(x,y,t)|/σ)2

, k = 0, . . . , 7 (4.9)

or

g(∣∣∇uα

k

(x,y, t

)∣∣) = 1

1 +(∣∣∇uα

k

(x,y, t

)∣∣/σ)2, k = 0, . . . , 7, (4.10)

where | · | is the absolute value of the number and the constant σ controls the sensitivity to

edges.

The new FPMD based on G-L fractional-order derivative is defined as

∂u(i, j, t

)∂t

= div

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

g0∇α0u(i, j, t

)g1∇α

1u(i, j, t

)g2∇α

2u(i, j, t

)g3∇α

3u(i, j, t

)g4∇α

4u(i, j, t

)g5∇α

5u(i, j, t

)g6∇α

6u(i, j, t

)g7∇α

7u(i, j, t

)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (4.11)

where the ∇αku(i, j, t), k = 0, . . . , 7, are the components of vector ∇αu(i, j, t) in (4.3) and gk,

k = 0, . . . , 7, defined in (4.8) are the components of g in (4.7).The above equation can be represented as

∂u(i, j, t

)∂t

=7∑

k=0

gk∇2αk u

(i, j, t

), (4.12)

where∑7

k=0 gk = 1 and ∇2αku(i, j, t) can be computed according to (4.5).

Thus the explicit form for solving (4.12) is

u(i, j, t + 1

)= u

(i, j, t

)+

7∑k=0

gk∇2αk u

(i, j, t

), (4.13)

where u(i, j, t + 1) is the gray level of (i, j) at time t + 1 and gk, ∇2αku(i, j, t) are the same as in

(4.12).

5. Experiments and Discussion

The main objective for smoothing L-CT images is to delete the noise while to preserve

anatomy details for the images.


Table 1: RMSE of different smoothing methods.

Noisy Median Wlener GaussianPMD

FPMD FPMD FPMD

image Filter Filter Filter α = 0.2 α = 0.5 α = 1.5

RMSE 0.0962 0.0804 0.0634 0.0963 0.0774 0.0603 0.0735 0.0752

In order to show the performance of FPMD, a 2-dimensional 256 × 256 Shepp-Logan

head phantom developed in MatLab. The number of bins per view is 888 with 984 views

evenly spanned on a circular orbit of 360◦. The detector arrays are on an arc concentric to

the X-ray source with a distance of 949.075 mm. The distance from the rotation center to the

X-ray source is 541 mm. The detector cell spacing is 1.0239 mm. The L-CT projection data

(sinogram) is simulated by adding Gaussian-dependent noise (GDN) whose analytic form

between its mean and variance has been shown in (2.4). In this paper, set fi = 4.0 and T =2e + 4. The projection data is reconstructed by standard Filtered Back Projection (FBP). Since

both the original projection data and sinogram have been provided, the evaluation based on

root-mean-square error (RMSE) between the ideal reconstructed image is and reconstructed

images defined as

√√√√ 1

256 × 256

256∑i=1

256∑j=1

(frecon

(i, j) − fPh

(i, j))2

, (5.1)

where frecon(i, j) denotes the reconstructed value on position (i, j) while fPh(i, j) denotes the

ideal reconstructed value on position (i, j).Two abdominal CT images of a 62-year-old woman with different doses were scanned

from a 16 multidetector row CT unit (Somatom Sensation 16; Siemens Medical Solutions)using 120 kVp and 5 mm slice thickness. Other remaining scanning parameters are gantry

rotation time, 0.5 second; detector configuration (number of detector rows section thickness),16 × 1.5 mm; table feed per gantry rotation, 24 mm; pitch, 1 : 1 and reconstruction method,

Filtered Back Projection (FBP) algorithm with the soft-tissue convolution kernel “B30f”.

Different CT doses were controlled by using two different fixed tube current 30 mAs and

150 mAs ((60 mA or 300 mAs) for L-CT and standard-dose CT (SDCT) protocols, resp.).The CT dose index volume (CTDIvol) for LDCT images and SDCT images are in positive

linear correlation to the tube current and are calculated to be approximately ranged between

15.32 mGy to 3.16 mGy [51] (see Figures 2(a) and 2(b)).On sinogram space, FPMD with α = 0.2, α = 0.5, and α = 1.5 is carried on two image

collections. Other compared methods include median filter with 5 × 5 window; wiener filter

with 5 × 5 window; Gaussian filter whose mean is 0 and its standard deviation is 1.8. The

diffusion coefficient for PMD and FPMDs is selected as a Gaussian function whose standard

deviation is 2. All smoothed projection data will be reconstructed by standard FBP.

Table 1 summarized RMSE between the ideal reconstructed image and filtered

reconstructed image. The FPMD with α = 0.2 has the best performance in RMSE, while other

FPMD with α = 0.5 and α = 1.5 also has better performance than almost other comparing

methods except for 5 × 5 wiener. In summary, the FPMD has a very good performance in

RMSE. Since FPMD provides a more flexible way for diffusion than PMD, FPMD has much

good performance in denoising while preserving structures.


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 1: Shepp-Logan phantoms. (a) Original ideal reconstructed phantom. (b) Simulated LDCTreconstructed phantom. (c) LDCT reconstructed phantom processed by 5 × 5 median filter. (d) LDCTreconstructed phantom processed by 5 × 5 wiener filter. (e) LDCT reconstructed phantom processedby Gaussian smoothing with [σ = 1.8, μ = 0]. (f) LDCT reconstructed phantom processed by PMDwith [σ = 2]. (g) LDCT reconstructed phantom processed by FPMD with [σ = 2, α = 0.2]. (h) LDCTreconstructed phantom processed by FPMD with [σ = 2, α = 0.5]. (i) LDCT reconstructed phantomprocessed by FPMD with [σ = 2, α = 1.5].

Comparing all the original SDCT images and L-CT images in Figures 1 and 2, we found

that the L-CT images were severely degraded by nonstationary noise and streak artifacts.

In Figures 2(g)–2(i), for the proposed FPMD approach, experiments with fractional-order

α gradually increased will obtain more smooth images. Both in Figure 1 and 2, we can

observe better noise/artifacts suppression and edge preservation when α = 0.2. Especially,

compared to their corresponding original SDCT images, the fine features representing the

intrahepatic bile duct dilatation and the hepatic cyst were well restored by using the


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

Figure 2: Abdominal CT images of a 62-year-old woman. (a) Original SDCT image with tube current timeproduct 150 mAs. (b) Original LDCT image with tube current time product 60 mAs. (c) LDCT imageprocessed by 5 × 5 median filter. (d) LDCT image processed by 5 × 5 wiener filter. (e) LDCT imageprocessed by Gaussian smoothing with [σ = 1.8, μ = 0]. (f) LDCT image processed by PMD with [σ = 2].(g) LDCT image processed by FPMD with [σ = 2, α = 0.2]. (h) LDCT image processed by FPMD with[σ = 2, α = 0.5]. (i) LDCT image processed by FPMD with [σ = 2, α = 1.5].

proposed FPMD. We can observe that, the noise grains and artifacts were significantly

reduced for the FPMD processed L-CT images with suitable α both in Figures 1 and 2. The

fine anatomical/pathological features can be well preserved compared to the original SDCT

images (Figures 1(a) and 2(a)) under standard dose conditions.

6. Conclusions

In this paper, we propose a new fractional-order PMD (FPMD) for L-CT sinogram imaging

based on G-L fractional-order derivative definition. Since FPMD is a interpolation between


integer-order Nonlinear Anisotropic Diffusion (NAD) and fourth-order partial differential

equations, it provides a more flexible way to balance the noise reducing and anatomical

details preserving. Smoothing results for phantoms and real sinograms show that FPMD

with suitable parameters can suppress the staircase effects and speckle effects efficiently. In

addition, FPMD also has good performance in visual quality and root mean square errors

(RMSE).

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (no. 60873102),Major State Basic Research Development Program (no. 2010CB732501), and Open Foundation

of Visual Computing and Virtual Reality Key Laboratory Of Sichuan Province (no. J2010N03).This paper was supported by a Grant from the National High Technology Research and

Development Program of China (no. 2009AA12Z140) and Open foundation of Key Labora-

tory of Land Resources Evaluation and Monitoring of Southwest Sichuan Normal University,

Ministry of Education.

References

[1] D. J. Brenner and E. J. Hall, “Computed tomography—an increasing source of radiation exposure,”New England Journal of Medicine, vol. 357, no. 22, pp. 2277–2284, 2007.

[2] J. Hansen and A. G. Jurik, “Survival and radiation risk in patients obtaining more than six CTexaminations during one year,” Acta Oncologica, vol. 48, no. 2, pp. 302–307, 2009.

[3] H. J. Brisse, J. Brenot, N. Pierrat et al., “The relevance of image quality indices for dose optimizationin abdominal multi-detector row CT in children: experimental assessment with pediatric phantoms,”Physics in Medicine and Biology, vol. 54, no. 7, pp. 1871–1892, 2009.

[4] L. Yu, “Radiation dose reduction in computed tomography: techniques and future perspective,”Imaging in Medicine, vol. 1, no. 1, pp. 65–84, 2009.

[5] J. Weidemann, G. Stamm, M. Galanski, and M. Keberle, “Comparison of the image quality of variousfixed and dose modulated protocols for soft tissue neck CT on a GE Lightspeed scanner,” EuropeanJournal of Radiology, vol. 69, no. 3, pp. 473–477, 2009.

[6] W. Qi, J. Li, and X. Du, “Method for automatic tube current selection for obtaining a consistent imagequality and dose optimization in a cardiac multidetector CT,” Korean Journal of Radiology, vol. 10, no.6, pp. 568–574, 2009.

[7] A. Kuettner, B. Gehann, J. Spolnik et al., “Strategies for dose-optimized imaging in pediatric cardiacdual source CT,” Rofo, vol. 181, no. 4, pp. 339–348, 2009.

[8] P. Kropil, R. S. Lanzman, C. Walther et al., “Dose reduction and image quality in MDCT of the upperabdomen: potential of an adaptive post-processing filter,” Rofo, vol. 182, no. 3, pp. 248–253, 2009.

[9] M. K. Kalra, M. M. Maher, M. A. Blake et al., “Detection and characterization of lesions on low-radiation-dose abdominal CT images postprocessed with noise reduction filters,” Radiology, vol. 232,no. 3, pp. 791–797, 2004.

[10] H. B. Lu, X. Li, L. H. Li et al., “Adaptive noise reduction toward low-dose computed tomography,” inMedical Imaging: Physics of Medical Imaging, vol. 5030 of Proceedings of the SPIE, pp. 759–766, 2003.

[11] M. K. Kalra, C. Wittram, M. M. Maher et al., “Can noise reduction filters improve low-radiation-dosechest CT images? Pilot study,” Radiology, vol. 228, no. 1, pp. 257–264, 2003.

[12] M. K. Kalra, M. M. Maher, D. V. Sahani et al., “Low-dose CT of the abdomen: evaluation of imageimprovement with use of noise reduction filters—Pilot study,” Radiology, vol. 228, no. 1, pp. 251–256,2003.

[13] J. C. Ramirez Giraldo, Z. S. Kelm, L. S. Guimaraes et al., “Comparative study of two image space noisereduction methods for computed tomography: bilateral filter and nonlocal means,” in Proceedingsof the 31st Annual International Conference of the IEEE Engineering in Medicine and Biology Society:Engineering the Future of Biomedicine (EMBC ’09), pp. 3529–3532, September 2009.


[14] A. Manduca, L. Yu, J. D. Trzasko et al., “Projection space denoising with bilateral filtering and CTnoise modeling for dose reduction in CT,” Medical Physics, vol. 36, no. 11, pp. 4911–4919, 2009.

[15] N. Mail, D. J. Moseley, J. H. Siewerdsen, and D. A. Jaffray, “The influence of bowtie filtration oncone-beam CT image quality,” Medical Physics, vol. 36, no. 1, pp. 22–32, 2009.

[16] M. Kachelrieß, O. Watzke, and W. A. Kalender, “Generalized multi-dimensional adaptive filtering forconventional and spiral single-slice, multi-slice, and cone-beam CT,” Medical Physics, vol. 28, no. 4,pp. 475–490, 2001.

[17] G. F. Rust, V. Aurich, and M. Reiser, “Noise/dose reduction and image improvements in screeningvirtual colonoscopy with tube currents of 20 mAs with nonlinear Gaussian filter chains,” in MedicalImaging: Physiology and Function from Multidimensional Images, vol. 4683 of Proceedings of the SPIE, pp.186–197, San Diego, Calif, USA, February 2002.

[18] Z. Liao, S. Hu, and W. Chen, “Determining neighborhoods of image pixels automatically for adaptiveimage denoising using nonlinear time series analysis,” Mathematical Problems in Engineering, vol. 2010,Article ID 914564, 14 pages, 2010.

[19] H.-C. Hsin, T.-Y. Sung, and Y.-S. Shieh, “An adaptive coding pass scanning algorithm for optimal ratecontrol in biomedical images,” Computational and Mathematical Methods in Medicine, vol. 2012, ArticleID 935914, 7 pages, 2012.

[20] H. Lu, I. T. Hsiao, X. Li, and Z. Liang, “Noise properties of low-dose CT projections and noisetreatment by scale transformations,” in Proceedings of the IEEE Nuclear Science Symposium ConferenceRecord, vol. 1–4, pp. 1662–1666, November 2001.

[21] J. Xu and B. M. W. Tsui, “Electronic noise modeling in statistical iterative reconstruction,” IEEETransactions on Image Processing, vol. 18, no. 6, pp. 1228–1238, 2009.

[22] I. A. Elbakri and J. A. Fessler, “Statistical image reconstruction for polyenergetic X-ray computedtomography,” IEEE Transactions on Medical Imaging, vol. 21, no. 2, pp. 89–99, 2002.

[23] P. J. La Riviere and D. M. Billmire, “Reduction of noise-induced streak artifacts in X-ray computedtomography through spline-based penalized-likelihood sinogram smoothing,” IEEE Transactions onMedical Imaging, vol. 24, no. 1, pp. 105–111, 2005.

[24] P. J. La Riviere, “Penalized-likelihood sinogram smoothing for low-dose CT,” Medical Physics, vol. 32,no. 6, pp. 1676–1683, 2005.

[25] J. Wang, H. Lu, J. Wen, and Z. Liang, “Multiscale penalized weighted least-squares sinogramrestoration for low-dose X-ray computed tomography,” IEEE Transactions on Biomedical Engineering,vol. 55, no. 3, pp. 1022–1031, 2008.

[26] P. Forthmann, T. Kohler, P. G. C. Begemann, and M. Defrise, “Penalized maximum-likelihoodsinogram restoration for dual focal spot computed tomography,” Physics in Medicine and Biology, vol.52, no. 15, pp. 4513–4523, 2007.

[27] J. Wang, T. Li, H. Lu, and Z. Liang, “Penalized weighted least-squares approach to sinogram noisereduction and image reconstruction for low-dose X-ray computed tomography,” IEEE Transactions onMedical Imaging, vol. 25, no. 10, pp. 1272–1283, 2006.

[28] Z. Liao, S. Hu, M. Li, and W. Chen, “Noise estimation for single- slice sinogram of low-dose X-raycomputed tomography using homogenous patch,” Mathematical Problems in Engineering, vol. 2012,Article ID 696212, 16 pages, 2012.

[29] H. B. Lu, X. Li, I. T. Hsiao, and Z. G. Liang, “Analytical noise treatment for low-dose CT projectiondata by penalized weighted least-square smoothing in the K-L domain,” in Medical Imaging: Physicsof Medical Imaging, vol. 4682 of Proceedings of the SPIE, pp. 146–152, 2002.

[30] P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Transactionson Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629–639, 1990.

[31] A. M. Mendrik, E. J. Vonken, A. Rutten, M. A. Viergever, and B. Van Ginneken, “Noise reduction incomputed tomography scans using 3-D anisotropic hybrid diffusion with continuous switch,” IEEETransactions on Medical Imaging, vol. 28, no. 10, pp. 1585–1594, 2009.

[32] T. S. Kim, C. Huang, J. W. Jeong, D. C. Shin, M. Singh, and V. Z. Marmarelis, “Sinogram enhancementfor high-resolution ultrasonic transmission tomography using nonlinear anisotropic coherencediffusion,” in Proceedings of the IEEE Ultrasonics Symposium, vol. 1, pp. 1816–1819, October 2003.

[33] M. Ceccarelli, V. De Simone, and A. Murli, “Well-posed anisotropic diffusion for image denoising,”IEE Proceedings, vol. 149, no. 4, pp. 244–252, 2002.

[34] V. B. Surya Prasath and A. Singh, “Well-posed inhomogeneous nonlinear diffusion scheme for digitalimage denoising,” Journal of Applied Mathematics, vol. 2010, Article ID 763847, 14 pages, 2010.


[35] Z. Liao, S. Hu, D. Sun, and W. Chen, “Enclosed laplacian operator of nonlinear anisotropic diffusionto preserve singularities and delete isolated points in image smoothing,” Mathematical Problems inEngineering, vol. 2011, Article ID 749456, 15 pages, 2011.

[36] T. Li, X. Li, J. Wang et al., “Nonlinear sinogram smoothing for low-dose X-ray CT,” IEEE Transactionson Nuclear Science, vol. 51, no. 5, pp. 2505–2513, 2004.

[37] S. Hu, Zh. Liao, D. Sun, and W. Chen, “A numerical method for preserving curve edges in nonlinearanisotropic smoothing,” Mathematical Problems in Engineering, vol. 2011, Article ID 186507, 14 pages,2011.

[38] Y. L. You and M. Kaveh, “Fourth-order partial differential equations for noise removal,” IEEETransactions on Image Processing, vol. 9, no. 10, pp. 1723–1730, 2000.

[39] Y. Zhang, H. D. Cheng, Y. Q. Chen, and J. Huang, “A novel noise removal method based on fractionalanisotropic diffusion and subpixel approach,” New Mathematics and Natural Computation, vol. 7, no. 1,pp. 173–185, 2011.

[40] D. Chen, Y. Q. Chen, and D. Xue, “Digital fractional order Savitzky-Golay differentiator and its appli-cation,” in Proceedings of the ASME International Design Engineering Technical Conferences & Computersand Information in Engineering Conference (IDETC/CIE ’11), Washington, DC, USA, August 2011.

[41] L.-T. Ko, J.-E. Chen, Y.-S. Shieh, M. Scalia, and T.-Y. Sung, “A novel fractional discrete cosine transformbased reversible watermarking for healthcare information management systems,” MathematicalProblems in Engineering, vol. 2012, Article ID 757018, 2012.

[42] T. F. Chan, S. Esedoglu, and F. Park, “A fourth order dual method for staircase reduction in textureextraction and image restoration problems,” in Proceedings of the 17th IEEE International Conference onImage Processing (ICIP ’10), pp. 4137–4140, September 2010.

[43] T. Chan, A. Marquina, and P. Mulet, “High-order total variation-based image restoration,” SIAMJournal on Scientific Computing, vol. 22, no. 2, pp. 503–516, 2001.

[44] Z. Jun and W. Zhihui, “A class of fractional-order multi-scale variational models and alternating pro-jection algorithm for image denoising,” Applied Mathematical Modelling, vol. 35, no. 5, pp. 2516–2528,2011.


[46] M. Li, C. Cattani, and S. Y. Chen, “Viewing sea level by a one-dimensional random function with longmemory,” Mathematical Problems in Engineering, vol. 2011, Article ID 654284, 2011.

[47] M. Li, “Fractal time series-a tutorial review,” Mathematical Problems in Engineering, vol. 2010, ArticleID 157264, 2010.

[48] Y. F. Pu, “Application of fractional differential approach to digital image processing,” Journal ofSichuan University, vol. 39, no. 3, pp. 124–132, 2007.

[49] J. Bai and X. C. Feng, “Fractional-order anisotropic diffusion for image denoising,” IEEE Transactionson Image Processing, vol. 16, no. 10, pp. 2492–2502, 2007.

[50] P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Transactionson Pattern Analysis and Machine Intelligence, vol. 12, no. 7, pp. 629–639, 1990.

[51] Y. Chen, W. Chen, X. Yin et al., “Improving low-dose abdominal CT images by Weighted IntensityAveraging over Large-scale Neighborhoods,” European Journal of Radiology, vol. 80, no. 2, pp. e42–e49,2011.


Research ArticleDifference-Equation-BasedDigital Frequency Synthesizer

Lu-Ting Ko,1 Jwu-E. Chen,1 Yaw-Shih Shieh,2Hsi-Chin Hsin,3 and Tze-Yun Sung2

1 Department of Electrical Engineering, National Central University, Chungli 320-01, Taiwan2 Department of Electronics Engineering, Chung Hua University, Hsinchu 300-12, Taiwan3 Department of Computer Science and Information Enginnering, National United University,Miaoli 360-03, Taiwan

Correspondence should be addressed to Tze-Yun Sung, [email protected]



Copyright q 2012 Lu-Ting Ko et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

This paper presents a novel algorithm and architecture for digital frequency synthesis (DFS). It isbased on a simple difference equation. Simulation results show that the proposed DFS algorithmis preferable to the conventional phase-locked-loop frequency synthesizer and the direct digitalfrequency synthesizer in terms of the spurious-free dynamic range (SFDR) and the peak-signal-to-noise ratio (PSNR). Specifically, the results of SFDR and PSNR are more than 186.91 dBc and127.74 dB, respectively. Moreover, an efficient DFS architecture for VLSI implementation is alsoproposed, which has the advantage of saving hardware cost and power consumption.

1. Introduction

Many modern devices, for example, radio receivers, ADSL (Asymmetric Digital Subscriber

Line), XDSL (X Digital Subscriber Line), 3G/4G mobile phones, walkie-talkies, CB radios,

satellite receivers, and GPS systems, require frequency synthesizers with fine resolutions,

fast channel switching, and large bandwidths. There are two types of frequency synthesizer

available: phase-locked loop (PLL) and direct digital frequency synthesis (DDFS).PLL is a control system, which generates an output signal with phase matched that

of the input reference signal. Figure 1 shows the conventional PLL frequency synthesizer

consisting of a phase detector, a charge bump, a lowpass filter, a voltage control oscillator, and

a frequency divider [1–8]. The lower frequency signal, Fdiv, obtained by dividing the output

signal via the frequency divider, is compared with the reference signal, Fref, in the phase


Phase detector Charge bump Low-pass filter

Voltage

controlled

oscillator

Frequency

divider

OutputFref

Fdiv

Figure 1: Block diagram of the conventional PLL system.

vv

Fclk

Phase

accumulator

Sine/cosine

generator

Digital to

analog

converter

Low pass filter

θ cos θ

sin θFCW

Figure 2: Block diagram of the conventional DDFS architecture.

detector to generate an error signal, which is proportional to the phase difference. The charge

bump converts the error signal pulse into analog current pulses, which are then integrated

by using the lowpass filter, and drives the voltage-controlled oscillator to obtain the desired

frequency.

The commonly used architecture of DDFS [9] shown in Figure 2 consists of a phase

accumulator, a sine/cosine generator, a digital-to-analog converter (DAC), and a lowpass

filter (LPF). It takes two inputs: a reference clock and a v-bit frequency control word (FCW).In each clock cycle, the phase accumulator integrates FCW with periodical overflow to

produce an angle in the range of [0, 2π), the sine/cosine generator computes its sinusoidal

value, which in practice is implemented digitally and, therefore, follows by DAC and LPF

[10–23]. Various fractional-order ideal filters and fractional oscillators were proposed in [24–

29].Instead of using the conventional methods above, we propose a novel digital

frequency synthesis (DFS) algorithm based on a simple difference equation. The rest of the

paper is organized as follows. In Section 2, a novel DFS algorithm is proposed. In Section 3,

the VLSI (very large-scale integration) digital frequency synthesizer is presented. In Section 4,

the FPGA implementation and the performance evaluation are given. Conclusion can be

found in Section 5.

2. The Proposed DFS Algorithm

The difference equation of DFS is as follows:

y[n − 2] − 2a1y[n − 1] + a2y[n] = x[n]. (2.1)


Thus, we have the following characteristic equation:

z−2 − 2a1z−1 + a2 = 0. (2.2)

The eigen-functions of (2.2) are represented as

z1,2 = re±jθ. (2.3)

The ZIR (zero-input response) of DFS can be written as

ZIR = rn(B1 cos(nθ) + B2 sin(nθ)), (2.4)

where B1 and B2 are determined by initial conditions, and n = 0, 1, 2, . . . .

For DFS with sine wave generator, we have

r = 1, B1 = 0, B2 = 1, ZIR = sin(nθ). (2.5)

The eigen-functions of DFS are therefore as follows:

z1,2 = e±jθ = cos θ ± j sin θ = c ± jd. (2.6)

Thus, the characteristic equation can be expressed as

z−2 − 2cz−1 + c2 + d2 = 0, (2.7)

where c2 + d2 = 1.

Equation (2.7) could be rewritten as

z−2 − 2cz−1 + 1 = 0, (2.8)

and the transfer function of DFS can be derived as

H(z) =1

z−2 − 2cz−1 + 1. (2.9)

According to (2.9), the corresponding difference equation could be derived as

y[n] = x[n] − y[n − 2] + 2c · y[n − 1], (2.10)

where

y[0] = B1 = 0,

y[1] = −B1 cos θ + B2 sin θ = d.(2.11)


As one can see, a rotation of angle φ in the circular coordinate system can be obtained

by performing a sequence of microrotations in an iterative manner. In particular, a vector

can be successively rotated through the use of a sequence of predetermined step angles:

α(i) = tan−1(2−i). This technique can be applied to generate many elementary functions,

in which only simple adders and shifters are required. Thus, the well-known coordinate

rotation digital computer (CORDIC) algorithm can be used for the DFS applications. The

conventional CORDIC in the circular coordinate system is as follows [36–39]:

u(i + 1) = u(i) − σ(i)2−iv(i),

v(i + 1) = v(i) + σ(i)2−iu(i),

w(i + 1) = w(i) − σ(i)α(i),

α(i) = tan−12−i,

(2.12)

where σ(i) ∈ {−1,+1} denotes the direction of the ith microrotation, σi = sign(w(i)) with

w(i) → 0 in the vector rotation mode, σi = − sign(u(i)) · sign(v(i)) with v(i) → 0 in the

angle accumulated mode, the corresponding scale factor k(i) is equal to√

1 + σ2(i)2−2i, and

i = 0, 1, . . . , l − 1. The product of all scale factors after l microrotations is given by

K1 =n−1∏i=0

k(i) =n−1∏i=0

√1 + 2−2i. (2.13)

In the vector rotation mode, sinφ and cosφ can be obtained, where the initial value (u(0),v(0)) = (1/K1, 0). In principle, uout and vout can be computed from the initial value (uin,vin) =(u(0),v(0)) by using the following equation:

[uout

vout

]= K1

[cosφ − sinφ

sinφ cosφ

][uin

vin

]. (2.14)

In order to evaluate the sinusoidal parameters: c and d for the proposed digital fre-

quency synthesizer, the inputs of the CORDIC processor are uin = 1/K1, vin = 0, and win = θ

as shown in Figure 3.

3. Proposed Architecture for Digital Frequency Synthesizer

In this section, the architecture and the terminology associated with the proposed digital

frequency synthesizer are presented. Our scheme is based on the proposed DFS algorithm

combined with a CORDIC processor. It consists mainly of the radian converter, the CORDIC

processor, and the sine generator as shown in Figure 4.

Figure 5 shows the radian converter. It is a constant multiplier, which converts the

input signal into radians. Figure 6 shows the CORDIC processor, which evaluates the sinus-

oidal value and consists of three adders and two shifters.

Figure 7 shows the architecture of sine generator, which is the core of the proposed

digital frequency synthesizer. It consists of one multiplier, one adder, and two latches only.


CORDIC

Angle accumulated mode

uin =

vin = 0

win = θvout = sin θ

uout = cos θ

1

K1

Figure 3: The CORDIC arithmetic for the proposed digital frequency synthesizer.

sin θ

cos θ

θRadian

converter

CORDIC

processor

Sine

generator

Fo

FS

Figure 4: The proposed digital frequency synthesizer.

The key terminologies associated with the proposed digital frequency synthesizer are

as follows.

3.1. Output Frequency

The output frequency of the proposed digital frequency synthesizer is determined by the

coefficients d and c, since

θ = ωTs = tan−1

(d

c

),

Fo =1

2π· tan−1

(d

c

)· Fs.

(3.1)

3.2. Frequency Resolution

For m-bit digital frequency synthesizer, the minimum change of the output frequency ΔFo,min

is expressed as

ΔFo,min =1

2π· tan−1

(2−(m−1)

)· Fs. (3.2)


Input

2 1 2

CSA(3,2) CSA(3,2)

CSA(4,2)

+

Output

5 9 15

Figure 5: The radian converter.

3.3. Bandwidth

The bandwidth of digital frequency synthesizer is defined as the difference between the

highest and lowest attainable output frequencies, which are expressed as follows:

Fo,max =1

2π· tan−1(1) · Fs,

Fo,min =1

2π· tan−1

(2−(m−1)

)· Fs.

(3.3)

3.4. Peak Signal-to-Noise Ratio (PSNR)

A good direct digital frequency synthesizer should have an output signal with low noise,

which can be evaluated by using the following signal-to-noise-ratio (PSNR) measured in dB:

PSNR = 20 log

(255√MSE

), (3.4)

where MSE is the mean square error.

3.5. Spurious-Free Dynamic Range (SFDR)

The spurious-free dynamic range (SFDR) is defined as the ratio of the amplitude of the

desired frequency component to that of the largest undesired frequency component at the

output of a DDFS. It is expressed in decibels (dBc) as follows:

SFDR = 20 log

(Ap

As

)= 20 log

(Ap

) − 20 log(As), (3.5)


Latch

10

10

+

+

+

Sign

detector

0

1LUT

Barrel

shifter

Barrel

shifter

clk

u(i)

i

i

α(i)

v(i + 1)

w(i + 1)

−1

−1

−1

u(i + 1)

v(i)

w(i)

i

Figure 6: The CORDIC processor (LUT: Lookup table).

Latch

Latch

2c = 2 cos θ

Output

Initial value: sin θ

clk

−1

+ ×

Figure 7: The sine generator.

where Ap is the amplitude of the desired frequency component, As is the amplitude of the

largest undesired frequency component, and the higher the better.

4. FPGA Implementation of Digital Frequency Synthesizer

In this section, the proposed high-performance architecture of digital frequency synthesizer

is presented. Figure 8 depicts the system block diagram. The PSNR and SFDR of the proposed


sin θ

cos θθRadian

converter

CORDIC

processor

Output

Control

CLA

Latch Latch

1 0 1-bit

shifter

Multiplier

Fo

FS

Figure 8: The proposed digital frequency synthesizer.

0 10 14 18 22 26 30 3460

80

100

120

140

160

180

200

220

240

Word length

PS

NR

(d

B)

Figure 9: The PSNR of the proposed digital frequency synthesizer at various word lengths (100 MHzsampling rate and the maximum output frequency 12.5 MHz).

digital frequency synthesizer at various word lengths at 100 MHz sampling rate and the

maximum output frequency 12.5 MHz are shown in Figures 9 and 10, respectively.

The platform for architecture development and verification has also been designed

and implemented to evaluate the development cost. The proposed architecture of digital

frequency synthesizer has been implemented on the field programmable gate array (FPGA)emulation board [40]. The FPGA has been integrated with the microcontroller (MCU)and I/O interface circuit (USB 2.0) to form the architecture development and verification

platform.

Figure 11 depicts the block diagram and circuit board of the architecture development

and evaluation platform. In which, the microcontroller reads data and commands from PC

and writes the results back to PC via USB 2.0 bus; the FPGA implements the proposed


0 10 14 18 22 26 30 340

100

200

300

400

500

600

700

Word length

SF

DR

(d

Bc)

Figure 10: The SFDR of the proposed digital frequency synthesizer at various word lengths (100 MHzsampling rate and the maximum output frequency 12.5 MHz).

MCU FPGA

Architecture evaluation

boardPC

USB 2.0

Figure 11: Block diagram and circuit board of the architecture development and verification platform.

architecture of digital frequency synthesizer. The hardware code in portable hardware

description language runs on PC with the logic circuit simulator [41] and FPGA compiler

[42]. It is noted that the throughput can be improved by using the proposed pipelined

architecture while the computation accuracy is the same as that obtained by using the

conventional architecture with the same word length. Thus, the proposed digital frequency

synthesizer improves the power consumption and performance significantly. Moreover,

all the control signals are internally generated onchip. The proposed digital frequency

synthesizer provides a high-performance sinusoid waveform.

5. Conclusion

In this paper, we present a novel digital frequency synthesizer based on a simple difference

equation with pipelined data path. Circuit emulation shows that the proposed high-

performance architecture has the advantages of high precision, high data rate, and simple

hardware. For 16-bit digital frequency synthesizer, the PSNR and SFDR obtained by using

the proposed architecture at the maximum output frequency are 127.74 dB and 186.91 dBc,

respectively. As shown in Table 1, the proposed digital frequency synthesizer is superior to

the previous works in terms of SFDR, PSNR, and hardware [18, 30–35]. The proposed digital


Table 1: Comparisons between the proposed DFS and other related works.

AuthorsItems

Outputresolution (bits) SFDR (dBc) PSNR (dB) ROM

(words) Adders Multipliers

Strollo et al. [18] 13 90.2 — 1344 8 0

Song and Kim [30] 16 100 — 270 6 6

Langlois and Al-Khalili[31] 14 96.2 — 1152 8 0

De Caro et al. [32] 12 80 — 0 2 2

De Caro and Strollo [33] 12 83.6 — 896 2 2

Yang et al. [34] 12 80 — 2176 6 0

Curticapean andNiittylahti [35] 14 85 — 832 2 2

Ko et al. (This work) 14 133 113 14 8 1

16 187 128 16 8 1

frequency synthesizer designed by portable hardware description language is a reusable IP,

which can be implemented in various VLSI processes with trade-offs of performance, area

and power consumption.

Acknowledgment

The National Science Council of Taiwan, under Grants NSC100-2628-E-239-002-MY2, sup-

ported this work.

References

[1] W. Chen, V. Pouget, H. J. Barnaby et al., “Investigation of single-event transients in voltage-controlledoscillators,” IEEE Transactions on Nuclear Science, vol. 50, no. 6, pp. 2081–2087, 2003.

[2] F. M. Gardner, Phaselock Techniques, Wiley, New York, NY, USA, 3rd edition, 2005.[3] J. Y. Chang, C. W. Fan, C. F. Liang, and S. I. Liu, “A single-PLL UWB frequency synthesizer using

multiphase coupled ring oscillator and current-reused multiplier,” IEEE Transactions on Circuits andSystems II: Express Briefs, vol. 56, no. 2, pp. 107–111, 2009.

[4] H. H. Chung, W. Chen, B. Bakkaloglu, H. J. Barnaby, B. Vermeire, and S. Kiaei, “Analysis of singleevents effects on monolithic PLL frequency synthesizers,” IEEE Transactions on Nuclear Science, vol.53, no. 6, pp. 3539–3543, 2006.

[5] T. Wu, P. K. Hanumolu, K. Mayaram, and U. K. Moon, “Method for a constant loop bandwidth inLC-VCO PLL frequency synthesizers,” IEEE Journal of Solid-State Circuits, vol. 44, no. 2, pp. 427–435,2009.

[6] R. B. Staszewski, “State-of-the-art and future directions of high-performance all-digital frequencysynthesis in nanometer CMOS,” IEEE Transactions on Circuits and Systems. I. Regular Papers, vol. 58,no. 7, pp. 1497–1510, 2011.

[7] M. Zanuso, P. Madoglio, S. Levantino, C. Samori, and A. L. Lacaita, “Time-to-digital converter forfrequency synthesis based on a digital bang-bang DLL,” IEEE Transactions on Circuits and Systems. I.Regular Papers, vol. 57, no. 3, pp. 548–555, 2010.

[8] R. B. Staszewski, S. Vemulapalli, P. Vallur, J. Wallberg, and P. T. Balsara, “1.3 V 20 ps time-to-digitalconverter for frequency synthesis in 90-nm CMOS,” IEEE Transactions on Circuits and Systems II:Express Briefs, vol. 53, no. 3, pp. 220–224, 2006.

[9] J. Tierney, C. Rader, and B. Gold, “A digital frequency synthesizer,” IEEE Transactions on Audio andElectroacoustics, vol. 19, no. 1, pp. 48–57, 1971.


[10] A. Bonfanti, D. De Caro, A. D. Grasso, S. Pennisi, C. Samori, and A. G. M. Strollo, “A 2.5-GHz DDFS-PLL with 1.8-MHz bandwidth in 0.35-μm CMOS,” IEEE Journal of Solid-State Circuits, vol. 43, no. 6,pp. 1403–1413, 2008.

[11] D. De Caro, E. Napoli, and A. G. M. Strollo, “Direct digital frequency synthesizers with polynomialhyperfolding technique,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 51, no. 7, pp.337–344, 2004.

[12] Y. H. Chen and Y. A. Chau, “A direct digital frequency synthesizer based on a new form of polynomialapproximations,” IEEE Transactions on Consumer Electronics, vol. 56, no. 2, pp. 436–440, 2010.

[13] C. Y. Kang and E. E. Swartzlander, “Digit-pipelined direct digital frequency synthesis based ondifferential CORDIC,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 53, no. 5, pp.1035–1044, 2006.

[14] D. De Caro, N. Petra, and A. G. M. Strollo, “High-performance special function unit for programmable3-D graphics processors,” IEEE Transactions on Circuits and Systems. I. Regular Papers, vol. 56, no. 9, pp.1968–1978, 2009.

[15] D. Yang, F. F. Dai, W. Ni, S. Yin, and R. C. Jaeger, “Delta-sigma modulation for direct digital frequencysynthesis,” IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 17, no. 6, pp. 793–802,2009.

[16] T. Rapinoja, K. Stadius, L. Xu et al., “A digital frequency synthesizer for cognitive radio spectrumsensing applications,” IEEE Transactions on Microwave Theory and Techniques, vol. 58, no. 5, pp. 1339–1348, 2010.

[17] A. McEwan and S. Collins, “Direct digital-frequency synthesis by analog interpolation,” IEEETransactions on Circuits and Systems II: Express Briefs, vol. 53, no. 11, pp. 1294–1298, 2006.

[18] A. G. M. Strollo, D. De Caro, and N. Petra, “A 630 MHz, 76 mW direct digital frequency synthesizerusing enhanced ROM compression technique,” IEEE Journal of Solid-State Circuits, vol. 42, no. 2, pp.350–360, 2007.

[19] Y. D. Wu, C. M. Lai, C. C. Lee, and P. C. Huang, “A quantization error minimization method usingDDS-DAC for wideband fractional-N frequency synthesizer,” IEEE Journal of Solid-State Circuits, vol.45, no. 11, pp. 2283–2291, 2010.

[20] S. E. Turner, R. T. Chan, and J. T. Feng, “ROM-based direct digital synthesizer at 24 GHz clockfrequency in InP DHBT technology,” IEEE Microwave and Wireless Components Letters, vol. 18, no. 8,pp. 566–568, 2008.

[21] T. Y. Sung, L. T. Ko, and H. C. Hsin, “Low-power and high-SFDR direct digital frequency synthesizerbased on hybrid CORDIC algorithm,” in Proceedings of the IEEE International Symposium on Circuitsand Systems (ISCAS ’09), pp. 249–252, May 2009.

[22] T. Y. Sung, H. C. Hsin, and L. T. Ko, “High-SFDR and multiplierless direct digital frequencysynthesizer,” WSEAS Transactions on Circuits and Systems, vol. 8, no. 6, pp. 455–464, 2009.

[23] Y. J. Cao, Y. Wang, and T. Y. Sung, “A ROM-Less direct digital frequency synthesizer based on ascaling-free CORDIC algorithm,” in Proceedings of the 6th International Forum on Strategic Technology,pp. 1186–1189, Harbin, China, 2011.

[24] M. Li, “Approximating ideal filters by systems of fractional order,” Computational and MathematicalMethods in Medicine, vol. 2012, Article ID 365054, 6 pages, 2012.

[25] M. Li, S. C. Lim, and S. Chen, “Exact solution of impulse response to a class of fractional oscillatorsand its stability,” Mathematical Problems in Engineering, vol. 2011, Article ID 657839, 2011.



[28] S. Y. Chen, J. Zhang, H. Zhang, N. M. Kwok, and Y. F. Li, “Intelligent lighting control for vision-basedrobotic manipulation,” IEEE Transactions on Industrial Electronics, vol. 59, no. 8, pp. 3254–3263, 2012.


[30] Y. Song and B. Kim, “Quadrature direct digital frequency synthesizers using interpolation-basedangle rotation,” IEEE Transactions on Very Large Scale Integration (VLSI) Systems, vol. 12, no. 7, pp.701–710, 2004.

[31] J. M. P. Langlois and D. Al-Khalili, “Novel approach to the design of direct digital frequencysynthesizers based on linear interpolation,” IEEE Transactions on Circuits and Systems II: Analog andDigital Signal Processing, vol. 50, no. 9, pp. 567–578, 2003.


[32] D. De Caro, E. Napoli, and A. G. M. Strollo, “Direct digital frequency synthesizers with polynomialhyperfolding technique,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 51, no. 7, pp.337–344, 2004.

[33] D. De Caro and A. G. M. Strollo, “High-performance direct digital frequency synthesizers usingpiecewise-polynomial approximation,” IEEE Transactions on Circuits and Systems. I. Regular Papers,vol. 52, no. 2, pp. 324–337, 2005.

[34] B. D. Yang, J. H. Choi, S. H. Han, L. S. Kim, and H. K. Yu, “An 800-MHz low-power direct digitalfrequency synthesizer with an on-chip D/A converter,” IEEE Journal of Solid-State Circuits, vol. 39, no.5, pp. 761–774, 2004.

[35] F. Curticapean and J. Niittylahti, “A hardware efficient direct digital frequency synthesizer,” inProceedings of the IEEE International Conference Electronics, Circuits and Systems (ICECS ’01), pp. 51–54,Malta, 2001.

[36] J. Volder, “The CORDIC trigonometric computing technique,” IRE Transactions on Electronic Comput-ers, vol. 8, no. 3, pp. 330–334, 1959.

[37] J.S. Walther, “A unified algorithm for elementary functions,” in Proceedings of the Spring Joint ComputerConference (AFIPS ’71), pp. 379–385, 1971.

[38] T. Y. Sung and H. C. Hsin, “Design and simulation of reusable IP CORDIC core for special-purposeprocessors,” IET Computers and Digital Techniques, vol. 1, no. 5, pp. 581–589, 2007.

[39] Y. H. Hu, “CORDIC-based VLSI architectures for digital signal processing,” IEEE Signal ProcessingMagazine, vol. 9, no. 3, pp. 16–35, 1992.

[40] SMIMS Technology Corp., http://www.smims.com.

[41] S. Palnitkar, Verilog HDL-A Guide to Digital Design and Synthesis, Prentice-Hall, 2nd edition, 2003.

[42] Xilinx FPGA products, 2011, http://www.xilinx.com/products/.


Research ArticleKernel Optimization for Blind Motion Deblurringwith Image Edge Prior

Jing Wang, Ke Lu, Qian Wang, and Jie Jia

College of Computing & Communication Engineering, Graduate University of Chinese Academy of Science,Beijing 100049, China

Correspondence should be addressed to Ke Lu, [email protected]

Received 10 January 2012; Accepted 20 February 2012


Copyright q 2012 Jing Wang et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Image motion deblurring with unknown blur kernel is an ill-posed problem. This paper proposesa blind motion deblurring approach that solves blur kernel and the latent image robustly. Forkernel optimization, an edge mask is used as an image prior to improve kernel update, then anedge selection mask is adopted to improve image update. In addition, an alternative iterativemethod is introduced to perform kernel optimization under a multiscale scheme. Moreover, forimage restoration, a total-variation-(TV-) based algorithm is proposed to recover the latent imagevia nonblind deconvolution. Experimental results demonstrate that our method obtains accurateblur kernel and achieves better deblurring results than previous works.

1. Introduction

Motion deblurring is a type of image restoration problems [1, 2]. Commonly, image motion

blur is caused by camera sensor motion, where the track of the sensor motion is represented

by a blur kernel [3]. Theoretically, the motion blur process is modeled as the convolution

of the latent image and a blur kernel with additive noise (Figure 1). Therefore, motion

deblurring not only solves for blur kernel but also recovers latent image. As a blind de-

convolution process, motion deblurring is always split into two stages: kernel estimation

and nonblind image deconvolution. Note that motion deblurring with single-input image is

more complicated than that with two-or-more-input images because multiple blurred images

always provide more information in solving the problem [4–6]. In this paper, we mainly focus

on the single-image-based motion deblurring.

To address such challenging problem, various theories and methods have been pro-

posed. In early days, blind deconvolution recovers sharp images by simple motion and

Gaussian blur based on frequency-domain constraints or assumptions [7]. Recently, many


Noise

∗ +

Figure 1: Image motion blur process.

researchers believe that the more accurate the obtained kernel is, the more clear the recovered

image will be. For this reason, kernel estimation becomes a principal task in motion

deblurring research. Several novel ideas based on image spatial domain priors are put

forward to solve for the blur kernel. For example, Fergus et al. [8] introduced a statistics

research of natural image as image gradient prior. Their method uses a Bayesian approach to

solve the kernel and then adopts the Richardson-Lucy deconvolution algorithm to reconstruct

the image with the estimated kernel. Jia [9] suggested that the blur kernel can be determined

by the transparency on the object boundary, and a maximum a posteriori (MAP) model

was implemented to estimate the blur kernel. Cai et al. [10] handled a joint optimization

problem with the linearized Bregman iteration method, which maximizes the sparsity of the

blur kernel and the latent image under curvelet system and framelet system, respectively.

Moreover, Joshi et al. [11] solved the simple motion blur and defocus blur kernel by a

predicted sharp edge of the blurry image. Xu and Jia [12] proposed a two-phase kernel es-

timation scheme, which uses a gradient selection process to measure the usefulness of image

edges [13].Even with the estimated kernel, the restoration of the latent image is still a tough

problem. In the process of motion blur, the latent image loses much high-frequency infor-

mation. The traditional methods (inverse filter, wiener filter, etc.) always give undesirable

restoration results because of the effect of the additive noise [14, 15]. To overcome such

difficulty, novel image restoration method with total variation regularization term was

proposed recently which removes the image noise and preserves edge details simultaneously

[16, 17]. To solve the total variation deconvolution problem, it is common to transform it into

a partial difference equation first. Rudin and Osher [16] proposed a time marching scheme to

solve the TV model. Vogel and Oman [18] used the fix point iteration method to optimize the

TV image deconvolution. According to the variable split method and half quadratic penalty

function method, Wang et al. [19] presented a fast total variation deconvolution (FTVd)algorithm to compute TV image deconvolution. Afonso et al. [20] proposed a split augmented

Lagrangian shrinkage algorithm (SALSA), where the augmented Lagrangian method is used

in computing. Similarly, Chan et al. [21] adopted the alternating direction method (ADM)which considered another variant of the augmented Lagrangian method.

In this paper, a complete blind deblurring algorithm is proposed to handle image

motion blur with image edge prior. In kernel optimization, an edge mask is used as im-

age prior to improve kernel update and an edge selection mask is adopted to improve im-

age update. Moreover, an alternative iterative method is introduced to implement kernel

optimization under a multiscale scheme. For image restoration, a total-variation-based image

nonblind deconvolution algorithm is proposed to restore latent image. The rest of the paper


is organized as follows. In Section 2, the edge character in the blurry image is analyzed and

an edge detection process is introduced. In Section 3, given the image edge prior, a complete

blind deblurring algorithm is described including a kernel estimation model and an image

restoration model. Then, the algorithm is implemented with motion blurred images and

the restored results are shown and discussed in Section 4. Finally, Section 5 concludes our

work.

2. Image Edge Prior

Motion deblurring is an ill-posed problem where the number of unknowns is more than the

number of observed measurements. Generally, the motion blur process can be modeled as

B = K ∗ I +N, (2.1)

where B is the blurry image, K is the motion blur kernel, I is the latent image, N is the

noise effect, and ∗ is the convolution operator. This model suggests that the blurry image is

the convolution of a blur kernel and the latent image. Accordingly, our goal is to solve the

kernel and the latent image inversely from a single blurred image, which is obviously an

ill-posed problem. Therefore, such inverse problem can be solved only with other necessary

prior knowledge being provided.

In image processing system, an edge is defined as the continuous boundary pixels that

connect two separate regions with changing image amplitude attributes [22–24]. It offers

information including magnitude and orientation, which has been widely used as image

prior knowledge in solving many image processing and computer vision problems, such as

image restoration [11, 13] and image superresolution [25, 26]. After edge detection, there

always exist some particular edges caused by blur or noise in edge map. Such edges are

called false edges, which can be further removed by other image processing techniques.

In motion deblurring problem, it can be seen that the edge in the blurry image usually

appears fuzzy or unsharp, as shown in Figures 2(c) and 2(d), while the latent image has

clear edges, as shown in Figures 2(a) and 2(b). If an edge map can be found from the motion

blurred image, which is also assumed to be closed enough to the edge map of the latent image,

then it might be used to refine the kernel estimation. In this paper, it is assumed that the fuzzy

edges in the blurry image are viewed as false edges, which are removed through an edge

detection process. In other words, sharp edges can be found through certain edge detection

process. This sharp edge map is taken as an edge prior to improve kernel estimation.

The edge detection process finds the presence and locations of the intensity transitions.

To find an ideal edge map from the blurry image, a modified edge detection process is used

and described as follows. First the blurry image is convolved by the derivatives of Gaussian.

Then, the magnitude and orientation of its gradient are computed. Thirdly, a nonmaxima

suppression method is used to get the thinned gradient magnitudes. Finally, hysteresis is

used to get the sharp edge map by doing threshold operation on the gradient magnitude.

Especially, two adaptive thresholds are used to suppress the false edges. As shown in Figures

2(e) and 2(f), the detected edge map and its close-up are sufficiently clear and sharp in detail.

According to this edge map, the edge locations are labeled in a mask, which is used to solve

for the blur kernel as described in the Section 3.


(a) (b)

(c) (d)

(e) (f)

Figure 2: Edge features analysis in both motion blurred image and latent image. (a) Edge map beforemotion blur, (b) close-up of (a), (c) edge map after motion blur, (d) close-up of (c), (e) edge map detectedfrom the blurry image by our edge detection process, and (f) close-up of (e).

3. Blind Deblurring Algorithm

3.1. Kernel Estimation Model

Before kernel estimation, the blurry image and the initial kernel, which are later used as the

inputs of our algorithm, need to be preprocessed. More specifically, the bilateral filter and the


shock filter are used to smooth the noise and keep the edge details, respectively [27]. The blur

kernel is defined as a smooth convolution mask with nonnegativity values and normalized.

So the initial kernel is set as a unit matrix with unit value at its central position.

For kernel optimization, an iterative optimization problem model is constructed, and

its task is to optimize the blur kernel and the latent image alternately under a multiscale

scheme. The edge map mentioned above is introduced as image prior, which adds mask on

both latent image and blurry image. On the other hand, the L1 norm of the kernel is used as

a regularization term to suppress the noise in the blur kernel. According to the motion blur

model, the minimization energy function for motion blur kernel is as follows:

minK

{‖K ∗ME(∇I) −ME(∇B)‖2 + α‖K‖1

}, (3.1)

where ‖K ∗ME(∇I) −ME(∇B)‖2 is the data fitting term and ‖K‖1 is the L1 regularization

term of K. ME is the edge location mask mentioned in Section 2, ∇I is the gradient of latent

image, ∇B is the gradient of blurry image, and parameter α controls the relative strength

of the data fitting and kernel regularization terms. Before adding the edge mask ME, lateral

filter is used to suppress noise in blurry image B. Here an unconstrained iterative reweighted

least squares (IRLS) system [28, 29] is adopted to solve this minimization problem, and the

conjugate gradient (CG) method is used to solve the inner IRLS system.

For latent image optimization, an edge selection mask mentioned in [13] is used to

recover a coarse latent image. Image edges do not always profit kernel estimation, so we

need an edge selection process to choose useful ones. The energy function is then modeled as

follows:

‖K ∗ I − B‖2 + β‖∇I −MS(∇I)‖2, (3.2)

where ‖K ∗ I − B‖2 is the data fitting term, β‖∇I −MS(∇I)‖2 is the regularization term, β

is playing the same role as α, and MS is the edge selection mask. According to Parseval’s

theorem, this equation has a closed-form solution by using FFTs:

I = F−1

⎛⎜⎝F(K)F(B) + β(F(∂x)F

(IxMs

)+ F

(∂y)F(Iy

Ms

))F(K)F(K) + β

(F(∂x)F(∂x) + F

(∂y)F(∂y))

⎞⎟⎠, (3.3)

where F denotes the FFT and F−1 denotes the inverse FFT.

To solve for the kernel accurately, a multiscale scheme is introduced to implement

the whole blind deblurring algorithm. Under this scheme, blur kernel and latent image are

estimated by using a coarse-to-fine pyramid of image resolutions. The number of scale levels

is computed by the size of blur kernel and the scale level step is√

2. The blurry image is

downsampled as the algorithm input. In each scale, the latent image is updated by solving

(3.2). Then the updated latent image is used to update the blur kernel by solving (3.1). Finally,

the updated kernel is used in the next scale.


3.2. Image Restoration Model

Given the estimated blur kernel, the latent image can be restored by using a nonblind image

deconvolution algorithm. As mentioned above, a clear image should have sharp edge details.

For this reason, an image restoration model with TV regularization term is built to recover

the latent image. This model is a minimum optimization problem:

minI

{λ

2‖K ∗ I − B‖2 + ‖DI‖1

}, (3.4)

where ‖K ∗ I − B‖2 is the data fitting term, ‖DI‖1 is the TV regularization term, D is the finite

difference operator, and λ is weight factor. When (3.4) is used to restore the latent image, the

TV regularization term can keep the image edge details satisfyingly.

It can be seen that (3.4) is an L1 norm regularization optimization problem. In this

paper, the split Bregman method [30, 31] is introduced to solve the problem. The split Breg-

man method, proposed by Goldstein and Osher, is a fast scheme to solve a type of optimiza-

tion problem with the form

minu

{‖l(u)‖1 + f(u)}, (3.5)

where l(u) and f(u) are both convex functions. According to the variable split method, the

split Bregman method transforms (3.5) into an unconstraint optimization problem with an

auxiliary variable and quadratic penalty function. Then this unconstraint optimization model

is divided into two or three optimization subproblems and solved alternatively with the

Bregman iteration.

At first, an auxiliary variable G takes the place of DI, and (3.4) is transformed into a

unconstrained optimization equation related to I and G as follows:

minG,I

{λ

2‖K ∗ I − B‖2 +

γ

2‖G −DI‖2 + ‖G‖1

}. (3.6)

Then, (3.6) is divided into two subproblems related to I and G, respectively. According to the

Bregman iteration, these two optimization subproblems are modeled as

minI

{λ

2‖K ∗ I − B‖2 +

γ

2‖G −DI − b‖2

}, (3.7)

minG

{λ

2‖G −DI − b‖2 + ‖G‖1

}, (3.8)

where b is an iteration parameter and b = b + (G −DI).To solve these two sub-problems, an alternative minimization method (AMD) is used

to optimize them. In each iteration, (3.7) is transformed into the equation as follows:

(DTD +

γ

λKTK

)I =

γ

λKTB +DT (G − b), (3.9)


(a)

(b)

(c)

Figure 3: Testing our algorithm with the synthetic images. (a)–(c) In order from left to right, the imagesare the original synthetic images, the synthetic images after motion blur, restored results and estimatedkernels by using our algorithm, and the close-ups of them (the red rectangle in the blurry image shows thelocation of close-ups).

where K and D are both block circulant matrices. So (3.9) is computed by FFTs. On the

other hand, (3.8) is optimized by the shrinkage technique, and it can be solved by using

the following equation:

G = max

{∥∥g∥∥2− 1

λ, 0

}g∥∥g∥∥

2

, (3.10)

where g = DI + b. Interested readers can refer to [32] for more details.

4. Experiments

In this Section, the proposed blind deblurring algorithm was tested with both synthetic

motion blurred images and real-life motion blurred images. In the kernel estimation process,


(a)

(b)

(c)

Figure 4: Testing our algorithm with real-life motion blurry images. (a)–(c) In order from left to right, theimages are original blurry images, restored results and estimated kernels by our algorithm, restored resultsand estimated kernels by using the algorithm in [28] and the close-ups of them (the red rectangle in theoriginal blurry image shows the location of close-up).

the parameters α and β were set as 1e − 4, 2e − 3, the initial minimum kernel size was 3 × 3,

and the initial maximum kernel size was 35 × 35. According to the multiscale scheme, the

outer iteration was controlled by the maximum kernel size and the inner iteration was set as

8 uniformly. In the image recovery process, the parameter λ was set as 2e + 3. Our algorithm

was implemented on Matlab experimental platform.

To verify the validity of our algorithm, the synthetic blurry images were generated by

convolving the synthetic images with a 15 × 15 synthetic kernel. The Gaussian white noise

was added whose standard deviation was 0.001. Figure 3 shows the experimental results of

several synthetic images. The deblurring results are extremely close to the original synthetic

images, and they manifest the significance of the proposed algorithm.

On the other hand, the proposed algorithm was compared with the approach

described in [28] by deblurring the real-life motion blur images. In Figure 4, the restored

results of some real-life images are given. As shown in recovery results, our method is

robust to restored sharp images and accurate kernels. In contrast to the approach in [28],the deblurring images and the close-ups show that our algorithm could obtain clearer image

detail information.


5. Conclusion

In this paper, a novel blind deblurring algorithm is presented for motion blur occurring in

photography. The approach consists of two stages: kernel estimation and image reconstruc-

tion. The edge information in blurry images is explored as an image prior for obtaining

accurate blur kernel and the use of total variation regularization keeps image details during

image recovery. The proposed algorithm was tested with synthetic and really captured

motion blur images. The experimental results demonstrated the efficacy of our algorithm

in image motion deblurring. On the other hand, there still exist some defects (cartoon effect

and unclear texture detail) in the restored images. Our future work is to extend the current

research by considering more complex blurs (such as blur with rotation and shift-variant

blur) and other image analysis problems [33–35].

Acknowledgments

This work was supported by the China Special Fund for Meteorological-scientific Research

in the Public Interest (GYHY201106044), NSFC (Grant nos. 61103130, 61070120, 61141014);National Program on Key basic research Project (973 Programs) (Grant nos. 2010CB731804-1,

2011CB706901-4).

References

[1] P. C. Hansen, J. G. Nagy, and D. P. O’Leary, Deblurring Images: Matrices, Spectra, and Filtering, vol. 3,Society for Industrial and Applied Mathematics SIAM, Philadelphia, Pa, USA, 2006.

[2] M. Ben-Ezra and S. K. Nayar, “Motion-based motion deblurring,” IEEE Transactions on Pattern Analysisand Machine Intelligence, vol. 26, no. 6, pp. 689–698, 2004.

[3] S. Y. Chen, J. H. Zhang, Y. F. Li, and J. W. Zhang, “A hierarchical model incorporating segmentedregions and pixel descriptors for video background subtraction,” IEEE Transactions on Industrial Infor-matics, vol. 8, no. 2, 2012.

[4] A. Rav-Acha and S. Peleg, “Two motion-blurred images are better than one,” Pattern RecognitionLetters, vol. 26, no. 3, pp. 311–317, 2005.


[6] W. Li, J. Zhang, and Q. H. Dai, “Exploring aligned complementary image pair for blind motiondeblurring,” in Proceedings of the IEEE International Conference Computer Vision and Pattern Recognition(CVPR ’11), pp. 273–280.

[7] D. Kundur and D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Processing Magazine, vol.13, no. 3, pp. 43–64, 1996.

[8] R. Fergus, B. Singh, A. Hertzmann, S. T. Roweis, and W. T. Freeman, “Removing camera shake froma single photograph,” in Proceedings of the Annual Meeting of the Association for Computing Machinery’sSpecial Interest Group on Graphics (SIGGRAPH ’06), pp. 787–794, August 2006.

[9] J. Jia, “Single image motion deblurring using transparency,” in Proceedings of the IEEE Computer SocietyConference on Computer Vision and Pattern Recognition (CVPR’07), pp. 1–8, June 2007.

[10] J. F. Cai, H. Ji, C. Liu, and Z. Shen, “Blind motion deblurring from a single image using sparseapproximation,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and PatternRecognition Workshops (CVPR ’09), pp. 104–111, June 2009.

[11] N. Joshi, R. Szeliski, and D. J. Kriegman, “PSF estimation using sharp edge prediction,” in Proceedingsof the 26th IEEE Conference on Computer Vision and Pattern Recognition (CVPR ’08), June 2008.

[12] L. Xu and J. Jia, “Two-phase kernel estimation for robust motion deblurring,” Computer Vision-ECCV,vol. 6311, no. 1, pp. 157–170, 2010.

[13] S. Y. Chen and Y. F. Li, “Determination of stripe edge blurring for depth sensing,” IEEE Sensors Journal,vol. 11, no. 2, Article ID 5585653, pp. 389–390, 2011.


[14] R. C. Gonzalez and R. E. Woods, Digital Image Processing, Prentice Hall, 2nd edition, 2002.

[15] M. Li, C. Cattani, and S.-Y. Chen, “Viewing sea level by a one-dimensional random function with longmemory,” Mathematical Problems in Engineering, vol. 2011, Article ID 654284, 13 pages, 2011.

[16] L. Rudin and S. Osher, “Total variation based image restoration with free local constraints,” inProceedings of the IEEE International Conference on Image Processing (ICIP ’94), pp. 31–35, Austin, Tex,USA, 1994.

[17] M. Li and W. Zhao, “Representation of a stochastic traffic bound,” IEEE Transactions on Parallel andDistributed Systems, vol. 21, no. 9, Article ID 5342414, pp. 1368–1372, 2010.

[18] C. R. Vogel and M. E. Oman, “Fast, robust total variation-based reconstruction of noisy, blurredimages,” IEEE Transactions on Image Processing, vol. 7, no. 6, pp. 813–824, 1998.

[19] Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variationimage reconstruction,” SIAM Journal on Imaging Sciences, vol. 1, no. 3, pp. 248–272, 2008.

[20] M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variablesplitting and constrained optimization,” IEEE Transactions on Image Processing, vol. 19, no. 9, ArticleID 5445028, pp. 2345–2356, 2010.

[21] S. H. Chan, P. E. Gill, and T. Q. Nguyen, “An augmented lagrangian method for total variation imagerestoration,” IEEE Transactions on Image Processing, vol. 20, no. 11, pp. 3097–3111, 2011.

[22] V. Torre and T. A. Poggio, “On edge detection,” IEEE Transactions on Pattern Analysis and MachineIntelligence, vol. 8, no. 2, pp. 147–163, 1986.

[23] T.-H. H. Lee, “Edge detection analysis,” 2007, http://disp.ee.ntu.edu.tw/.

[24] S. Y. Chen, G. J. Luo, X. Li, S. M. Ji, and B. W. Zhang, “The specular exponent as a criterion forappearance quality assessment of pearl-like objects by artificial vision,” IEEE Transactions on IndustrialElectronics, vol. 59, no. 99, 2012.

[25] J. Sun, Z. Xu, and H. Y. Shum, “Image super-resolution using gradient profile prior,” in Proceedings ofthe 26th IEEE Conference on Computer Vision and Pattern Recognition (CVPR ’08), pp. 1–8, June 2008.

[26] S. Y. Chen, H. Tong, Z. Wang, S. Liu, M. Li, and B. Zhang, “Improved generalized belief propagationfor vision processing,” Mathematical Problems in Engineering, Article ID 416963, 12 pages, 2011.

[27] S. Cho and S. Lee, “Fast motion deblurring,” in Proceedings of the Annual Meeting of the Association forComputing Machinery’s Special Interest Group on Graphics, vol. 28, p. 145, 2009.

[28] D. Krishnan, T. Tay, and R. Fergus, “Blind deconvolution using a normalized sparsity measure,” inProceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR ’11), pp. 233–240,Providence, RI, USA, 2011.

[29] A. Levin, R. Fergus, F. Durand, and W. T. Freeman, “Image and depth from a conventional camerawith a coded aperture,” in Proceedings of the 34th Annual Meeting of the Association for ComputingMachinery’s Special Interest Group on Graphics, August 2007.

[30] T. Goldstein and S. Osher, “The split bregman algorithm for L1 regularized problems,” SIAM Journalon Imaging Sciences, vol. 2, no. 2, pp. 323–343, 2009.

[31] S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for totalvariation-based image restoration,” Multiscale Modeling and Simulation, vol. 4, no. 2, pp. 460–489, 2005.

[32] J. Wang, K. Lu, and N. He, “Total variant image deblurring based on split bregman method,” ChineseJournal of Electronics. In press.

[33] W. S. Chen, P. C. Yuen, and X. H. Xie, “Kernel machine based rank-lifting regularized Discriminantanalysis method for face recognition,” neurocomputing, vol. 74, no. 17, pp. 2953–2960, 2011.

[34] Z. Liao, S. Hu, M. Li, and W. Chen, “Noise estimation for single-slice sinogram of low-dose X-Raycomputed tomography using homogenous patch,” Mathematical Problems in Engineering, vol. 2012,Article ID 696212, 16 pages, 2012.

[35] J. Yang, Z. Chen, W.-S. Chen, and Y. Chen, “Robust affine invariant descriptors,” Mathematical Problemsin Engineering, Article ID 185303, 15 pages, 2011.


Research ArticleDual-EKF-Based Real-Time Celestial Navigationfor Lunar Rover

Li Xie,1, 2 Peng Yang,1, 2 Thomas Yang,3 and Ming Li4

1 Department of Information Science and Electronic Engineering, Zhejiang University,Hangzhou 310027, China

2 Zhejiang Provincial Key Laboratory of Information Network Technology, Hangzhou 310027, China3 The Department of Electrical, Computer, Software, and Systems Engineering, Embry-RiddleAeronautical University, Daytona Beach, FL 32114, USA

4 School of Information Science and Technology, East China Normal University,Shanghai 200241, China

Correspondence should be addressed to Li Xie, [email protected]

Received 27 December 2011; Accepted 14 February 2012


Copyright q 2012 Li Xie et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

A key requirement of lunar rover autonomous navigation is to acquire state information accuratelyin real-time during its motion and set up a gradual parameter-based nonlinear kinematics modelfor the rover. In this paper, we propose a dual-extended-Kalman-filter- (dual-EKF-) based real-time celestial navigation (RCN) method. The proposed method considers the rover position andvelocity on the lunar surface as the system parameters and establishes a constant velocity (CV)model. In addition, the attitude quaternion is considered as the system state, and the quaterniondifferential equation is established as the state equation, which incorporates the output of angularrate gyroscope. Therefore, the measurement equation can be established with sun direction vectorfrom the sun sensor and speed observation from the speedometer. The gyro continuous outputensures the algorithm real-time operation. Finally, we use the dual-EKF method to solve the systemequations. Simulation results show that the proposed method can acquire the rover positionand heading information in real time and greatly improve the navigation accuracy. Our methodovercomes the disadvantage of the cumulative error in inertial navigation.

1. Introduction

In order to conduct scientific exploration on the lunar surface, lunar rover must have the

ability to execute tasks in unstructured environment. Its navigation system must have a

high degree of autonomy and the capabilities of high-accuracy real-time positioning and

orientation. On lunar surface, some commonly used navigation methods on the earth are not


applicable. There is no GPS system on the moon. If we use radio navigation, the rover control

may fail because of the two-way communication delay. The moon rotation is very slow, so we

cannot use north seeking gyro. Also, lunar magnetic field is very weak, so magnetic sensor-

based methods are ineffective.

Lunar rover navigation techniques mainly include absolute positioning and relative

positioning. For absolute positioning, such as autonomous celestial navigation [1], position

and heading errors are bounded and do not accumulate over time, and the output is discrete.

The initial positioning is generally absolute positioning, and its accuracy directly affects

relative positioning accuracy. Relative positioning, such as inertial navigation, achieves high

accuracy of position and heading in short time, but the errors accumulate over time (which

may lead to divergence), and the output is continuous. The current trend for lunar rover

navigation is integrated navigation, which combines the advantages of celestial navigation

and inertial navigation.

The earliest researchers [2–4] carried out celestial navigation by the altitude difference

method through observing the sun, earth, and fixed stars. Kuroda et al. [5] utilized celestial

navigation and dead-reckoning-based integrated navigation method to obtain lunar rover’s

absolute position and heading, which is achieved by observing the altitude and azimuth of

the sun and the earth. However, on the moon, the time period during which the sun and

the earth appear simultaneously is very short. Therefore, the application of this method is

limited. Altitude difference method is very sensitive to measurement noise, and positioning

accuracy [6] is low. Vision-based navigation is often used in robotics (Chen 2012, see [7, 8]),but it has difficulty in determining the absolute location and attitude.

Recent researchers use vector-observations-based quaternion estimation (QUEST) to

get the rover heading angle [9–12]. Ashitey proposed an absolute heading detection method

for the field integrated, design and operation (FIDO) rover [9]. When stopped, it uses sun

sensor and accelerometer to sense the sun orientation and the local gravity orientation, supply

absolute heading for rover with QUEST, and correct the gyroscope cumulative error. Ali

described that the US Mars rovers (the “Hope” and the “Spirit”) utilized this method to self-

correct the heading information [10]. Some recent technologies used in robotics can be found

in the works of Chen et al. [13]. Methods in Chinese literature are similar to the method

by Ashitey and they also calculate the heading through QUEST [11, 12]. Thein analyzed

the relationship between lunar rover positioning accuracy and astronomical instrument

measurement noise [14]. If we want to limit the position error within 50 m, measurement

noise should be less than 5.93 arcsec.

The above celestial navigation methods (except [5]) do not combine celestial

positioning with orientation and cannot get the absolute heading and location information

in real time. Ning established a position and attitude determination method based on

celestial observations [15], but its reference frame is moon fixed coordinate system rather

than local level coordinate system, and it does not consider the impact of the position and

speed changes on the gyro angular rate output. Ning proposed a lunar rover kinematics

model-based augmented unscented particle filter (ASUPF) as a new autonomous celestial

navigation method for dealing with systematic errors and measurement noise [16]. However,

the altitude angle measurements in this method are based on the local level provided by

the inertial measurement unit (IMU), assuming the rover is keeping static or in constant

motion. When the rover is moving, it needs the support of attitude update algorithm in

inertial navigation, because the gyro accumulates error and the local level precision is low. Pei

proposed a strapdown inertial navigation and celestial-navigation-based integrated method

for lunar rovers [17].


Since lunar rover position change on the lunar surface is very slow, in order to reduce

the dimension of the system, we can set position as a gradual system parameter to estimate

the rover state, that is, heading and attitude. To correct the position of the lunar rover, the

velocity observation is introduced. Meanwhile, in order to obtain real-time navigation output,

the output of the gyro is needed in integrated navigation. In this paper, a method of real-time

celestial navigation is proposed, in which positioning and orientation are simultaneous. Also,

the error bounded sun sensor output and high accuracy rate gyro output are fused, which

ensures the navigation output to be both real-time and of higher accuracy when the rover

is moving. Because there is no accelerometer in the system, the impact of the accelerometer

error and gravity anomaly on navigation is avoided.

The organization of the paper is as follows. Section 1 describes the principles of

celestial navigation and attitude quaternion kinematics. Section 2 describes the dual EKF-

based real-time celestial navigation method. Section 3 presents the results of computer

simulations and compares the accuracy of the results obtained with and without velocity

observation. Section 4 presents conclusions and discussions.

2. Celestial Navigation Principle and Attitude Kinematics

2.1. Principle of Celestial Navigation

Set the selenocenter celestial coordinate system as the inertial coordinate system (i), the moon

fixed coordinate system as m, geographic coordinate system (NED) as n, the lunar rover body

coordinate system as b, the local level coordinate system as l, and the sun sensor coordinate

system as c. After installation, the sensor coordinate systems b and c coincide with each other.

Celestial navigation system can detect the rover geographical position and heading

provided that the local gravity datum (level posture) is known. The outputs are lunar rover

position (latitude and longitude) on the moon and the attitude, including the heading, pitch,

and roll. State vector of the system x = [λ, L,An] is used to describe the lunar rover position

and heading information, in which (λ, L) is the lunar rover longitude and astronomical

latitude and An is just heading relative to North Pole of the moon.

In Figure 1, the moon fixed coordinate system, after rotating λ (east longitude is

positive) around the Z axis, becomes coordinate system O − x1y1z1. After further rotating

by −L − π/2 (north latitude is positive) around the Oy1 axis, the navigation coordinate w is

obtained. The attitude matrix about the latitude and longitude is as follows [18]:

nm = y

(−L − π

2

) z(λ). (2.1)

The attitude matrix about the navigation coordinate system and the lunar body coordinate

system is:

bn = x

(ϕ) y

(ψ) z(θ). (2.2)

Here, θ is the heading, ψ is the pitch angle, and ϕ is the roll. To prevent the risk of rollover,

lunar rover pitch and roll should between ±45◦. The attitude matrix about the moon fixed

coordinate system and the local level coordinate system is A = lm (called target matrix


Moon-fixedframe

The moon

The sun

North-Earth-downframe

Zm

O m

Ym

X n

Yn

O nZn

X m

L

Figure 1: Moon-fixed (m) and navigation (n) coordinate system.

here). Substituting z(θ), y(−L − π/2), z(λ) into the above formula, we get the following

equation:

lm = z(θ) y

(−L − π

2

) z(λ). (2.3)

2.2. Quaternion Attitude Kinematics

Attitude can be expressed in several mathematical parameters: quaternion, attitude matrix,

Euler angles, Rodrigues parameters, and so on. The attitude matrix contains a total of nine

parameters, but because it is orthogonal matrix, only three components are independent. One

of the most useful parameters is the attitude quaternion, which is a four-dimensional vector,

defined as q = [ρT q4]T , where ρ = [q1 q2 q3]

T = e sin(ϑ/2) and q4 = cos(ϑ/2). Here, e is the

rotation axis and ϑ is the rotation angle. When using a four-dimensional vector to describe

the three-dimensional rotation, the four parameters of quaternion are not independent, and

they are subject to the constraint qTq = 1. The relationship between the attitude matrix and

the quaternion from the inertial coordinate system i to the body coordinate system b is

biA(q) = ΞT (q)Ψ(q), (2.4)

where

Ξ(q) ≡[q4I3×3 + [ρ×]

−ρT

], Ψ(q) ≡

[q4I3×3 − [ρ×]

−ρT

]· Ξ(q) ≡

[q4I3×3 + [ρ×]

−ρT

],

Ψ(q) ≡[q4I3×3 − [ρ×]

−ρT

].

(2.5)


Here [ρ×] is the cross-product matrix, defined as [ρ×] =[

0 −q3 q2

q3 0 −q1

−q2 q1 0

]. One advantage of using

quaternion is that the attitude matrix is quadratic equation of the parameter and thus does

not include any transcendental function. For small angles, the vector part of the quaternion

is approximately half of the rotation angle, and therefore ρ ≈ α/2, q4 ≈ 1, where the 3-

dimensional vector α includes roll, pitch, and heading. Therefore, the attitude matrix can be

approximated as biA ≈ I3×3 − [α×], which is effective in the first-order approximation.

The attitude kinematics equation is

bi A = −

[ωb

ib×]biA . (2.6)

Here, ωbib

is the angular velocity of the b frame relative to i frame expressed in b coordinates.

The quaternion differential equation is

q =1

2Ξ(q)ωb

ib =1

2Ω(ωb

ib

)q, (2.7)

where

Ω(ωb

ib

)≡

⎡⎢⎣−[ωb

ib×]

ωbib

−(ωb

ib

)T0

⎤⎥⎦. (2.8)

The main advantage of using the quaternion is that the kinematics equation is linear

and there is no singularity. Another advantage is that continuous rotation of coordinate

frames can be expressed as the quaternion multiplication. Suppose a continuous rotation can

be expressed as

A(q′)A(q) = A

(q′ ⊗ q

). (2.9)

The composition of the quaternion is bilinear, with

q′ ⊗ q =[Ψ(q′) q′]q = [Ξ(q) q]q′, (2.10)

and the inverse quaternion is defined by q−1 =[ −ρq4

]. Note that q ⊗ q−1 = [ 0 0 0 1 ]T is the

identity quaternion.

3. Dual-EKF-Based RCPO Method

Assume the state vector of the navigation system is xs, the system parameter vector is xp, and

the observation vector is yk. According to the problem, a continuous-discrete nonlinear state

space model can be derived:

xs(t) = f{t, xs(t),u(t), xp(t)

}+w(t),

yk = hk

(xs,k, xp,k

)+ vk,

(3.1)


where f(·), h(·) are implicit vector functions, w(t) is the continuous process noise, and vkis the discrete measurement noise. In the state vector xs = [qT , βT ]T , q is the heading and

attitude quaternion in the navigation frame (w) for the lunar rover and β is the constant bias

for gyro. In parameter vector xp = [pT , vT ]T , p = [L λ]T is the rover position, which is the

latitude and longitude; V = [vL vλ]T is the north speed and east speed on the lunar surface.

3.1. System Parameter and State Equations

The lunar rover position and velocity equations constitute the system parameter equations:

xp = Fpxp +wp (3.2)

with the parameter vector xp = [pT VT ]T , the state transition matrix Fp =[

0 0 1 00 0 0 10 0 0 00 0 0 0

], the

parameters process noise wp =[

00wLwλ

], and the noise covariance Qp = diag[0 0 σ2

L σ2λ].

The lunar rover attitude constitutes the system state equations, and the quaternion

differential equation is expressed by

q2 =1

2Ω(ωb

nb

)q2. (3.3)

Here, ωbnb is the angular velocity of the b frame relative to n frame expressed in b coordinates.

The gyro measurement model is

ωbib = ωb

ib + β + ηv,

β = ηu.(3.4)

Here, ωbib

is the angular velocity of the b frame relative to i frame expressed in b coordinates.

β is the constant bias of the gyro, ηv and ηu are zero mean Gaussian white noise processs, and

their spectral density functions are σ2vI3×3 and σ2

uI3×3, respectively.

Because the selenocenter celestial coordinate system is the inertial coordinate system

here, so

ωbnb = ωb

ib − bnA(q2)ωn

in. (3.5)

Also, ωnin is the angular velocity of the n frame relative to i frame expressed in n coordinates

ωnin = ωim

⎡⎣ cosL

0

− sinL

⎤⎦ +

⎡⎢⎢⎢⎢⎢⎢⎢⎣

VE

R

−VN

R

−VE tanL

R

⎤⎥⎥⎥⎥⎥⎥⎥⎦, VE = vλR cosL, VN = vLR. (3.6)


In (3.6), ωim is the angular velocity of the m frame relative to i frame, and the second

expression on the right side is the angular velocity of the n frame relative to m frame. The

angular velocity of the m frame relative to i frame ωim is

ωim = ωgz + mi Aωzz . (3.7)

In (3.7), ωgz is the revolution angular velocity of the moon around the earth, ωzz is the moon

spin velocity, and mi A is the attitude matrix from the inertial reference frame i to the moon

fixed frame m, which can be calculated after querying ephemeris [18].

3.2. Celestial and Speed Observation Equations

The measurement principle of vector observation attitude sensor can be expressed as bi =A(q)ri + vi, i = 1, . . . , n. If n celestial bodies are observable simultaneously, we can get n

vector pairs, so the measurement equation at time k is

bk =

⎡⎢⎢⎢⎣A(q2)A(p1)r1

A(q2)A(p1)r2

...

A(q2)A(p1)rn

⎤⎥⎥⎥⎦∣∣∣∣∣∣∣∣ tk

+

⎡⎢⎢⎢⎣v1

v2

...

vn

⎤⎥⎥⎥⎦∣∣∣∣∣∣∣∣ tk

, (3.8)

where A(q2) = bnA, A(p1) = n

mA = y(−L − π/2) z(λ).

Set vk =

⎡⎣ v1v2

...vn

⎤⎦∣∣∣∣ tk , its variance is R = diag[σ21I3×3, σ

22I3×3, . . . , σ

2nI3×3], where diag[· · ·]

is the diagonal matrix. In this paper, n = 1, r = is, b = bs, where is is the sun unit vector in

inertial frame and bs is the sun unit vector in the body frame.

The speed observation equation of the speedometer is

Vk = Vk + uk, (3.9)

where Vk is speed measurement at time k, uk is the measurement noise, and its covariance

matrix is Ru = σ2uI2×2.

3.3. Dual Continuous-Discrete EKF

Dual-EKF algorithm uses two mutual coupling extended Kalman filters working in parallel

and a state estimator working between the system parameter time update process and the

measurement update process [19]. Dual-EKF can estimate the system state and parameter

online. Using the above model, a continuous-discrete extended Kalman filter can be derived

(Chen 2012, [20]). The process equation about the system parameter is a continuous linear

equation, which can be discretized directly. The process equations about the system state


are nonlinear equations, and the Jacobian matrix needs to be calculated. Finally, we get the

discrete linear state space model (without considering the control input uk):

xs,k+1 = f{xs,k, xp,k

}+wk,

yk = hk

(xs,k, xp,k

)+ vk.

(3.10)

3.3.1. Linearization of State Process Equations

In order to maintain the quaternion normalization constraint, we use the multiplicative error

quaternion in the body frame to express the attitude error:

δq = q ⊗ q−1, (3.11)

where q−1 is the inverse of the quaternion estimate and δq ≡ [δρT δq4]T . If the error

quaternion δq is very small, we can use the small angle approximation. After a series of

derivation, the linear kinematic model of the attitude error [21] is obtained:

δα = −[ωb

nb×]δα + δωb

ib −A(q2)δωnin, δq4 = 0, (3.12)

where δωbib= ωb

ib− ωb

iband δωn

in = ωnin − ωn

in = 0. Also, δωbib= −(Δβ + ηv) is available by the

above gyro model, in which Δβ ≡ β − β. So the above formula becomes

δα = −[ωb

nb×]δα − (Δβ + ηv

). (3.13)

The remaining error equation can be obtained by similar methods. The state vector,

the state error vector, and the process noise vector and covariance in this EKF are defined as

xs ≡[qβ

], Δxs ≡

[δα

Δβ

], ws ≡

[ηvηu

], Qs =

[σ2vI3×3 03×3

03×3 σ2uI3×3

]. (3.14)

The error dynamics of time update in the EKF is Δx = FΔx + Gw. Here, the state

transition matrix F and the noise coefficient matrix G are

F ≡[−[ωb

nb×]

−I3×3

03×3 03×3

], G ≡

[−I3×3 03×3

03×3 I3×3

]S. (3.15)

3.3.2. Linearization of Measurement Equations

Next we determine the sensitive matrix Hs(x−s ) of the system state observation equation. The

true value and the estimate of the celestial bodies vector in the body coordinate system are

b = A(q2)A(p−

1

)r, b− = A

(q−

2

)A(p−

1

)r. (3.16)


According to (2.6),

A(q2) = A(δq2)A(q−

2

)= (I3×3 − [δα2×])A

(q−

2

). (3.17)

From (3.16), we have

Δb = b − b− =[A(q−

2

)A(p−

1

)r×]δα2. (3.18)

Note that Hsq = [A(q−2 )A(p−

1 )r×], so the sensitivity matrix for all measurements is

Hs

(x−s)=

⎡⎢⎢⎢⎣Hsq1 03×3

Hsq2 03×3

......

Hsqn 03×3

⎤⎥⎥⎥⎦∣∣∣∣∣∣∣∣ tk

. (3.19)

Next we determine the sensitive matrix Hp(x−p) of the system parameter observation

equation.

The true value and the estimate of the celestial bodies vector in the body coordinate

system are

b = A(q−

2

)A(p)r, b− = A

(q−

2

)A(p−)r. (3.20)

Function A(p) is expanded as a Taylor series, which is

A(p) ≈ A(p−) + 2∑

j=1

A−j Δpj , (3.21)

where A−1 = ∂A/∂L|L− , A

−2 = ∂A/∂L|λ− .

Finally, we have

Δb = b − b− =2∑

j=1

A(q−

2

)A−

j rΔpj . (3.22)

Note that Hp =[A(q−

2 )A−1 r A(q−

2 )A−2 r]. Combined with the speed observations, the

sensitivity matrix of all measurements is

Hp

(x−p)=

⎡⎢⎢⎢⎢⎢⎢⎣Hp1 03×2

Hp2 03×2

......

Hpn 03×2

02×2 I2×2

⎤⎥⎥⎥⎥⎥⎥⎦

∣∣∣∣∣∣∣∣∣∣∣ tk. (3.23)


Table 1: Dual-EKF algorithm.

InitializationParameter: xp(t0) = xp,0, Pp(t0) = Pp,0

State: xs(t0) = xs,0, Ps(t0) = Ps,0

State measurement update

Ks,k = P−s,kHT

s,k[Hs,kP

−s,kHT

s,k+ R]−1

εk = bs,k − hk(x−s,k, x−

p,k)

Δx+s,k

= Ks,kεk

q+2,k

= q−2,k

+1

2Ξ(q−

2,k)δα+

2,k, normalization

β+k = β

−k + Δβ

+k

P+s,k

= [I −Ks,kHs,k]P−s,k

Parameter measurement update

Kp,k = P−p,k

HTp,k

[Hp,kP−p,k

HTp,k

+ R′]−1

R′ = diag([R,Ru])x+p,k

= x−p,k

+Kp,k[εk ; (Vk − V−k)]

P+p,k

= [I −Kp,kHp,k]P−p,k+1

Parameter time updatex−p,k+1

= Φpx+p,k

P−p,k+1

= ΦpP+p,k

ΦTp +Qp

State time update

ωbnb

= (ωbib− β+

k) −A(q+

2,k)ωn

in(x−p,k+1

)

˙q2 =1

2Ω(ωb

nb)q2

˙β = 0

Ps = FsPs + PsFsT +GQsG

T

3.3.3. Dual-EKF Algorithm

Finally the proposed algorithm of dual-EKF is shown in Table 1.

4. Simulations and Discussions

4.1. Simulation Conditions

Specific simulation parameters are shown in Table 2.

4.2. Simulation of Moving Lunar Rover

In this paper, we carried out lunar rover simulation under various moving conditions

described in Table 3, and navigation accuracy with and without the speed observation is

compared. The lunar rover movement includes rotational and translational movements,

where the former can be sensed by the gyro angular velocity and the latter can be measured

by the speedometer line speed.

The simulation results of the lunar rover are shown in Figure 2, with the left diagram

on each figure representing the simulation result without speed observation and the right

diagram representing the simulation result with speed observation.

Figure 2 shows the position error and its 3σ boundary, and we see the latitude and

longitude errors in the left diagram diverge at last. After the uniform motion error expands,

we mainly have the lunar rover speed changes, so the constant velocity (CV) model is no


0 200 400 600

0

50

100

−100

−50

Time (s)

Lati

tud

e(a

rcse

c)

(a)

0 200 400 600

0

50

100

−100

−50Lati

tud

e(a

rcse

c)

Time (s)

(b)

0 200 400 600

0

50

100

Time (s)

−100

−50Lo

ng

itu

de(a

rcse

c)

(c)

0 200 400 600

0

50

100

Lo

ng

itu

de(a

rcse

c)

Time (s)

−100

−50

(d)

Figure 2: Position error and 3σ boundary.

Table 2: Simulation parameters.

Beginning time 2011-01-01 00:00:00

Sampling interval Δt = 1 s

Initial origin λ(t0) = 0◦, L(t0) = 0◦

Initial velocity vL = −0.1m/(s · R),vλ = 0.1m/(s · R)Initial attitude q(t0) = [ 0 0 0 1 ]T

Gyro biases β(t0) = 0.1[ 1 1 1 ]T deg/hr

Initial covariance

Pp

0 = 0.052 deg2

PV0 = 0.12(m/s)2

Pα0 = 0.12 deg2

Pβ

0 = 0.22(deg/hr)2

Gyro noise (Qs)σgv =

√10 × 10−7 rad/s1/2

σgu =√

10 × 10−10 rad/s3/2

CV model (Qp)σL = σL = 0.0001m/(s · R)(R: moon radius, the same below)

Sun sensor (R) 1′(3σs)

Velocity sensor (Ru) σu = 0.001m/(s · R)


0 200 400 600

0

0.2

0.4

−0.4

−0.2

Time (s)

Vel

oci

ty l

ati

tud

e(m

/s)

(a)

Time (s)

Vel

oci

ty l

ati

tud

e(m

/s)

0 200 400 600−0.04

−0.02

0

0.02

0.04

(b)

0 200 400 600

0

0.2

0.4

Vel

oci

ty l

on

git

ud

e(m

/s)

−0.4

−0.2

Time (s)

(c)

0 200 400 600

0

0.02

0.04

Vel

oci

ty l

on

git

ud

e(m

/s)

−0.04

−0.02

Time (s)

(d)

Figure 3: The speed error and 3σ boundary.

Table 3: Lunar rover motion.

Motion Time (s) Angular velocity( ◦/s)

Linear velocity(m/(s · R))

(1) Static 1∼100 0 0

(2) Rotation 101∼200 ωz = 1 0

(3) Uniform motion 201∼300 0vL = 0.2

vλ = −0.25

(4) Rotation and uniform motion 301∼500 ωz = 1vL = 0.2

vλ = −0.25

(5) Static again 501∼600 0 0

longer applicable. The navigation error in the right diagram is kept within the 3σ boundary,

and it does not diverge. Because of the speedometer line speeds information, the absolute

position of the rover can be adjusted in real time. The mean of the latitude error is 3.97′′,

and the standard deviation is 0.83′′; the mean of longitude error is 1.07

′′, and the standard

deviation is 1.42′′. Converted into the line error according to the lunar radius, the error is

35.51 m.


0 200 400 600

0

100

−100

Ro

ll(a

rcse

c)

Time (s)

(a)

0 200 400 600

0

100

−100

Ro

ll(a

rcse

c)

Time (s)

(b)

0 200 400 600

0

100

Pit

ch (

arc

sec)

−100

Time (s)

(c)

0 200 400 600

0

100

Pit

ch (

arc

sec)

Time (s)

−100

(d)

0 200 400 600

0

100

Yaw

(arc

sec)

−100

Time (s)

(e)

0 200 400 600

0

100

Yaw

(arc

sec)

Time (s)

−100

(f)

Figure 4: Attitude, heading error, and 3σ boundary.

Figure 3 shows the speed error and its 3σ boundary. The initial velocity is not accurate.

In the left diagram, when uniform motion speed changes cannot be sensed any longer, the

error shape exhibits phase steps. While the speed observation is available, the navigation

system can sense it after the speed change. We see the two speed changes in the lunar rover

movement are in zigzag fashions on the speed error figure and then quickly disappear.

Figure 4 shows the attitude, heading error, and its 3σ boundary, but the heading

information is of main interest in the navigation. The mean of the heading error in the left

diagram is −5.55” with a standard deviation of 3.03”. The mean of the heading error in the

right diagram is 1.71”, with a standard deviation of 3.53”.

Figure 5 shows the constant gyro bias error and its 3σ boundary. As can be seen from

the graph, the 3-channel constant bias basically converges in the Motion 1 stage, that is, static,

and completes the initial alignment of the gyroscope.


0 200 400 600

0

0.1

0.2

−0.1

Time (s)

x(d

eg/

hr)

(a)

−0.1

x(d

eg/

hr)

0 200 400 600

0

0.1

0.2

Time (s)

(b)

0 200 400 600

0

0.1

0.2

−0.1

Time (s)

y(d

eg/

hr)

(c)

0

0.1

0.2

0 200 400 600−0.1

Time (s)y(d

eg/

hr)

(d)

−0.1

Time (s)

0 200 400 600

0

0.1

0.2

z(d

eg/

hr)

(e)

0 200 400 600

0

0.1

0.2

−0.1

Time (s)

z(d

eg/

hr)

(f)

Figure 5: The constant gyro bias error and 3σ boundary.

4.3. Discussions and Remarks

From the above analysis and simulation, it can be seen that the significance of this work is to

combine celestial and inertial sensor data to obtain the attitude and heading information for

the real-time navigation of the lunar rover. The simulation results indicate that the dual-EKF

method is valid in this field. To obtain better results, the following two properties are worth

of being further investigated in the future work on navigation.

Computational accuracy: the technology of imaging processing plays a role in the

celestial navigation. The performance of noise filtering and feature extraction for

the astronomical images will affect the navigation precision directly (Liao et al.,

see [22, 23]; Yang et al., see [24, 25]). In addition, the nonlinear properties, such as

fractals [26, 27], in the astronomical images can affect the navigation effect also.

Computational complexity: though the Kalman filter is the most widely used attitude

estimation algorithm for navigation and it offers the optimal recursive solution to

the nonlinear estimation problem, the implementation efficiency of the recursive


Kalman estimator has been an issue. Correlation is a useful technique in the field.

Real-time navigation may use it to help in Kalman filtering [28, 29].

5. Discussion and Conclusions

In this paper, a sun-orientation-and-speed-observations-based lunar rover real-time celestial

navigation method is proposed, using dual-EKF to estimate system parameters and state.

The method treats the position and velocity as system parameters and establishes a position,

velocity differential equation. Further, the rover attitude quaternion is treated as the system

state, and the quaternion differential equation is established as the state equation. To establish

the measurement equation, the sun direction vector is obtained from the sun sensor and the

speed observation is obtained from the speedometer. Finally, the rover position and heading

information is obtained in real-time through the dual-extended Kalman filter (Dual-EKF).The proposed system does not use accelerometers and thus avoids the acceleration noises.

Also, the system uses a high-precision gyro to improve the navigation accuracy.

Simulation results show that the proposed technique is able to obtain the rover

navigation information in real time, and it overcomes the two shortcomings of more

traditional navigation methods: the discrete output (of pure celestial navigation) and

cumulative error (of inertial navigation).

Acknowledgments

L. Xie and P. Yang were supported by the National Natural Science Foundation of China

(NSFC) under Grant no. 60534070, Zhejiang Provincial Program of Science and Technology

under Grant no. 2009C33085, Wenzhou Program of Science and Technology under Grant no.

S20100029. M. Li would like to acknowledge the support from the 973 plan under the project

no. 2011CB302802 and from the National Natural Science Foundation of China under Project

Grant no. 61070214 and 60873264.

References

[1] U. Henning, “A short guide to celestial navigation,” 2006, Germany, http://www.celnav.de.[2] E. Krotkov, M. Hebert, M. Bufa et al., “Stereo driving and position estimation for autonomous

planetary rovers,” in Proceedings of the IARP Workshop on Robotics in Space, Montreal, Canada, 1994.[3] R. Volpe, “Mars rover navigation results using sun sensor heading determination,” in Proceedings of

the IEEE/RSJ International Conference on Intelligent Robot and Systems, pp. 460–467, Kyongju, Korea,1999.

[4] P. M. Benjamin, Celestial Navigation on the Surface of Mars, Naval Academy, Annapolis, Md, USA, 2001.[5] Y. Kuroda, T. Kurosawa, A. Tsuchiya, and T. Kubota, “Accurate localization in combination with

planet observation and dead reckoning for lunar rover,” in Proceedings of the IEEE InternationalConference on Robotics and Automation (ICRA ’04), pp. 2092–2097, New Orleans, La, USA, May 2004.

[6] J. C. Fang, X. L. Ning, and Y. L. Tian, Spacecraft Autonomous Celestial Navigation Principles and Methods,National Defense Industry Press, Beijing, China, 2006.

[7] S. Y. Chen, Y. F. Li, and J. W. Zhang, “Vision processing for realtime 3D data acquisition based oncoded structured light,” IEEE Transactions on Image Processing, vol. 17, no. 2, pp. 167–176, 2008.

[8] S. Y. Chen, Y. H. Wang, and C. Cattani, “Key issues in modeling of complex 3D structures from videosequences,” Mathematical Problems in Engineering, vol. 2012, Article ID 856523, 17 pages, 2012.

[9] A. Trebi-Ollennu, T. Huntsberger, Y. Cheng, E. T. Baumgartner, B. Kennedy, and P. Schenker, “Designand analysis of a sun sensor for planetary rover absolute heading detection,” IEEE Transactions onRobotics and Automation, vol. 17, no. 6, pp. 939–947, 2001.


[10] K. S. Ali, C. A. Vanelli, J. J. Biesiadecki et al., “Attitude and position estimation on the Mars explorationrovers,” in Proceedings of the IEEE Systems, Man and Cybernetics Society, International Conference onSystems, pp. 20–27, The Big Island, Hawaii, USA, October 2005.

[11] F. Z. Yue, P. Y. Cui, H. T. Cui, and H. H. Ju, “Algorithm research on lunar rover autonomous headingdetection,” Acta Aeronautica et Astronautica Sinica, vol. 27, no. 3, pp. 501–504, 2006 (Chinese).

[12] F. Z. Yue, P. Y. Cui, H. T. Cui, and H. H. Ju, “Earth sensor and accelerometer based autonomousheading detection algorithm research of lunar rover,” Journal of Astronautics, vol. 26, no. 5, pp. 553–557, 2005 (Chinese).

[13] S. Y. Chen, Y. F. Li, and M. K. Ngai, “Active vision in robotic systems: a survey of recentdevelopments,” The International Journal of Robotics Research, vol. 30, no. 11, pp. 1343–1377, 2011.

[14] M. W. L. Thein, D. A. Quinn, and D. C. Folta, “Celestial navigation (CelNav): lunar surfacenavigation,” in Proceedings of the AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Honolulu,Hawaii, USA, August 2008.

[15] X. L. Ning and J. C. Fang, “Position and pose estimation by celestial observation for lunar rovers,”Journal of Beijing University of Aeronautics and Astronautics, vol. 32, no. 7, pp. 756–759, 2006 (Chinese).

[16] X. L. Ning and J. C. Fang, “A new autonomous celestial navigation method for the lunar rover,”Robotics and Autonomous Systems, vol. 57, no. 1, pp. 48–54, 2009.

[17] F. J. Pei, H. H. Ju, and P. Y. Cui, “A long-range autonomous navigation method for lunar rovers,” HighTechnology Letters, vol. 19, no. 10, pp. 1072–1077, 2009 (Chinese).

[18] X. N. Xi, Lunar Probe Orbit Design, National Defense Industry, Beijing, China, 2001.[19] E. A. Wan and A. T. Nelson, “Dual extended kalman filter methods,” in Kalman Filtering and Neural

Networks, John Wiley & Sons, New York, NY, USA, 2001.[20] S. Y. Chen, “Kalman filter for robot vision: a survey,” IEEE Transactions on Industrial Electronics, vol.

59, no. 99, 2012.[21] S. G. Kim, J. L. Crassidis, Y. Cheng, A. M. Fosbury, and J. L. Junkins, “Kalman filtering for relative

spacecraft attitude and position estimation,” in Proceedings of the AIAA Guidance, Navigation, andControl Conference, pp. 2518–2535, San Francisco, Calif, USA, August 2005.

[22] Z. W. Liao, S. X. Hu, D. Sun, and W. Chen, “Enclosed laplacian operator of nonlinear anisotropicdiffusion to preserve singularities and delete isolated points in image smoothing,” MathematicalProblems in Engineering, vol. 2011, Article ID 749456, 15 pages, 2011.

[23] Z. W. Liao, S. X. Hu, M. Li et al., “Noise estimation for single-slice sinogram of low-dose x-raycomputed tomography using homogenous patch,” Mathematical Problems in Engineering, vol. 2012,Article ID 696212, 16 pages, 2012.

[24] J. W. Yang, Z. Chen, W. S. Chen, and Y. Chen, “Robust affine invariant descriptors,” MathematicalProblems in Engineering, vol. 2011, Article ID 185303, 2011.

[25] J. W. Yang, M. Li, Z. Chen et al., “Cutting affine invariant moments,” Mathematical Problems inEngineering. In press.

[26] M. Li, “Fractal time series—a tutorial review,” Mathematical Problems in Engineering, vol. 2010, ArticleID 157264, 26 pages, 2010.

[27] C. Cattani, “Fractals and hidden symmetries in DNA,” Mathematical Problems in Engineering, vol. 2010,Article ID 507056, 31 pages, 2010.

[28] L. Gottschalk, E. Leblois, and J. O. Skøien, “Correlation and covariance of runoff revisited,” Journal ofHydrology, vol. 398, no. 1-2, pp. 76–90, 2011.

[29] E. Pardo-Iguzquiza, K. V. Mardia, and M. Chica-Olmo, “MLMATERN: a computer program formaximum likelihood inference with the spatial Maern covariance model,” Computers and Geosciences,vol. 35, no. 6, pp. 1139–1150, 2009.


Research ArticleHidden-Markov-Models-Based DynamicHand Gesture Recognition

Xiaoyan Wang,1 Ming Xia,1 Huiwen Cai,2Yong Gao,3 and Carlo Cattani4

1 College of Computer Science and Technology, Zhejiang University of Technology, Hangzhou 310023, China2 Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China3 Zhejiang Jieshang Vision Science and Technology Cooperation, Hangzhou 310013, China4 Department of Mathematics, University of Salerno, Via Ponte Don Melillo, 84084 Fisciano, Italy

Correspondence should be addressed to Xiaoyan Wang, [email protected]

Received 12 January 2012; Accepted 3 February 2012


Copyright q 2012 Xiaoyan Wang et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

This paper is concerned with the recognition of dynamic hand gestures. A method based onHidden Markov Models (HMMs) is presented for dynamic gesture trajectory modeling andrecognition. Adaboost algorithm is used to detect the user’s hand and a contour-based handtracker is formed combining condensation and partitioned sampling. Cubic B-spline is adopted toapproximately fit the trajectory points into a curve. Invariant curve moments as global features andorientation as local features are computed to represent the trajectory of hand gesture. The proposedmethod can achieve automatic hand gesture online recognition and can successfully reject atypicalgestures. The experimental results show that the proposed algorithm can reach better recognitionresults than the traditional hand recognition method.

1. Introduction

The goal of Human Computer Interaction (HCI) is to bring the performance of human

machine interaction similar to human-human interaction [1]. Gestures play an important part

in our daily life, and they can help people convey information and express their feelings.

Among different body parts, the hand is the most effective, general-purpose interaction

tool. Therefore, hand gesture tracking and recognition becomes an active area of research

in human computer interaction and digital entertainment industry [2–4]. A gesture can be

static or dynamic or both. According to this, there are three types of gesture recognition: static

hand posture recognition, dynamic hand gesture recognition, and complicated hand gesture

recognition. Our work in this paper concentrates on dynamic gesture recognition, which


characterizes the hand movements. Tracking frameworks have been used to handle dynamic

gestures. Isard and Blake [5] established a hand tracking approach based on 2D deformable

contour model and Kalman filter [6]. However, it is inefficient to track an articulated object

which has a high dimension state space using condensation alone. MacCormick and Blake

[7] introduced a partition sampling method to track more than one object. MacCormick and

Isard [8] implemented a vision-based articulated hand tracker using this technique after that.

Their tracker is able to track position, rotation, and scale of the user’s hand while maintaining

a pointing gesture. Based on Blake’s work, Tosas [9] makes some technique extensions and

implements a full articulated hand tracker.

Several methods on hand gesture recognition have been proposed [10–13], which

differ from one another in their models, just like Neural Network, Fuzzy Systems and

Hidden Markov Models (HMMs) [14]. The most challenging problem of dynamic gesture

recognition is its spatial-temporal variability, when the same gesture can differ in velocity,

shape, duration, and integrality. These characteristics make it more difficult to recognize

dynamic hand gestures than to recognize static ones. HMM is a statistical model widely

used in hand writing, speech, and character recognition [13, 15] because of its capability of

modeling spatial-temporal time series. HMM has also been successfully used in hand gesture

recognition [13, 16–18], in respect that it can preserve the spatial-temporal identity of hand

gesture and have an ability to do the segmentation automatically. Motion features of each

time point have been modeled in most of the dynamic hand gesture recognition methods

using HMM, nevertheless, the whole trajectory shape characters are not considered at the

same time. The recognition based on local features is very sensitive to sampling period and

velocity, and the continuous local process of gesture will cause false recognition.

Researches on psychology indicate that human brains lean to perceive object from a

whole, and then apprehend its details, which illustrates that an object can only be described

perfectly when the local and global information are integrated. In this paper, we propose

a dynamic gesture trajectory modeling and recognition method based on HMM. Cubic B-

spline is adopted to approximately fit the trajectory points into a curve, and invariant curve

moments as global features and orientation as local features are computed to represent the

trajectory of hand gesture. Threshold model is used to model all the atypical gesture patterns,

and automatically segment and recognize the dynamic gesture trajectory. The proposed

method can achieve automatic hand gesture online recognition and can successfully reject

atypical gestures. Meanwhile, the experiment results show that the recognition performance

of the proposed algorithm can be greatly improved by combining the global invariant curve

features with local orientation features.

The rest of the paper is organized as follows: Section 2 describes the dynamic gesture

representation and the global and local features we used. Section 3 gives the continuous

hand gesture recognition procedure, which contains hand detection, tracking, and gesture

recognition based HMM. The experimental results are shown in Section 4. Finally, Section 5

and ends the paper with a summary.

2. Dynamic Gesture Representation

A dynamic hand gesture is a spatial-temporal pattern and has four basic features: velocity,

shape, location, and orientation. The motion of the hand can be described as a temporal

sequence of points with respect to the hand centroid of the person performing the gesture.


50

40

30

20

10

0400

300

200

100 150200

250300

XY

Tim

e450

400

350

300

250

200

150

100

50

00 100 200 300 400 500 600

X

Y

Figure 1: A dynamic hand gesture instance.

In this paper, the hand shape is not considered and each dynamic hand gesture instance is

represented by a time series of the hand’s location:

pt =(xt,yt

), (t = 1, 2, . . . , T), (2.1)

where T represents the length of gesture path and varies across different gesture instances.

Consequently, a gesture containing an ordered set of points can be regarded as a mapping

from time to location. Figure 1 shows a dynamic hand gesture instance and gives its

projection along the time axis onto the image plane.

2.1. Local Feature Representation

There is no doubt that selecting good features plays significant role in hand gesture

recognition performance. The orientation feature is proved to be the best local representation

in terms of accuracy results [19–21] and it is considered as the most important feature in

dynamic gesture recognition using HMM [22, 23]. Therefore, we will rely upon it as a main

local feature in our system. The orientation of hand movement is computed between two

consecutive points of the hand gesture trajectory:

θt = arctan

(yt+1 − yt

xt+1 − xt

), (t = 1, 2, . . . , T). (2.2)

A feature vector will be determined by converting the orientation to directional

codewords by a vector quantizer. For example, in Figure 2 the orientation is quantized to

generate the codewords from 1 to 20 by dividing it by 20 degree. Thereby, the discrete feature

vector will be used as an input to discrete HMM.

2.2. Global Feature Presentation

The human brain is inclined to sense object from a whole, and people also try to understand

a gesture as integrity. Accordingly, we try to connect all the discrete points of gesture using


1

2

3

456

78

9

10

11

12

1314

15 1617

18

19

20

90

180

270

dy

dx

(xt+1, yt+1)

(xt, yt)

θt

Figure 2: The orientation and its codewords.

a slippery line. Cubic B-spline function is adopted to approximately fit the trajectory points

into a curve:

p(t) =3∑

m=0

Bm(t)CPm, (2.3)

where B0(t) = (1 − t)3, B1(t) = 3t(1 − t)2, B2(t) = 3t2(1 − t), B3(t) = t3, CPm are control

points. After the curve is shaped, an issue to be addressed is the variation of speed of the

same gesture. To overcome this problem, all curves are scaled such that they lie within the

same range. Those curves for faster moves are relatively expanded by interpolation and those

of slower moves are contracted.

The trajectories of a same gesture vary in size and shape. We use invariant curve

moments as global features to represent the trajectory [24]. The advantage of moment

methods is that they are mathematically concise and invariant to translation, rotation, and

scale. Furthermore, they reflect not only the shape but also the density distribution within

the curve.

The (p + q)th-order moments of plane curve l are defined as

mpq =∫xpyq ds,

(p, q = 0, 1, 2, . . .

), (2.4)

where ds is the arc differentiation of curve l. The (p + q)th-order central moments are defined

as:

μpq =∫(x − x)p

(y − y

)qds,

(p, q = 0, 1, 2, . . .

), (2.5)

where x = m10/m00, y = m01/m00.


For a digital image f(x,y),

mpq =∑x,y

xpyqf(x,y

),

μpq =∑x,y

(x − x)p(y − y

)qf(x,y

).

(2.6)

This paper defines f(x,y) as

f(x,y

)=

{1,

(x,y

) ∈ l,

0,(x,y

)/∈ l.

(2.7)

Thus, the global descriptors of hand gestures have been calculated using the central

moments of the curve. As we use discrete HMM, all the features extracted need to be

represented as an integer. The statistical distributions of the central moments are calculated

and then a feature is denoted as one or two digits.

3. The Continuous Hand Gesture Recognition Scheme

In this paper, we consider online-continuous-handed dynamic gestures based on discrete

HMM. The hand gesture recognition system consists of three major parts: palm detection,

hand tracking, and trajectory recognition. Figure 3 shows the whole process. The hand

tracking function is trigged when the system detects an opened hand before the camera; the

hand gesture classification based on HMM is activated when the user finishing the gesture.

The basic algorithmic framework for our recognition process is the following.

(1) Detect the palm from video and initialize the tracker with the template of hand

shape.

(2) Track the hand motion using a contour-based tracker and record the trajectory of

palm center.

(3) Extract the discrete vector feature from gesture path by the global and local feature

quantization.

(4) Classify the gesture using HMM which gives maximum probability of occurrence

of observation sequence.

3.1. Hand Detection and Tracking

We use Adaboost algorithm with (histograms of gradient) HOG feature to detect the user’s

hand. The shape information of an opening hand is relatively unique in the scene. We

calculate the HOG features of a new observed image to detect the opened hand at different

scales and location. When the hand is detected, we update the hand color model which will

be used in hand tracking. The system requires user to keep his palm opened vertically and

statically before the palm is captured by the detection algorithm. In this paper, we have

considered single handed dynamic gestures. A gesture is composed of a sequence of epochs.

Each epoch is characterized by the motion of distinct hand shapes.


Image input

Tracking Detection

2D mappingHand

trajectory

B-spline fit

Global

featureLocal

feature

Quantization the features

discrete vector

Feature extraction

using one integer

Combine the features to

Initialize the HMM

Gesture Nongesture

Satisfy the

gesture

ending

condition?

Satisfy the

gesture

ending

condition?

Gesture

model

Nongesture

model

Fit which model?

HMM classification

endend

Figure 3: Overview of the hand gesture recognition process.

Figure 4: Hand contour.

We have implemented a contour-based hand tracker, which combines two techniques

called condensation and partitioned sampling. During tracking, we record the trajectory of

the hand which will be used in the hand recognition stage. The hand contour is represented

with B-Splines, as shown in Figure 4. A fourteen-dimension state vector is used to describe

the dynamics of the hand contour:

χ =(tx, ty, α, s, θL, lL, θR, lR, θM, lM, θI , lI , θTh1, θTh2

), (3.1)

where the subvector (tx, ty, α, s) is a nonlinear representation of a Euclidean similarity

transform applied to the whole hand contour template, (tx, ty) is the palm center. (θL, lL)represents the nonrigid movement of the little finger, θL means the little finger’s angle with

respect to the palm, and lL means the little finger’s length relative to its original length in


the hand template. (θR, lR), (θM, lM), and (θI, lI) have the same meaning as the subvector

(θL, lL), but for different fingers. θTh1 represents the angle of the first segment of the thumb

with respect to the palm, and the last part θTh2 represents the angle of the second segment of

the thumb with respect to the first segment of the thumb.

We use a second-order autoregressive processes to predict the motion of the hand

contour:

xt = A1xt−1 +A2xt−2 + Bωt, (3.2)

where A1 and A2 are fixed matrices representing the deterministic components of the

dynamics, B is another fixed matrix representing the stochastic component of the dynamics,

and ωt is a vector of independent random normal N(0, 1) variants.

In prediction, lots of candidate contours will be produced. We choose the one which

matches the image feature (edges, boundaries of regions in skin color) best. Usually,

more dimensions of the state space are required to make the condensation filter achieve

considerable performance. However, this will increase computation complexity. In order to

alleviate the problem, partitioned sampling is used, which divides the hand contour tracking

into two steps: first, track the rigid movement of the whole hand, which is represented by

(tx, ty, α, s); second, track the nonrigid movement of the each finger, which is represented by

angle and length of each finger. The above operations can reduce the amount of candidate

contours and improve the efficiency of tracking.

3.2. Recognition Based on HMM

After the trajectory is obtained from the tracking algorithm, features are abstracted and used

to compute the probability of each gesture type with HMM. We use a vector to describe those

features and as the input of the HMM.

There are three main problems for HMM: evaluation, decoding, and training, which

are solved by using Forward algorithm, Viterbi algorithm, and Baum-Welch algorithm,

respectively [25]. The gesture models are trained using BW re-estimation algorithm and the

numbers of states are set depending on the complexity of the gesture shape.

We choose left-right banded model (Figure 5(a)) as the HMM topology, because the

left-right banded model is good for modeling-order-constrained time-series whose properties

sequentially change over time [26]. Since the model has no backward path, the state index

either increases or stays unchanged as time increases. After finishing the training process

by computing the HMM parameters for each type of gesture, a given gesture is recognized

corresponding to the maximal likelihood of seven HMM models by using viterbi algorithm.

Although the HMM recognizer chooses a model with the best likelihood, we cannot

guarantee that the pattern is really similar to the reference gesture unless the likelihood is

high enough. A simple threshold for the likelihood often does not work well. Therefore, we

produce a threshold model [22] that yields the likelihood value to be used as a threshold.

The threshold model is a weak model for all trained gestures in the sense that its likelihood

is smaller than that of the dedicated gesture model for a given gesture and is constructed

by collecting the states of all gesture models in the system using an ergodic topology shown

in Figure 5(b). A gesture is then recognized only if the likelihood of the best gesture model

is higher than that of the threshold model; otherwise, it is recognized as nongesture type.

Therefore, we can segment the online gestures using the threshold model.


S1 S2 S3 S4

(a) Left-right banded topology

S1 S2

S3 S4

(b) Ergodic topolgy

Figure 5: HMM topologies.

4. Experiments

For experimentation, we develop a human machine interaction interface based on hand

gesture. It can work with regular webcams that is connected to PC, which is used to capture

live images of the users’ hand movement. The minimum requirements of webcams are (1)frame rate up to 25 frames per second and (2) capture capability up to 640 × 480 pixels. The

interface can be deployed in indoors environment, which generally has static background

and less light changes. Hand gestures are those articulated with poses and movement with

hands. The interface is able to track and recognize the following predefined hand gestures:

(1) user drawing three circles continuously in a line horizontally with hand movement

in the air,

(2) user drawing a question mark (?) with hand movement in the air,

(3) user drawing three circles continuously in a line vertically with hand movement in

the air,

(4) hand being vertically lifted upwards,

(5) hand waving from left to right,

(6) hand waving from right to left,

(7) user drawing an exclamation mark (!) with hand movement in the air.

For the quantification of local oriental features, we pick 18 as the codeword number

from experience. Figure 6 shows the distribution histogram of central moment μ11 of the

seven gestures as our global feature, where all the sample amounts are 450. We can set

the number of the vector quantizer of global features to 20 according to the distribution.

It can also be seen that the central moment feature can express the shape characteristic of

trajectories. For example, gesture 1 and gesture 3, gesture 4 and gesture 7, gesture 5 and

gesture 6 are close in their integral form, respectively, and it can be separated easily using the

global feature.

We choose the state number of HMM for each gesture according to the experiment

results and find that the recognition rate cannot be promoted when the state numbers of

gestures 1 and gesture 3 are 10, and the other state numbers are set to 8. Therefore, we use

this setting in the following experiments.

We collected more than 800 trajectory samples of each isolated gesture from seven

people for training and more than 330 trajectory samples of each isolated gesture from eight

different users for testing. The recognition results are listed in Table 1. It can be seen that the

proposed method can greatly improve the recognition process, especially for those relatively

complicated gestures such as predefined gesture 1 and gesture 3. It is difficult to separate


180

160

140

120

100

80

60

40

20

00 5 10 15 20 25 30 35

Sam

ple

nu

mb

er

Gesture 1

Gesture 2

Gesture 3

Gesture 4

Gesture 5

Gesture 6

Gesture 7

μ11

Figure 6: The distribution of μ11.

Table 1: Recognition results comparison.

Gestures Test sets’ numbers Our method (%) Traditional method (%)1 339 84.1 94.7

2 407 95.1 98.2

3 372 73.4 89.7

4 454 95.9 98

5 424 98.1 100

6 476 95.8 99.8

7 474 98.9 99.6

gesture 1 and gesture 3 only using local features, because their motions resemble temporally.

Our algorithm can resolve this problem effectively.

5. Conlusion

We have implemented an automatic dynamic hand gesture recognition system in this paper.

The user’s hand is detected using Adaboost algorithm with HOG features and tracked using

condensation and partitioned sampling. The trajectory of hand gesture is represented by both

local and global features. Then, we take a discrete HMM method to recognize the gestures.

The experimental results show that the proposed algorithm can reach better recognition

results than the traditional hand recognition method. However, the tracking algorithm is still

very sensitive to light and the system can only report the detection until a gesture reaches its

end. Therefore, our future work will focus on improving the tracking algorithm and making

the recognition more natural.


Acknowledgments

This work was supported by the Research Project of Department of Education of Zhe-

jiang Province (Y201018160), and the Natural Science Foundation of Zhejiang Province

(Y1110649).

References

[1] S. Chen, Y. Li, and N. M. Kwok, “Active vision in robotic systems: a survey of recent developments,”International Journal of Robotics Research, vol. 30, no. 11, pp. 1343–1377, 2011.

[2] T. Gu, L. Wang, Z. Wu, X. Tao, and J. Lu, “A pattern mining approach to sensor-based human activityrecognition,” IEEE Transactions on Knowledge and Data Engineering, vol. 23, no. 9, pp. 1359–1372, 2011.

[3] X. Zhang, X. Chen, Y. Li, V. Lantz, K. Wang, and J. Yang, “A framework for hand gesture recognitionbased on accelerometer and EMG sensors,” IEEE Transactions on Systems, Man, and Cybernetics Part A,vol. 41, no. 6, pp. 1064–1076, 2011.

[4] I. N. Junejo, E. Dexter, I. Laptev, and P. Perez, “View-independent action recognition from temporalself-similarities,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 33, no. 1, pp. 172–185, 2011.

[5] M. Isard and A. Blake, “Condensation—conditional density propagation for visual tracking,”International Journal of Computer Vision, vol. 29, no. 1, pp. 5–8, 1998.

[6] S. Chen, “Kalman filter for robot vision: a survey,” IEEE Transactions on Industrial Electronics, vol. 59,Article ID 814356, 18 pages, 2012.

[7] J. MacCormick and A. Blake, “Probabilistic exclusion principle for tracking multiple objects,” inProceedings of the 7th IEEE International Conference on Computer Vision (ICCV ’99), pp. 572–578,September 1999.

[8] J. MacCormick and M. Isard, “Partitioned sampling, articulated objects, and interface-quality handtracking,” in Proceedings of the European Conferene Computer Vision, 2000.

[9] M. Tosas, Visual articulated hand tracking for interactive surfaces, Ph.D. thesis, University of Nottingham,2006.

[10] X. Deyou, “A neural network approach for hand gesture recognition in virtual reality driving trainingsystem of SPG,” in Proceedings of the 18th International Conference on Pattern Recognition (ICPR ’06), pp.519–522, August 2006.

[11] D. B. Nguyen, S. Enokida, and E. Toshiaki, “Real-time hand tracking and gesturerecognition system,”in Proceedings of the International Conference on on Graphics, Vision and Image Processing (IGVIP ’05 ), pp.362–368, CICC, 2005.

[12] E. Holden, R. Owens, and G. Roy, “Hand movement classification using an adaptive fuzzy expertsystem,” International Journal of Expert Systems, vol. 9, no. 4, pp. 465–480, 1996.

[13] M. Elmezain, A. Al-Hamadi, and B. Michaelis, “Real-time capable system for handgesture recognitionusing hidden markov models in stereo color image sequences,” Journal of WSCG, vol. 16, pp. 65–72,2008.

[14] G. Saon and J. T. Chien, “Bayesian sensing hidden markov models,” IEEE Transactions on Audio,Speech, and Language Processing, vol. 20, pp. 43–54, 2012.

[15] M. Li, C. Cattani, and S. Y. Chen, “Viewing sea level by a one-dimensional random function with longmemory,” Mathematical Problems in Engineering, vol. 2011, Article ID 654284, 2011.

[16] S. Eickeler, A. Kosmala, and G. Rigoll, “Hidden markov model based continuous online gesturerecognition,” in Proceedings of 14th International Conference on Pattern Recognition, vol. 2, pp. 1206–1208,1998.

[17] N. D. Binh and T. Ejima, “Real-time hand gesture recognition using pseudo 3-d Hidden MarkovModel,” in Proceedings of the 5th IEEE International Conference on Cognitive Informatics (ICCI ’06), pp.820–824, July 2006.

[18] L. Shi, Y. Wang, and J. Li, “A real time vision-based hand gestures recognition system,” in Proceedingsof the 5th International Symposium on Advances in Computation and Intelligence (ISICA ’10), vol. 6382, no.M4D, pp. 349–358, 2010.

[19] M. Elmezain, A. Al-Hamadi, and B. Michaelis, “Real-time capable system for handgesture recognitionusing hidden markov models in stereo color image sequences,” The Journal of WSCG, vol. 16, pp. 65–72, 2008.


[20] N. Liu, B. C. Lovell, P. J. Kootsookos, and R. I. A. Davis, “Model structure selection & trainingalgorithms for an HMM gesture recognition system,” in Proceedings of the 9th International Workshopon Frontiers in Handwriting Recognition (IWFHR-9 ’04), pp. 100–105, October 2004.

[21] S. Y. Chen and Y. F. Li, “Determination of stripe edge blurring for depth sensing,” IEEE Sensors Journal,vol. 11, no. 2, pp. 389–390, 2011.

[22] H. K. Lee and J. H. Kim, “An HMM-Based threshold model approach for gesture recognition,” IEEETransactions on Pattern Analysis and Machine Intelligence, vol. 21, no. 10, pp. 961–973, 1999.


[24] S. Chen, J. Zhang, Q. Guan, and S. Liu, “Detection and amendment of shape distor-tions based onmoment invariants for active shape models,” IET Image Processing, vol. 5, no. 3, pp. 273–285, 2011.

[25] S. Chen, H. Tong, Z. Wang, S. Liu, M. Li, and B. Zhang, “Improved generalizedbelief propagation forvision processing,” Mathematical Problems in Engineering, vol. 2011, Article ID 416963, 12 pages, 2011.

[26] M. Elmezain, A. Al-Hamadi, J. Appenrodt, and B. Michaelis, “A hidden markovmodel-based isolatedand meaningful hand gesture recognition,” Proceedings of World Academy of Science, Engineering andTechnology, vol. 31, pp. 1307–6884, 2008.


Research ArticleStable One-Dimensional Periodic Wave inKerr-Type and Quadratic Nonlinear Media

Roxana Savastru, Simona Dontu, Dan Savastru,Marina Tautan, and Vasile Babin

Department of Constructive and Technological Engineering—Lasers and Fibre Optic Communications,National Institute of R&D for Optoelectronics INOE 2000, 409 Atomistilor Street, P.O. Box MG-5,077125 Magurele, Ilfov, Romania

Correspondence should be addressed to Simona Dontu, [email protected]

Received 6 December 2011; Revised 9 February 2012; Accepted 13 February 2012


Copyright q 2012 Roxana Savastru et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We present the propagation of optical beams and the properties of one-dimensional (1D) spatialsolitons (“bright” and “dark”) in saturated Kerr-type and quadratic nonlinear media. Specialattention is paid to the recent advances of the theory of soliton stability. We show that thestabilization of bright periodic waves occurs above a certain threshold power level and thedark periodic waves can be destabilized by the saturation of the nonlinear response, while thedark quadratic waves turn out to be metastable in the broad range of material parameters. Thepropagation of (1+1) a dimension-optical field on saturated Kerr media using nonlinearSchrodinger equations is described. A model for the envelope one-dimensional evolution equationis built up using the Laplace transform.

1. Introduction

The discrete spatial optical solitons have been introduced and studied theoretically as

spatially localized modes of periodic optical structures [1]. A standard theoretical approach

in the study of the discrete spatial optical solitons is based on the derivation of an effective

discrete nonlinear Schrodinger equation and the analysis of its stationary localized solitons-

discrete localized modes [1, 2].The spatial solitons may exist in a broad branch of nonlinear materials, such as cubic

Kerr, saturable, thermal, reorientation, photorefractive, and quadratic media, and periodic

systems. Furthermore, the existence of solitons varies in topologies and dimensions [3].The theory of spatial optical solitons has been based on the nonlinear Schrodinger

(NLS) equation with a cubic nonlinearity, which is exactly integrable by means of the inverse


scattering (IST) technique. From the physical point of view, the integrable NLS equation

describes the (1+1)-dimensional beams in a Kerr (cubic) nonlinear medium in the framework

of the so-called paraxial approximation [4].Bright solitons are formed due to the diffraction or dispersion compensated by self-

focusing nonlinearity and appear as an intensity hump in a zero background. Solitons, which

appear as intensity dips with a CW background, are called dark soliton [3].Kerr solitons rely primarily on a physical effect, which produces an intensity-

dependent change in refractive index [3].The periodic wave structures play an important role in the nonlinear wave domain

so that they are core of instability modulation development and optics chaos on continuous

nonlinear media, modes of quasidiscrete systems or discrete system on mechanic and electric

domain. Thus, periodic wave structures are unstable in the propagation process. For example,

photorefractive crystals accept relatively high nonlinearity of saturated character at an

already known intensity for He-Ne laser in continuous regime.

2. Methodology

The propagation of the optical radiation in (1+1) dimensions in saturable Kerr-type medium

is described by the nonlinear Schrodinger equation for the varying field amplitude Φ(ς, ρ)[5]:

2i∂Φ

(ς, ρ

)∂ς

+∂2Φ

(ς, ρ

)∂ρ2

− 2Φ(ς, ρ

)∣∣Φ(ς, ρ)∣∣2

1 + S∣∣Φ(ς, ρ)∣∣2

= 0. (2.1)

The transverse ς and the longitudinal ρ coordinates are scaled in terms of the

characteristic pulse (beam) width and dispersion (diffraction) length, respectively; S is the

saturation parameter; σ = −1 (+1) stands for focusing (defocusing) media [5]

ς = σKZ,

ρ =√σK

√X2 + Y 2,

ϕ = arctg

(Y

X

),

η = ρ sinϕ,

ξ = ρ cosϕ.

(2.2)

The simplest periodic stationary solutions of (2.1) have the following form:

Φ(ς, ρ

)= U

(ρ)e+2ihς, (2.3)

where h is the propagation constant.

By replacing the field in such a form into (2.1), one gets

∂2U(ρ)

∂ρ2− 2hU

(ρ) − 2U3

(ρ)

1 + SU2(ρ) = 0. (2.4)


To perform the linear stability analysis of periodic waves in the saturable medium,

we use the mathematical formalism initially developed for periodic waves in cubic nonlinear

media [5].We consider an analytic model, which used the Laplace transform of (2.4):

(α(U(ρ))

=∫+∞

0

U(ρ)e−pρdρ = U

(p))

p = u1 + iv1

−(

p2

2− 2h

)U(p)+

⎡⎣p

2U(0) +

1

2

(∂U

(ρ)

∂ρ

)ρ=0

⎤⎦ +∫∞

0

U3(ρ)

1 + SU2(ρ)e−pρdρ = 0.

(2.5)

With the boundary conditions,

U(ρ)∣∣

ρ=0= U(0) = U0,

∂U(ρ)

∂ρ

∣∣∣∣∣ρ=0

= 0.(2.6)

From (2.5) we get the Laplace transform of the field:

(i) direct form:

U(p)=

pUo + 2∫+∞

0

((U3(ρ))/(1 + SU2

(ρ)))

e−pρdρ((p2/2

) − 2h) (2.7)

(ii) inverse transformation form:

U(ρ)=

1

2πi

∫u+i∞

u−i∞

pU0 + 2∫+∞

0

(U3(ρ)/1 + SU2

(ρ))e−pρdρ(

p2 − 4h) e+pρdp, (2.8)

where u is a finite number.

For the integration on real (h > 0) and imaginary (h < 0) poles, we calculated the

complex amplitude of nonlinear equation such as

U(ρ)= U0ch

(2√hρ)− 4

∫+∞

0

d

⎛⎜⎝sh2√h(ρ − ρ′

)(2√h)2

⎞⎟⎠(U3(ρ′)

1 + SU2(ρ′)),

U(ρ)= U0 cos

(2√hρ)+ 4

∫+∞

0

dρ′(

U3(ρ′)

1 + SU2(ρ′))(cos 2

√h(ρ − ρ′

)2√h

).

(2.9)


For the harmonic case (h < 0) integration form of the complex amplitude is

U(ρ)= U0 cos

(2√hρ)+

1

hcos

(2√hρ)∫+∞

0

U3(ρ′)

1 + SU2(ρ′)d(sin 2

√hρ′

)− 1

hsin

(2√hρ)∫+∞

0

U3(ρ′)

1 + SU2(ρ′)d(cos 2

√hρ′

).

(2.10)

By using the integration, we get

U(ρ)= U0 cos

(2√hρ)+

1

h

U3(ρ)

1 + SU2(ρ) sin

(2√hρ)

(2.11)

or

U(ρ)=

√√√√U2

0 +

(1

h

U3(ρ)

1 + SU2(ρ))2

sin(

2√hρ + ϕ1

),

ϕ1 = arctg

(U0

(1/h)(U3(ρ)/1 + SU2

(ρ))).

(2.12)

The total phase of the optical field envelope is as follows:

ϕT = 2√hρ + arctg

(U0

(1/h)(U3(ρ)/1 + SU2

(ρ))). (2.13)

We assume a frequency (ω) as a speed variation of total phase such as

ωDef=

dϕT

dρ=(

2√h)+

d

dρ

{arctg

(hU0

1 + SU2(ρ)

U3(ρ) )}

. (2.14)

We have the complex amplitude of envelope field with the following form:

U(ρ)= A

(ω, ρ

)cos

(2√hρ)+ B

(ω, ρ

)sin

(2√hρ). (2.15)


un2, un21, un22

0

0.5

1

1.5

2

2.5

un2un21un22

nh

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

(a)

nh

−90

−75

−60

−45

−30

−15

0

15

30

45

60

75

90ϕn2, ϕn21

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

jn2

jn21

(b)

Figure 1: Numerical simulations of complex amplitude and phase.

The hyperbolic secant plays this equation resulting in a conservative effect. The longi-

tudinal component is

A(ω, ρ

)= U0 − 5ω

2(Sh)3/2(1 + (ω3/4h))ch(ωρ

) ,B(ω, ρ

)=

(1

h

U30

1 + SU2(0)

)−

(5ω(ω2/2

√h))

+ h(ωρ

)2(Sh)3/2(1 + (ω3/4h))ch

(ωρ

) .(2.16)

Some numerical simulations of the complex amplitude of the nonlinear equation and

the total phase of the optical field depending on the propagation constant and an integer

number n are illustrated in Figure 1.


Figure 1 represents the model amplitude and the phase functions of the complex total

number, which explained the theoretical model presented. Thanks to the complex model, the

initial solution includes the hyperbolic secant and the conjugate complex part

Φ(ξ, ρ

)=

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

[U0 − 5ω

2(Sh)3/2(1 + (ω3/4h))ch(ωρ

)] cos(

2√hρ)

+

⎡⎢⎣( 1

h

U30

1 + SU20

)−

5ω(ω2/2

√h)+ h

(ωρ

)2(Sh)3/2(1 + (ω3/4h))ch

(ωρ

)⎤⎥⎦ sin

(2√h)ρ

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭e2ihξ,

ω0 = 2√h,

ω =dϕT

dρ,

ω(ρ)= ω0 +

e−4ω0ρ

4((ω2

0/4)U0S3/2

)2

1

ch2(ω0ρ) ,

ω(0) −ω0 =16

U20S

3

1

ω40

.

(2.17)

3. Conclusions

We have described the propagation in quadratic nonlinear media of the periodic waves in sat-

urated Kerr type. The analytic solution for one-dimensional, bright and dark spatial solitons

was found. To describe the spatial optical solitons in saturated Kerr type and the quadratic

nonlinear media, we propose an analytical model based on Laplace transform. The theoretical

model consists in solving analytically the Schrodinger equation with photonic network

using Laplace transform. The propagation properties were found by using different forms

of saturable nonlinearity. However, an exact analytic solution of the propagation problem

presented herein creates possibilities for further theoretical investigation. As a result, it is a

useful structure, which obtains one-dimensional “bright” and “dark” solitons with transver-

sal structure and transversal one-dimensional periodic waves.

References

[1] B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,”Journal of the Optical Society of America B, vol. 14, no. 11, pp. 2980–2993, 1997.

[2] F. Lederer, S. Darmanyan, and A. Kobyakov, Spatial Solitons, springer, Berlin, Germany, 2001.[3] X. u. Zhiyong, All-optical Soliton Control in Photonic Lattices, Master work, Universitat Politecnica de

Catalunya, Barcelona, Spain, 2007.[4] Y. S. Kivshar, “Bright and dark spatial solitons in non-Kerr media,” Optical and Quantum Electronics,

vol. 30, no. 7–10, pp. 571–614, 1998.[5] Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, “Stable one-dimensional periodic waves

in Kerr-type saturable and quadratic nonlinear media,” Journal of Optics B, vol. 6, no. 5, pp. S279–S287,2004.


Research ArticleCutting Affine Moment Invariants

Jianwei Yang,1 Ming Li,2 Zirun Chen,1 and Yunjie Chen1

1 School of Math and Statistics, Nanjing University of Information Science and Technology,Nanjing 210044, China

2 School of Information Science and Technology, East China Normal University, no. 500 Dong-Chuan Road,Shanghai 200241, China

Correspondence should be addressed to Jianwei Yang, [email protected]

Received 18 December 2011; Accepted 26 January 2012


Copyright q 2012 Jianwei Yang et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The extraction of affine invariant features plays an important role in many fields of imageprocessing. In this paper, the original image is transformed into new images to extract more affineinvariant features. To construct new images, the original image is cut in two areas by a closed curve,which is called general contour (GC). GC is obtained by performing projections along lines withdifferent polar angles. New image is obtained by changing gray value of pixels in inside area. Thetraditional affine moment invariants (AMIs) method is applied to the new image. Consequently,cutting affine moment invariants (CAMIs) are derived. Several experiments have been conductedto evaluate the proposed method. Experimental results show that CAMIs can be used in objectclassification tasks.

1. Introduction

The extraction of affine invariant features plays a very important role in object recognition

and has been found applicable in many fields such as shape recognition and retrieval [1, 2],watermarking [3], identification of aircrafts [4, 5], texture classification [6], image registration

[7], and contour matching [8].Many algorithms have been developed for affine invariant features extraction. Based

on whether the features are extracted from the contour only or from the whole shape region,

the approaches can be classified into two main categories: region-based methods and contour-

based methods [9]. For good overviews of the various techniques, refer to [9–12].Contour-based methods [4, 5, 13–18] provide better data reduction and the contour

usually offers more shape information than interior content [9]. But these methods are unap-

plicable to objects with several separable components (like some Chinese characters).


In contrast to contour-based methods, region-based techniques take all pixels within

a shape region into account to obtain the shape representation. Moment invariant methods

are the most widely used techniques. The commonly used affine moment invariants (AMIs)[19–21] are extensions of the classical moment invariants firstly developed by Hu [22].Although the moment-based methods can be applicable to binary or gray-scale images with

low computational demands, they would be sensitive to noise. Hence, only a few low-order

moment invariants can be used and limit the ability of object classification with a large-sized

database [18].A number of new region-based methods have also been introduced, such as Ben-

Arie’s frequency domain technique [23, 24], cross-weighted moment (CWM) [25], and Trace

transform [26]. A novel approach called multi scale autoconvolution (MSA) was derived by

Rahtu et al. [27]. These new methods give high accuracy, but usually at the expense of high

complexity and computational demands [27]. It is reported in [27] that one needs O(N4)and O(N2log2N) operations for computing CWM and MSA, respectively. It can be shown

that some of these methods are sensitive to noise in the background. To derive robust affine

invariant features, in [28], we cut the object into slices by division curves which are derived

from the object based on the obtained general contour (GC). The affine invariant descriptors

are constructed by summing up the gray value associated with every pixels in each slice.

However, the maximum of the division quantity τ is hard to be determined. To cut object into

small slices, the computational complexity is very large.

Recently, structure moment invariants have been introduced in [29, 30]. These invari-

ants are very efficient in object classification tasks for gray level images or color images, but

they are unapplicable to binary images. The density of binary images can not be changed

only by squaring.

All in all, contour-based methods can only be used to objects with single boundary;

whereas some region-based methods can achieve high accuracy but usually at the expense

of high computational demands, and some region-based methods are unapplicable to binary

images.

To extract affine invariant features more efficiency, we transform the original image

into new images in this paper. Affine invariants are extracted from new images. In order

to construct new images, the original image is cut in two areas: the inside area and the

outside area. To establish correspondence between areas of an image and those of its

affine transformed image, as in [28], general contour (GC) of the image is constructed by

performing projection along lines with different polar angles. A nonnegative constant is

added to the gray value associated with every pixel of inside area. As a result, new images

are obtained. Consequently, affine invariant features can be derived from these new images.

In this paper, AMIs method is applied to the obtained new images. More affine invariant

features, cutting affine moment invariants (CAMIs), are extracted. Furthermore, we combine

CAMIs with the original AMIs (we call the obtained affine invariants as CCAMIs). To test

and evaluate the proposed method, several experiments have been conducted. Experimental

results show that CAMIs and CCAMIs can be used in object classification tasks.

The rest of the paper is organized as follows: in Section 2, the GC of an image is

introduced. Consequently, the image is cut in two areas by putting GC on the image. New

image is formed by changing gray value of the inside area. We apply AMIs method to the new

image in Section 3. The performance of the proposed method is evaluated experimentally in

Section 4. Finally, some conclusion remarks are provided in Section 5.


2. The Construction of New Images

To derive affine invariant features, we construct new images by cutting the original image in

two areas. New images can be obtained by changing the gray value associated with pixels in

one of these areas.

2.1. GC of an Image

Suppose that an image is represented by I(x,y) in the 2D plane. Firstly, the origin of the

reference system is transformed to the centroid of the image. To derive general contour of

an image, the Cartesian coordinate system should be converted to polar coordinate system.

Hence, the shape can be represented by a function f of r and θ, namely,

I(x,y

)= f(r, θ), (2.1)

where r ∈ [0,∞), and θ ∈ [0, 2π). Take projection along lines from the centroid with different

angles by computing the following integral:

g(θ) =∫∞

0

f(r, θ)dr, (2.2)

where θ ∈ [0, 2π).

Definition 2.1. For an angle θ ∈ R, if g(θ) is given in (2.2), then (θ, g(θ)) denotes a point in the

plane of R2. Let θ go from 0 to 2π , then {(θ, g(θ)) | θ ∈ [0, 2π)} forms a closed curve. We call

this closed curve the general contour (GC) of the image.

By (2.2), a single value is correspond to an angle θ ∈ R. Consequently, a single closed

curve can be derived from any image. For an image I, we denote the GC extracted from

it as ∂I. Equation (2.2) is called central projection transform in [31–33]. It has been used in

those papers to extract rotation invariant signature by combining wavelet analysis and fractal

theory. Satisfying classification rates have been achieved in the recognition of rotated English

letters, Chinese characters, handwritten signatures, and so forth. As aforementioned, in [28],by employing GC, we derive division curves to cut object into slices. The affine invariant

descriptors are constructed by summing up the gray value associated with every pixels in

each slice. However, the maximum of the division quantity is hard to be determined. In this

paper, we use GC to construct new images. Affine invariant features are extracted from these

new images.

2.2. The Affine Property of GC

An affine transformation A of coordinates x ∈ R2 is defined as

x′ = Ax + b, (2.3)

where b =(

b1

b2

)∈ R2, and A = ( a11 a12

a21 a22) is a 2-by-2 nonsingular matrix with real entries.


Affine maps parallel lines onto parallel lines, intersecting lines into intersecting lines.

Based on these facts, it can be shown that the GC extracted from the affine transformed image

is also the same affine transformed version of GC extracted from the original image. In other

words, if two images I and I ′ are related by an affine transformation A,

�′ ={x′ | x′ = Ax + b, x ∈ �

}, (2.4)

where � and �′ are supports of I and I ′, respectively. Then ∂I and ∂I ′, GCs of I, and I ′ are

related by the same affine transformation A too:

∂I ′ ={x′ | x′ = Ax + b, x ∈ ∂I

}. (2.5)

2.3. The Construction of New Images

To construct new images, we put the GC on the original image. The image is cut in two areas:

the inside area (denoted as Dinside) and the outside area (denoted as Doutside). In Figure 1(b),we put the GC of Figure 1(a) on the image. Figure 1(c) is the inside area of the image, and

Figure 1(d) is the outside area of the image.

As aforementioned, GC preserves the affine transformation signature. As a result, the

inside area preserves affine transformation too. If two images I and I ′ are related by an affine

transformation A as in (2.4), then DIinside

and DI ′inside

, inside areas of I and I ′, are related by the

same affine transformation A too:

DI ′inside =

{x′ | x′ = Ax + b, x ∈ DI

inside

}. (2.6)

For example, Figure 2(a) is an affine transform version of Figure 1(a). Put the GC of

Figure 2(a) on the image (as shown in Figure 2(b)). Figure 2(c) is the inside area of the image

of Figure 2(b). Figure 2(d) is the outside area of the image of Figure 2(b). We observe that

Figures 2(c) and 2(d) are affine transformed versions of Figures 1(c) and 1(d). The affine

transformation is the same as that of Figure 2(a) to Figure 1(a).Consequently, new images can be constructed by changing gray value associated with

pixels in Dinside. For an image, a constant d (d ≥ 0) is added to the gray value associated with

every pixels in Dinside. The obtained new image is denoted as I(d)(x,y):

I(d)(x,y

)=

⎧⎨⎩I(x,y

),

(x,y

) ∈ Doutside,

I(x,y

)+ d,

(x,y

) ∈ Dinside.(2.7)

For different d, various new images can be derived. It is obvious that I(d)(x,y) is the original

image if d = 0.

Suppose that I(x,y) is an affine transformed image of the original image I(x,y).

I(d)(x,y) is the new image constructed from I(x,y) by (2.7). ˜I(d)(x,y) is the new image

constructed from I(x,y) by (2.7). Then ˜I(d)(x,y) is the same version affine transformed image

of I(d)(x,y). For example, we add 0.1 to the inside area of Figure 1(a); the obtained new image

is shown in Figure 3(a). The gray value of the inside area of Figure 2(a) is also added 0.1;


(a) (b)

(c) (d)

Figure 1: (a) Chinese character “Fu”. (b) Put the GC of Figure 1(a) on the image. (c) The inside area ofFigure 1(a). (d) The outside area of Figure 1(a).

the obtained image is shown in Figure 3(b). We observed that Figure 3(b) is the same affine

transform version of Figure 3(a) as that of Figure 2(a) to Figure 1(a).Some well-developed methods can be applied to the derived new images. More affine

invariant features can be constructed. As aforementioned, only a few low-order moment

invariants can be used for object classification. We can apply AMIs method to the derived

new images. More low-order moment invariants can be extracted. We construct new affine

moment invariants in the next section.

3. Cutting Affine Moment Invariants

By applying various region-based methods to the derived new image, some affine invariant

features can be extracted. As aforementioned, AMIs method is region-based method with low

computational demands. We apply AMIs to the constructed new image.

Geometric moment m(d)pq of the new image I(d)(x,y) is defined as

m(d)pq =

∫xpyqI(d)

(x,y

)dx dy, (3.1)


(a) (b)

(c) (d)

Figure 2: (a) An affine transformation version of Figure 1(a). (b) Put the GC of Figure 2(a) on the image.(c) The inside area of Figure 2(a). (d) The outside area of Figure 2(a).

(a) (b)

Figure 3: (a) New image constructed from Figure 1(a). (b) New image constructed from Figure 2(a).

where p, q are nonnegative integers. μ(d)pq is the central moments:

μ(d)pq =

∫(x − x0)p

(y − y0

)qI(d)

(x,y

)dx dy, (3.2)

where x0, y0 are the coordinates of the centroid of the image.


For two points x1 = (x1,y1)T , x2 = (x2,y2)

T ∈ R2, we denote the cross product C12 as

C12 = x1y2 − x2y1. (3.3)

After an affine transform, the following equation holds:

C12 = JC12, (3.4)

where J denotes the Jacobian of affine transformation: J = det(A).For N points (N ≥ 2): Ui = (xi,yi)

T ∈ R2, i = 1, 2, . . .N, and non-negative integers

nkj(1 ≤ k < j ≤ N), we define RCAMI(d) of the form:

RCAMI(d) =∫ ∏

1≤k<j≤NC

nkj

kj

N∏i=1

I(d)(xi,yi

)dxi dyi. (3.5)

Denote w =∑

1≤k<j≤N nkj . We normalized RCAMI(d) as follows:

CAMI(d) =RCAMI(d)(μ(d)00

)w+N. (3.6)

Using similar argument with that of affine moment invariants (see [20], etc.), it can be shown

that CAMI(d) is affine invariant. We call these invariants as cutting affine moment invariants(CAMIs). If d = 0, these invariants are the same as moment invariants given in [20].

By expanding Ckj in (3.5), RCAMI(d) becomes a polynomial of moments given in

(3.2). Consequently, we can compute CAMIs from moments given in (3.2). Invariants can

be derived by replacing moments in AMIs with the moments given in (3.2). Here, we use the

well-developed theory for the AMIs as described in [19]. The following form invariants are

used in this paper:

F(d)1 =

(μ(d)20 μ

(d)02 −

(μ(d)11

)2)

(μ(d)00

)4,

F(d)2 =

((μ(d)30

)2(μ(d)03

)2 − 6μ(d)30 μ

(d)21 μ

(d)12 μ

(d)03 + 4μ

(d)30

(μ(d)12

)3+ 4

(μ(d)21

)3μ(d)03 − 3

(μ(d)21

)2(μ(d)12

)2)

(μ(d)00

)10

F(d)3 =

(μ(d)20

(μ(d)21 μ

(d)03 −

(μ(d)12

)2)

− μ(d)11

(μ(d)30 μ

(d)03 − μ

(d)21 μ

(d)12

)+ μ

(d)02

(μ(d)31 μ

(d)12 −

(μ(d)21

)2))

(μ(d)00

)7.

,

(3.7)


If we set d = 0, (3.7) results in AMIs used in [19]. By changing the constant d, different

invariants can be constructed. Consequently, more low-order moment invariants can be

extracted. We will show that the obtained CAMIs can be used in object classification.

Furthermore, we will combine the obtained CAMIs with the traditional AMIs. The obtained

features (we call them CCAMIs) are also used in object classification.

4. Experiments

In this section, we evaluate the proposed method in object classification tasks. We will show

that the derived affine invariants (CAMIs) can be used in object classification. Furthermore,

CAMIs can be combined with the original AMIs (we call the obtained affine invariants as

CCAMIs). We denote AMIs used in [19] as: f1, f2, f3 (d = 0 in (3.7)).In the first experiment, some binary images of Chinese characters are used. The CAMIs

used in this experiment are obtained by setting d equal to 10% of the maximum gray

value in the image. Hence, d is set to 0.1. These CAMIs are denoted as: F(0.1)1 , F

(0.1)2 , F

(0.1)3 .

Figure 4(a) shows the original six Chinese characters. Some of these characters are very

similar. Figure 4(b) shows the same set of characters deformed by affine transforms. The

values of invariants AMIs: f1, f2, f3 and CAMIs: F(0.1)1 , F

(0.1)2 , F

(0.1)3 are given in Table 1. It can

be seen clearly that CAMIs really are invariant under affine transform. Furthermore, CAMIs

are different with the original AMIs.

In the second experiment, we test the combined invariants (CCAMIs): f1, f2, f3,

F(0.1)1 , F

(0.1)2 , F

(0.1)3 . Two groups of Chinese characters, shown in Figures 5(a) and 5(b), are

chosen as databases. Each group include 40 Chinese characters with regular script font. The

images in Figure 5(a) have size of 128 × 128, and those in Figure 5(b) have size of 256 × 256.

Some characters in these databases have the same structures, but the number of stokes or the

shape of specific stokes may be a little different. The affine transformations are generated by

the following transformation matrix [4]:

T = l

(cos θ − sin θ

sin θ cos θ

)⎛⎜⎝a b

01

a

⎞⎟⎠, (4.1)

where a ∈ {1, 2}, b ∈ {−1.5, −1, −0.5, 0, −0.5, 1, 1.5}, θ ∈ {0◦, 72◦, 144◦, 216◦, 288◦}, and

l ∈ {0.8, 1.2}. l, θ denote the scaling, rotation transformation, respectively, and a, b denote

the skewing transformation.

Each character will be transformed 140 times as described above. With these affine

transformations and the database, 5600 tests run using the proposed method for each group.

In our experiments, the classification accuracy is defined as

η =γ

η× 100%, (4.2)

where γ denotes the number of correctly classified images, and η denotes the total number of

images applied in the test.

The AMIs, CAMIs, and the combined invariants CCAMIs are applied to databases in

Figures 5(a) and 5(b). Classification is performed by the method used in [19]. Table 2 shows


Da Quan Tai Tian Yao Fu

(a)

Test1

Test2

Test3

Test4

Test5Test6

(b)

Figure 4: (a) The original six model Chinese characters. (b) Deformed Chinese characters to be recognized.

Table 1: AMIs and CAMIs for some similar Chinese characters.

f1 · 104 f2 · 108 f3 · 106 F(0.1)1 · 104 F

(0.1)2 · 108 F

(0.1)3 · 106

Da 1033 −29586 −6040 889 −20096 −4579

Test1 1029 −29457 −6013 894 −20689 −4645

Quan 1132 −27041 −6406 980 −18884 −4915

Test2 1131 −26560 −6336 1000 −19504 −5051

Tai 1080 −50285 −8123 904 −31880 −5861

Test3 1081 −51263 −8201 915 −33214 −6019

Tian 939 −22855 −5612 805 −15556 −4220

Test4 939 −23053 −5627 816 −16216 −4341

Yao 1037 −29694 −6538 907 −21196 −5118

Test5 1031 −28759 −6415 915 −20846 −5153

Fu 850 −23926 −5847 726 −15462 −4351

Test6 846 −23686 −5795 726 −15324 −4354

the results. For the first group of Chinese characters, we observe that the performance of

CAMIs is a little better than that of AMIs, and the combined invariants CCAMIs have better

performance than the original AMIs and CAMIs. For the other group of Chinese characters,

we observe that the performance of the traditional AMIs is better than that of CAMIs, and

the combined invariants CCAMIs have also better performance than the original AMIs and

CAMIs. Hence, the original AMIs can be combined with CAMIs, more shape information

may be extracted.

5. Conclusions

In this paper, an approach is developed for the extraction of affine invariant features

by cutting image into areas: the inside area and the outside area. In order to establish

correspondence between areas of an image and those of its affine transformed version,


(a)

(b)

Figure 5: (a) First group of 40 characters. (b) Second group of 40 characters.

Table 2: Classification accuracies of AMIs, CAMIs, and CCAMIs in case of different affine transformations.

AMIs CAMIs CCAMIs

Group one 86.46% 87.62% 90.14%

Group two 95.55% 89.64% 96.13%

general contour (GC) of the object is employed. A nonnegative constant is added to the

gray value associated with every pixel of inside area. Consequently, new image is obtained,

and CAMIs are constructed from the new image. To test and evaluate the proposed method,

several experiments have been conducted. Experimental results show that CAMIs can be

used in object classification tasks.

Acknowledgments

This work was supported in part by the National Science Foundation under Grant

60973157, 61003209 in part by the Natural Science Foundation of Jiangsu Province Education


Department under Grant 08KJB520004. Ming Li acknowledges the 973 plan under the Project

no. 2011CB302802 and the NSFC under the Project Grant nos. 61070214 and 60873264.

References

[1] M. R. Daliri and V. Torre, “Robust symbolic representation for shape recognition and retrieval,”Pattern Recognition, vol. 41, no. 5, pp. 1799–1815, 2008.

[2] P. L. E. Ekombo, N. Ennahnahi, M. Oumsis, and M. Meknassi, “Application of affine invariant fourierdescriptor to shape based image retrieval,” International Journal of Computer Science and NetworkSecurity, vol. 9, no. 7, pp. 240–247, 2009.

[3] X. Gao, C. Deng, X. Li, and D. Tao, “Geometric distortion insensitive image watermarking in affinecovariant regions,” IEEE Transactions on Systems, Man and Cybernetics Part C, vol. 40, no. 3, Article ID5378648, pp. 278–286, 2010.

[4] M. I. Khalil and M. M. Bayoumi, “A dyadic wavelet affine invariant function for 2D shaperecognition,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 23, no. 10, pp. 1152–1164, 2001.

[5] M. I. Khalil and M. M. Bayoumi, “Affine invariants for object recognition using the wavelettransform,” Pattern Recognition Letters, vol. 23, no. 1-3, pp. 57–72, 2002.

[6] G. Liu, Z. Lin, and Y. Yu, “Radon representation-based feature descriptor for texture classification,”IEEE Transactions on Image Processing, vol. 18, no. 5, pp. 921–928, 2009.

[7] R. Matungka, Y. F. Zheng, and R. L. Ewing, “Image registration using adaptive polar transform,” IEEETransactions on Image Processing, vol. 18, no. 10, pp. 2340–2354, 2009.

[8] Y. Wang and E. K. Teoh, “2D affine-invariant contour matching using B-Spline model,” IEEETransactions on Pattern Analysis and Machine Intelligence, vol. 29, no. 10, pp. 1853–1858, 2007.

[9] D. Zhang and G. Lu, “Review of shape representation and description techniques,” PatternRecognition, vol. 37, no. 1, pp. 1–19, 2004.

[10] E. Rahtu, A multiscale framework for affine invariant pattern recognition and registration, Ph.D. thesis,University of Oulu, Oulu, Finland, 2007.

[11] R. Veltkamp and M. Hagedoorn, “State-of the art in shape matching,” Tech. Rep. UU-CS-1999, 1999.

[12] I. Weiss, “Geometric invariants and object recognition,” International Journal of Computer Vision, vol.10, no. 3, pp. 207–231, 1993.

[13] K. Arbter, W. E. Snyder, H. Burkhardt, and G. Hirzinger, “Application of affine-invariant Fourierdescriptors to recognition of 3-D objects,” IEEE Transactions on Pattern Analysis and MachineIntelligence, vol. 12, no. 7, pp. 640–647, 1990.

[14] W. S. Lin and C. H. Fang, “Synthesized affine invariant function for 2D shape recognition,” PatternRecognition, vol. 40, no. 7, pp. 1921–1928, 2007.

[15] F. Mokhtarian and S. Abbasi, “Curvature scale space for shape similarity retrieval under affinetransforms,” in Computer Analysis of Images and Patterns, Lecture Notes in Computer Science, pp. 65–72, Springer-Verlag, New York, NY, USA, 1999.

[16] F. Mokhtarian and S. Abbasi, “Affine curvature scale space with affine length parametrisation,”Pattern Analysis and Applications, vol. 4, no. 1, pp. 1–8, 2001.

[17] I. El Rube, M. Ahmed, and M. Kamel, “Wavelet approximation-based affine invariant shaperepresentation functions,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 28, no.2, pp. 323–327, 2006.

[18] Q. M. Tieng and W. W. Boles, “Wavelet-based affiine invariant representation: a tool for recognizingplanar objects in 3 D space,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 19, no.8, pp. 1287–1296, 1997.

[19] J. Flusser and T. Suk, “Pattern recognition by affine moment invariants,” The Journal of the PatternRecognition Society, vol. 26, no. 1, pp. 167–174, 1993.

[20] T. Suk and J. Flusser, “Affine moment invariants generated by graph method,” Pattern Recognition,vol. 44, no. 9, pp. 2047–2056, 2011.

[21] T. H. Reiss, “The revised fundamental theorem of moment invariants,” IEEE Transactions on PatternAnalysis and Machine Intelligence, vol. 13, no. 8, pp. 830–834, 1991.


[22] M. K. Hu, “Visual pattern recognition by moment invariants,” IEEE Transactions on Information Theory,vol. 8, no. 2, pp. 179–187, 1962.

[23] J. Ben-Arie and Z. Wang, “Pictorial recognition of objects employing affine invariance in the frequencydomain,” IEEE Transactions on Pattern Analysis andMachine Intelligence, vol. 20, no. 6, pp. 604–618, 1998.

[24] S. Y. Chen, J. Zhang, Q. Guan, and S. Liu, “Detection and amendment of shape distortions based onmoment invariants for active shape models,” IET Image Processing, vol. 5, no. 3, pp. 273–285, 2011.

[25] Z. Yang and F. S. Cohen, “Image registration and object recognition using affine invariants and convexhulls,” IEEE Transactions on Image Processing, vol. 8, no. 7, pp. 934–946, 1999.

[26] M. Petrou and A. Kadyrov, “Affine Invariant Features from the Trace Transform,” IEEE Transactionson Pattern Analysis and Machine Intelligence, vol. 26, no. 1, pp. 30–44, 2004.

[27] E. Rahtu, M. Salo, and J. Heikkila, “Affine invariant pattern recognition using multiscale autocon-volution,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 27, no. 6, pp. 908–918,2005.

[28] J. Yang, Z. Chen, W.-S. Chen, and Y. Chen, “Robust affine invariant descriptors,” Mathematical Problemsin Engineering, vol. 2011, Article ID 185303, 15 pages, 2011.

[29] Z. M. Li, K. P. Hou, Y. J. Liu, L. H. Diao, and H. Li, “The shape recognition based on structure momentinvariants,” International Journal of Information Technology, vol. 12, no. 2, pp. 97–105, 2006.

[30] Z. Li, Y. Zhang, K. Hou, and H. Li, “3D polar-radius invariant moments and structure momentinvariants,” Lecture Notes in Computer Science, vol. 3611, pp. 483–492, 2005.

[31] Y. Y. Tang, Y. Tao, and E. C. M. Lam, “New method for feature extraction based on fractal behavior,”Pattern Recognition, vol. 35, no. 5, pp. 1071–1081, 2002.

[32] Y. Tao, E. C. M. Lam, C. S. Huang, and Y. Y. Tang, “Information distribution of the central projectionmethod for Chinese character recognition,” Journal of Information Science and Engineering, vol. 16, no.1, pp. 127–139, 2000.

[33] Y. Tao, E. C. M. Lam, and Y. Y. Tang, “Feature extraction using wavelet and fractal,” Pattern RecognitionLetters, vol. 22, no. 3-4, pp. 271–287, 2001.


Research ArticleHomotopy Perturbation Method and VariationalIteration Method for Harmonic Waves Propagationin Nonlinear Magneto-Thermoelasticitywith Rotation

Khaled A. Gepreel,1, 2 S. M. Abo-Dahab,2, 3 and T. A. Nofal2, 4

1 Math. Department, Faculty of Science, Zagazig University, Zagazig 44519, Egypt2 Math. Department, Faculty of Science, Taif University, Saudi Arabia3 Math. Department, Faculty of Science, SVU, Qena 83523, Egypt4 Math. Department, Faculty of Science, El-Minia University, Egypt

Correspondence should be addressed to Khaled A. Gepreel, [email protected] and

S. M. Abo-Dahab, [email protected]

Received 17 August 2011; Accepted 3 October 2011


Copyright q 2012 Khaled A. Gepreel et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The homotopy perturbation method and variational iteration method are applied to obtain theapproximate solution of the harmonic waves propagation in a nonlinear magneto-thermoelasticityunder influence of rotation. The problem is solved in one-dimensional elastic half-space model sub-jected initially to a prescribed harmonic displacement and the temperature of the medium. The dis-placement and temperature are calculated for the methods with the variations of the magnetic fieldand the rotation. The results obtained are displayed graphically to show the influences of the newparameters and the difference between the methods’ technique. It is obvious that the homotopyperturbation method is more effective and powerful than the variational iteration method.

1. Introduction

In the past recent years, much attentions have been devoted to simulate some real-life pro-

blems which can be described by nonlinear coupled differential equations using reliable

and more efficient methods. The nonlinear coupled system of partial differential equations

often appear in the study of circled fuel reactor, high-temperature hydrodynamics, and

thermoelasticity problems, see [1–4]. From the analytical point of view, lots of work have been

done for such systems. With the rapid development of nanotechnology, there appears an ever-

increasing interest of scientists and researchers in this field of science. Nanomaterials, because

of their exceptional mechanical, physical, and chemical properties, have been the main topic


of research in many scientific publications. Wave generation in nonlinear thermoelasticity

problems has gained a considerable interest for its utilitarian aspects in understanding the

nature of interaction between the elastic and thermal fields as well as for its applications.

A lot of applications was paid on existence, uniqueness, and stability of the solution of the

problem, see [5–7].Much attention has been devoted to numerical methods, which do not require dis-

cretization of space-time variables or linearization of the nonlinear equations, among which

the variational iteration method (VIM) suggested in [8–20] shows its remarkable merits

over others. The method was successfully applied to a nonlinear one dimensional coupled

equations in thermoelasticity [21], revealing that the method is very convenient, efficient, and

accurate. The basic idea of variational iteration method is to construct a correction functional

with a general Lagrange multiplier which can be identified optimally via variational theory.

The homotopy perturbation method [8, 22] has the merits of simplicity and easy

execution. Unlike the traditional numerical methods, the HPM does not need discretization

and linearization. Most perturbation methods assume that a small parameter exists, but

most nonlinear problems have no small parameter at all. Many new methods have been

proposed to eliminate the small parameter. Recently, the applications of homotopy theory

among scientists appeared, and the homotopy theory becomes a powerful mathematical

tool, when it is successfully coupled with perturbation theory. Sweilam and Khader [1]investigated variational iteration method for one dimensional nonlinear thermoelasticity.

Applying He’s variational iteration method for solving differential-difference equation is

discussed by Yildirim [23]. Noor and Mohyud-Din [24], Mohyud-Din et al. [25–27] used

He’s polynomials or Pade approximants to solve solving higher-order nonlinear boundary

value problems, second-order singular problems, and nonlinear boundary value problems.

Mohyud-Din et al. [28] applied the modified variational iteration method for free-convective

boundary-layer equation using Pade approximation. Mohyud-Din and Noor [29, 30] used

Homotopy perturbation method for solving some new boundary value problems. Mohyud-

Din et al. [31] investigated some relatively new techniques for nonlinear problems.

In this paper, the homotopy perturbation method and variational iteration method

are used to solve the coupled harmonic waves nonlinear magneto-thermoelasticity equations

under influence of rotation. The Maple and Mathematica software packages are used to

obtain the approximate solutions in one-dimensional half-space. The displacement and tem-

perature which obtained have been calculated numerically and presented graphically.

2. Basic Idea of He’s Homotopy Perturbation Method

We illustrate the following nonlinear differential equation [8, 22]:

A(u) − f(r) = 0, r ∈ Λ, (2.1)

with the boundary conditions:

B

(u,

∂u

∂n

)= 0, r ∈ Γ, (2.2)


where A is a general differential operator, B is a boundary operator, f(r) is an analytic

function, and Γ is the boundary of the domain Λ. Generally speaking, the operator A can

be divided into two parts which are L and N, where L is linear operator but N is nonlinear

operator. Equation (2.1) can therefore be rewritten as follows:

L(u) +N(u) − f(r) = 0. (2.3)

By the homotopy technique, we construct a homotopy V (r, p): Λ × [0, 1] → R which satisfies

H(V, p

)=(1 − p

)[L(V ) − L(u0)] + p

[A(V ) − f(r)

]= 0, r ∈ Λ, (2.4)

or

H(V, p

)= L(V ) − L(u0) + pL(u0) + p

(N(V ) − f(r)

)= 0, r ∈ Λ, (2.5)

where p ∈ [0, 1] is an embedding parameter and u0 is an initial approximation of (2.1) which

satisfies the boundary conditions (2.2). Obviously, from (2.4) and (2.5) we have

H(V, 0) = L(V ) − L(u0) = 0,

H(V, 1) = A(V ) − f(r) = 0.(2.6)

The changing process of p from zero to unity is just that of V (r, p) from u0(r) to u(r). In

topology, this is called deformation, and L(V ) − L(u0) and A(V ) − f(r) are called homotopy.

According to the homotopy perturbation method, we can first use the embedding parameter

“p” as a small parameter and assume that the solution of (2.4) and (2.5) can be written as a

power series in “p” as follows:

V = V0 + pV1 + p2V2 + · · · . (2.7)

On setting p = 1 results in the approximate solution of (2.3), we have

u = limp→ 1

V = V0 + V1 + V2 + · · · . (2.8)

The combination of the perturbation method and the homotopy method is called the homo-

topy perturbation method, which has eliminated the limitations of the traditional per-

turbation methods. On the other hand, this technique can have full advantage of the

traditional perturbation techniques. The series (2.8) is convergent to most cases. However,

the convergent rate depends on the nonlinear operator A(V ).

(1) The second derivative of N(V ) with respect to V must be small because the

parameter may be relatively large, that is, p → 1.

(2) The norm of L−1(∂N/∂V ) must be smaller than one so that the series converges.


3. Application of Homotopy Perturbation Method on the NonlinearMagneto-Thermoelastic with Rotation Equations

In this section, we use the homotopy perturbation method to calculate the approximate

solutions of the following nonlinear magneto-thermoelastic with rotation equations:

(1 + σ1)utt + Ωut − uxx

(1 − σ2 + 2γux + 3δu2

x

)− β1θx − β2(θux)x = 0,(

θ − aux − 1

2bu2

x

)t

− [(1 + αux)θx]x = 0,

(3.1)

where γ, β1, β2, a, b, α are arbitrary constants, σ1, σ2 are the sensitive parts of the magnetic

field, and Ω is the rotation parameter, with the initial conditions

u(x, 0) = θ(x, 0) = A(1 − cos(x)), ut(x, 0) = θt(x, 0) = 0, (3.2)

where A is an arbitrary constant and the boundary conditions

u(0, t) = θ(0, t) = 0, ut(0, t) = θt(0, t) = 0. (3.3)

To investigate the traveling wave solution of (3.1), we first construct a homotopy perturbation

method as follows:

(1 − p

)[(1 + σ1)(Vtt − V0tt)] + p

[(1 + σ1)Vtt + ΩVt − Vxx

(1 − σ2 + 2γVx + 3δV 2

x

)−β1Θx − β2(ΘVx)x

]= 0,(

1 − p)(Θt −Θ0t) + p[Θt − aVxt − bVxVxt −Θxx − αVxxΘx − αVxΘxx] = 0,

(3.4)

where the initial approximations take the following form:

V0(x, t) = u0(x, t) = u(x, 0) = A(1 − cos(x)),

Θ0(x, t) = θ0(x, t) = θ(x, 0) = A(1 − cos(x)).(3.5)

According to the homotopy perturbation method, we can first use the embedding parameter

“p” as a small parameter and assume that the solution of (3.4) can be written as a power

series in “p” as the following:

V = V0(x, t) + pV1(x, t) + p2V2(x, t) + p3V3(x, t) + · · · ,

Θ(x, t) = Θ0(x, t) + pΘ1(x, t) + p2Θ2(x, t) + p3Θ3(x, t) + · · · ,(3.6)

where Vj and Θj , j = 1, 2, 3, . . . are functions to be determined.


Substituting from (3.6) into (3.4) and arranging the coefficients of “p” powers, we have

(1 + σ1)V0,tt +(σ1V1,tt − 3δV 2

0,xV0,xx − β2V0,xxΘ0 − 2γV0,xV0,xx + σ2V0,xx − β1Θ0,x

−V0,xx − β2V0,xΘ0,x + ΩV0,t + V1,tt

)p

+(σ1V2,tt − 2γV1,xV0,xx + V2,tt − 2γV0,xV1,xx − 6δV0,xV1,xV0,xx − V1,xx − β2V1,xxΘ0

−β1Θ1,x − β2V0,xΘ1,x − β2V1,xΘ0,x − β2V0,xxΘ1 + ΩV1,t − 3δV 20,xV1,xx + σ2V1,xx

)p2

+(V3,tt − β2V2,xΘ0,x − β2V2,xxΘ0 − 2γV2,xV0,xx − 2γV1,xV1,xx + σ1V3,tt − β1Θ2,x

− 6δV0,xV2,xV0,xx − 3δV 21,xV0,xx + σ2V2,xx + ΩV2,t − V2,xx − 3δV 2

0,xV2,xx − 2γV0,xV2,xx

−β2V1,xxΘ1 − β2V0,xxΘ2 − β2V0,xΘ2,x − 6δV0,xV1,xV1,xx − β2V1,xΘ1,x

)p3 + · · · = 0,

Θ0,t + (−aV0,xt − αV0,xΘ0,xx + Θ1,t − αV0,xxΘ0,x − 2bV0,xV0,xt −Θ0,xx)p

+ (Θ2,t −Θ1,xx − aV1,xt − αV1,xxΘ0,x − 2bV1,xV0,xt − αV0,xΘ1,xx − αV1,xΘ0,xx

−2bV0,xV1,xt − αV0,xxΘ1,x)p2

× (−αV1,xxΘ1,x − αV2,xxΘ0,x − 2bV2,xV0,xt − αV0,xxΘ2,x − 2bV1,xV1,xt − aV2,xt

+Θ3,t − 2bV0,xV2,xt − αV0,xΘ2,xx − αV1,xΘ1,xx − αV2,xΘ0,xx −Θ2,xx)p3 + · · · = 0.

(3.7)

In order to obtain the unknowns of Vj and Θj , (j = 1, 2, 3, . . .), we construct and solve the

following system considering the initial conditions (3.2):

σ1V1,tt − 3δV 20,xV0,xx − β2V0,xxΘ0 − 2γV0,xV0,xx + σ2V0,xx − β1Θ0,x − V0,xx

− β2V0,xΘ0,x + ΩV0,t + V1,tt = 0,

σ1V2,tt − 2γV1,xV0,xx + V2,tt − 2γV0,xV1,xx − 6δV0,xV1,xV0,xx − V1,xx − β2V1,xxΘ0

− β1Θ1,x − β2V0,xΘ1,x − β2V1,xΘ0,x − β2V0,xxΘ1 + ΩV1,t − 3δV 20,xV1,xx + σ2V1,xx = 0,

V3,tt − β2V2,xΘ0,x − β2V2,xxΘ0 − 2γV2,xV0,xx − 2γV1,xV1,xx + σ1V3,tt − β1Θ2,x

− 6δV0,xV2,xV0,xx − 3δV 21,xV0,xx + σ2V2,xx + ΩV2,t − V2,xx − 3δV 2

0,xV2,xx − 2γV0,xV2,xx

− β2V1,xxΘ1 − β2V0,xxΘ2 − β2V0,xΘ2,x − 6δV0,xV1,xV1,xx − β2V1,xΘ1,x = 0,

− aV0,xt − αV0,xΘ0,xx + Θ1,t − αV0,xxΘ0,x − 2bV0,xV0,xt −Θ0,xx = 0,

Θ2,t −Θ1,xx − aV1,xt − αV1,xxΘ0,x − 2bV1,xV0,xt − αV0,xΘ1,xx − αV1,xΘ0,xx

− 2bV0,xV1,xt − αV0,xxΘ1,x = 0,

− αV1,xxΘ1,x − αV2,xxΘ0,x − 2bV2,xV0,xt − αV0,xxΘ2,x − 2bV1,xV1,xt − aV2,xt + Θ3,t

− 2bV0,xV2,xt − αV0,xΘ2,xx − αV1,xΘ1,xx − αV2,xΘ0,xx −Θ2,xx = 0.

(3.8)


Consequently, we deduce after some calculations the following results:

u = limp→ 1

V = V0 + V1 + V2 + · · · ,

θ = limp→ 1

Θ = Θ0 + Θ1 + Θ2 + · · · ,(3.9)

where

V0 = A(1 − cosx),

V1 =t2

2(4 + 4σ1)

[4γA2 sin 2x + 3δA3 cosx − 3δA3 cos 3x + 4β2A

2 cosx

−4β2A2 cos 2x4σ2A cosx + 4β1A sinx + 4A cosx

],

V2 =t4

32(σ1 + 1)2

{(−8

3β2

2A3 − 8

3β2A

2 − 8

3γ2A3 − 4δA3 + 4A3σ2δ +

8

3σ2A − 9

2δ2A5 − 4

3Aσ2

2

−4

3A +

8

3A2σ2β2 − 4β2A

4δ

)cosx

−(

4

3β1A + δA3β1 +

4

3γA3β2 − 4

3Aσ2β1 +

4

3β2A

2β1

)sinx

+(−8γA2 + 8A2σ2γ − 8β2A

3γ +4

3β2A

2β1 − 16δA4γ

)sin 2x

+(

12γA3β2 + 3δA3β1

)sin 3x + 20δA4γ sin 4x

+(−20

3A2σ2β2 +

20

3β2

2A3 +

8

3γA2β1 +

20

3β2A

2 + 6δA4β2

)cos 2x

+(

63

4δ2A5 − 4β2

2A3 + 12δA3 + 12β2A

4δ + 8γ2A3 − 12A3σ2δ

)cos 3x

−14β2A4δ cos 4x − 45

4δ2A5 cos 5x

}

+t3

32(σ1 + 1)2

{(−16

3ΩAβ1 − 16

3β1Aσ1 − 8

3β2ασ1A

3 − 16

3β1A − 8

3β2A

3α

)sinx

− 16

3ΩA2γ sin 2x +

(8β2A

3ασ1 + 8β2A3α)

sin 3x

+(

16

3ΩAσ2 − 4ΩA3δ − 16

3ΩA2β2 − 16

3ΩA

)cos(x)

+(

16

3β2A

2 +32

3β1A

2ασ1 +16

3ΩA2β2 +

16

3β2A

2σ1 +32

3β1A

2α

)cos 2x

+ 4ΩA3δ cos 3x

},


Θ0 = A(1 − cos(x)),

Θ1 = A2tα sin(2x) +A cos(x)t,

Θ2 =A

2(1 + σ1)

(t3

3

(12αA2γcos3(x) + 2αAcos2(x)β1 + 36αA3δ sin(x)cos3(x)

+ 12αA2β2 sin(x)cos2(x) + 2αAσ2 sin(x) cos(x) − 24αA3δ sin(x) cos(x)

− 2αA2β2 sin(x) cos(x) − 2αA sin(x) cos(x) − αAβ1 − 10αγA2 cos(x)

−4αβ2A2 sin(x)

)

+t2

2

(− 2 cos(x) + 18aδA2 sin(x)cos2(x) + 16bA2γ sin(x)cos2(x)

+ 8aβ2A sin(x) cos(x) + 2aσ2 sin(x) + 2aβ1 cos(x) + 24α2A2σ1cos3(x)

− 36bA3δcos4(x) − 16bA2β2cos3(x) + 48bA3δcos2(x) + 4bA2β2cos2(x)

+ 8cos2(x)aγA − 4 cos (x)2bAσ2 − 20α2A2 cos(x) + 24 cos (x)3α2A2

+ 4bAcos2(x) − 4bA − 6aδA2 sin(x) − 20αA sin(x) cos(x)

− 2aβ2A sin(x) − 4bA2β2 − 12bA3δ + 4bAσ2 − 8bA2 sin(x)γ

+ 16bA2β2 cos(x) − 20α2A2 cos(x) σ1 − 4aγA − 2 cos(x)σ1

−2a sin(x) + 4bAβ1 sin(x) cos(x) − 20αA sin(x) cos(x)σ1

)).

(3.10)

Now we make calculations for the results obtained by the homotopy perturbation method

using the Maple software package with the following arbitrary constants:

a = 0.5, A = 0.001, b = 0.5, α = 1, β1 = β2 = 0.05, γ = 1, δ = 0.8. (3.11)

The results obtained in (3.9) are displayed graphically in Figures 1–4.

3.1. Special Cases

(1) If we take into our consideration the first iteration (i.e., u = V0+V1 and θ = Θ0+Θ1).See Figures 5, 6, 7, and 8.

(2) If the magnetic field and rotation are neglected, the components of the displacement

u and temperature θ take the following forms. See Figures 9 and 10.


−8000

−4000

0

4000

8000

u

xt

02

46

810

1214 0

1020

3040

50

(a)

xt

02

46

810

1214 0

1020

3040

50

−60

−20

20

60

(b)

Figure 1: Variations of the displacement u and temperature θ for various values of the axis x and time twhen Ω = 0.1, σ1 = 0.2, σ2 = 0.1.

u

x

02

46

810

1214 0

12

34

5

Ω

0

0.0005

0.001

0.0015

0.002

(a)

x

02

46

810

1214 0

12

34

5

Ω

0.0002

0.0006

0.0014

0.0018

0.001

(b)

Figure 2: Variations of the displacement u and temperature θ for various values of the axis x and rotationΩ when t = 0.1, σ1 = 0.2, σ2 = 0.1.

1

u

x

02

46

810

1214 0

24

68

100

0.0005

0.001

0.0015

0.002

(a)

x

02

46

810

1214 1

02

46

810

0.0002

0.0006

0.0014

0.0018

0.001

(b)

Figure 3: Variations of the displacement u and temperature θ for various values of the axis x and magneticfield σ1 when t = 0.1, Ω = 0.1, σ2 = 0.1.


u

x

02

46

810

1214 0

24

68

10

0

0.001

0.002

2

(a)

x

02

46

810

1214 0

24

68

10

2

0.0002

0.0006

0.0014

0.0018

0.001

(b)


u

xt

02

46

810

1214 0

1020

3040

50

−0.8

−0.4

0

0.4

0.8

(a)

xt

02

46

810

1214 0

1020

3040

50

−0.4

0

0.4

0.2

−0.2

(b)

Figure 5: Variations of the displacement u and temperature θ for various values of the axis x and time twhen Ω = 0.1, σ1 = 0.2, σ2 = 0.1.

u

x

02

46

810

1214 0

12

34

5

Ω

0

0.0005

0.001

0.0015

0.002

(a)

01

23

45

Ω

0.0002

0.0006

0.0014

0.0018

0.001

x

02

46

810

1214

(b)

Figure 6: Variations of the displacement u and temperature θ for various values of the axis x and rotationΩ when t = 0.1, σ1 = 0.2, σ2 = 0.1.


1

02

46

810

u

x

02

46

810

1214

0

0.0005

0.001

0.0015

0.002

(a)

0.0002

0.0006

0.0014

0.0018

0.001

x

02

46

810

1214

1

02

46

810

(b)


x

02

46

810

1214 0

24

68

10

2

u

0

0.0005

0.001

0.0015

0.002

(a)

x

02

46

810

1214 0

24

68

10

2

0.0002

0.0006

0.0014

0.0018

0.001

(b)


4. Basic Idea of Variational Iteration Method

Consider the following nonhomogeneous nonlinear system of partial differential equations:

L1u(x, t) +N1(u(x, t), θ(x, t)) = f(x, t), (4.1)

L2θ(x, t) +N2(u(x, t), θ(x, t)) = g(x, t), (4.2)


u

x t

02

46

810

1214 0

1020

3040

50

−1

−0.5

0

0.5

1

(a)

xt

02

46

810

1214 0

1020

3040

50

−0.04

−0.02

0

0.02

0.04

(b)

Figure 9: Variations of the displacement u and temperature θ for various values of the axis x and time t(u = V0 + V1 + V2 and θ = Θ0 + Θ1 + Θ2) when Ω = σ1 = σ2 = 0.

x t

02

46

810

1214 0

1020

3040

50

u

−0.04

−0.02

0

0.02

0.04

(a)

x t

02

46

810

1214 0

0

10

1

2030

4050

−1

(b)

Figure 10: Variations of the displacement u and temperature Θ for various values of the axis x and time t(u = V0 + V1 and θ = Θ0 + Θ1) when Ω = σ1 = σ2 = 0.

where L1, L2 are linear differential operators with respect to time, N1, N2 are nonlinear

operators, and f(x, t), g(x, t) are given functions.

According to the variational iteration method, we can construct correct functionals as

follows:

un+1(x, t) = un(x, t) +∫ t

0

λ1(τ)[L1un(x, τ) +N1

(un(x, τ), θn(x, τ)

)− f(x, τ)

]dτ, (4.3)

θn+1(x, t) = θn(x, t) +∫ t

0

λ2(τ)[L2θn(x, τ) +N2

(un(x, τ), θn(x, τ)

)− g(x, τ)

]dτ, (4.4)


where λ1 and λ2 are general Lagrange multipliers, which can be identified optimally via

variational theory [8–20]. The second term on the right-hand side in (4.3) and (4.4) is called

the corrections, and the subscript n denotes the nth order approximation, un and θn are

restricted variations. We can assume that the above correctional functionals are stationary

(i.e., δun+1 = 0 and δθn+1 = 0), then the Lagrange multipliers can be identified. Now we

can start with the given initial approximation and by the previous iteration formulas we can

obtain the approximate solutions.

5. Application of the Variational Iteration Method on the NonlinearMagneto-Thermoelastic with Rotation Equations

According to the variational iteration method and after some manipulation of (4.3) and (4.4),the correct functionals are as follows:

un+1(x, t) = un(x, t) +∫ t

0

λ1(τ)[(1 + σ1)un,tt(x, τ) + Ωun,t(x, τ)

− un,xx

(1 − σ2 + 2γun,x(x, τ) + 3δu2

n,x(x, τ))

−β1θn,x(x, τ) − β2

(θn(x, τ)un,x(x, τ)

)x

]dτ,

θn+1(x, t) = θn(x, t) +∫ t

0

λ2(τ)[θn,t(x, τ) − aun,xt(x, τ) − bun,x(x, τ)un,xt(x, τ)

−θn,xx(x, τ) − αun,xx(x, τ)θn,x(x, τ) − αun,x(x, τ)θn,xx(x, τ)]dτ,

(5.1)

where un and θn are considered as a restricted variation, that is, δun+1 = 0 and δθn+1 = 0.

Consequently, the general Lagrange multipliers λ1 and λ2 take the following form:

λ1(τ) =τ − t

1 + σ1, λ2(τ) = −1. (5.2)

By the substitution of the identified Lagrange multipliers (5.2) into (5.1), we have the

following iteration relations:

un+1(x, t) = un(x, t) +∫ t

0

τ − t

1 + σ1

[(1 + σ1)un,tt(x, τ) + Ωun,t(x, τ)

− un,xx

(1 − σ2 + 2γun,x(x, τ) + 3δu2

n,x(x, τ))

−β1θn,x(x, τ) − β2(θn(x, τ)un,x(x, τ))x]dτ,


θn+1(x, t) = θn(x, t) −∫ t

0

[θn(x, τ) − aun,xt(x, τ) − bun,x(x, τ)un,xt(x, τ) − θn,xx(x, τ)

−αun,xx(x, τ)θn,x(x, τ) − αun,x(x, τ)θn,xx(x, τ)]dτ, n ≥ 0.

(5.3)

With help of Maple or Mathematica, we get the following results:

u0 = θ0 = A(1 − cosx),

u1 = − t2

2(1 + σ1)

[−β1 sinx − 2Aγ cosx sin(x) + 2β2Acos2x − cosx − 3δA2 cosx

−β2A + σ2 cosx − β2A cosx + 3δA2cos3x]+A(1 − cosx),

θ1 = A[cosx + 2αA cosx sinx]t +A(1 − cosx),

u2 =

{− Acos6(x)

6720(1 + σ1)4

(−19440A5δ2β1γ + 5760A5δβ3

2 − 247860A7δ3β2 − 19440A6δ2β22

−19440A5δ2β2 − 17280δβ2γ2A5 + 19440A5δ2β2σ2

)+

[− Acos2(x)

6720(1 + σ1)4

(−6480A5δβ2

2γ + 2025A4δβ1β22 + 270A3δβ1σ2β2 + 3240A4δ2β1σ2

− 38880A6δ2β2γ − 8505A6δ3β1 − 135A2δβ1σ22 − 6480A4γβ2δ

− 135A2δβ1 + 6480A4δβ2γσ2 − 3240A5δ2β1β2 + 270A2δβ1σ2

− 270A3δβ1β2 − 3240A4δ2β1 − 2160A4δβ1γ2 + 45A2δβ3

1

)− Acos4(x)

6720(1 + σ1)4

(7200A4γβ2δ + 118800A6δ2β2γ − 4050A4δ2β1σ2 + 4050A4δ2β1

+ 4050A5δ2β1β2 + 3600A4δβ1γ2 − 3600A4δβ1β

22

+ 7200A5δβ22γ + 30375A6δ3β1 − 7200A4δβ2γσ2

)− Acos3(x)

6720(1 + σ1)4

(17280A5δ2β1β2 − 1440A4γβ2δ − 93960A7δ3γ + 16560A5δβ2

2γ

+ 1440A4δβ2γσ2 − 23760A6δ2β2γ + 1440A4δβ1β22 − 720A3γδ

+ 1440A3δβ1β2 + 23760A5δ2γσ2 − 720A3δγσ22 − 5760A5δγ3

− 23760A5γδ2 − 1440A3δβ1σ2β2 + 720A3δβ21γ + 1440A3γδσ2

)− Acos5(x)

6720(1 + σ1)4

(−19440A5δ2β1β2 + 189540A7δ3γ + 19440A6δ2β2γ − 17280A5δβ2

2γ

+ 19440A5γδ2 + 5760A5δγ3 − 19440A5δ2γσ2

)


− A cos(x)

6720(1 + σ1)4

(− 1080A3γδσ2 + 540A3γδ + 540A3δγσ2

2 − 180A3δβ21γ

− 1080A4δβ2γσ2 − 720A4δβ1β22 − 2160A5δ2β1β2 + 1080A4γβ2δ

+ 720A3δβ1σ2β2 + 6480A6δ2β2γ − 2340A5δβ22γ − 6480A5δ2γσ2

− 720A3δβ1β2 + 14580A7δ3γ + 1440A5δγ3 + 6480A5γδ2)

e899

+243A8δ3γ

141(1 + σ1)4cos7(x) − Acos6(x)

6720(1 + σ1)4

(−90720A6δ2β2γ − 25515A6δ3β1

)

− A

6720(1 + σ1)4

(720A5δβ2

2γ + 45A2δβ1σ22 + 405A6δ3β1 + 720A4γβ2δ

− 720A4δβ2γσ2 + 90A3δβ1β2 + 45A4δβ1β22 + 2160A6δ2β2γ

+ 270A5δ2β1β2 + 45A2δβ1 − 90A3δβ1σ2β2 + 270A4δ2β1

− 270A4δ2β1σ2 − 90A2δβ1σ2 + 180A4δβ1γ2)]

sin(x)

− Acos5(x)

6720(1 + σ1)4

(204120A8δ4 − 7200A4δβ1γβ2 + 52245A6δ3 + 2025δ2A4

− 52245A6δ3σ2 + 3600A4γ2δ − 80055A6δ2β22 − 3600A5δβ3

2

+ 3600δβ2γ2A5 + 52245A7δ3β2 + 82080δ2γ2A6 + 2025A4δ2σ2

2

+ 3600A4δβ22σ2 − 4050A5δ2β2σ2 − 4050A4δ2σ2 − 2025A4δ2β2

1

+ 4050A5δ2β2 − 3600A4δβ22 − 3600A4δγ2σ2

)− 243A8δ3β2cos8(x)

141(1 + σ1)4

− Acos4(x)

6720(1 + σ1)4

(25920δβ2γ

2A5 + 1440A4δβ22 + 166860A7δ3β2 + 33480A5δ2β2

− 1440A4δβ22σ2 + 1440A3γβ1δ − 720A3δβ2

1β2 + 33480A6δ2β22

− 1440A3δβ1γσ2 − 7920A5δβ32 + 1440A4δβ1γβ2 + 27000A5δ2β1γ

− 33480A5δ2β2σ2 − 1440A3δβ2σ2 + 720A3δβ2σ22 + 720δβ2A

3)

− Acos3(x)

6720(1 + σ1)4

(2025A4δ2β2

1 + 6480A4δ2σ2 + 5040A4δγ2σ2 + 135A3δβ21β2

− 3240δ2A4 + 7920A4δβ1γβ2 − 70470A8δ4 − 32805A6δ3

− 4905A4δβ22σ2 + 135A2δβ2

1 + 270A3δβ2σ2 + 6480A5δ2β2σ2

− 135A3δβ2σ22 − 135A2δσ2

2 + 135A2σ2δ + 45A2δσ32 − 45A2δ

− 32805A7δ3β2 − 45900δ2γ2A6 + 32805A6δ3σ2 + 37800A6δ2β22


− 135δβ2A3+4995A5δβ3

2−135A2δβ21σ2−6480A5δ2β2−5040δβ2γ

2A5

+ 4905A4δβ22 − 5040A4γ2δ − 3240A4δ2σ2

2

)− 6561A9δ4cos9(x)

4481(1 + σ1)4

− Acos2(x)

6720(1 + σ1)4

(540A3δβ2

1β2 + 1980A5δβ32 − 9180A5δ2β1γ − 900δβ2A

3

+ 1800A4δβ22σ2 − 15120A5δ2β2 − 10080δβ2γ

2A5 − 15120A6δ2β22

+ 1440A3δβ1γσ2 − 1440A3γβ1δ − 37260A7δ3β2 − 1440A4δβ1γβ2

+ 15120A5δ2β2σ2 + 1800A3δβ2σ2 − 900A3δβ2σ22 − 1800A4δβ2

2

)− A

6720(1 + σ1)4

(180A4δβ1γβ2 + 360A4δβ2

2 + 180δβ2A3 − 360A4δβ2

2σ2

+ 1080A6δ2β22 + 1080A5δ2β2 + 1620A7δ3β2 − 180A3δβ1γσ2

+ 180A3γβ1δ + 180A5δβ32 + 540A5δ2β1γ − 1080A5δ2β2σ2

− 360A3δβ2σ2 + 180A3δβ2σ22 + 720δβ2γ

2A5)

− Acos7(x)

6720(1 + σ1)4

(45360A6δ2β2

2 + 25515A6δ3σ2 − 25515A6δ3 − 45360δ2γ2A6

− 240570A8δ4 − 25515A7δ3β2

)− A cos(x)

6720(1 + σ1)4

(− 270A4δ2β2

1−2430A4δ2σ2−1620A4δγ2σ2−90A3δβ21β2+1215δ2A4

− 1440A4δβ1γβ2 + 8505A8δ4 + 6075A6δ3 + 1305A4δβ22σ2

− 90A2δβ21 − 270A3δβ2σ2 − 2430A5δ2β2σ2 + 135A3δβ2σ

22

+ 135A2δσ22 − 135A2σ2δ − 45A2δσ3

2 + 45A2δ + 6075A7δ3β2

+ 8100δ2γ2A6 − 6075A6δ3σ2 − 3105A6δ2β22 + 135δβ2A

3

− 1395A5δβ32 + 90A2δβ2

1σ2 + 2430A5δ2β2 + 1620δβ2γ2A5

− 1305A4δβ22 + 1620A4γ2δ + 1215A4δ2σ2

2

)}t8

+

{− Acos6(x)

6720(1 + σ1)4

(72576A5δ2β2σ1 + 72576A5δ2β2

)

+

[− Acos5(x)

6720(1 + σ1)4

(−99792A5γδ2σ1 − 99792A5γδ2

)

− Acos2(x)

6720(1 + σ1)4

(1008A3δβ1β2 + 1344A2γ2β1 − 1344A2γβ2σ2σ1 − 1008A2δβ1σ2


+ 1344A3γβ22 + 1344A2γβ2σ1 + 1344A3γβ2

2σ1 + 1344A2γβ2

− 1008A2δβ1σ1σ2 + 1344A2γ2β1σ1 + 12096A4δ2β1

+ 1008A3δβ1σ1β2 + 12096A4δ2β1σ1 − 1344A2γβ2σ2

+ 40320A4γβ2δσ1 + 40320A4γβ2δ + 1008A2δβ1

+ 1008A2δβ1σ1

)− A

6720(1 + σ1)4

(− 448A2γβ2 + 448A2γβ2σ2σ1 − 448A2γβ2σ1 − 448A3γβ2

2σ1

+ 448A2γβ2σ2 + 336A2δβ1σ2 − 1008A4δ2β1 − 336A3δβ1β2

− 336A2δβ1 − 336A3δβ1σ1β2 + 336A2δβ1σ1σ2 − 336A2δβ1σ1

− 4032A4γβ2δ − 224A2γ2β1 − 448A3γβ22 − 4032A4γβ2δσ1

− 224A2γ2β1σ1 − 1008A4δ2β1σ1

)− A cos(x)

6720(1 + σ1)4

(2688A3δβ1β2 + 1680A3γβ2

2σ1 − 1792A3γ3 + 2688A3δβ1σ1β2

+ 6720A3γδσ2σ1 − 6720A3γδσ1 + 6720A3γδσ2 − 31248A5γδ2

− 224A2γβ2σ1 + 224A2γβ2σ2σ1 + 1680A3γβ22 + 224A2γβ2σ2

− 1792A3γ3σ1 − 6720A3γδ − 224A2γβ2 − 112Aγ

+ 112Aγβ21 + 112Aγβ2

1σ1 − 112Aγσ22σ1 + 224Aγσ2σ1

− 112Aγσ22 + 224Aγσ2 − 112Aγσ1 − 6720A4γβ2δσ1

− 31248A5γδ2σ1 − 6720A4γβ2δ)

− Acos3(x)

6720/(1 + σ1)4

(− 3584A3γβ2

2σ1 + 9408A4γβ2δσ1 + 118944A5γδ2 + 9408A4γβ2δ

+ 3584A3γ3σ1 + 9408A3γδ + 9408A3γδσ1 + 118944A5γδ2σ1

− 3584A3γβ22 − 5376A3δβ1β2 + 3584A3γ3 − 9408A3γδσ2

− 5376A3δβ1σ1β2 − 9408A3γδσ2σ1

)− Acos4(x)

6720(1 + σ1)4

(−15120A4δ2β1 − 47040A4γβ2δ − 15120A4δ2β1σ1

− 47040A4γβ2δσ1

)]sin(x)

− Acos7(x)

6720(1 + σ1)4

(95256A6δ3σ1 + 95256A6δ3

)


− Acos3(x)

6720(1 + σ1)4

(− 24192A4δ2σ2 − 24192A4δ2σ1σ2 + 24192δ2A4 − 1344A2γβ1β2σ1

+ 122472A6δ3 + 1008A3β2δσ1 − 1344A2γβ1β2 − 18312A4δβ22σ1

+ 1344A3γ2β2σ1 − 504A2δβ21 + 1344A2γ2σ1 − 1008A3δβ2σ2

+ 24192A5δ2β2σ1 + 504A2δσ1 + 24192A4δ2σ1 + 54024A4γ2δσ1

− 1344A2γ2σ2 + 504A2δσ22 − 1008A2σ2δ + 1008A2σ2δσ1

+ 504A2δ − 1008A3δβ2σ1σ2 − 1344A2γ2σ2σ1 − 122472A6δ3σ1

+ 1008δβ2A3 + 1344A2γ2 + 504A2δσ1σ

22 − 504A2δβ2

1σ1

+ 24192A5δ2β2 − 18312A4δβ22 + 45024A4γ2δ + 1344A3γ2β2

)− Acos5(x)

6720(1 + σ1)4

(− 195048A6δ3 + 15120A4δ2σ1σ2 + 15120A4δ2σ2 − 195048A6δ3σ1

− 15120A5δ2β2 + 13440A4δβ22 − 33600A4γ2δσ1

− 33600A4γ2δ − 15120δ2A4 − 15120A4δ2σ1 − 15120A5δ2β2σ1

+ 13440A4δβ22σ1

)− A

6720(1 + σ1)4

(− 112Aγβ1σ1 − 1344δβ2A

3 − 112A2γβ1β2 − 1008A3γβ1δσ1

+ 112Aγβ1σ2σ1 + 1344A3δβ2σ1σ2 − 896A3γ2β2 − 112A2γβ1β2σ1

− 1344A3β2δσ1 − 1344A4δβ22σ1 − 1008A3γβ1δ − 112Aγβ1

+ 112Aγβ1σ2 − 896A3γ2β2σ1 − 4032A5δ2β2 − 1344A4δβ22

+ 1344A3δβ2σ2 − 4032A5δ2β2σ1

)− Acos4(x)

6720(1 + σ1)4

(− 7168A3γ2β2 − 5376A4δβ2

2σ1 − 9408A3γβ1δ − 9408A3γβ1δσ1

− 5376A3β2δσ1 − 7168A3γ2β2σ1 − 5376δβ2A3 − 124992A5δ2β2

− 5376A4δβ22 + 5376A3δβ2σ2 + 5376A3δβ2σ1σ2−124992A5δ2β2σ1

)− A cos(x)

6720(1 + σ1)4

(9072A4δ2σ2 + 9072A4δ2σ1σ2 − 9072δ2A4 + 896A2γβ1β2σ1

− 22680A6δ3 − 1008A3β2δσ1 + 896A2γβ1β2 + 4872A4δβ22σ1

− 1120A3γ2β2σ1 + 336A2δβ21 − 1120A2γ2σ1 + 1008A3δβ2σ2

− 9072A5δ2β2σ1 − 504A2δσ1 − 9072A4δ2σ1 − 13440A4γ2δσ1


+ 1120A2γ2σ2 − 504A2δσ22 + 1008A2σ2δ + 1008A2σ2δσ1

− 504A2δ + 1008A3δβ2σ1σ2 + 1120A2γ2σ2σ1 − 22680A6δ3σ1

− 1008δβ2A3 − 1120A2γ2 − 504A2δσ1σ

22 + 336A2δβ2

1σ1

− 9072A5δ2β2 + 4872A4δβ22 − 13440A4γ2δ − 1120A3γ2β2

)− Acos2(x)

6720(1 + σ1)4

(− 224Aγβ1σ2σ1 + 224Aγβ1σ1 + 224A2γβ1β2 + 56448A5δ2β2

+ 6720A4δβ22 + 9072A3γβ1δσ1 + 224Aγβ1 + 6720δβ2A

3

+ 9072A3γβ1δ + 224A2γβ1β2σ1 + 7168A3γ2β2 + 56448A5δ2β2σ1

+ 6720A3β2δσ1 + 7168A3γ2β2σ1 − 6720A3δβ2σ1σ2

− 224Aγβ1σ2 + 6720A4δβ22σ1 − 6720A3δβ2σ2

)}t6

+

{− Acos3(x)

6720(1 + σ1)4

(− 1008A2β2β1ασ

21 − 4032A2β2

2σ1 − 1008A2β2β1α − 2016A2β2β1ασ1

− 2016A2β22 − 2016A2β2

2σ21

)− A cos(x)

6720(1 + σ1)4

(1344A2β2

2σ21 + 672A2αβ2β1 + 1344A2β2

2 + 2688A2β22σ1

+ 1344A2β2β1ασ1 + 672A2β2β1ασ21

)− A

6720(1 + σ1)4

(− 504δβ2A

3 − 336A2β22σ1 − 168Aβ2σ

21 + 336Aβ2σ2σ1

+ 168Aβ2σ2 − 1008A3β2δσ1 − 168β2A − 504A3β2δσ21

− 168A2β22 − 672A3β2γασ

21 − 168A2β2

2σ21 − 336Aβ2σ1

+ 168Aβ2σ2σ21 − 672A3β2γα − 1344A3β2γασ1

)− Acos2(x)

6720(1 + σ1)4

(5544A3β2δσ

21 + 5376A3β2γασ

21 − 672Aβ2σ2σ1 − 336Aβ2σ2

+ 336β2A + 672A2β22σ1 + 336A2β2

2 + 5376A3β2γα

+ 672Aβ2σ1 − 336Aβ2σ2σ21 + 336Aβ2σ

21 + 10752A3β2γασ1

+ 336A2β22σ

21 + 5544δβ2A

3 + 11088A3β2δσ1

)+

[− A cos(x)

6720(1 + σ1)4

(672β2Aβ1σ1 + 336β2Aβ1σ

21 + 2688A3β2

2ασ21

+ 2688A3β22α + 336β2Aβ1 + 5376A3β2

2ασ1

)


− Acos2(x)

6720(1 + σ1)4

(2016A2γβ2 + 24192A4β2δασ1 + 2016A2γβ2σ

21 + 2016A2β2σ1α

+ 1008A3β22α + 12096A4β2δασ

21 + 1008A3β2

2ασ21 + 4032A2γβ2σ1

− 1008A2β2σ2ασ21 + 2016A3β2

2ασ1 − 2016A2β2σ2ασ1

+ 12096A4β2δα + 1008A2β2σ21α + 1008A2β2α − 1008A2β2σ2α

)− A

6720(1 + σ1)4

(− 336A2γβ2 − 336A2β2α − 2016A4β2δασ1 − 1008A4β2δασ

21

− 1008A4β2δα − 336A2β2σ21α − 672A2γβ2σ1 + 336A2β2σ2α

− 336A3β22ασ

21 + 672A2β2σ2ασ1 + 336A2β2σ2ασ

21 − 336A3β2

2α

− 672A3β22ασ1 − 336A2γβ2σ

21 − 672A2β2σ1α

)− Acos4(x)

6720(1 + σ1)4

(−30240A4β2δασ1 − 15120A4β2δα − 15120A4β2δασ

21

)

− Acos3(x)

6720(1 + σ1)4

(−5376A3β2

2ασ21 − 5376A3β2

2α − 10752A3β22ασ1

)]sin(x)

− Acos4(x)

6720(1 + σ1)4

(− 5376A3β2γα − 12096A3β2δσ1 − 6048δβ2A

3 − 6048A3β2δσ21

− 10752A3β2γασ1 − 5376A3β2γασ21

)}t5

+

{− cos2(x)

6720(1 + σ1)4

(5600Aβ2σ2σ1 − 26040A3β2δσ

21 − 5600Aβ2σ1 − 2800A2β2

2

− 2800A2β22σ

21 − 2800β2A − 52080A3β2δσ1 + 2800Aβ2σ2σ

21

− 26040δβ2A3 − 2240Aγβ1σ1 − 1120Aγβ1 − 5600A2β2

2σ1

− 2800Aβ2σ21 − 1120Aγβ1σ

21 + 2800Aβ2σ2

)− A cos(x)

6720(1 + σ1)4

(280 + 560σ1 − 560σ2 + 22680δ2A4 − 560σ2σ

21 + 560β2A

+ 1120Aβ2σ1 − 560Aβ2σ2 + 560Aβ2σ21 − 1960A2β2

2σ21 +8400A3β2δσ

21

+ 16800A3β2δσ1 + 5600A2γ2σ21 + 560σ2

2σ1 + 280σ22σ

21 −560Aβ2σ2σ

21

+ 280σ22 +11200A2γ2σ1+16800A2δσ1+45360A4δ2σ1+22680A4δ2σ2

1

− 8400A2σ2δ − 8400A2σ2δσ21 − 16800A2σ2δσ1+8400A2δσ2

1 +280σ21

+ 8400A2δ − 1120σ2σ1 + 8400δβ2A3 + 5600A2γ2

− 1960A2β22 − 1120Aβ2σ2σ1 − 3920A2β2

2σ1

)


− Acos4(x)

6720(1 + σ1)4

(23520A3β2δσ

21 + 47040A3β2δσ1 + 23520δβ2A

3)

− Acos3(x)

6720(1 + σ1)4

(10080A2σ2δ − 6720A2γ2 − 13440A2γ2σ1 + 10080A2σ2δσ

21

+ 3360A2β22 − 10080A2δσ2

1 − 60480δ2A4 − 10080A2δ

− 6720A2γ2σ21 − 120960A4δ2σ1 − 20160A3β2δσ1 − 10080δβ2A

3

− 20160A2δσ1 + 20160A2σ2δσ1 + 6720A2β22σ1 + 3360A2β2

2σ21

− 60480A4δ2σ21 − 10080A3β2δσ

21

)+

[− A cos(x)

6720(1 + σ1)4

(− 560β2Aβ1 + 23520A3γδσ2

1 + 23520A3γδ − 1120β2Aβ1σ1

+ 3360Aγσ21 − 3360Aγσ2σ

21 + 6720Aγσ1 − 560β2Aβ1σ

21

+ 47040A3γδσ1 + 3360A2γβ2σ21 − 3360Aγσ2 + 3360A2γβ2

+ 6720A2γβ2σ1 − 6720Aγσ2σ1 + 3360Aγ)

− A

6720(1 + σ1)4

(560β1σ1 − 280σ2β1 − 280σ2β1σ

21 − 560σ2β1σ1 + 280β1

+ 1680A2δβ1σ1 + 280β2Aβ1σ21 + 560β2Aβ1σ1 + 2800A2γβ2

+ 280β1σ21 + 2800A2γβ2σ

21 + 5600A2γβ2σ1 + 840A2δβ1σ

21

+ 280β2Aβ1 + 840A2δβ1

)− Acos3(x)

6720(1 + σ1)4

(−33600A3γδ − 67200A3γδσ1 − 33600A3γδσ2

1

)

− Acos2(x)

6720(1 + σ1)4

(− 10080A2γβ2 − 20160A2γβ2σ1 − 5040A2δβ1σ1 − 2520A2δβ1σ

21

− 10080A2γβ2σ21 − 2520A2δβ1

)]sin(x)

− Acos5(x)

6720(1 + σ1)4

(37800δ2A4 + 37800A4δ2σ2

1 + 75600A4δ2σ1

)

− A

6720(1 + σ1)4

(4200A3β2δσ

21 + 2800A2β2

2σ1 − 1400Aβ2σ2σ21 + 1400Aβ2σ

21

− 2800Aβ2σ2σ1 + 1400β2A + 1400A2β22σ

21 + 560Aγβ1σ

21

+ 1400A2β22 + 8400A3β2δσ1 + 4200δβ2A

3 − 1400Aβ2σ2

+ 2800Aβ2σ1 + 1120Aγβ1σ1 + 560Aγβ1

)}t4


+

{[− A cos(x)

6720(1 + σ1)4

(2240ΩAγ + 4480ΩAγσ1 + 2240ΩAγσ2

1

)

− Acos2(x)

6720(1 + σ1)4

(−6720A2β2σ

31α − 20160A2β2σ

21α − 6720A2β2α − 20160A2β2σ1α

)

− A

6720(1 + σ1)4

(6720A2β2σ1α + 1120Ωβ1σ

21 + 1120β1σ

31 + 1120β1

+ 3360β1σ21 + 2240Ωβ1σ1 + 3360β1σ1 + 1120Ωβ1

+ 6720A2β2σ21α + 2240A2β2σ

31α + 2240A2β2α

)]sin(x)

− Acos2(x)

6720(1 + σ1)4

(− 4480ΩAβ2σ1 − 2240ΩAβ2σ

21 − 2240β2A − 6720Aβ2σ

21

− 13440Aβ1ασ21 − 2240Aβ2σ

31 − 4480Aβ1α − 2240ΩAβ2

− 13440Aβ1ασ1 − 6720Aβ2σ1 − 4480Aβ1ασ31

)− A

6720(1 + σ1)4

(1120β2A + 1120ΩAβ2σ

21 + 1120Aβ2σ

31 + 2240Aβ1α

+ 2240Aβ1ασ31 + 1120ΩAβ2 + 3360Aβ2σ

21 + 2240ΩAβ2σ1

+ 6720Aβ1ασ1 + 3360Aβ2σ1 + 6720Aβ1ασ21

)− A cos(x)

6720(1 + σ1)4

(3360ΩA2δ + 1120ΩAβ2 − 2240Ωσ2σ1 − 1120Ωσ2

− 1120Ωσ2σ21 + 1120Ω + 2240Ωσ1 + 2240ΩAβ2σ1

+ 1120Ωσ21 + 6720ΩA2δσ1 + 3360ΩA2δσ2

1 + 1120ΩAβ2σ21

)− Acos3(x)

6720(1 + σ1)4

(−6720ΩA2δσ1 − 3360ΩA2δσ2

1 − 3360ΩA2δ)}

t3

+

{− Acos3(x)

6720(1 + σ1)4

(30240A2δσ2

1 + 10080A2δσ31 + 10080A2δ + 30240A2δσ1

)

− A

6720(1 + σ1)4

(−10080Aβ2σ

21 − 3360β2A − 3360Aβ2σ

31 − 10080Aβ2σ1

)

+

[− A

6720(1 + σ1)4

(−10080β1σ

21 − 3360β1 − 3360β1σ

31 − 10080β1σ1

)

− A cos(x)

6720(1 + σ1)4

(−20160Aγσ1 − 6720Aγσ3

1 − 6720Aγ − 20160Aγσ21

)]sin(x)

− Acos2(x)

6720(1 + σ1)4

(20160Aβ2σ1 + 20160Aβ2σ

21 + 6720Aβ2σ

31 + 6720β2A

)


− A/ cos(x)

6720(1 + σ1)4

(− 30240A2δσ2

1 − 10080Aβ2σ1 + 3360σ2σ31 − 10080A2δ

+ 10080σ2σ1 − 3360Aβ2σ31 − 3360 − 30240A2δσ1 − 3360β2A

+ 10080σ2σ21 + 3360σ2 − 10080σ2

1 − 10080A2δσ31

− 10080Aβ2σ21 − 3360σ3

1 − 10080σ1

)}t2

− A cos(x)

6720(1 + σ1)4

(6720 + 6720σ4

1 + 40320σ21 + 26880σ3

1 + 26880σ1

)− A

6720(1 + σ1)4

(−6720 − 26880σ1 − 26880σ3

1 − 6720σ41 − 40320σ2

1

),

(5.4)

θ2 =

{− A cos (x)5

24(1 + σ1)2

(−540A4δα2 − 540A4δα2σ1 + 216A4bδβ2

)

− A cos (x)3

24(1 + σ1)2

(− 24A3bβ2

2 + 36A3β2α2 + 24A2bσ2β2 + 36A3β2α

2σ1 + 36A2α2σ1

+ 36A2αγσ1 + 702A4δα2σ1 − 36A2σ2α2σ1 − 36A2σ2α

2 + 36A2α2

− 24A2bβ1γ − 288A4bδβ2 − 24A2bβ2 + 36A2αγ + 702A4δα2)

+

[− Acos2(x)

24(1 + σ1)2

(− 24A2 bγσ2 + 36A2β1α

2σ1 − 24A2bβ1β2 + 36A2β1α2 + 36A2β2α

+ 24A3bγβ2 + 24A2bγ + 180A4bγδ + 36A2β2σ1α)+

9A5bγδcos4(x)

(1 + σ1)2

− A cos(x)

24(1 + σ1)2

(− 6αAσ1 + 6bAβ1 + 18A3bβ1δ − 72αA3δ − 96A3γα2σ1 + 48A3bγβ2

− 6A2β2σ1α − 6A2β2α − 96A3γα2 + 6αAσ2 − 6αA − 6Abβ1σ2

− 72αA3δσ1 + 6αAσ2σ1 + 6A2bβ1β2

)− Acos3(x)

24(1 + σ1)2

(192A3γα2σ1 + 192A3γα2 + 108αA3δ − 96A3bγβ2

+ 108αA3δσ1 − 54A3bβ1δ)

− A

24(1 + σ1)2

(− 12A2bγ − 12A2β2α − 12A3bγβ2 − 36A4bγδ − 6A2β1α

2 − 12A2β2σ1α

+12A2bγσ2 − 6A2β1α2σ1

)]sin(x)


− Acos2(x)

24(1 + σ1)2

(3Ab + 72A4bδβ2 + 48A3bγ2 + 3Abσ2

2 + 189A5bδ2 + 6Aβ1ασ1

− 6A2bσ2β2 + 6Aβ1α − 3Abβ21 + 192A3β2α

2 + 192A3β2α2σ1

+ 72A3bδ + 6A2bβ2 − 45A3bβ22 − 72A3bδσ2 − 6Abσ2

)− A cos(x)

24(1 + σ1)2

(72A4bδβ2 − 198A4δα2σ1 − 30A3β2α

2 − 30A2α2σ1 + 24A3bβ22

− 30A3β2α2σ1 + 30A2σ2α

2 − 24A2bσ2β2 + 12A2bβ1γ + 30A2σ2α2σ1

− 198A4δα2 − 30A2αγσ1 + 24A2bβ2 − 30A2αγ − 30A2α2)

− Acos4(x)

24(1 + σ1)2

(− 405A5bδ2 − 192A3β2α

2 + 54A3bδσ2 − 54A4bδβ2 − 48A3bγ2

+ 48A3bβ22 − 54A3bδ − 192A3β2α

2σ1

)− 81A6bδ2cos6(x)

8(1 + σ1)2

− A

24(1 + σ1)2

(− 12A3bγ2 − 18A4bδβ2 − 3Ab − 3Abσ2

2 − 3A3bβ22

+ 6A2bσ2β2 − 3Aβ1α − 27A5bδ2 − 3Aβ1ασ1 − 24A3β2α2σ1 − 18A3bδ

− 6A2bβ2 + 18A3bδσ2 − 24A3β2α2 + 6Abσ2

)}t4

+

{−A

(−48A2αγ − 48A2αγσ1

)cos3(x)

24(1 + σ1)2− A

(40A2αγ + 40A2αγσ1

)cos(x)

24(1 + σ1)2

− A(4Aβ1α + 4Aβ1ασ1

)24(1 + σ1)2

− A(−8Aβ1α − 8Aβ1ασ1

)cos2(x)

24(1 + σ1)2

+

[−A

(−48A2β2σ1α − 48A2β2α)

24(1 + σ1)2cos2(x) − A

(−144αA3δσ1 − 144αA3δ)cos3(x)

24/(1 + σ1)2

− A cos(x)

24(1 + σ1)2

(8αA + 96αA3δσ1 − 8αAσ2σ1 − 8αAσ2 + 8A2β2α + 96αA3δ

+ 8A2β2σ1α + 8αAσ1

)− A

24(1 + σ1)2

(16A2β2α + 16A2β2σ1α

)]sin(x)

}t3

+

{− A cos(x)

24(1 + σ1)2

(12 − 48A2bβ2 + 120A2α2σ2

1 − 48A2σ1bβ2 + 24σ1 + 12σ21

− 12σ1aβ1 + 240A2α2σ1 − 12aβ1 + 120A2α2)

− Acos3(x)

24(1 + σ1)2

(48A2bβ2 − 144A2α2σ2

1 − 288A2α2σ1 − 144A2α2 + 48A2σ1bβ2

)


− A

24(1 + σ1)2

(24Aaγ + 36A3bδ + 12A2bβ2 − 12Abσ2 + 12Ab + 36A3σ1bδ

+ 24Aσ1aγ + 12Aσ1b − 12Aσ1bσ2 + 12A2σ1bβ2

)+

[− Acos2(x)

24(1 + σ1)2

(−48A2bγ − 108aδA2 − 108σ1aδA

2 − 48A2σ1bγ)

− A

24(1 + σ1)2

(12σ1aβ2A + 36σ1aδA

2 − 12aσ2 + 12aβ2A − 12σ1aσ2

+ 12σ1a + 24A2bγ + 36aδA2 + 12a + 24A2σ1bγ)

− A cos(x)

24(1 + σ1)2

(240αAσ1 + 120αA − 48aβ2A − 12bAβ1 − 48σ1aβ2A

+ 120αAσ21 − 12σ1bAβ1

)]sin(x)

− Acos2(x)

24(1 + σ1)2

(− 12Ab + 12Abσ2 − 144A3σ1bδ − 144A3bδ − 12A2bβ2 − 48Aaγ

− 12A2σ1bβ2 − 48Aσ1aγ − 12Aσ1b + 12Aσ1bσ2

)− A

(108A3bδ + 108A3σ1bδ

)cos4(x)

24(1 + σ1)2

}t2

×{A(−48αA − 96αAσ1 − 48αAσ2

1

)cos(x) sin(x)

24(1 + σ1)2+ −A

(−24 − 48σ1 − 24σ21

)cos(x)

24(1 + σ1)2

}t

− A(24 + 48σ1 + 24σ2

1

)cos(x)

24(1 + σ1)2− A

(−24 − 48σ1 − 24σ21

)24(1 + σ1)2

.

(5.5)

Now we make calculations for the results obtained by the variational iteration method using

the Maple software package with the following arbitrary constants:

a = 0.5, A = 0.001, b = 0.5, α = 1, β1 = β2 = 0.05, γ = 1, δ = 0.8. (5.6)

5.1. Special Case

(1) If we take into our consideration the first iteration (i.e., u = u1 and θ = θ1). See

Figures 15, 16, 17, and 18.

(2) If the magnetic field and rotation are neglected, the components of the displacement

u2 and temperature θ2 take the following forms. See Figures 19 and 20.


xt

02

46

810

1214 0

1020

3040

50

−150−100−50

050

100150

u

(a)

xt

02

46

810

1214 0

1020

3040

50

−1

−0.5

0

0.5

1

1.5

(b)

Figure 11: Variations of the displacement u2 and temperature θ2 for various values of the axis x and time twhen Ω = 0.1, σ1 = 0.2, σ2 = 0.1.

x

02

46

810

1214 0

12

34

5

Ω

u

0

0.0005

0.001

0.0015

0.002

(a)

x

02

46

810

1214

01

23

45

Ω

0.0002

0.0006

0.0014

0.0018

0.001

(b)

Figure 12: Variations of the displacement u2 and temperature θ2 for various values of the axis x and rotationΩ when t = 0.1, σ1 = 0.2, σ2 = 0.1.

6. Discussion

With the view of illustrating the theoretical results obtained in the preceding sections, a

numerical result is calculated for the homotopy perturbation method and variational iteration

method.

Figures (1–10) illustrate the influences of time t, rotation Ω, and sensitive pats of the

magnetic field σ1 and σ2 for the iterations (u = V0 + V1 + V2 and θ = Θ0 + Θ1 + Θ2) and

(u = V0 + V1 and θ = Θ0 + Θ1), and if the rotation and magnetic field neglected, respectively,

respect to the coordinate x for the homotopy perturbation method. Figures (11–20) illustrate

the influences of time t, rotation Ω, and sensitive pats of the magnetic field σ1 and σ2 for the

iterations (u = V2, θ = θ2 and u = V1 and θ = θ1), and if the rotation and magnetic field have

been neglected, respectively, respect to the coordinate x for the variational iteration method.

From Figures 1 and 11, it is concluded that the displacement u and temperature θ

start from their maximum values, decrease and increase periodically with an increasing of


1

02

46

810

u

x

02

46

810

1214

0

0.0005

0.001

0.0015

0.002

(a)

0.0002

0.0006

0.0014

0.0018

0.001

x

02

46

810

1214 1

02

46

810

(b)

Figure 13: Variations of the displacement u2 and temperature θ2 for various values of the axis x andmagnetic field σ1 when t = 0.1, Ω = 0.1, σ2 = 0.1.

x

02

46

810

1214 0

24

68

10

2

u

0

0.001

0.002

(a)

x

02

46

810

1214 0

24

68

10

2

0.0002

0.0006

0.0014

0.0018

0.001

(b)


x t

02

46

810

1214 0

1020

3040

50

u

−0.8−0.4

00.4

0.8

(a)

x t

0 2 4 6 8 10 12 14 010

2030

4050

−0.04

−0.02

0

0.02

0.04

(b)

Figure 15: Variations of the displacement u1 and temperature θ1 for various values of the axis x and time twhen Ω = 0.1, σ1 = 0.2, σ2 = 0.1.


x

02

46

810

1214 0

12

34

5

Ω

u

0

0.0005

0.001

0.0015

0.002

(a)

x

02

46

810

1214

0.0002

0.0006

0.0014

0.0018

0.001

01

23

45

Ω

(b)

Figure 16: Variations of the displacement u1 and temperature θ1 for various values of the axis x and rotationΩ when t = 0.1, σ1 = 0.2, σ2 = 0.1.

10

24

68

10

u

x

02

46

810

1214

0

0.0005

0.001

0.0015

0.002

(a)

0.0002

0.0006

0.0014

0.0018

0.001

x0

24

68

1012

14

1

02

46

810

(b)


the coordinate x, also, it is obvious that their values take the minimum values and increases

with the increasing values of the time t. From Figures 2, 3, 4, 12, 13, 14, it is seen that the

components of the displacement u and temperature θ begin from the minimum values near

zero increase and then decrease periodically with the coordinate x, it is clear also that there

are a sligh increasing with an increasing of the sensitive parts of the magnetic field, also, one

can see that u and θ decrease with an increasing of the rotation Ω.

Figures 5–8 and 15–18 display the first iteration with respect to the homotopy

perturbation method and variational iteration method on the influences of the parameters

time t, rotation Ω, and sensitive pats of the magnetic field σ1 and σ2 to obtain the displacement

and the temperature components on the medium due to the harmonic wave propagation. It

is shown that the increasing of the coordinate x sensitive an increasing and dereasing on

them periodically due to appearance of the pairs (cos, sin) in the initial condition and the

approximate solutions; it is also clear that the components begin from their minimum values

and increase absolutely with the variation of the time t. With the variations of the rotation

and magnetic field tends to slightly affect on the displacment and the temperature.


x

02

46

810

1214 0

24

68

10

2

u

0

0.0005

0.001

0.0015

0.002

(a)

x

02

46

810

1214 0

24

68

10

2

0.0002

0.0006

0.0014

0.0018

0.001

(b)


−400

−200

0

200

400

u

x t

02

46

810

1214 0

1020

3040

50

(a)

−1−0.5

00.5

1.52

1

xt

02

46

810

1214 0

1020

3040

50

(b)

Figure 19: Variations of the displacement u2 and temperature θ2 for various values of the axis x and time t(u = u2 and θ = θ2) when Ω = σ1 = σ2 = 0.

u

x t

02

46

810

1214 0

1020

3040

50

−1

−0.5

0

0.5

1

(a)

xt

02

46

810

1214 0

1020

3040

50

−0.04

−0.02

0

0.02

0.04

(b)

Figure 20: Variations of the displacement u1 and temperature θ1 for various values of the axis x and time t(u = u1 and θ = θ1) when Ω = σ1 = σ2 = 0.


It seems too that there are a clear differs between the results obtained by the HPM

and the corresponding results obtained by VIM resultant to the appearance of the high order

of time in VIM tends to the high values of the approximate solution comparing with the

results obtained by HPM. Because of the results obtained, we concluded that the homotopy

perturbation method is more effective and powerful than the variational iteration method.

On the other hand, from Figures 9, 10, 19, and 20, it is obvious that if the rotaion and

magnetic field are neglected, the approximate solutions by HPM and VIM in first iteration

are the same in both methods and agree with the results obtained by Sweilam and Khader

[1].

References

[1] N. H. Sweilam and M. M. Khader, “Variational iteration method for one dimensional nonlinearthermoelasticity,” Chaos, Solitons and Fractals, vol. 32, no. 1, pp. 145–149, 2007.

[2] N. H. Sweilam, “Harmonic wave generation in non linear thermoelasticity by variational iterationmethod and Adomian’s method,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp.64–72, 2007.

[3] H. N. Sweilam, M. M. Khader, and F. R. Al-Bar, “On the numerical simulation of population dynamicswith density-dependent migrations and the Allee effects,” Journal of Physics: Conference Series, vol. 96,no. 1, Article ID 012008, 2008.

[4] N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, “Nonlinear focusing Manakov systems by variationaliteration method and adomian decomposition method,” Journal of Physics: Conference Series, vol. 96,no. 1, Article ID 012164, 2008.

[5] A. N. Abd-Alla, A. F. Ghaleb, and G. A. Maugin, “Harmonic wave generation in nonlinearthermoelasticity,” International Journal of Engineering Science, vol. 32, no. 7, pp. 1103–1116, 1994.

[6] C. A. De Moura, “A linear uncoupling numerical scheme for the nonlinear coupled thermodynamicsequations,” in Lecture Notes in Mathematics, V. Pereyra and A. Reinoze, Eds., vol. 1005, pp. 204–211,Springer, Berlin, Germany, 1983.

[7] M. Slemrod, “Global existence, uniqueness, and asymptotic stability of classical smooth solutions inone-dimensional nonlinear thermoelasticity,” Archive for Rational Mechanics and Analysis, vol. 76, no.2, pp. 97–133, 1981.

[8] J.H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of ModernPhysics B, vol. 20, no. 10, pp. 1141–1199, 2006.

[9] J. H. He, Non-Perturbative Methods for Strongly Nonlinear Problems, Dissertation.de Verlag im InternetGmbH, Berlin, Germany, 2006.

[10] J. H. He and X. H. Wu, “Construction of solitary solution and compacton-like solution by variationaliteration method,” Chaos, Solitons and Fractals, vol. 29, no. 1, pp. 108–113, 2006.

[11] J. H. He, Y. Q. Wan, and Q. Guo, “An iteration formulation for normalized diode characteristics,”International Journal of Circuit Theory and Applications, vol. 32, no. 6, pp. 629–632, 2004.

[12] J. H. He, “A simple perturbation approach to Blasius equation,” Applied Mathematics and Computation,vol. 140, no. 2-3, pp. 217–222, 2003.

[13] J. H. He, “Variational iteration method for autonomous ordinary differential systems,” AppliedMathematics and Computation, vol. 114, no. 2-3, pp. 115–123, 2000.

[14] J. H. He, “Variational iteration method-a kind of non-linear analytical technique: some examples,”International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999.

[15] J. H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porousmedia,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57–68, 1998.

[16] A. M. Wazwaz, “The variational iteration method for solving nonlinear singular boundary valueproblems arising in various physical models,” Communications in Nonlinear Science and NumericalSimulation, vol. 16, no. 10, pp. 3881–3886, 2011.

[17] M. A. Noor, K. I. Noor, and S. T. Din, “Variational iteration method for solving sixth-order boundaryvalue problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 6, pp.2571–2580, 2009.


[18] F. Soltanian, S. M. Karbassi, and M. M. Hosseini, “Application of He’s variational iteration methodfor solution of differential-algebraic equations,” Chaos, Solitons and Fractals, vol. 41, no. 1, pp. 436–445,2009.

[19] S. M. Hassan and N. M. Alotaibi, “Solitary wave solutions of the improved KdV equation by VIM,”Applied Mathematics and Computation, vol. 217, no. 6, pp. 2397–2403, 2010.

[20] E. Yusufoglu, “Two convergence theorems of variational iteration method for ordinary differentialequations,” Applied Mathematics Letters. In press.

[21] J. H. He, “Approximate solution of nonlinear differential equations with convolution productnonlinearities,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 69–73,1998.

[22] J. H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering,vol. 178, no. 3-4, pp. 257–262, 1999.

[23] A. Yildirim, “Applying he’s variational iteration method for solving differential-difference equation,”Mathematical Problems in Engineering, vol. 2008, Article ID 869614, 7 pages, 2008.

[24] M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for solving higher-order nonlinearboundary value problems using He’s polynomials,” International Journal of Nonlinear Sciences andNumerical Simulation, vol. 9, no. 2, pp. 141–156, 2008.

[25] S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Solving second-order singular problems using He’spolynomials,” World Applied Sciences Journal, vol. 6, no. 6, pp. 769–775, 2009.

[26] S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Traveling wave solutions of seventh-order generalizedKdV equations using he’s polynomials,” International Journal of Nonlinear Sciences and NumericalSimulation, vol. 10, no. 2, pp. 227–233, 2009.

[27] S. T. Mohyud-Din and A. Yildirim, “Solving nonlinear boundary value problems using He’s poly-nomials and Pade approximants,” Mathematical Problems in Engineering, vol. 2009, Article ID 690547,17 pages, 2009.

[28] S. T. Mohyud-Din, A. Yildirim, S. Anl Sezer, and M. Usman, “Modified variational iteration methodfor free-convective boundary-layer equation using pade approximation,” Mathematical Problems inEngineering, vol. 2010, Article ID 318298, 11 pages, 2010.

[29] S. T. Mohyud-Din and M. A. Noor, “Homotopy perturbation method for solving fourth-orderboundary value problems,” Mathematical Problems in Engineering, Article ID 98602, 15 pages, 2007.

[30] S. T. Mohyud-Din and M. A. Noor, “Homotopy perturbation method for solving partial differentialequations,” Zeitschrift fur Naturforschung, vol. 64, no. 3-4, pp. 157–170, 2009.

[31] S. T. Mohyud-Din, M. A. Noor, and K. I. Noor, “Some relatively new techniques for nonlinearproblems,” Mathematical Problems in Engineering, vol. 2009, Article ID 234849, 25 pages, 2009.


Research ArticleSimplicial Approach to Fractal Structures

Carlo Cattani, Ettore Laserra, and Ivana Bochicchio

Department of Mathematics, University of Salerno, Via Ponte don Melillo, 84084 Fisciano, Italy

Correspondence should be addressed to Ivana Bochicchio, [email protected]

Received 28 July 2011; Accepted 16 November 2011


Copyright q 2012 Carlo Cattani et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

A fractal lattice is defined by iterative maps on a simplex. In particular, Sierpinski gasket and vonKoch flake are explicitly obtained by simplex transformations.

1. Introduction

Simplicial calculus [1–3] has been since the beginning a suitable tool for investigating discrete

models in many physical problems such as discrete models in space-time [4–9] complex

networks [10–13], molecular crystals, aggregates and diamond lattices [14–17], computer

graphics [18, 19], and more recently signal processing and computer vision, such as stereo

matching and image segmentation [20, 21].In some recent papers [22–25], fractals [26–29] generated by simplexes, also called

fractal lattices, were proposed for the analysis of nonconventional materials as some kind of

polymers [24, 25] or nanocomposites [22, 23, 30, 31] having extreme physical and chemical

properties. Moreover, the analysis of complex traffic on networks [32, 33] and image analysis

[20, 21] based on fractal geometry and simplicial lattices has focussed on the importance of

these methods in handling modern challenging problems.

However, only a few attempts were made in order to define the fractal lattice

(structure) by an iterated system of functions on simplexes [34, 35]. The main scheme for

affine contraction has been given in [35], whereas some generation of fractals by simplicial

maps can be found in [34].In this paper, we define a method based on simple algorithms for the generation

of fractal-like structures by continuously deforming a simplex. This algorithm is based on

a well-defined analytical map, which can be used to finitely describe fractals. Instead of

recursive law, or nested maps (see, e.g., [1, 2, 15]), we propose a method which can be more

easily implemented.


In the following, we will study an m-dimensional fractal structure defined by the

transformation group of a simplicial complex. Starting from a simplex, it will define the group

of transformation on it, so that the intrinsic (affine) metric remains scale invariant. The group

of transformations (isometries and homotheties) will be characterized by matrices acting on

the skeleton of the simplex. We will derive the basic properties of the fractal lattice and give

a suitable definition of self-similarity on lattices. The concept of self-similarity is shown to

be fulfilled by some classical transformation on simplices (homothety) and, simplicial based,

fractals as the Sierpinski tessellations and the von Koch flake.

2. Euclidean Simplexes

In the ordinary Euclidean space Rn, we assume that there exists a triangulation of Rn, in the

sense that there is at least a finite set of n+1 points geometrically independent (simplexes). A

simplex will be considered both as a set of points and as the convex subspace of Rn, defined

by the geometrical support of the simplex. Union of n-adjacent simplexes is an n-polyhedron

P [4, 18, 19].The euclidean m-simplex σm, of independent vertices V0, V1, . . . , Vm, is defined [1–3]

as the subset of Rn,

σm def=

{P ∈ Rn | P

m∑i=0

λiVi withm∑i=0

λi = 1, 0 ≤ λi ≤ 1

}. (2.1)

Let us denote with [σm] = [V0, V1, . . . , Vm] the set of points which form the skeleton of σm, and

let #σm = m + 1 be the cardinality of the set of points. The p-face of σm, with p ≤ m, is any

simplex σp such that [σp] ∩ [σm]/= ∅, and we write σp � σm.

The number of p-faces of σm is (m+1p+1 ).

The m-dimensional simplicial complex Σm is defined as the finite set of p simplexes

(p ≤ m) such that

(1) for all σk ∈ Σm if σh � σk, then σh ∈ Σm,

(2) for all σk, σh ∈ Σm, then either [σh] ∩ [σk] = ∅ or [σh] ∩ [σk] = [σj] with σj ∈ Σm.

The set of points P such that P ∈ σp, p ≤ m, and σp ∈ Σm is the geometric support

of Σm also called m-polyhedron Mm. The p-skeleton of Σm is [Σm]p def= [σp] for all σp ∈ Σm.

The boundary ∂Σm of Σm is the complex Σm−1 such that each σm−1 ∈ Σm−1 is face of only one

m-simplex of Σm. A finite set of simplexes is also called lattice (or tessellation).

2.1. Barycentric Coordinates and Barycentric Bases

In each simplex, it is possible to define the barycentric basis as follows: given the m-simplex

σm with vertices V0, . . . , Vm, the barycentric basis is the set of (m + 1) vectors

eidef= Vi − Gm, (2.2)


based on the barycenter

Gm def= G(σm) =m∑i=0

1

m + 1Vi. (2.3)

These vectors ei belong to the n-dimensional vector space E isomorphic to Rn. Moreover, they

are linearly dependent, since according to their definition, it is

m∑i=0

ei = 0. (2.4)

Each point P ∈ σm can be characterized by a set of barycentric coordinates (λ0, . . . , λm)such that

0 ≤ λi ≤ 1,m∑i=0

λi = 1, i = 0, . . . , m, (2.5)

and P − Gm =∑m

i=0 λiei

(2.2),(2.4)=⇒ P =

∑mi=0 λ

iVi. Therefore, each point of σm can be formally

expressed as a linear combination of the skeleton [σm].The dual space is defined as the linear map of the vector space E into R as

⟨ei, ek

⟩=δi

k, (2.6)

with [14]

δik

def= δik −

1

m + 1=

⎧⎪⎪⎨⎪⎪⎩− 1

m + 1, i /= k,

+m

m + 1, i = k,

(2.7)

δik

being the Kroneker symbol. According to the definition (2.7), it is

m∑i=0

δik =

m∑k=0

δik = 0. (2.8)

In addition, the metric tensor in σm is defined as [5]

gijdef= −1

2δhi δ

kj

2hk,

(i, j, h, k = 0, 1, . . . , m

)(2.9)

being 2hk

def= (Vk − Vh)2 = (ek − eh)

2.


2.2. Measures of the m-Simplex

Let

Lijdef= Vj − Vi

(= ej − ei

), lij

def=⟨Lij ,Lij

⟩, (2.10)

by using the ordinary wedge product of the vectors ej1 , . . . , ejp , we can define the p-form ω,

ω =1

p!

∑j1,...,jm

ωj1...jpej1 ∧ · · · ∧ ejp , (2.11)

whose affine components are ωj1...jp def= 〈ω, ej1 ∧ ej2 ∧ · · · ∧ ejp〉 [14].The euclidean measure of the m-simplex σ (volume) is [14]

εΩ2 def=1

m!|L01 ∧ · · · ∧ L0m|, (2.12)

from where, it follows that

Ω2 =(

1

m!

)3 ∑j1 ,...,jm

k1,...,km

εj1...jmεk1...kmm∏a=1

l2jaka , (2.13)

being

εj1...jmdef= ±1, (2.14)

according to the even/odd permutation j0, j1, . . . , jm of the indices 0, 1, . . . , m.

In particular, the volume of each p-face σi1...im−p (see also [9]) is

Ω2i1...im−p =

(−1

2

)p( 1

p!

)3 ∑j1 ,...,jp

k1,...,kp

εj1...jp εk1...kp

p∏a=1

l2jaka(0 < p ≤ m

), (2.15)

where j1, . . . , jp, k1, . . . , kp /= i1, . . . , im−p.

3. m-Dimensional Homothety

Let I(σi) be the subspace of Rm to which σi belongs; it can be easily proved that [14]

∀ v ∈ I(σi)⟨ni,v

⟩= 0 ( i fixed ), (3.1)


where the normal vector ni is defined as

ni def= −mΩΩi

ei, hi def=mΩΩi

. (3.2)

The above definition of vector orthogonal to a (m − 1)-face allows us to characterize

the m-parallelism of simplexes as follows. Let σ,�σ be two simplexes in Rm; let σi,

�σi be the ith

(m − 1)-faces of σ and�σ, respectively, and let ni,

�ni

be their normal vectors, then we say that

σ is m-parallel to�σ ( σ‖m�

σ) if and if only σi‖�σi, that is, ni =�ni(i = 0, . . . , m).

Let ϕ be a map

ϕ : Rm −→ Rm, σϕ −→ �

σ (3.3)

such that

(1) ϕ is a bijective simplicial map on σ,

(2) the s-adjacent faces of σ correspond (under the map ϕ) to s-adjacent faces of�σ,

(3) σ and�σ are m-parallel.

We also assume that this transformation depends on the edge vectors and in particular

on the edge lengths, so that any quantity, defined on the simplex, transforming under the action ofϕ, is a function of the edge lengths. Furthermore, we assume the following conditions:

(4) there exists a fixed point under the action of ϕ:

∃O ∈ Rm | ϕ(O) ≡ O, (3.4)

(5) each (m − 1)-face σi translates of an amount t ∈ [0,∞).Let us choose as a fixed point one of the vertices, for example, V0. We define this

bijective simplicial map applying any P ∈ σ into�

P ∈ �σ (t ∈ [0,∞)) as

�

Pdef= P + t

Ω0

mΩ

m∑i=0

λiL0i; (3.5)

in particular, this function acts on any vertex Vi as

�

V i = Vi + tΩ0

mΩL0i,

(�

V 0 = V0

), (3.6)

so that we can easily prove that all the previous conditions are easily satisfied [14]. According

to the above equations, each edge transforms as

�

Lij =(

1 + tΩ0

mΩ

)Lij , (3.7)

where�

Lij =�

V j −�

V i.


3.1. Variation Law of the p-Faces of σ

The variation law of the edge lengths, resulting from (3.7), is given by the formula

�

l ij =(

1 + tΩ0

mΩ

)lij , (3.8)

where lij is the length of the edge Lij , and�

l ij is the length of the edge�

Lij .

According to (2.13), the volume Ω is a homogeneous function of degree m of the m(m+1)/2 variable {l2ij}i<j , so that its variation law is

�

Ω =(

1 + tΩ0

mΩ

)m

Ω, (3.9)

and for any p-face,

�

Ωi1...im−p =(

1 + tΩ0

mΩ

)p

Ωi1...im−p(0 < p < m

); (3.10)

analogously, taking into account the definition (5.5)2, we have the transformation law of hi:

�

hi = m

(1 + t

Ω0

mΩ

)hi. (3.11)

There follows, for the fundamental vectors of�σ, that

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

�e i

def=�

V i −�

G =(

1 + tΩ0

mΩ

)ei,

�ni

= ni,

�ei

=

�

Ωi/Ωi�

Ω/Ωei =

(1 + t

Ω0

mΩ

)−1

ei.

(3.12)

4. Self-Similar Structure

Let (Rn, d) be the complete metric space with the standard Euclidean metric d, and let K(Rn)be the set

K(Rn) ={K ⊆ Rn : K is a nonempty compact set

}. (4.1)

The iterated function system (IFS)

{wi} = (Rn, d,w1, w2, . . . , wn) (4.2)


is the finite set of contractions wi on the complete metric space (Rn, d), being the contraction

w defined as

d(w(x), w

(y)) ≤ cd

(x,y

), ∀x,y ∈ Rn, (4.3)

with c contraction coefficient.

For each A ∈ K(Rn), the (IFS) contracting mapping is

w : A ∈ K(Rn) −→ w1(A)⋃

· · ·⋃

wn(A) ∈ K(Rn), (4.4)

with contraction coefficient c = max{c1, . . . , cn}. Each function wi usually is linear, or more

generally an affine transformation, but sometimes it can be nonlinear, including projective

and Mobius transformations [27].According to the Banach fixed-point theorem (see, e.g., [36]), every contraction

mapping on a nonempty complete metric space has a unique fixed point, so that there exists

a unique compact (i.e., closed and bounded) fixed set A such that A = w(A). The set A is

also known as the fixed set of the Hutchinson operator [28].One way of constructing such fixed set is to start with an initial set A and by iterating

the actions of w. Hence,

A =⋃

i1,...,ih=1,...,n

wi1 ◦ · · · ◦wih(A), (4.5)

so that A is a self-similar set, expressed as the finite union of its conformal copies, each one

reduced by a factor ch.

The attractor A of IFS is characterized by a similarity dimension as follows.

Definition 4.1. Given an IFS of n contraction mappings with the same contraction coefficient

c, the similarity dimension is defined as

s =log n

log 1/c

(= − log n

log c

). (4.6)

Sets having noninteger similarity dimensions are called fractal sets, or simply fractals.

There follows that the iterated function systems are a method of constructing fractals; the

resulting constructions are always self-similar such that w(μx) = μHw(x). Hence, each map

w is also called a self-similar map [27].

5. Fractal Structures from Simplicial Maps

In this section, some examples of self-similar (scale invariant) structures obtained by IFS on

simplexes are given in R2. In particular, the IFS will be defined by affine transformations, as

conformal maps of the affine metrics.


In the following, we will introduce some self-similar maps defined both on 2-simplexes

and 1-simplexes, so that, from (4.1),

K(R2)={σ2; σ1;σ0

},

w : K(R2) −→ K

(R2).

(5.1)

In particular, let σ2 be the simplex [V1, V2, V3], then it is

K(R2)= {[V1, V2, V3]; [V1, V2], [V1, V3], [V2, V3]; [V1], [V2], [V3]}, (5.2)

so that a map w on K(R2) could be the more general function defined on any face of σ2.

Examples. If the skeleton of σ2 is the set of vertices {V1, V2, V3} with V1 = (x1,y1),V2 = (x2,y2),and V3 = (x3,y3), the affine map w is defined by the matrix

W =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

a11 a12 a13 a14 a15 a16

a21 a22 a23 a24 a25 a26

a31 a32 a33 a34 a35 a36

a41 a42 a43 a44 a45 a46

a51 a52 a53 a54 a55 a56

a61 a62 a63 a64 a65 a66

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(5.3)

and the constant vector

U = (u1, u2, u3, u4, u5, u6). (5.4)

The function w maps a 2-simplex into a 2-simplex whereas, by a matrix product, the vector

X =(x1,y1,x2,y2,x3,y3

)(5.5)

is mapped into the vector

WX +U, (5.6)


so that the skeleton of w(σ2) is given by the vector WX + U. For instance, a rotation with

fixed point V1 is given by the matrix

W =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 0 0

0 1 0 0 0 0

0 0 a33 a34 0 0

0 0 a43 a44 0 0

0 0 0 0 a55 a56

0 0 0 0 a65 a66

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (5.7)

with

a33a44 − a34a43 = ±1, a55a66 − a56a65 = ±1, (5.8)

and the vector U = {0, 0, 0, 0, 0, 0}.

Some more special maps will be given in the following where, in particular, we

consider, without restriction, some special maps on the 1-faces of σ2 such that

w(σ2)= w1

(σ2

1

)∪w2

(σ2

2

)∪w3

(σ2

3

), #w

(σ2)= 3. (5.9)

In this case, the matrix W , acting on σ2, follows from the direct sum of lower-order matrices

acting on σ1 simplexes, as follows:

(a) the first vertex V1 remains fixed, and the map w on σ2 is a consequence of the

transformation of the simplex σ1 = [V2, V3], that is, by defining

I =

(1 0

0 1

), W1 =

⎛⎜⎜⎜⎜⎜⎝a33 a34 a35 a36

a43 a44 a45 a46

a53 a54 a55 a56

a63 a64 a65 a66

⎞⎟⎟⎟⎟⎟⎠, (5.10)

it is

W = I ⊕W1 =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 0 0

0 1 0 0 0 0

0 0 a33 a34 a35 a36

0 0 a43 a44 a45 a46

0 0 a53 a54 a55 a56

0 0 a63 a64 a65 a66

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (5.11)


(b) the second vertex V2 remains fixed, and the map w on σ2 is a consequence of the

transformation of the simplex σ1 = [V1, V3], so that

W2 =

⎛⎜⎜⎜⎜⎜⎝a11 a12 a15 a16

a21 a22 a25 a26

a51 a52 a55 a56

a61 a62 a65 a66

⎞⎟⎟⎟⎟⎟⎠ ,

W =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

a11 a12 0 0 a15 a16

a21 a22 0 0 a25 a26

0 0 1 0 0 0

0 0 0 1 0 0

a51 a52 0 0 a55 a56

a61 a62 0 0 a65 a66

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,

(5.12)

(c) the third vertex V3 remains fixed, and the map w on σ2 is a consequence of the

transformation of the simplex σ1 = [V1, V2], that is,

W = W3 ⊕ I =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

a11 a12 a13 a14 0 0

a21 a22 a23 a24 0 0

a31 a32 a33 a34 0 0

a41 a42 a43 a44 0 0

0 0 0 0 1 0

0 0 0 0 0 1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠, (5.13)

being

W3 =

⎛⎜⎜⎜⎜⎜⎝a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41 a42 a43 a44

⎞⎟⎟⎟⎟⎟⎠. (5.14)

In the following, we will characterize the transformation on a 2-simplex as a result

of iterative maps on its boundary 1-simplexes. These maps on 1-simplexes are defined by

the matrices W1,W2, and W3, applied to the vectors of coordinates of [V2, V3], [V1, V3], and

[V1, V2], respectively.


C C

B

A

B

C

B

A C

B

Figure 1: Homothety map.

5.1. Homothety

Let us consider the 2-simplex σ2 = {A,B,C} and the map (Figure 1)

σ2 = {A,B,C} =⇒ w(σ2)={A,B′, C′}, (5.15)

such that nC = ±nC′ . This map, according to (5.9), is obtained as a combination of 3 maps

acting on the faces of σ2, since

w(σ2)= w1([A,B]) ∪w2([B,C]) ∪w3([A,C]). (5.16)

This map is a scale invariant, since there results

2AB = λ 2

A′B′ , (0 ≤ λ), (5.17)

2BC = λ 2

B′C′ , and 2AC = λ 2

A′C′ , as well.

So that when λ < 1, we have a contraction and a dilation when λ > 1.

Moreover, according to (2.9), the metric g ′ij of the transformed simplex is given by a

conformal transformation g ′ij = λgij .

5.2. Sierpinski Gasket

As a first example of fractal defined by IFS on simplexes, we will consider the Sierpinski

gasket. To this end, let us introduce an orthogonal coordinate system 0xy in R2 and three

homothety maps w1, w2, and w3. Each wi is uniquely and completely determined once we

know as it acts on the paired points A = (xA,yA), B = (xB, yB), and C = (xC, yC), vertices of

the 2-simplex [σ2] = [A,B,C].


In order to define the Sierpinski gasket by IFS of maps, we consider a sequence of

maps that, at each step, shrink the area of σ2 by a factor 0.25 and move the edges by a suitable

homothety (Figure 2). In particular, the 3 maps are explicitly defined as follows:

w1 :

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

xA

yA

xB

yB

xC

yC

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⇒ M ·

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

xA

yA

xB

yB

xC

yC

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,

w2 :

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

xA

yA

xB

yB

xC

yC

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⇒ M ·

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

xA

yA

xB

yB

xC

yC

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0

1/2

0

1/2

0

1/2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,

w3 :

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

xA

yA

xB

yB

xC

yC

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⇒ M ·

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

xA

yA

xB

yB

xC

yC

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠+

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1/2

0

1/2

0

1/2

0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,

(5.18)

where M is the matrix

M =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1/2 0 0 0 0 0

0 1/2 0 0 0 0

0 0 1/2 0 0 0

0 0 0 1/2 0 0

0 0 0 0 1/2 0

0 0 0 0 0 1/2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (5.19)

Once we get the vertices of w(σ2), we can easily define the map for each point P of the

σ2 convex domain.


Figure 2: Fundamental maps.

Comment. In fact, let λ1, λ2, and λ3 be the barycentric coordinates of a given point P inside

σ2, as given by (2.1), then we can write the barycentric expansion of P ≡ (x,y) in terms of the

coordinates of vertices A,B, and C as

x = λ1xA + λ2xB + λ3xC,

y = λ1yA + λ2yB + λ3yC.(5.20)

Substituting λ3 = 1 − λ1 − λ2 into the above and rearranging, this linear transformation can be

written as

H ·Λ = P − C, (5.21)


where Λ is the vector of barycentric coordinates, and H is the matrix

H =

(xA − xC xB − xC

yA − yC yB − yC

). (5.22)

Since H is invertible, we can easily obtain the barycentric coordinates of P = (x,y):

λ1 =

(yB − yC

)(x − xC) + (xC − xB)

(y − yC

)(yB − yC

)(xA − xC) + (xC − xB)

(yA − yC

) ,λ2 =

(yC − yA

)(x − xC) + (xA − xC)

(y − yC

)(yC − yA

)(xB − xC) + (xA − xC)

(yB − yC

) ,λ3 = 1 − λ1 − λ2.

(5.23)

According to (5.9) each map wi, i = {1, 2, 3} is a contraction (dilation) of the σ2 faces,

such that the union gives rise to a 2-simplex (Figure 2). Any P ∈ [σ2] is mapped into�

P ∈ �σ =

wi([σ2]) as

�

P = P − 1

2

m∑i=0

λiL0i. (5.24)

Moreover, each vertex in the wi([σ2]), i = 1, 2, 3 can be expressed as in (3.6)

�

V i = Vi − 1

2L0i, (5.25)

so that

�

Lij =1

2Lij ,

�

l ij =1

2lij ,

�

Ω =1

4Ω.

(5.26)

Reiterating this process for each remaining triangle, at the step k, we will obtain the

compact set Tk given by 3k triangles whose edges are contracted by (1/2k). In other words,

�

Lij =1

2kLij ;

�

l ij =1

2klij ,

�

Ω =1

22kΩ.

(5.27)

Finally, we note that through the three simplicial maps in R2, provided with the natural

metric d, we are able to construct the IFS (R2, d,w1, w2w3) that has the well-known Sierpinski


Figure 3: Sierpinski gasket.

gasket T =⋂

k Tk as fractal attractor. So, we have obtained the Sierpinski gasket, as the

combination of homothety maps (Figure 2). The iterating function will generate the known

fractal-shaped curve (Figure 3).The Sierpinski gasket supplies one of the most simple cases of construction of fractals

through simplicial maps. In fact, the fractal structure is obtained acting on the 2-simplex only

with homothetic transformations. Sometimes a fractal object can be constructed not only

acting on simplexes with one map, but considering the compositions of different suitable

transformations. Hereafter, in order to obtain another fractal object, we will consider, in

details, some more elementary maps: the translation and the rotation (which are special cases

of the matrix W).

6. Von Koch Curve

The von Koch curve [27, 28] can be obtained as a combination of homothety, translation, and

rotation maps, so that the von Koch snowflake is obtained by their iteration.

6.1. Translation

Let the translation operator be defined as the operator

T : Rm −→ Rm, σT −→ �

σ (6.1)

such that

T(P) = P + v, (6.2)


where v = (v1,v2, . . . ,vm) is a given vector of Rm, then the image of a simplex σ under the

function T is the translation of σ by T so that any vertex Vi is transformed into

�

V i = Vi + v. (6.3)

Since in a Euclidean space, any translation is an isometry, we have no variation of the edge

lengths of σ.

According to the definitions (5.3), (5.4) in R2, it is

U = (v1, v2,v1, v2,v1, v2), v1 = Cnst., v2 = Cnst., (6.4)

being W the zero matrix.

6.2. Rotation

Rotation is characterized by having a fixed point, however, like the translation which is an

isometry. This is like the previous maps on simplexes that can be defined by a suitable matrix

(5.3). R2 rotation is defined by (5.7), which however can be expressed by a single parameter

(rotation angle). Hence, in two dimensions, a rotation with fixed point V0 is the operator

R : R2 −→ R2, σR −→ �

σ (6.5)

such that

R(P) = V0 + Rθ(P − V0), (6.6)

where Rθ is the matrix

Rθ =

(cos θ − sin θ

sin θ cos θ

), (6.7)

so that (5.7), when applied to the simplex σ2 = [V0, V1, V2] with one fixed vertex, becomes

W =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 0 0 0 0 0

0 1 0 0 0 0

0 0 cos θ − sin θ 0 0

0 0 sin θ cos θ 0 0

0 0 0 0 cos θ − sin θ

0 0 0 0 sin θ cos θ

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠. (6.8)


With respect to an orthogonal coordinate system with origin O, for any P ∈ σ, we

define the rotation as the bijective simplicial map which applies P ∈ σ into�

P ∈ �σ,

�

Pdef= [Rθ(P − v)] + v, (6.9)

where v = V0 −O; in particular, the vertices V0, V1, and V2 are transformed into

�

V 0 = V0,�

V i = [Rθ(L0i +O)] + (V0 −O), (i = 1, 2). (6.10)

6.3. Von Koch Snowflake

Let 0xy be an orthogonal coordinate system for R2, and let σ2 = [A,B,C] be a two simplex

under the homothety map. According to (5.16), this map can be realized by a composition of

maps on the 1-simplexes σ1 = [A,B], σ1 = [B,C], and σ1 = [A,C]. The coordinates of vertices

are A = (xA,yA), B = (xB,yB), and C = (xC,yC), respectively. In the following we will

give both the construction of the Koch curve as IFS on σ1 and the construction of the Koch

snowflake as IFS on σ2.

6.3.1. Von Koch Curve

Koch curve can be classically constructed by starting with a line segment, then recursively

altering the shape as follows: divide the line segment into three segments of equal length;

draw an equilateral triangle that has the middle segment from step 1 as its base and points

outward; remove the line segment that is the base of the triangle from step 2 (see Figure 5).Following the classical construction, we consider the following maps on 1-simplexes:

w1 :

⎛⎜⎜⎜⎜⎜⎝xA

yA

xB

yB

⎞⎟⎟⎟⎟⎟⎠ =⇒ M ·

⎛⎜⎜⎜⎜⎜⎝xA

yA

xB

yB

⎞⎟⎟⎟⎟⎟⎠,

w2 :

⎛⎜⎜⎜⎜⎜⎝xA

yA

xB

yB

⎞⎟⎟⎟⎟⎟⎠ =⇒ M ·

⎛⎜⎜⎜⎜⎜⎝xA

yA

xB

yB

⎞⎟⎟⎟⎟⎟⎠ +

⎛⎜⎜⎜⎜⎜⎝2/3

0

2/3

0

⎞⎟⎟⎟⎟⎟⎠,

(6.11)

where M is the matrix

M =

⎛⎜⎜⎜⎜⎜⎝1/3 0 0 0

0 1/3 0 0

0 0 1/3 0

0 0 0 1/3

⎞⎟⎟⎟⎟⎟⎠. (6.12)


Hence, w1 is a factor of the homothety w having A as a fixed vertex, while w2 leaves B

unchanged:

w1(P) = P − 2

3λiL0i, L0i = Vi −A,

w2(P) = P − 2

3λiL0i, L0i = Vi − B.

(6.13)

Moreover, each vertex in the wi([σ1]), i = 1, 2, can be expressed as in(3.6)

�

V i = Vi − 2

3L0i. (6.14)

Let us now consider the transformation, on two steps, which first rotates w1([A,B])of an angle θ = 60◦ around the fixed point A, and then it translates the rotated simplex by a

vector v = (1/3, 0).So that we obtain

w3 :

⎛⎜⎜⎜⎜⎜⎝xA

yA

xB

yB

⎞⎟⎟⎟⎟⎟⎠ =⇒ M′ ·

⎛⎜⎜⎜⎜⎜⎝xA

yA

xB

yB

⎞⎟⎟⎟⎟⎟⎠ +

⎛⎜⎜⎜⎜⎜⎝1/3

0

1/3

0

⎞⎟⎟⎟⎟⎟⎠, (6.15)

where M′ is the matrix

M′ =

⎛⎜⎜⎜⎜⎜⎝1/6 −√3/6 0 0√

3/6 1/6 0 0

0 0 1/6 −√3/6

0 0√

3/6 1/6

⎞⎟⎟⎟⎟⎟⎠. (6.16)

Finally, let us apply the transformation which first rotates w1([A,B]) of an angle θ =120◦ around the fixed point A, and then it translates the rotated simplex by the vector v =(2/3, 0). Accordingly, it is

w4 :

⎛⎜⎜⎜⎜⎜⎝xA

yA

xB

yB

⎞⎟⎟⎟⎟⎟⎠ =⇒ M′′ ·

⎛⎜⎜⎜⎜⎜⎝xA

yA

xB

yB

⎞⎟⎟⎟⎟⎟⎠ +

⎛⎜⎜⎜⎜⎜⎝2/3

0

2/3

0

⎞⎟⎟⎟⎟⎟⎠, (6.17)


Figure 4: Image of w([σ1]) =⋃

i = 1,...,4 wi([σ1]), where the 1-simplex σ1 is the unitary interval.

where M′′ is the matrix

M′′ =

⎛⎜⎜⎜⎜⎜⎝−1/6 −√3/6 0 0√

3/6 −1/6 0 0

0 0 −1/6 −√3/6

0 0√

3/6 −1/6

⎞⎟⎟⎟⎟⎟⎠. (6.18)

Since, as previously shown, rotation and translation are isometries, for each wi([σ1]),i = 1, 2, 3, 4, we obtain

�

Lij =1

3Lij ,

�

l ij =1

3lij ,

�

Ω =1

3Ω.

(6.19)

In order to visualize the von Koch pattern, let us consider the 1-simplex {A,B} ={{0, 0}, {1, 0}}; since the point A has been chosen as the origin of the reference system, and

w3 and w4 are obtained as rotation leaving fixed the origin, the transformed instances can be

easily computed so that, at the first step, the IFS maps on 1-simplexes can be drawn (Figure 4).Reiterating this process for each remaining segment, at the step k we will obtain the

compact set Tk made of 22k segments whose sides are contracted by a factor (1/3)k. The IFS

map (R1, d,w1, w2, w3, w4) gives us the Koch curve L =⋂

k Lk with similarity dimension

equal to

s =log 4

log 1/(1/3)=

log 4

log 3. (6.20)

This is also the similarity dimension of the Koch snowflake [27, 28]. So, the Koch curve

(Figure 5) is obtained as a combination of IFS simplicial maps generating the known fractal-

shaped curve.

6.3.2. Koch Flake

According to (5.16) and to the examples previously given, Koch flake (snowflake) can be

constructed in a non-classical approach as IFS of maps on a 2-simplex. Koch snowflake can

be seen as the image of a suitable system of iterated homotheties acting on a 2-simplex, given

by suitable translations of the boundary 1-simplexes.


Figure 5: Koch curve.

In this process, the total length of each side of a triangle increases by one-third, and

thus, the total length at the kth step will be (4/3)k of the original triangle perimeter.

7. Conclusion

In this paper, a nonclassical approach to fractal generation based on IFS of maps on simplexes

has been given. Some of the most popular fractals, as the Sierpinski gasket and the von Koch

flake, were obtained by iterative maps on simplexes. All maps were also intrinsically defined

by using the affine (barycentric) coordinates and some basic measures on simplexes. The

method proposed in this paper could be used to generate some new classes of fractals in any

dimension, by simply defining suitable IFS on simplexes, thus opening new perspectives in

fractal lattice geometry.

References

[1] A. T. Fomenko, Differential Geometry and Topology, Contemporary Soviet Mathematics, ConsultantsBureau, New York, NY, USA, 1987.

[2] G. L. Naber, Topological Methods in Euclidean Spaces, Cambridge University Press, Cambridge, UK,1980.

[3] I. M. Singer and J. A. Thorpe, Lecture notes on elementary topology and geometry, Scott, Foresman andCo., Glenview, Ill, USA, 1967.

[4] C. Cattani, “Variational method and Regge equations,” in Proceedings of the 8th Marcel GrossmannMeeting, Gerusalem, Palestine, June 1997.

[5] C. Cattani and E. Laserra, “Discrete electromagnetic action on a gravitational simplicial net,” Journalof Interdisciplinary Mathematics, vol. 3, no. 2-3, pp. 123–132, 2000.

[6] I. T. Drummond, “Regge-Palatini calculus,” Nuclear Physics. B, vol. 273, no. 1, pp. 125–136, 1986.[7] C. W. Misner, K. S. Thorne, and J. A. Wheeler, “Regge Calculus,” in Gravitation, p. ii+xxvi+1279+iipp,

W. H. Freeman and Co., San Francisco, Calif, USA, 1973.[8] T. Regge, “General relativity without coordinates,” Il Nuovo Cimento Series 10, vol. 19, no. 3, pp. 558–

571, 1961.[9] R. Sorkin, “The electromagnetic field on a simplicial net,” Journal of Mathematical Physics, vol. 16, no.

12, pp. 2432–2440, 1975.[10] A. L. Barabasi, Linked: How Everything is Connected to Everything Else and What It Means for Business,

Science and Everyday Life, Plume Books, 2003.[11] A. L. Barabasi, E. Ravasz, and T. Vicsek, “Deterministic scale-free networks,” Physica A, vol. 299, no.

3-4, pp. 559–564, 2001.


[12] S. H. Strogatz, “Exploring complex networks,” Nature, vol. 410, no. 6825, pp. 268–276, 2001.[13] D. J. Watts and S. H. Strogatz, “Collective dynamics of ’small-world9 networks,” Nature, vol. 393, no.

6684, pp. 440–442, 1998.[14] C. Cattani and E. Laserra, “Simplicial geometry of crystals,” Journal of Interdisciplinary Mathematics,

vol. 2, no. 2-3, pp. 143–151, 1999.[15] N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, Addison-Wesley, Reading,

Mass, USA, 1992.[16] V. M. Kenkre and P. Reineker, Exciton Dynamics in Molecular Crystals and Aggregates, Springer, Berlin,

Germany, 1982.[17] F. Vogtle, Ed., Dendrimers, Springer, Berlin, Germany, 1998.[18] C. Cattani and A. Paoluzzi, “Boundary integration over linear polyhedra,” Computer-Aided Design,

vol. 22, no. 2, pp. 130–135, 1990.[19] C. Cattani and A. Paoluzzi, “Symbolic analysis of linear polyhedra,” Engineering with Computers, vol.

6, no. 1, pp. 17–29, 1990.[20] S. Y. Chen, H. Tong, and C. Cattani, “Markov models for image labeling,” Mathematical Problems in

Engineering, vol. 2012, Article ID 814356, 18 pages, 2012.[21] S. Y. Chen, H. Tong, Z. Wang, S. Liu, M. Li, and B. Zhang, “Improved generalized belief propagation

for vision processing,” Mathematical Problems in Engineering, vol. 2011, Article ID 416963, 12 pages,2011.

[22] J. P. Blondeau, C. Orieux, and L. Allam, “Morphological and fractal studies of silicon nanoaggregatesstructures prepared by thermal activated reaction,” Materials Science and Engineering B, vol. 122, no. 1,pp. 41–48, 2005.

[23] C. C. Doumanidis, “Nanomanufacturing of random branching material architectures,” MicroelectronicEngineering, vol. 86, no. 4-6, pp. 467–478, 2009.

[24] D. Lekic and S. Elezovic-Hadizic, “Semi-flexible compact polymers on fractal lattices,” Physica A, vol.390, no. 11, pp. 1941–1952, 2011.

[25] R. Lua, A. L. Borovinskiy, and A. Y. Grosberg, “Fractal and statistical properties of large compactpolymers: a computational study,” Polymer, vol. 45, no. 2, pp. 717–731, 2004.

[26] M. F. Barnsley, Fractal Everywhere, Academic Press, 1988.[27] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons,

Chichester, UK, 1990.[28] J. E. Hutchinson, “Fractals and self-similarity,” Indiana University Mathematics Journal, vol. 30, no. 5,

pp. 713–747, 1981.[29] B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman and Co., San Francisco, Calif, USA,

1982.[30] C. Cattani and J. Rushchitsky, Wavelet and Wave Analysis as Applied to Materials with Micro or

Nanostructure, vol. 74 of Series on Advances in Mathematics for Applied Sciences, World Scientific,Hackensack, NJ, USA, 2007.

[31] E. V. Pashkova, E. D. Solovyova, I. E. Kotenko, T. V. Kolodiazhnyi, and A. G. Belous, “Effect ofpreparation conditions on fractal structure and phase transformations in the synthesis of nanoscaleM-type barium hexaferrite,” Journal of Magnetism and Magnetic Materials, vol. 323, no. 20, pp. 2497–2503, 2011.

[32] M. Li, “Fractal time series—a tutorial review,” Mathematical Problems in Engineering, vol. 2010, ArticleID 157264, 26 pages, 2010.

[33] M. Li, S. C. Lim, and S. Chen, “Exact solution of impulse response to a class of fractional oscillatorsand its stability,” Mathematical Problems in Engineering, vol. 2011, Article ID 657839, 9 pages, 2011.

[34] C. Cattani and E. Laserra, “Self-similar hierarchical regular lattices,” in the International Conference onComputational Science and Its Applications (ICCSA ’10), D. Taniar et al., Ed., vol. 6017 of Lecture Notes inComputer Science, pp. 225–240, 2010.

[35] L. Kocic, S. Gegovska-Zajkova, and L. Stefanovska, “Affine invariant contractions of simplices,”Kragujevac Journal of Mathematics, vol. 30, pp. 171–179, 2007.

[36] M. A. Khamsi and W. A. Kirk, An Introduction toMetric Spaces and Fixed Point Theory, Pure and AppliedMathematics, Wiley-Interscience, New York, NY, USA, 2001.


Research ArticleGaussian Curvature in Propagation Problems inPhysics and Engineering

Ezzat G. Bakhoum

Department of Electrical and Computer Engineering, University of West Florida,11000 University Parkway, Pensacola, FL 32514, USA

Correspondence should be addressed to Ezzat G. Bakhoum, [email protected]

Received 1 September 2011; Accepted 9 October 2011


Copyright q 2012 Ezzat G. Bakhoum. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The computation of the Gaussian curvature of a surface is a requirement in many propagationproblems in physics and engineering. A formula is developed for the calculation of the Gaussiancurvature by knowledge of two close geodesics on the surface, or alternatively from the projection(i.e., image) of such geodesics. The formula will be very useful for problems in general relativity,civil engineering, and robotic navigation.

1. Introduction

In many propagation problems in physics and engineering, it becomes necessary to compute

the Gaussian curvature of a two-dimensional surface. In physics, this becomes necessary in

the applications of general relativity, where it is sometimes desired to calculate the Gaussian

curvature at a point in space from the observed geodesic paths of planets or light rays [1, 2]. In

engineering, engineers who are involved in the design of structures such as geodesic domes

frequently require a practical formula for computing the Gaussian curvature, where relations

exist between the Gaussian curvature at any point on the surface of the structure and the

stability of such a structure [3]. In certain other engineering applications, such as computer

vision and robotic navigation, engineers sometimes find themselves facing the complicated

problem of having to compute the Gaussian curvature of a surface in order to calculate 3-

dimensional depth data (or range) [4–6].From the basic principles of differential geometry, the Gaussian curvature G at any

point of a two-dimensional surface S is given by

G = k1k2, (1.1)


S

S′

O

p

Figure 1: A general curve S that is embedded in a surface of revolution and a copy S′ that is separatedfrom S by a small rotation.

where k1 and k2 are the maximum and the minimum normal curvatures [6]. Unfortunately,

in many practical situations, k1 and k2 are simply unknown. In the following section, we

will derive a formula for computing the Gaussian curvature at any point on a surface by

knowledge of two close geodesics on the surface, or alternatively from the projection (i.e.,

image) of such geodesics (this is very important in applications such as general relativity and

robotic navigation, where no direct knowledge of the geodesics exists, but only an image of

the geodesics is available). A simple test of the formula is given in Section 3 (the test shows

that G, as computed from the formula, must vanish in an Euclidean 2-space). In Section 4, it

is proven that the Gaussian curvature is a projective invariant and hence can be calculated

from any projected image of two geodesics.

2. Calculation of the Gaussian Curvature from Geodesic Deviation

It is well known that any general 2-dimensional surface is topologically equivalent at any

given point to a surface of revolution [6]. Hence, two close geodesics on the surface, when

considered only within a small surface patch, can be treated as embedded in a surface

of revolution. Such curves, however, will not necessarily be geodesics in the surface of

revolution. Consider now a surface of revolution, where the smooth curve S is a general

curve that is embedded in the surface (Figure 1). S′ is a copy of S that is obtained by rotating

S through a small angle θ.

pr is a position vector, defined over a circular ring passing by S-S′. Let us select two

parameters u and v, such that u varies as we travel along the curve S, but v is constant, and

v varies as we pass from one curve to another, but u is constant. Obviously,

u = s, v = θ, (2.1)


where θ is the rotation angle of the axis from S to S′. Given such parameters on any surface

in space, it can be shown that [7]

(∇2

sθδρ)dθ −

(∇2

θsδρ)dθ =

∑ζ,μ,ν

Rρ

ζμνδζδνpμ, (2.2)

where δr is the unit tangent vector to the curve, Rabcd

is the mixed curvature tensor [7], and

the symbol ∇ is the covariant derivative operator [6, 7]. If the curve S was a geodesic in the

surface, we must have had [6, 7]

∇sδr = 0, (2.3)

since the covariant derivative of the unit tangent vector to a geodesic vanishes along the curve

[6, 7]. Since S is a general curve, however, then ∇sδr will be the components of a vector of

finite length, normal to the vector δr [6]. On the other hand, due to circular symmetry in a

surface of revolution, the vector ∇sδr , clearly, is parallel transported [6, 7] along a circular

ring in the surface. Hence, we must conclude that

∇θ(∇sδr) = ∇2

θsδr = 0, (2.4)

at any point on S. Further, given the parameters s and θ, it can be shown that [7]

(∇2

sθδr)dθ = ∇2

spr, (2.5)

for any 2-dimensional surface. From (2.4) and (2.5), (2.2) is rewritten as

∇2sp

ρ =∑ζ,μ,ν

Rρ

ζμνδζδμpν, (2.6)

where ζ, μ, ν = 1, 2. Moreover, it has be shown that [7]

Rabcd

= G(δac gbd − δa

dgbc

), (2.7)

for a smooth 2-dimensional manifold, where G is the Gaussian curvature, δab

is the Kronecker

delta, and gab are the components of the metric tensor at any point on the surface. Substituting

from (2.7) into (2.6) and carrying out the summation, we obtain

∇2sp

ρ +Gpρ = Gδρ∑μ,ν

gμνδμpν, (2.8)

where we have used the identity

∑μ,ν

gμνδμδν = 1. (2.9)


Axis of rotation

O

S′

S

r′

r

Figure 2: Unit tangent vectors to S and S′, respectively.

Equation (2.8) is analogous to the equation of geodesic deviation [6, 7]. Once again, if S was

a geodesic in the surface, we must have had an orthogonality condition

∑μ,ν

gμνδμpν = 0, (2.10)

and (2.8) would have reduced to the well-known equation of deviation of two geodesics in

a Riemannian 2-manifold. Equation (2.8), in its given form, will not allow the computation

of the Gaussian curvature G, since the metric tensor components, as well as all the covariant

derivatives on the surface, are unknown. However, (2.8) can be further reduced as follows:

for an infinitesimal rotation dθ,

pr =∂xr

∂θdθ. (2.11)

Thus,

∂pr

∂s=

∂δr

∂θdθ. (2.12)

Now,

δrS′ = δr

S +∂δr

S

∂θdθ

= δrS +

∂pr

∂s,

(2.13)

where δrS, δr

S′ are unit tangent vectors at S and S′, respectively, separated by a rotation dθ

(Figure 2).


Given that, for any vector pr on the surface [7],

∇spρ =

∂pρ

∂s+∑μ,ν

Γρμνpμ∂xν

∂s, (2.14)

where Γabc

is a Christoffel symbol of the second kind, we can always select coordinates such

that Christoffel symbols vanish at the origin [6, 7] (e.g., we can select coordinates on the

surface, at the location of the vector pr). Then, the vector∑

μ,νΓρμνp

μδν is generally very small

in the vicinity of the origin, and can be neglected (i.e., a linear approximation of ∇spρ is

assumed here. This approximation will be further justified in the following discussion and in

Section 3). Therefore, let

ηr =(δrS′ − δr

S

)=

∂pr

∂s≈ ∇sp

r. (2.15)

Further, let us define the deviation angle, ψ, as the angle between the two unit tangent vectors

δrS, δr

S′ , at any point along the curve S. Generally, the angle between two curves is given by

[7]

cosψ =∑μ,ν

gμνdxμ

ds· dx

ν

ds′, (2.16)

but since s = s′ is the length of the curve, and having δr = dxr/ds, we can write

cosψ =∑μ,ν

gμν(δμ)S(δν)S′ . (2.17)

Hence, from (2.15) and (2.17),

∑μ,ν

gμνημην =

∑μ,ν

gμν[δμ

S′δνS′ + δ

μ

SδνS − 2δ

μ

S′δνS

]= 2

(1 − cosψ

)≈∑μ,ν

gμν(∇sp

μ)(∇sp

ν).

(2.18)

We also see that

cosψ ≈∑μ,ν

gμνδμ[δν +∇sp

ν]

≈ 1 +∑μ,ν

gμνδμ(∇sp

ν).

(2.19)


Now, consider (2.8) and the summation

∑μ,ν

gμνpμ(∇2

spν)+G

∑μ,ν

gμνpμpν = G

∑α,β

gαβpαδβ

[∑μ,ν

gμνδμpν

]

= G

[∑μ,ν

gμνδμpν

]2

,

(2.20)

and let

P =√∑

μ,ν

gμνpμpν (2.21)

denote the Euclidean norm of the vector pr ; thus,

d

dsP 2 = 2

∑μ,ν

gμνpμ(∇sp

ν), (2.22)

d2

ds2P 2 = 2

∑μ,ν

gμν[pμ(∇2

spν)+(∇sp

μ)(∇sp

ν)]. (2.23)

Substitution from (2.18), (2.21), and (2.23) into (2.20) gives

1

2

d2P 2

ds2− 2

(1 − cosψ

)+GP 2 = G

[∑μ,ν

gμνδμpν

]2

. (2.24)

To evaluate the last term in (2.24), we rewrite (2.8) as

pρ = δρ∑μ,ν

gμνδμpν − 1

G∇2

spρ. (2.25)

Now, from (2.22) and (2.25), we have

dP 2

ds= 2

∑μ,ν

gμν

⎡⎣δμ∑α,β

gαβpαδβ − 1

G∇2

spμ

⎤⎦(∇spν)

= 2

(∑μ,ν

gμνpμδν

)(∑μ,ν

gμνδμ(∇sp

ν))

− 2

G

∑μ,ν

gμν(∇2

spμ)(

δνs′ − δν

s

).

(2.26)


Each of the components in the last term of (2.26) vanishes identically. To prove this, we

evaluate each of the components for each of the curves, S and S′, by substitution from (2.6).We have

∑μ,ν

gμνδμ(∇2

spν)=∑α

δα

⎡⎣ ∑ρ,ζ,μ,ν

gαρRρ

ζμνδζδμpν

⎤⎦=

∑α,ζ,μ,ν

Rαζμνδαδζδμpν

=∑

α,ζ,μ,ν

G(gαμgζν − gανgζμ

)δαδζδμpν.

(2.27)

A straightforward summation shows that the right-hand side of (2.27) vanishes. We therefore

conclude that ∇2sp

r is in the direction normal to the curve. In plus ∇spr is in the direction of

the tangent to the curve.

Finally, substitution from (2.19) into the first term of (2.26) gives

dP 2

ds= 2

(∑μ,ν

gμνpμδν

)(cosψ − 1

), (2.28)

or

∑μ,ν

gμνpμδν =

−dP 2/ds

2(1 − cosψ

) , (2.29)

and hence (2.24) is further reduced to

G =(1/2)d2P 2/ds2 − 2

(1 − cosψ

)[(dP 2/ds)/2(1 − cosψ)

]2 − P 2, (2.30)

where G is the Gaussian curvature of the surface at the location of the vector P . In the

following section, we prove that the Gaussian curvature given by (2.30) must vanish in an

Euclidean 2-space. In Section 4, it is further proven that G is a projective invariant and hence

can be calculated from any projected image of the curves S and S′.

3. Investigation of the Behavior of G in an Euclidean Space

Here, we illustrate by a simple example that the Gaussian curvature G, given by (2.30), must

vanish in an Euclidean 2-space.

Consider a right circular cone, shown in Figure 3 .

P is the Euclidean norm of the position vector, and θ is the rotation angle (as discussed

in the above text).


r

r

P

s

ss

r

r

Figure 3: A right circular cone and the corresponding geometry.

From the Figure 3 , we see that

P 2 = 2r2(1 − cos θ),

r = s sinα,(3.1)

where s is the length of the generator. Hence,

P 2 = 2(1 − cos θ)(s2sin2α

),

dP 2

ds= 4ssin2α(1 − cos θ).

(3.2)

Thus

d2P 2

ds2= 4sin2α(1 − cos θ). (3.3)

For an infinitesimal rotation, cos θ is expressed by the first two terms of its power series, that

is,

cos θ ≈ 1 − θ2

2!, (3.4)

and thus, (3.3) is written as

d2P 2

ds2= 2sin2αθ2. (3.5)


Furthermore, we can see that

r2θ2 = 2s2(1 − cosψ

), (3.6)

where ψ is the deviation angle, or

(1 − cosψ

)=

1

2sin2αθ2. (3.7)

From (3.5) and (3.7), we have

d2P 2

ds2= 4

(1 − cosψ

). (3.8)

By comparison of (2.30) and (3.8), we immediately see that G must vanish in an Euclidean

2-space. This proves the correctness of (2.30).

4. Proof That the Gaussian Curvature G Is a Projective Invariant

Now, we will reach our final goal by demonstrating that G, formulated by (2.30), can be

measured directly in the image plane.

We rewrite (2.8) as

G = − ∇2sp

ρ

pρ − δρ∑μ,νgμνδμpν

. (4.1)

From (2.27), we saw that ∇2sp

r is a vector in the direction normal to the curve. Now, by using

(2.9) and taking the summation

∑μ,ν

gμνδμpν −

∑μ,ν

gμνδμδν

[∑μ,ν

gμνδμpν

]= 0, (4.2)

it is easy to see that the denominator in (4.1) is also a vector in the direction normal to the

curve.

If we now let

∇2sp

ρ = αvρ,

pρ − δρ∑μ,ν

gμνδμpν = βvρ,

(4.3)

where α, β are scalars, and vρ is a vector in the direction normal to the curve, then

G = −αβ. (4.4)


Now, consider the orthographic projection of the two vectors in (4.3), written as

∇2sp

ρ =∑σ

Jρσ

(∇2

spσ)= α

∑σ

Jρσv

σ,

pρ − δρ∑μ,ν

gμνδμpν =∑σ

Jρσ

(pσ − δσ

∑μ,ν

gμνδμpν

)

= β∑σ

Jρσv

σ,

(4.5)

where Jab

is a transformation Jacobian between a coordinate system on the surface and a

coordinate system in the image plane.

If measured in the image plane, the Gaussian curvature G is now given by

G = − ∇2sp

ρ

pρ − δρ∑μ,νgμνδμpν

= −αβ, (4.6)

as we can easily see from (4.5).Hence,

G = G. (4.7)

The Gaussian curvature is therefore a projective invariant. It should be noted that, while

orthographic projection is assumed, the image plane may be placed in any arbitrary position

with respect to the curve S, and for all such positions, the Gaussian curvature K holds the

same numerical value. Equation (2.30) should be used in the image plane to obtain a correct

measurement of K.

References

[1] S. K. Blau, “Gravity probe B concludes its 50-year quest,” Physics Today, vol. 64, no. 7, pp. 14–16, 2011.[2] E. G. Bakhoum and C. Toma, “Relativistic short range phenomena and space-time aspects of pulse

measurements,” Mathematical Problems in Engineering, vol. 2008, Article ID 410156, 20 pages, 2008.[3] T. Rothman, “Geodesics, domes, and spacetime,” in Science a la Mode, Princeton University Press, 1989.[4] R. O. Duda and P. E. Hart, Pattern Classification and Scene Analysis, Wiley, New York, NY, USA, 1973.[5] B. K. Horn, Robot Vision, The MIT Press, 1987.[6] M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, NJ, USA,

1976.[7] J. L. Synge and A. Schild, Tensor Calculus, Dover, New York, NY, USA, 1978.


Research ArticleSolving Linear Coupled FractionalDifferential Equations by Direct OperationalMethod and Some Applications

S. C. Lim,1 Chai Hok Eab,2 K. H. Mak,1 Ming Li,3 and S. Y. Chen4

1 Faculty of Engineering, Multimedia University, Selangor Darul Ehsan, 63100 Cyberjaya, Malaysia2 Department of Chemistry, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand3 School of Information Science & Technology, East China Normal University, No. 500, Dong-Chuan Road,Shanghai 200241, China

4 College of Computer Science, Zhejiang University of Technology, Hangzhou 310023, China

Correspondence should be addressed to S. C. Lim, [email protected]

Received 20 July 2011; Accepted 7 September 2011


Copyright q 2012 S. C. Lim et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

A new direct operational inversion method is introduced for solving coupled linear systems ofordinary fractional differential equations. The solutions so-obtained can be expressed explicitly interms of multivariate Mittag-Leffler functions. In the case where the multiorders are multiples ofa common real positive number, the solutions can be reduced to linear combinations of Mittag-Leffler functions of a single variable. The solutions can be shown to be asymptotically oscillatoryunder certain conditions. This technique is illustrated in detail by two concrete examples, namely,the coupled harmonic oscillator and the fractional Wien bridge circuit. Stability conditions andsimulations of the corresponding solutions are given.

1. Introduction

Fractional differential equations are well suited to model physical systems with memory or

fractal attributes. This is particularly true in the fields of condensed matter physics, where

fractional differential equations have been used to model various anomalous transport and

relaxation phenomena [1–9]. Coupled fractional differential equations (CFDEs) of nonlinear

type are widely used in studying various chaotic systems [10] such as the Lorentz system

[11], fractional Chuah’s circuit [12], fractional Rossler system [13], and fractional Duffing

system [14]. Since in most cases no analytic solutions for such nonlinear CFDEs exist, it is

necessary to resort to numerical approximations and simulations [15–21]. Even for linear


CFDEs with unequal multiorders, analytic solutions, if they exist, are difficult to obtain and

very often numerical methods have to be used.

In this paper we introduce a direct operational method to solve a system of linear

inhomogeneous CFDEs. We will restrict our discussion to a system of linear nonhomoge-

neous ordinary differential equations of arbitrary fractional-orders. These equations based

on two types of fractional derivatives will be considered, namely, the Caputo and Riemann-

Liouville fractional derivatives. The main idea is to reexpress the coupled fractional equations

by incorporating the initial conditions based on the definitions of these derivatives. The

solutions obtained can be expressed in terms of multivariate Mittag-Leffler functions. When

each order of the CFDEs is an integer multiple of a certain common real positive number, it

is possible to further reduce the solutions to the single-variate Mittag-Leffler functions. For

such cases, we study the conditions for the existence of asymptotic periodic solutions.

In the next section we consider two types of coupled fractional differential equations of

Caputo and Riemann-Liouville type, and a direct operational method is introduced to solve

these equations. Subsequent sections deal with the applications of the coupled fractional

differential equations to two physical systems, namely, the coupled fractional oscillator and

the fractional Wien bridge circuit, as examples to illustrate the proposed method.

2. Linear-Coupled Fractional Differential Equations

We consider two types of fractional derivatives [22–26]:

Caputo Dα∗f(t) = Im−αDmf(t), (2.1a)

Riemann-Liouville Dα#f(t) = DmIm−αf(t), (2.1b)

where the fractional integral is defined for γ > 0 as

Iγf(t) =1

Γ(γ) ∫∞

0

(t − τ)γ−1f(τ)dτ. (2.2)

When referring to either definition, we simply use the notation Dα.

Let us consider a linear-coupled system of inhomogeneous fractional differential equa-

tions of the form

DαX(t) = B(t) +AX(t), (2.3)

where X = (x1, . . . ,xn) and B = (b1, . . . , bn) are vectors of dimension n, A = (aij), i, j = 1, . . . , n

is an (n×n)-matrix, and Dα is the fractional differential operator given by the diagonal matrix

operator:

Dα =

⎛⎜⎜⎜⎜⎜⎜⎝Dα1 0 · · · 0

0 Dα2 · · · 0

......

. . ....

0 0 · · · Dαn

⎞⎟⎟⎟⎟⎟⎟⎠. (2.4)


The orders αi are real positive numbers with mi − 1 < αi < mi, where mi is a positive integer

for each i = 1, 2, . . . , n. The boundary conditions for (2.3) are given by

[Dki∗ xi

](0) = cki∗i for ki = 0, 1, 2, . . . , (mi − 1), (2.5a)[

Dαi−ki# xi

](0) = cki#i for ki = 1, 2, . . . , mi, (2.5b)

and i = 1, 2, . . . , n. We remark that the mi’s generally differ in value. Let its maximum and

minimum values be denoted by m and mo, respectively.

For a single fractional differential equation, its solution can be obtained by integral

transform methods such as the Fourier, Laplace, and Mellin transforms (see, e.g., [24, Chapter

4]). However, in the case of a system of CFDEs, it is necessary to employ specific techniques

appropriate to the given problem, that is, the form of matrix A and the type of fractional

derivative involved. There exist several methods (see [24, Chapters 5 and 6]) for solving

such problems. Here we want to develop a technique which is more direct, similar to Green’s

function method.

Let the operator be

L = Dα −A. (2.6)

The solution of (2.3) can then be expressed as

X(t) = L−1B. (2.7)

Unfortunately, the inverse of L may not exist. However, the right-inverse G does exist for

both the Riemann-Liouville and Caputo cases with

LG = 1/=GL. (2.8)

The main task now is to determine the right-inverse of L associated with (2.3) for both

Riemann-Liouville and Caputo fractional derivatives. We remark that our treatment is rather

formal, aiming mainly to provide an alternative direct operational method to the usual

Laplace transform technique in solving CFDEs. In particular, the existence of solutions will be

assumed, and all operators considered are assumed to be well defined in a certain appropriate

function space.

2.1. The Right-Inverse Operator

For the Caputo derivative Dα∗ with arbitrary α and m,

IαDα∗ f(t) = IαIm−αDmf(t) = ImDmf(t)

= f(t) −m−1∑k=0

tk

Γ(k + 1)

[Dkf

](0),

(2.9a)


and similarly for the Riemann-Liouville derivative,

IαDα#f(t) = IαDmIm−αf(t)

= f(t) −m∑k=1

tα−k

Γ(α − k + 1)

[Dα−k

# f](0).

(2.9b)

Applying the fractional integral operator Iα to (2.3) and using the initial conditions (2.5a) and

(2.5b) gives

X = IaAX + IaB +W, (2.10)

where W is given by

w∗i =mi−1∑k=0

tk

Γ(k + 1)ck∗i,

w#i =mi∑k=1

tα−k

Γ(α − k + 1)ck#i.

(2.11)

Let

Q = [1 − IαA]. (2.12)

Now by rearranging (2.10) one gets

QX = IαB +W. (2.13)

The operator Q has an inverse K. One possible representation of K is given by

K =∞∑0

(IαA)p. (2.14)

This form may not be a simple one, since in most cases Iα and A do not commute. However,

the verification is straightforward:

QK = KQ =∞∑p=0

(IαA)p −∞∑p=0

(IαA)p+1 = 1. (2.15)

The other possible representation is

K = ΨQ∗, (2.16)


where Q∗ is the adjoint of Q, that is, Q∗Q = detQ, and the Ψ is the inverse operator of detQsuch that ΨdetQf(t) = f(t). It is quite simple to verify that this representation is the inverse

of Q, and this will be shown in Section 4.

Now we define

G = KIα. (2.17)

One can easily verify that G is the right inverse of L. Thus, the solution is given by

X = GB +KW. (2.18)

The detailed evaluation can be carried out by using different techniques, a few of which will

be considered here.

2.2. System with Constant Inhomogeneous Terms

When the inhomogeneous term is a constant, we can absorb this term in the following way,

though it may not be immediately obvious. For the Caputo case, let

X∗ = X∗ −A−1B, (2.19)

with

x∗i(0) = x∗i(0) +(a−1

)ijbj . (2.20)

The initial conditions have to be transformed accordingly and they become

c0∗i = c0

∗i +(a−1

)ijbj . (2.21)

In the Riemann-Liouville case, we cannot absorb the term B into X as in the Caputo

case. However, if one compares the initial condition terms w#i and the term IαB, one can see

that if we modify

w#i =mi∑k=0

tα−k

Γ(α − k + 1)ck#i, (2.22)

with c0#i = bi, then the solution can be written as if it is a solution of a homogeneous linear

equation.

Thus the inhomogeneous linear fractional differential equation with constant source

term bi can be transformed into a homogeneous linear fractional differential equation. The

solution of the transformed equation can then be written as

X = KW. (2.23)


When bi is time dependent with power up to mi, the above modification can still apply to

the Caputo case. In the Riemann-Liouville case, if bi(·) are analytic functions, then the same

modification as above holds if (2.22) is altered with the summation from −∞ to mi − 1.

In subsequent sections, we will consider the solution of (2.3) according to the above

modifications.

3. System with Equal Fractional Orders

In the case where all αi = α, then Iα = Iα1, and

K =∞∑p=0

AnInα. (3.1)

It would be convenient if we introduce the Mittag-Leffler function with matrix argument

[27–29]:

Eα,β(Z) =∞∑n=0

Zn

Γ(nα + β

) . (3.2)

Then

Kδ(t) = t−1Eα,0(Atα). (3.3)

Here we have used the following definition of the Dirac delta function:

δ(t) = limε→ 0

tε−1+

Γ(ε). (3.4)

The matrix A can always be decomposed into Jordan normal form. However, we consider

only the case where it can be diagonalized:

Λ = P−1AP,

K = P−1KP,(3.5)

with eigenvalues λj as the diagonal elements of Λ, and

Kijδ(t) = δijt−1Eα,0

(λjt

α), (3.6)

X∗ =m−1∑k=0

PtkEα,k+1(Λtα)P−1Ck∗ ,

X# =m−1∑k=−1

Ptα−k−1Eα,α−k(Λtα)P−1Ck# .

(3.7)


Thus, the solutions of the fractional differential equation under consideration based on both

types of fractional derivatives are simply linear combinations of Mittag-Leffler functions.

3.1. Coupled Oscillator with Equal Fractional Orders

Here we demonstrate our method by considering a linear-coupled oscillator system given by

Dα1x1(t) = −ω2x1(t) +ω2(x2(t) − x1(t)),

Dα2x2(t) = −ω2x2(t) +ω2(x1(t) − x2(t)),(3.8)

where ω2 and ω2 are nonnegative real numbers. The initial conditions are

x∗j(0) = c0∗j , Dx∗j(0) = c1

∗j ,

Dαj−1x#j(0) = c0#j , Dαj−2x#j(0) = c1

#j .(3.9)

Thus

A = −(ω2 +ω2 −ω2

−ω2 ω2 +ω2

). (3.10)

For simplicity, we use the following notation:

a11 = a22 = −(ω2 +ω2

)= −η,

a12 = a21 = ω2 = ε.

(3.11)

We first consider the simpler case with α1 = α2 = α in this section. Since A is symmetric, it can

be diagonalized to give two eigenvalues:

λ± = −η ± ε =

⎧⎨⎩−ω2

−ω2 −ω2.(3.12)

Both eigenvalues are nonpositive and

P =1√2

(1 1

−1 1

). (3.13)


0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

t

= 1.3x(t)

(a)

0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

t

= 1.5

x(t)

(b)

= 1.7

0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

t

x(t)

(c)

= 1.9

0 5 10 15 20 25 30

−1

−0.5

0

0.5

1

t

x(t)

(d)

Figure 1: Coupled oscillator with equal fractional orders. Parameters: ω = 1, ω = 0.5, x∗1(0) = 1.0, x∗2(0) =0, Dx∗1(0) = 0, Dx∗2(0) = 0.1. Legend: x∗1 (solid line), x∗2 (dashed line).

Just as we have shown in (3.7), the solutions are again linear combinations of Mittag-Leffler

functions:

x∗1 =(c0∗1 + c0

∗2

)Eα,1(λ−tα) +

(c0∗1 − c0

∗2

)Eα,1(λ+t

α) +(c1∗1 + c1

∗2

)tEα,2(λ−tα)

+(c1∗1 − c1

∗2

)tEα,2(λ+t

α),(3.14a)

x#1 =(c1

#1 + c1#2

)tα−1Eα,α(λ−tα) +

(c1

#1 − c1#2

)tα−1Eα,α(λ+t

α) +(c2

#1 + c2#2

)tα−2Eα,α−1(λ−tα)

+(c2

#1 − c2#2

)tα−2Eα,α−1(λ+t

α).

(3.14b)

Figure 1 shows simulations of the Caputo solution (3.14a) for orders 1 < α ≤ 2. An

interesting feature of fractional oscillators in general is the presence of damping internal to the

system, that is, an inherent decay in the amplitude which is not associated with any external

friction. The variation in the amount of internal damping can be clearly seen as the order

increases. In the limiting cases we obtain exponential decay (α = 1) and undamped oscillation

(α = 2).


4. The Adjoint Method

As we mentioned in Section 2.1, (2.16), the main task now is to calculate the inverse of detQ.

Note that (2.16) can be reexpressed as

K = ψ[1 − IαA]∗, (4.1)

so that

G = ψ[1 − IαA]∗Iα. (4.2)

Also recall that

detQ = det(1 − IαA). (4.3)

We know that any finite dimension determinant can be evaluated; however, it can be easily

obtained only for a few lower-dimensional cases. We will compute it explicitly for a two-

dimensional system.

4.1. Two-Dimensional System

In a two-dimensional system the determinant is easy to calculate, and we have

detQ = 1 − a11Iα1 − a22I

α2 + Iα1+α2 detA. (4.4)

The inverse is given by

ψ =∞∑r=0

[a11Iα1 + a22I

α2 − Iα1+α2 detA]r (4.5)

=∞∑r=0

∑k1+k2+k3=r

r!

k1!k2!k3![a11I

α1]k1[a22Iα2]k2[−Iα1+α2 detA]k3 . (4.6)

Its kernel is the multivariate Mittag-Leffler function of the kind given by (B.3), Appendix B:

ψδ(t) = t−1Eα1,α2,α1+α2,0(a11tα1 , a22t

α2 ,−detAtα1+α2)

= εα1,α2,α1+α2,0(a11, a22,−detA : t).(4.7)

The adjoint of Q is

Q∗ =

(1 − a22I

α2 a12Iα1

a21Iα2 1 − a11I

α1

). (4.8)


In the following we evaluate explicitly only for the 11- and 12-elements, while the 22- and

21-elements can be obtained by just interchanging subscripts. The kernels of K are given by

K11δ(t) = εα1,α2,α1+α2,0(a11, a22,−detA : t) − a22εα1,α2,α1+α2,α2(a11, a22,−detA : t),

K12δ(t) = a12εα1,α2,α1+α2,α1(a11, a22,−detA : t).(4.9)

4.2. The Solutions

The solutions based on the adjoint method are given by

x∗1 =m1−1∑k=0

ck∗1[εα1,α2,α1+α2,k+1(a11, a22,−detA : t)

−a22εα1,α2,α1+α2,α2+k+1(a11, a22,−detA : t)]

+ a12

m2−1∑k=0

ck∗2εα1,α2,α1+α2,α1+k+1(a11, a22,−detA : t),

(4.10a)

x#1 =m1−1∑k=0

ck#1[εα1,α2,α1+α2,α1−k+1(a11, a22,−detA : t)

−a22εα1,α2,α1+α2,α2+α1−k+1(a11, a22,−detA : t)]

+ a12

m2−1∑k=0

ck#2εα1,α2,α1+α2,α1+α2−k+1(a11, a22,−detA : t).

(4.10b)

5. Laplace Transform Method

In this section we briefly discuss how the widely used Laplace transform technique can be

employed to determine Green’s function G and the operator K. There is no intention here to

provide a detailed discussion of this method. Instead, it will be discussed as a complement to

the adjoint method introduced above, so as to allow one to see the relation between the direct

operational method presented here and usual Laplace transform method.

Without loss of generality (see Section 2.2) the system of CFDEs is assumed to be

homogeneous. We begin by calculating the Laplace transform of detQδ(t) using (4.5):

∫∞

0

detQδ(t)e−stdt = 1 − a11s−α1 − a22s

−α2 + detAs−(α1+α2) (5.1)

and the Laplace transform of the adjoint kernel Q∗δ(t):

∫∞

0

Q∗11δ(t)e

−stdt = 1 − a22s−α2 ,

∫∞

0

Q∗12δ(t)e

−stdt = a12s−α1 ,

∫∞

0

Q∗21δ(t)e

−stdt = a21s−α2 ,

∫∞

0

Q∗22δ(t)e

−stdt = 1 − a11s−α1 .

(5.2)


Next, the Laplace transforms of the part related to the initial conditions from (2.11) are

∫∞

0

w∗i(t)e−stdt =mi−1∑k=0

s−k−1ck∗i,

∫∞

0

w#i(t)e−stdt =mi∑k=1

s−αi+k−1ck#i.

(5.3)

Thus the Laplace transforms of the solutions become

x∗1(s) =[1 − a22s

−α2]∑m1−1

k=0s−k−1ck∗1 + a12s

−α1∑m2−1

k=0s−k−1ck∗2

1 − a11s−α1 − a22s−α2 + det As−(α1+α2), (5.4a)

x#1(s) =[1 − a22s

−α2]∑m1

k=1s−α1+k−1ck#1 + a12s

−α1∑m2

k=1s−α2+k−1ck#2

1 − a11s−α1 − a22s−α2 + det As−(α1+α2). (5.4b)

x∗2(s) and x#2(s) can be obtained just by interchanging 1 ↔ 2. From the complexity of the

Laplace transforms, one sees that it is virtually impossible to obtain the analytic solutions

by direct application of the inverse Laplace transform. To obtain the solution of this type

one has to use the Laplace transform of the multivariate Mittag-Leffler function [27–29],which then gives the identity for getting the Laplace inversion of (5.4a) and (5.4b). This is

one main advantage of the direct operational inversion method proposed here as it will give

the solution directly.

Clearly, it is important that both methods produce equivalent solutions. This is verified

explicitly in Appendix C for a 2-dimensional system.

6. Multiple Fractional-Order System

In physics and engineering problems the fractional-orders can often be approximated by

rational numbers, that is, αi = pi/qi, for some pi, qi ∈ . Thus one gets αi = μi/q, where

q is the least common multiple of q1, q2, . . . , qn with some μi ∈ . However, we can also

consider the more general case with αj = μjα0, for some μj ∈ . Here, α0 ∈ + can be either

rational or irrational. In this section we show how such a system of CFDEs with these multiple

fractional-orders can be solved.

Referring to (4.4), if we assign the symbol ξ = Iα0 , the expansion of the determinant

will be a polynomial of order μ = μ1 + μ2 + μ3 + · · · + μn in ξ. By the fundamental theorem of

algebra, it must have in general μ complex roots, that is, ζj for j = 1, 2, 3, . . . , μ. Note that since

all coefficients of the polynomial are real, if any root ζp is complex, its complex conjugate ζpis also a root. That means that the complex roots occur in pairs. For convenience we write

λj = 1/ζj , for j = 1, 2, 3, . . . , μ. We can then factorize the polynomial

detQ =μ∏j=1

(1 − λjξ

). (6.1)


If all roots are distinct, the inverse can be written as a partial fraction:

ψ =μ∑j=1

hj

1 − λjξ

=μ∑j=1

hj

∞∑k=0

λkj I

kα0 .

(6.2)

6.1. Two-Dimensional System

We will explore this method further for two dimensions. Using (4.5) and the adjoint in (4.8),the solution is given by

x∗1 =μ1+μ2∑j=1

hj

{m1−1∑k=0

ck∗1

[tkEα0,k+1

(λjt

α0) − a22t

μ2α0+kEα0 ,μ2α0+k+1

(λjt

α0)]

+a12

m2−1∑k=0

ck∗2tμ1α0+kEα0,μ1α0+k+1

(λjt

α0)}

,

x#1 =μ1+μ2∑j=1

hj

{m1∑k=1

ck#1

[tμ1α0−kEα0 ,μ1α0−k+1

(λjt

α0) − a22t

(μ1+μ2)α0−kEα0,(μ1+μ2)α0−k+1

(λjt

α0)]

+a12

m2∑k=1

ck#2t(μ1+μ2)α0−kEα0 ,(μ1+μ2)α0−k+1

(λjt

α0)}

.

(6.3)

To simplify the problem we consider the case where 0 < max(μ1α0, μ2α0) ≤ 1, that is,

m1 = m2 = 1. The extension to the general case as above is straightforward:

x∗1 =μ1+μ2∑j=1

hj

{c0∗1

[Eα0,1

(λjt

α0) − a22t

μ2α0Eα0,μ2α0+1

(λjt

α0)]

+ a12c0∗2t

μ1α0Eα0,μ1α0+1

(λjt

α0)}

,

x#1 =μ1+μ2∑j=1

hj

{c1

#1tμ1α0−1Eα0,μ1α0

(λjt

α0) − (a22c

1#1 − a12c

1#2

)t(μ1+μ2)α0−1Eα0,(μ1+μ2)α0

(λjt

α0)}

.

(6.4)

Using the following formula:

zqEα,qα+γ (z) = Eα,γ(z) −q−1∑p=0

zp

Γ(pα + γ

) , (6.5)


(6.4) can be written as

x∗1 = f∗1 + g∗1,

x#1 = f#1 + g#1,(6.6)

where

f∗1 =μ1+μ2∑

j

hjfj

∗1, g∗1 =μ1+μ2∑

j

hjgj

∗1, (6.7a)

f#1 =μ1+μ2∑

j

hjfj

#1, g#1 =μ1+μ2∑

j

hjgj

#1. (6.7b)

fj

∗1 =[c0∗1

(1 − a22λ

−μ2

j

)+ a12c

0∗2λ

−μ1

j

]Eα0,1

(λjt

α0), (6.8a)

fj

#1 =[c1

#1λ1−μ1

j −(a22c

01 − a12c

1#2

)λ

1−μ1−μ2

j

]tα0−1Eα0,α0

(λjt

α0). (6.8b)

gj

∗1 = a22c0∗1λ

−μ2

j

μ2−1∑p=0

λp

j tpα0

Γ(pα0 + 1

) − a12c0∗2λ

−μ1

j

μ1−1∑p=0

λp

j tpα0

Γ(pα0 + 1

) , (6.9a)

gj

#1 = −c1#1λ

1−μ1

j

μ1−2∑p=0

λp

j tpα0+α0−1

Γ(pα0 + α0

) +(a22c

1#1 − a12c

1#2

)λ

1−μ1−μ2

j

μ1+μ2−2∑p=0

λp

j tpα0+α0−1

Γ(pα0 + α0

) . (6.9b)

6.2. Solutions with Asymptotic Oscillations

It is clear from the previous expansion of the Caputo terms that for the solution x∗1, as t → ∞,

any possible oscillation arises from the pair of complex roots (λj, λj), while all (negative) real

roots must result in an asymptotic decay. Note that each term in (6.9a) with p > 0 will grow

asymptotically as the power law tpα0 . However, when we combine the terms and reexpress

the equation explicitly as

g∗1 =μ1+μ2∑j=1

hjgj

∗1

= a22c0∗1

μ2−1∑p=0

⎡⎣μ1+μ2∑j=1

hjλp−μ2

j

⎤⎦ tpα0

Γ(pα0 + 1

) − a12c0∗2

μ1−1∑p=0

⎡⎣μ1+μ2∑j=1

hjλp−μ1

j

⎤⎦ tpα0

Γ(pα0 + 1

) ,(6.10)


for the particular case with μ1 = 1, μ2 = 2, one has

g∗1 = a22c0∗1

1∑p=0

⎡⎣ 3∑j=1

hjλp−2

j

⎤⎦ tpα0

Γ(pα0 + 1

) − a12c0∗2

⎡⎣ 3∑j=1

hjλ−1j

⎤⎦ 1

Γ(1)(6.11)

= a22c0∗1

⎧⎨⎩⎡⎣ 3∑

j=1

hjλ−2j

⎤⎦ 1

Γ(1)+

⎡⎣ 3∑j=1

hjλ−1j

⎤⎦ tα0

Γ(α0 + 1)

⎫⎬⎭ − a12c0∗2

⎡⎣ 3∑j=1

hjλ−1j

⎤⎦ 1

Γ(1).

(6.12)

The second term is the only term that grows asymptotically as ∼ tα0 , which does not

contribute since [∑3

j=1 hjλ−1j ] is zero. The verification of this result is given here for the general

case of n-roots. Let us consider general partial fractions:

1

(1 − λ1x)(1 − λ2x) · · · (1 − λnx)=

h1

(1 − λ1x)+

h2

(1 − λ2x)+ · · · + hn

(1 − λnx). (6.13)

We have

h1 + h2 + · · · + hn = 1, (6.14.0)

h1(λ2 + λ3 + · · ·λn) + h2(λ3 + λ4 + · · ·λn + λ1) + · · · + hn(λ1 + λ2 + · · ·λn−1) = 0,

(6.14.1)

... (...)

h1λ2λ3 · · ·λn + h2λ3λ4 · · ·λnλ1 + · · · + hnλ1λ2 · · ·λn−1 = 0. (6.14.n−1)

We can rewrite (6.14.n−1) as

h1

λ1+h2

λ2+h3

λ3+ · · · + hn

λn= 0. (6.15)

Using (6.15) with n = 3 in (6.12), we have

g∗1 = a22c0∗1

⎡⎣ 3∑j=1

hjλ−2j

⎤⎦, (6.16)

which is a constant. Thus for this particular case with the Caputo derivative one can have

asymptotic oscillations. In general, CFDEs based on the Caputo derivatives will not oscillate

asymptotically, since one cannot find any rule for the power law terms to cancel out.

However, this is possible for some special cases under suitable conditions on the elements

of A.

Similar consideration can be given to Riemann-Liouville system. However, now gj

#1

approaches a constant or possible zero as t → ∞ if 0 < max(μ1α0, μ2α0) ≤ 1, and


min(μ1, μ2) ≤ 2. Thus the possible asymptotically stable oscillations are due to the term fj

#1

with its corresponding complex conjugate. If one looks at (6.8b), one can write explicitly the

asymptotic expansion:

tα0−1Eα0,α0

(λjt

α0)=λ(1−α0)/α0

j

α0eλ

1/α0j t + tα0−1O

(1

tα0

)−→

λ(1−α0)/α0

j

α0eλ

1/α0j t +O

(1

t

)

−→λ(1−α0)/α0

j

α0eλ

1/α0j t

.

(6.17)

One has to impose the condition that λ1/α0

j be a purely imaginary number. Let us denote the

inversion of the root by λj = |λj |eiθ; we must have

θ = ±α0π

2. (6.18)

Furthermore, all the other roots that will not give rise to oscillation must have a negative real

part so that their contributions will be asymptotically zero.

Assume that there is only one pair of complex roots that satisfies (6.18). The coefficient

of the exponential in (6.17) after substitution into (6.8b) gives the jth term of (6.7b) as

hj

[c1

#1λ1−μ1

j −(a22c

1#1 − a12c

1#2

)λ

1−μ1−μ2

j

]λ(1−α0)/α0

j

α0= rje

iϕ, (6.19)

and we then have the asymptotic solution:

x1 −→ rjei|λj |t+iϕ + rje

−i|λj |t−iϕ = 2rj cos(∣∣λj

∣∣t + ϕ). (6.20a)

x2 can be evaluated in a similar way; it has the same period but different modulus and phase:

x2 −→ 2r ′j cos(∣∣λj

∣∣t + ϕ′). (6.20b)

We omit the determination of the roots for each system, which can be computed without any

difficulty.

The oscillation condition (6.18) was first derived by Matignon [30] who also showed

that an identical condition existed for the eigenfunction Eα,1(z) of the Caputo derivative. This

will be elaborated in the context of a physical system in Section 7.

7. Wien Bridge System

In this section we apply the solution methods discussed earlier to model a fractional-order

Wien bridge oscillator (Figure 2). The Wien bridge is a common electronic circuit that can

generate a sinusoidal output signal without requiring an oscillatory input. The resistor-

capacitor pairs form a frequency-selective network, hence allowing the selection of output


−

+

R

RC

vo

C

(a)

R

R

C

C

vo

v1(t)

+

−

v2(t)

+

−

(b)

Figure 2: Wien bridge oscillator: (a) circuit schematic with operational amplifier and (b) simplified circuitdiagram for voltage analysis.

sine wave frequency by varying the circuit parameters. Ahmad et al. [31] first proposed

a generalization of this circuit using fractional-order capacitors. Since the authors did not

obtain the solutions explicitly, we briefly show how analytic solutions for such a system

can be obtained within our present framework. Also, we show solutions based on both the

Caputo and the Riemann-Liouville derivatives. (Note that reference [31] does not mention

the type of derivative used; we were informed by one of the authors, Professor Ahmad, that

they used the Riemann-Liouville fractional derivative.)It is well known that a fractional differential equation of order 0 < α < 1 is usually used

to describe relaxation phenomena [2]. In the case of Wien bridge system, however, oscillation

is achieved via the active elements and feedback provided in the circuit (see Section 7.3).In the following we use normalized voltages xi = vi/Vsat where Vsat is the amplifier

saturation voltage and time axes (normalized with respect to time constant τ = RC).Using basic circuit analysis, it can be shown that the capacitor voltages are related via a 2-

dimensional CFDE:

DαX = AX + B, (7.1)

where

A =

(a − 2 −1

a − 1 −1

), B =

(b

b

), (7.2a)

(a, b) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩(0, 1), Kx1 ≥ 1,

(K, 0), −1 < Kx1 < 1,

(0, 1), Kx1 ≤ 1.

(7.2b)


Here K is the amplifier gain (i.e., vo = Kv1). In the linear region of the amplifier, −1 < Kx1 < 1

and (7.2a) simplifies to

A =

(K − 2 −1

K − 1 −1

), B =

(0

0

), (7.3)

Thus we have to solve the homogeneous linear fractional-order differential system with

a11 = K − 2, a21 = K − 1, a12 = −1, a22 = −1. (7.4)

For fractional capacitors, the real orders are restricted to

0 < α1 ≤ 1, 0 < α2 ≤ 1. (7.5)

We remark that the boundary conditions associated with the Caputo derivative seem more

“physical” as they can be verified by experiments, whereas for the Riemann-Liouville case,

the fractional derivative boundary conditions cannot be measured. However, the Riemann-

Liouville operators are popular with mathematicians and theoretical physicists. We can write

the initial conditions explicitly as

x1(0) = c0∗1, x2(0) = c0

∗2,

Dα1−1# x1(0) = c1

#1, Dα2−2# x2(0) = c1

#2.(7.6)

7.1. Solution Using the Adjoint Method

Substituting (7.4) into the solution (4.10a) we get for the Caputo case

x∗1 = c0∗1[εα1,α2,α1+α2,1(K − 2,−1,−1 : t) + εα1,α2,α1+α2,α2(K − 2,−1,−1 : t)]

− c0∗2εα1,α2,α1+α2,α1(K − 2,−1,−1 : t),

x∗2 = c0∗2[εα1,α2,α1+α2,1(K − 2,−1,−1 : t) − (K − 2)εα1,α2,α1+α2,α1(K − 2,−1,−1 : t)]

+ (K − 1)c0∗1εα1,α2,α1+α2,α2(K − 2,−1,−1 : t).

(7.7)

Similarly, for the Riemann-Liouville case, substituting (7.4) into the solution (4.10b), we get

x#1 = c1#1[εα1,α2,α1+α2,α1(K − 2,−1,−1 : t) + εα1,α2,α1+α2,α2+α1−1(K − 2,−1,−1 : t)]

− c1#2εα1,α2,α1+α2,α1+α2−1(K − 2,−1,−1 : t),

x#2 = c1#2[εα1,α2,α1+α2,α2(K − 2,−1,−1 : t) − (K − 2)εα1,α2,α1+α2,α2+α1−1(K − 2,−1,−1 : t)]

− c1#1(K − 1)εα1,α2,α1+α2,α1+α2−1(K − 2,−1,−1 : t).

(7.8)


7.2. Solution Using the Laplace Transform

Substituting (7.4) into the Laplace transform solution (5.4a), we obtain

x∗1(s) =[1 + s−α2]s−1c0

∗1 − s−α1−1c0∗2

1 − (K − 2)s−α1 + s−α2 + s−(α1+α2), (7.9a)

x∗2(s) =[1 − (K − 2)s−α1]s−1c0

∗2 + (K − 1)s−α2−1c0∗1

1 − (K − 2)s−α1 + s−α2 + s−(α1+α2). (7.9b)

Similarly, for the Laplace transform of the solution (5.4b),

x#1(s) =[1 + s−α2]s−α1c1

#1 − s−(α1+α2)c1#2

1 − (K − 2)s−α1 + s−α2 + s−(α1+α2),

x#2(s) =[1 − (K − 2)s−α1]s−α2c1

#2 + (K − 1)s−(α1+α2)c1#1

1 − (K − 2)s−α1 + s−α2 + s−(α1+α2).

(7.10)

In the following subsections, we study the conditions under which asymptotically stable

oscillations are possible for a fractional Wien bridge oscillator and also present numerical

simulations of the capacitor voltages.

7.3. Equal-Order Fractional Wien Bridge

The classical Wien bridge oscillator produces a stable sinusoidal output when its amplifier

gain K = 3. The amplitude and frequency of the sinusoid are a function of the initial

capacitor voltages and the circuit time constant. For a fractional capacitor, the current-voltage

relationship is dependent on both capacitor value and order; hence an additional degree of

freedom is introduced into the Wien bridge circuit. We consider first the simple case with

equal real fractional orders α1 = α2 = α. Equation (7.7) then simplifies to

x∗1 = c0∗1εα,α,2α,1(K − 2,−1,−1 : t) +

(c0∗1 − c0

∗2

)εα,α,2α,α(K − 2,−1,−1 : t),

x∗2 = c0∗2εα,α,2α,1(K − 2,−1,−1 : t) +

[c0∗1(K − 1) − c0

∗2(K − 2)]εα,α,2α,α(K − 2,−1,−1 : t).

(7.11)

To gain further insight into the system’s behaviour, it is advantageous to express the solution

in a simpler form using only 1-parameter Mittag-Leffler functions Eα,1(λitα). From (7.9a),

x∗1(s) = sα−1c0∗1s

α +(c0∗1 − c0

∗2

)s2α − (K − 3)sα + 1

=σ1s

α−1

sα − λ1+

σ2sα−1

sα − λ2, (7.12)

where the σi ∈ are constants to be determined from partial fraction decomposition (see

equivalent method in Section 6). The inverse transform yields

x∗1(t) = σ1Eα,1(λ1tα) + σ2Eα,1(λ2t

α). (7.13)


0 5 10 15 20

−0.4

−0.2

0

0.2

0.4

t

= 0.3

x(t)

(a)

= 0.5

0 5 10 15 20

−0.4

−0.2

0

0.2

0.4

t

x(t)

(b)

= 0.7

0 5 10 15 20

−0.4

−0.2

0

0.2

0.4

t

x(t)

(c)

0 5 10 15 20

−0.4

−0.2

0

0.2

0.4

t

= 1

x(t)

(d)

Figure 3: Caputo model of fractional Wien bridge with equal orders—comparison of phase and amplitudefor capacitor voltages. Parameters: x∗1(0) = x∗2(0) = 0.03. Legend: x∗1 (solid line); x∗2 (dashed line).

The solution for x∗2(t) can be found in a similar manner. For brevity, we present only

numerical simulations of the Caputo solutions (one plot of the Riemann-Liouville solution

is presented for comparison). In order for the Wien bridge to produce sustained oscillations,

we need to impose condition (6.18) on the complex roots λi. For the current system, this

translates to the following expression for K:

K = 3 + 2 cos(απ

2

). (7.14)

Hence, the amplifier gain K is no longer a constant as in the case of the classical Wien bridge

but a function of capacitor order α. Simulations of (7.13) were plotted using Mathematica.

Figure 3 shows plots of x∗1 and x∗2 for α = 0.3, 0.5, 0.7, and 1.0. There is a clear

dependence of waveform amplitude on the fractional-order. The plot for α = 1 corresponds

to the classical Wien bridge (with ordinary capacitors) and is included for comparison. It is

important to keep in mind that the time axes are normalized and oscillation frequency ωα

actually varies with order as

ωα = (RC)−1/α. (7.15)


0 5 10 15 20

−0.4

−0.2

0

0.2

0.4

t

x1(t)

RC = 0.9

(a)

0 5 10 15 20

−0.4

−0.2

0

0.2

0.4

t

x1(t)

= 0.3

(b)

Figure 4: Variation of waveform characteristics for x1(t). Parameters: x∗1(0) = x∗2(0) = 0.05, (a) Capacitororder affects both frequency and amplitude. Legend: α = 0.3 (solid), 0.5 (dashed), 0.7 (dash-dotted), 1.0(dotted) and (b) Time constant affects frequency while amplitude remains constant. Legend: RC = 0.8(solid), 0.9 (dashed), and 1.0 (dash-dotted), and 1.1 (dotted).

0 5 10 15 20

t

= 0.3

−0.4

−0.2

0

0.2

0.4

x1(t)

Figure 5: Comparison of x1(t) waveform for both derivative models. Solution for Riemann-Liouville modeldiverges as t → 0. Parameters: x∗1(0) = x∗2(0) = 0.03. Legend: Caputo (solid line), and Riemann-Liouville(dashed line).

This can be seen in Figure 4(a). It is interesting to note that the values of resistance and

capacitance have no effect on the output waveform amplitude (Figure 4(b)). Hence, the

frequency of oscillation can be controlled by both the value C and order α of the capacitors.

As noted in [31], a clear advantage of this is that high frequencies can be obtained by reducing

the order of the capacitors rather than their value, which can remain sufficiently large.

In Figure 5 we see that the Riemann-Liouville solutions (7.8) are very similar to the

Caputo solutions in terms of frequency and amplitude but differ in phase due to the second

parameter β of the Mittag-Leffler function. Of particular concern is the fact that the former

tends to diverge at the origin since the fractional initial conditions (2.5b) do not correspond to

measurable physical quantities. This is an important distinction between the two definitions

and has to be taken into consideration when modeling physical systems.


7.4. Multiorder Fractional Wien Bridge

The Wien bridge circuit can be further generalized by allowing the orders to assume values

αj = μjα as detailed in Section 6. We use the case α2 = 2α1 as a starting point. Substituting

α1 = α and α2 = 2α into (7.7), (7.9a) and (7.9b), we have

x∗1 = c0∗1[εα,2α,3α,1(K − 2,−1,−1 : t) + εα,2α,3α,2α(K − 2,−1,−1 : t)]

− c0∗2εα,2α,3α,α(K − 2,−1,−1 : t),

x∗2 = c0∗2[εα,2α,3α,1(K − 2,−1,−1 : t) − (K − 2)εα,2α,3α,α(K − 2,−1,−1 : t)]

+ (K − 1)c0∗1εα,2α,3α,2α(K − 2,−1,−1 : t).

(7.16)

As with the equal-order bridge, we express the solution in Laplace domain and use partial

fractions to obtain a more tractable form:

x∗1(s) = sα−1

(s2α + 1

)c0∗1 − sαc0

∗2

s3α − (K − 2)s2α + sα + 1=

3∑k=1

σksα−1

sα − λk, (7.17)

x∗1(t) =3∑

k=1

σkEα,1(λktα). (7.18)

Only solutions with one negative real and two complex-conjugate roots will be of

concern to our present discussion. To justify this, we note that the alternative case of three real

roots is of no physical interest as it does not produce oscillatory solutions. With the exception

of the exponentially decaying term (due to the negative real root, see asymptotic analysis

in Section 6.2), the solution is hence similar to the case with equal capacitor orders; that is,

the output of the fractional Wien bridge can be expressed as a linear combination of Mittag-

Leffler functions. Unfortunately, the relationship between K and α is not as simple as in the

equal-order case (7.14). Although K is still a function of α, its form is sufficiently complex

that a more convenient alternative is to define K implicitly, that is, find α = φ(K), and use

polynomial curve-fitting as shown in Figure 6.

Using the method of least-squares, we obtain a third-order approximation for K:

K ≈ 4.611 − 4.821α2 + 0.008α3. (7.19)

Two restrictions apply to the usable range of K and α. The first is the requirement that

the real root be negative. Plotting the denominator of (7.17) as a function of sα for various

amplifier gains, we obtain the relationship in Figure 7. For 0 < K < K0 ≈ 4.611, we have

λ1 ∈ − and λ2 = λ3 ∈ as required. Therefore, this serves as an upper limit to the amplifier

gain. The lower limit can be determined by recalling that 0 < α2 = 2α1 < 1 so that 0 < α < 0.5.

The result of these restrictions is also shown in Figure 6.

Within the stipulated range, the values of gain calculated from the polynomial curve

(7.19) are sufficiently accurate to create oscillatory solutions, as demonstrated in the following

simulations (Figure 8).


0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

KPossible values for , K

(a)

0 0.1 0.2 0.3 0.4 0.53.4

3.6

3.8

4

4.2

4.4

4.6

K()

(b)

Figure 6: Determination of amplifier gain K: (a) restrictions on possible values of K and α; (b) third-orderleast-squares approximation of K(α) for x ∈ (0, 0.5).

K = 0

K 0

2

4

−2 −1 0 1 2 3

s

Ch

ara

cter

isti

ceq

uati

on

−20

−10

0

10

20

Figure 7: Properties of roots of (7.19) (shown as horizontal intercepts) for parameter K.

As mentioned earlier, it is possible to extend this procedure to any case where one

order is an integer multiple of the other. Consider the solution of x∗1 for α1 = α, α2 = υα:

x∗1(s) = sα−1(sυα + 1)c0

∗1 − s(υ−1)αc0∗2

s(υ+1)α − (K − 2)sυα + sα + 1=

υ+1∑k=1

σksα−1

sα − λk, (7.20)

x∗1(t) =μ+1∑k=1

σkEα,1(λktα). (7.21)

Indeed, unless we are concerned about obtaining the actual Laplace solution in partial

fraction form, the value of K that produces asymptotic oscillations can be found by simply

studying the roots of the denominator in (7.20) (a polynomial in sα of order υ + 1) and

imposing suitable conditions as previously shown so that at least one pair of Mittag-Leffler

terms has an eigenvalue that satisfies (6.18). For example, when υ = 3, a possible solution

contains 4 roots in 2 complex-conjugate pairs. One can adjust the amplifier gain such that


−0.2

−0.1

0

0.1

0.2

t

1 = 0.3, 2 = 0.6, K = 4.18952

0 2 4 6 8 10 12 14

x(t)

−0.2

−0.1

0

0.1

0.2

t

0 2 4 6 8 10 12 14

x(t)

= 0.35, 2 = 0.7, K = 4.04346

−0.2

−0.1

0

0.1

0.2

t

0 2 4 6 8 10 12 14

x(t)

1 = 0.4, 2 = 0.8, K = 3.87881

−0.2

−0.1

0

0.1

0.2

t

0 2 4 6 8 10 12 14

x(t)

1 = 0.45, 2 = 0.9, K = 3.6971

(a)

−0.2

−0.1

0

0.1

0.2

t

0 2 4 6 8 10 12 14

x(t)

1 = 0.5, 2 = 1, K = 3.5

(b)

Figure 8: Caputo model of fractional Wien bridge with α2 = 2α1. The effect of the nonoscillatory termin (7.20) can be observed as an initial offset that decays asymptotically as t → ∞. Parameters: x∗1(0) =x∗2(0) = 0.03, (a) Comparison of phase and amplitude for capacitor voltages. (b) The limiting case of oneordinary capacitor and one semicapacitor (order 1/2). Legend: and x∗1 (solid line), x∗2 (dashed line).

the roots satisfy |θ| = α0π/2 for the first pair and |θ| > α0π/2 for the second pair, hence

resulting in sustained oscillation and asymptotically decaying oscillation, respectively.

8. Concluding Remarks

We have proposed a new direct operational method for solving coupled linear fractional

differential equations of multiorders. This technique provides an alternative way for solving


some linear CFDEs, and the solutions so obtained can be expressed in terms of multivariate

Mittag-Leffler functions. For the special cases where each of the multiorders is an integer

multiple of a real positive number, the solutions can be further reduced to linear combinations

of Mittag-Leffler functions of a single variable. Conditions for asymptotically oscillatory

solutions are considered. Two examples, namely, the coupled fractional harmonic oscillator

and the fractional Wien bridge circuit, are given to illustrate our method. Simulations of

solutions and stability conditions are given. Note that to obtain the solution based on our

method requires the use of the Laplace transform of the multivariate Mittag-Leffler function,

which then gives the identity for getting the Laplace inversion for the solution. This is one

main advantage of the direct operational inversion method proposed here as it will give the

solution directly. We remark that our method does not actually simplify the computational

aspect of obtaining solutions, though intuitively it allows one to obtain the solution in explicit

form.

Here we would like to remark that there were attempts recently to transform CFDEs

with different multiorders into an equivalent system of CFDEs of a single order [32, 33].Such a method again does not reduce the amount of computation necessary to obtain the

solutions; instead, due to the increase in the number of the auxiliary equations in the latter

system, it is actually more tedious to obtain the full solutions. Our view on CFDEs is that, in

general, one still has to use numerical methods to obtain approximate solutions. The point

is to find a method that provides a more efficient way of doing so. We hope to look into

this aspect in a future work. Finally, it will be interesting to consider whether the above

method can be extended to nonlinear CFDEs. One expects that such a generalization will

not be straightforward.

Appendices

A. Mittag-Leffler Function and Related Functions

The Mittag-Leffler function [26, 28] and its generalizations are defined as follows:

Eα(z) =∞∑n=0

zn

Γ(nα + 1),

Eα,β(z) =∞∑n=0

zn

Γ(nα + β

) ,Eγ

α,β(z) =∞∑n=0

(γ)nzn

Γ(nα + β

)n!,

(A.1)

where

(γ)n= γ

(γ + 1

)(γ + 2

) · · · (γ + n − 1)=Γ(γ + n

)Γ(γ) . (A.2)


Note that

(γ)

0= 1, (0)n = 0 for n = 0, (0)0 = 1. (A.3)

Thus we have

E0α,β(z) =

1

Γ(β) . (A.4)

For convenience we define the following functions:

εα,β(λ : t) = tβ−1Eα,β(λtα), (A.5a)

εγ

α,β(λ : t) = tβ−1E

γ

α,β(λtα). (A.5b)

A.1. Asymptotic Expansion of Mittag-Leffler Function [8, 24, 26]

For 0 < α < 2,

Eα(z) = −N∑n=1

z−n

Γ(1 − nα)+ 0

(|z|−N+1

), z −→ ∞,

απ

2< arg(z) < 2π − απ

2,

Eα(z) =e1/α

α−

N∑n=1

z−n

Γ(1 − nα)+ 0

(|z|−N+1

), z −→ ∞,

∣∣argz∣∣ < απ

2.

(A.6)

Similarly one has

Eα,β(z) = −N∑n=1

z−n

Γ(β − nα

) + 0(|z|−N+1

), z −→ ∞,

απ

2< arg(z) < 2π − απ

2,

Eα,β(z) =exp

(z1/α

)α

−N∑n=1

z−n

Γ(1 − nα)+ 0

(|z|−N+1

), z −→ ∞,

∣∣argz∣∣ < απ

2.

(A.7)

B. Multivariate Mittag-Leffler Functions [27–29]

Let us adopt the following notations:

αi ∈ , β ∈ , zi ∈ , pi ∈ 0 = ∪ {0},α = (α1, α2, α3, . . . , αn), z = (z1, z2, z3, . . . , zn), p =

(p1, p2, p3, . . . , pn

),

zp =n∏i=1

zpii , p · α =

n∑i=1

piαi,[p]=

n∑i=1

pi,

(B.1)


and the binomial coefficient is generalized to

αi ∈ ,(k

p

)=

k!∏ni=1pi!

.(B.2)

The multivariate Mittag-Leffler functions is defined as:

Eα,β(z) =∞∑k=0

∑[p]=k

(k

p

)zp

Γ(p · α + β

) . (B.3)

C. Equivalence of Adjoint Method and Laplace Transform Solutions

We demonstrate the equivalence of the time-domain and frequency-domain (Laplace) solu-

tions for the 2-dimensional system as given by (4.10a) and (4.10b) and (5.4a) and (5.4b),respectively. The generalization to systems of higher dimension is straightforward and will

be omitted for brevity. The Laplace transform of the multivariate Mittag-Leffler function in

(A.5a) is

L[εα1,...,αn,β(a1, . . . , an : t)

]=

s−β

1 −∑ni=1 ais−αi

, (C.1)

with β > 0. The transform of (4.10a) and (4.10b) is then

L[x∗1] =

∑m1−1k=0

ck∗1s−(k+1) −∑m1−1

k=0ck∗1a22s

−(α2+k+1) +∑m2−1

k=0ck∗2a12s

−(α1+k+1)

1 − a11s−α1 − a22s−α2 + detAs−(α1+α2),

L[x#1] =

∑m1

k=1ck#1s

−(α1−k+1) −∑m1

k=1ck#1a22s

−(α1+α2−k+1) +∑m2

k=1ck#2a12s

−(α1+α2−k+1)

1 − a11s−α1 − a22s−α2 + detAs−(α1+α2),

(C.2)

which agrees precisely with (5.4a) and (5.4b).

Acknowledgments

The authors would like to thank Professor W. Ahmad for correspondence. S. C. Lim would

like to thank the Malaysian Ministry of Science, Technology and Innovation for the support

under its Brain Gain Malaysia (Back to Lab) Program. Li acknowledges the 973 plan under

the project no. 2011CB302802, and the NSFC under the project Grant nos. 60873264, 61070214.

S. Y. Chen acknowledges the NSFC under the project Grant no. 60870002 and Zhejiang

Provincial Natural Science Foundation (R1110679).


References

[1] R. Kutner, R. A. Pekalski, and K. Sznajd-Weron, Anomalous Diffusion: From Basics to Applications,Springer, New York, NY, USA, 1999.

[2] R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.[3] R. Metzler and J. Klafter, “The random walk’s guide to anomalous diffusion: a fractional dynamics

approach,” Physics Reports, vol. 339, no. 1, pp. 1–77, 2000.[4] P. Arena, R. Caponetto, L. Fortuna, and D. Porto, Nonlinear Noninteger Order Circuits and Systems—An

Introduction, World Scientific, Singapore, 2000.[5] G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, UK,

2008.[6] R. Magin, Fractional Calculus in Bioengineering, Begell House, Redding, Calif, USA, 2006.[7] R. Klages, G. Radons, and I. M. Sokolov, Eds., Anomalous Transport; Foundations and Applications, Wiley-

VCH, New York, NY, USA, 2008.[8] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, UK,

2010.[9] J. Klafter, S. C. Lim, and R. Metzler, Eds., Fractional Dynamics: Recent Advances, World Scientific,

Singapore, 2011.[10] W. M. Ahmad and J. C. Sprott, “Chaos in fractional-order autonomous nonlinear systems,” Chaos,

Solitons and Fractals, vol. 16, no. 2, pp. 339–351, 2003.[11] T. T. Hartley, C. F. Lorenzo, and Qammer, “Chaos in a fractional order Chua’s system,” IEEE

Transactions on Circuits and Systems I, vol. 42, no. 8, pp. 485–490, 1995.[12] I. Grigorenko and E. Grigorenko, “Chaotic dynamics of the fractional Lorenz system,” Physical Review

Letters, vol. 91, no. 3, pp. 034101/1–034101/4, 2003.[13] C. Li and G. Chen, “Chaos and hyperchaos in the fractional-order Rossler equations,” Physica A, vol.

341, no. 1–4, pp. 55–61, 2004.[14] Z. M. Ge and C. Y. Qu, “Chaos in a fractional order modified Duffing system,” Chaos, Solitons and

Fractals, vol. 34, no. 2, pp. 262–291, 2007.[15] K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical

Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002.[16] V. Daftardar-Gejji and A. Babakhani, “Analysis of a system of fractional differential equations,”

Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp. 511–522, 2004.[17] K. Diethelm and N. J. Ford, “Multi-order fractional differential equations and their numerical

solutions,” Computer Methods in Applied Mechanics and Engineering, vol. 154, no. 3, pp. 621–640, 2004.[18] V. Daftardar-Gejji and H. Jafari, “Adomian decomposition: a tool for solving a system of fractional

differential equations,” Journal of Mathematical Analysis and Applications, vol. 301, no. 2, pp. 508–518,2005.

[19] S. Momani and Z. Odibat, “Numerical approach to differential equations of fractional order,” Journalof Computational and Applied Mathematics, vol. 207, no. 1, pp. 96–110, 2007.

[20] V. Daftardar-Gejji and H. Jafari, “Analysis of a system of nonautonomous fractional differentialequations involving Caputo derivatives,” Journal of Mathematical Analysis and Applications, vol. 328,no. 2, pp. 1026–1033, 2007.

[21] Y. Hu, Y. Luo, and Z. Lu, “Analytical solution of the linear fractional differential equation by Adomiandecomposition method,” Journal of Computational and Applied Mathematics, vol. 215, no. 1, pp. 220–229,2008.

[22] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations,John Wiley & Sons, New York, NY, USA, 1993.

[23] S. Samko, A. A. Kilbas, and D. I. Maritchev, Integrals and Derivatives of the Fractional Order and Some oftheir Applications, Gordon and Breach, Armsterdam, the Netherlands, 1993.

[24] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.[25] B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer-Verlag, New York, NY,

USA, 2003.[26] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential

Equations, Elsevier Science, Amsterdam, the Netherlands, 2006.[27] V. Kiryakova, “The multi-index Mittag-Leffler functions as an important class of special functions of

fractional calculus,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1885–1895, 2010.[28] A. M. Mathai, “Some properties of Mittag-Leffler functions and matrix-variate analogues: a statistical

perspective,” Fractional Calculus & Applied Analysis, vol. 13, no. 2, pp. 113–132, 2010.


[29] H. J. Haubold, A. M. Mathai, and R. K. Saxena, “Mittag-Leffler functions and their applications,”Journal of Applied Mathematics, vol. 2011, Article ID 298628, 51 pages, 2011.

[30] D. Matignon, “Stability results for fractional differential equations with applications to controlprocessing,” in Proceedings of the IMACS Multiconference in Computational Engineering in SystemsApplications, vol. 2, pp. 963–968, IEEE-SMC, Lille, France, 1996.

[31] W. Ahmad, R. El-khazali, and A. S. Elwakil, “Fractional-order Wien-bridge oscillator,” ElectronicsLetters, vol. 37, no. 18, pp. 1110–1112, 2001.

[32] W. Deng, C. Li, and Q. Guo, “Analysis of fractional differential equations with multi-orders,” Fractals,vol. 15, no. 2, pp. 173–182, 2007.

[33] C. H. Eab, S. C. Lim, and K. H. Mak, “Coupled fractional differential equations of multi-orders,”Fractals, vol. 17, no. 4, pp. 467–472, 2009.


Research ArticleStudy of the Fractal and Multifractal ScalingIntervening in the Description of FractureExperimental Data Reported by the Classical Work:Nature 308, 721–722(1984)

Liliana Violeta Constantin1 and Dan Alexandru Iordache2

1 Physics Faculty, University of Bucharest, P.O. Box MG-11, 077125 Bucharest, Romania2 Physics Department, University “Politehnica” of Bucharest, Splaiul Independentei,060042 Bucharest, Romania

Correspondence should be addressed to Liliana Violeta Constantin,

[email protected]

Received 9 September 2011; Accepted 4 October 2011


Copyright q 2012 L. V. Constantin and D. A. Iordache. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

Starting from the experimental data referring to the main parameters of the fracture surfacesof some 300-grade maraging steel reported by the classical work published in Nature 308, 721–722(1984), this work studied (a) the multifractal scaling by the main parameters of the slit islandsof fracture surfaces produced by a uniaxial tensile loading and (b) the dependence of the impactenergy to fracture and of the fractal dimensional increment on the temperature of the studiedsteels heat treatment, for the fracture surfaces produced by Charpy impact. The obtained resultswere analyzed, pointing out the spectral (size) distribution of the found slit islands in the frame ofsome specific clusters (fractal components of the multifractal scaling) of representative points ofthe logarithms of the slit islands areas and perimeters, respectively.

1. Introduction: Complexity, Universality, Power Laws, andFractal Scaling

As it is well-known, one of the most important present topics refers to the obtainment of

scientific information about the complex materials and systems [1–3].The main founders of the complexity theory in physics have pointed out [4–7] (see

also the synthesis review [8, 9]) that several completely different complex systems (computer

arrays, complex random (Internet, particularly), robots, networks, social sciences, biology

(with some specific topics: colonies, swarms, immunology, brain, genetics, and proteomics),


economics, mathematics, glasses, agents, and cognition [10–13], etc.) have some common

features centered on their statistical behavior and the corresponding phase transforms [4, 5, 8, 9]and chemical reactions, particularly, as well as of some dynamic aspects [14–16], nonlinear

effects [17], and so forth. It results that these complex systems have certain universality

properties, which—due to their generality (see, e.g., [8])—can be described only by some

specific numbers (the so-called similitude numbers, or criteria [18–20]).How could it be possible to describe dimensional (physical, particularly) quantities

only by numbers? The answer is obtained from the examination of: (a) predictions of

Anderson [4, 5, 8, 9] relative to the “explosive” autocatalytic (exponential) growth following

the spontaneous symmetry breaking inside the specific complex systems (one finds that a

certain dimensional parameter p has to be described by its logarithm: ln p), (b) Dalton’s law

of “defined proportions”, intervening in the theory of chemical reactions (somewhat similar

to the phase transforms) [3]: dξ = −dν1/ν1 = −dν2/ν2 = · · · = +dνN/νN , where the sign

“−” corresponds to substances that disappear during the considered chemical reaction, while

the sign “+” corresponds to the appearing substances, finding that the degree of advance ξ

of the considered reaction can be expressed by means of ln νj , where νj is the amount (e.g.,

number of moles) of one of the substances participating in the chemical reaction, (c) statistical

expression of the thermodynamic entropy (describing the dissipative processes), given by

the Planck-Boltzmann’s expression: S = −k · ln℘, where k is the the Boltzmann’s constant,

where ℘ is the probability density, (d) Claude Shannon’s expression [21–23] of the individual

information quantity: = −a · ln℘ (a = constant).The simplest expression (the zero-order approximation) of the relation between a test

physical parameter t(u) and the uniqueness one u is, of course, the linear expression:

ln t = ln t1 + s · lnu, equivalent to the power law: t(u) = t1 · us. (1.1)

If the uniqueness parameter u corresponds to the size L of the considered complex system,

then the power law (1.1) particularizes into the fractal scaling

t(L) = t1 · Ls. (1.2)

When the relation ln t = f(lnL) is more intricate than the linear one, presenting, for example,

a certain curvature, then the existing experimental data can be divided in some groups of

pairs {tk1, Lk1; . . . tkn, Lkn} so that for each group, a specific linear relation is valid: ln tki =ln t1k+sk ln uki, equivalent to the fractal scaling: tki = t1k ·usk

ki. Because the prepower coefficient

t1k and the power exponent sk depend on the group k of chosen data, it results that the set of

relations {tki = t1k · uskki| k = 1,N} corresponds to a multifractal scaling [24, 25].

Some additional detailed studies of the different types of fractal and multifractal

scaling were accomplished in the frame of works [26–28].

2. Critical Findings Referring to the Work Nature, 308, 721-722(1984)

In 1984, Mandelbrot et al. [29] claimed that the fracture surfaces of metals are fractal (self-

similar) over a wide range of sizes, and introduced the experimental methods named “slit

island analysis” (SIA) and “fracture profile analysis” (FPA). As the large majority of papers

published by Nature (average impact factor 12.86 in 1985 and 24.82 in 1996), the above-

indicated work had a high scientific impact: we identified [30, 31] at least 26 papers and


books published only in the following 10 years (up to 1993, inclusively [30]), studying

the fractal character of the fracture surfaces. Despite of its large impact, the hypothesis of

Mandelbrot et al. [29] was somewhat restricted by the following studies: (a) even the papers

of Underwood [32], Pande et al. [33], Lung and Mu [34], and Huang et al. [35] affirmed

that the fracture surfaces of metals can be approximately considered to possess a certain fractal

character, (b) Underwood and Banerji [32] concluded that the slit island analysis itself was

imperfect in nature as a method for measuring the fractal dimension of fractured surfaces,

(c) Lung and Mu [34] found that the fractal dimension was largely affected by the measuring

ruler employed and postulated the concept of inherent measuring ruler, (d) Huang et al.

[35] pointed out that how to determine the fractal dimension of a fractured surface has

always been a problem of “argument”, and (e) Williford [36] tried to explain the obtained

results in terms of multifractals, but this explanation seemed not to be satisfactory for some

experimental results [37, 38], and so forth.

The detailed numerical analysis accomplished in the frame of this work pointed out

that the main missing elements of work [29] are the following:

(a) no justification of the indicated values of fractal dimensional increment from the

capture of Figure 1 [29],

(b) no analysis of the multifractal scaling of the logA = f(log P) dependence corre-

sponding to the slit islands areas and perimeters, respectively,

(c) the regression line: impact energy = f (fractal dimensional increment) from

Figure 3 [29] is obviously inexact, and it does not consider the corresponding

possible nonlinear dependence,

(d) the dependence of the fractal dimensional increment on the temperature of the heat-

treatment of the 300-grade maraging steel Charpy impact specimens studied by

Figure 3 [29] was not studied.

3. Procedure Intended to the Evaluation of the Fractal Dimension ofthe Slit Islands

In order to evaluate the slope of the regression line logA = f(log P), the numerical values

of the decimal logarithms logA, logP of the slit islands areas and perimeters, respectively,

(indicated by Figure 1 [29]) were firstly evaluated by means of the scanning procedure [39].We obtained s ≡ D′ ∼= 1.6225 = D − 1 = iF , in considerable disagreement with the values 1.28

and 1.26 indicated by the capture of Figure 1 [29].Starting from the interpretation provided by the monograph [40, pages 64–65] of

the experimental data obtained by means of the slit island method, according to whom

(a) the cross-section of area A of the fractured material is not fractal; therefore, this area is

proportional to the square of the slit island average radius R: A ∝ R2, while (b) the perimeter

P of the slit island is really fractal (of dimension D−1, where D is the dimension of the fracture

surface); therefore, P ∝ RD−1, and we have found that A ∝ P 2/(D−1) and the slope of the

logA = f(log P) plot is: s = 2/(D − 1). From this relation, we obtained, in good quantitative

agreement with the indicated fractal dimensional increment, iF = D − 1 values indicated in

the caption of Figure 1 [29] as well as with the results obtained by other similar works (e.g.,

[41]).


Table 1: Main features of the fractal: logA = c0+c1 logP and multifractal (parabolic): logA = co+c1 logP +c2(logP)2 scalings of the parameters A, P of the slit islands of fracture surfaces reported by Figure 1 [29].

The type of the logA = f(logP) correlation(scaling)

Regression line(fractal scaling)

Parabolic correlation(multifractal scaling)

c0 −1.776 −1.5686

c1 1.6265 (average slope) +2.39074

c2 0 −0.16965

Correlation coefficient 0.9655 0.9792

Average relative error (%) for all 41 studied slitislands

7.540% 7.346%

Average relative error for the 6 extreme (first 3and last 3) representative points of Figure 1[24, 25]

10.134% 8.733%

Apparent fracture surface fractal dimension:DM = 1 + slope

2.6265 2.4487 · · · 3.0514

Fracture surface fractal dimension according toour considerations (this work): Ds = 1 + 2/slope

2.23 1.975 · · · 2.380

4. Study of the Multifractal Scaling of the logA = f(logP) Dependence

Taking into account that all 6 extreme (first 3 and last 3) representative points of Figure 1

[29] are located under the regression line, we assumed that a nonlinear (even a parabolic)logA = f(log P) expression could agree better with the experimental data reported by this

figure. To check this assumption, we used the procedures of the well-known classical gradient

method [42–44] in order to find the parameters of the parabolic correlation

logA = c2

(log P

)2 + c1 logP + c0, (4.1)

which ensure the best fit of the experimental data of Figure 1 [29].The obtained results are synthesized by Table 1.

One finds that the explanation given by Williford [36], in terms of multifractals, of the

experimental data referring to the fracture surfaces is more realistic than the initial Mandel-

brot’s hypothesis. We have to underline that this explanation (multifractals) is supported also

by the results obtained by Carpinteri and Chiaia [24, 25] especially for concrete samples.

The new versions of Figures 1 and 3 [29], after our numerical conversion (using the

method of work [39]) of the experimental data indicated by these figures and the following

parabolic fit (for the logA = f(log P) pairs), and the least-squares fit (for the fractal dimen-

sional increment = f (impact energy)) are presented below in the frame of our Figures 1 and

2.

5. Towards the Fractal Components of the Multifractal Set of FractureSurfaces Slit Islands of the Maraging Steels Studied by [29]

Taking into account the practical continuous change of the slope of the logA = f(log P)plot, the definition of the fractal components of the multifractal set of fracture surfaces

slit islands is strongly related to the experimental accuracy of the logA, log P parameters. As

the accuracy of these parameters is not known, a certain image on these fractal components


100

101

102

103

104

105

100 101 102 103 104 105

Are

a(μ

m2)

Perimeter (μm)

Figure 1: The new (improved) version of Figure 1 [29] after [29], the numerical conversion (using method[39]) of the corresponding experimental data and the parabolic fit of the logA = f(logP) pairs.

0 50 100 150 200

0

0.1

0.2

0.3

Fra

ctal

dim

ensi

on

al

incr

emen

t

Impact energy (J)

430 C

370 C

340 C

360 C

315 C

300 C

Figure 2: The new (corrected) version of Figure 3 [29] after the numerical conversion (using the method[39]) of the corresponding experimental data, and the least-squares fit of the fractal dimensional incre-ment = f (impact energy) dependence data.

can be obtained starting from the identification of clusters of representative points logA,

log P .

We defined the logA, log P clusters starting from the distances between the nearest

representative points in the space logA, log P . If the distance between the nearest represen-

tative points belonging to 2 neighbor sets is considerably larger than that for the nearest such

points belonging to each set, these neighbor sets correspond to the desired clusters.

Using this procedure, we have identified 6 clusters in the logA, log P space of

Figure 1 [29], defined by the pairs of logA, log P coordinates corresponding to the marginal

representative points of each cluster.

These clusters of representative points in the logA, log P space are gathered around

some average (logP)i values (i = 1,N). For each cluster of representative points, the local


Table 2: The “spectral” (size) distribution of the clusters of representative points in the logA, logP plotinvolved by Figure 1 [26–29] as representing the fractal components of the multifractal scaling of thelogA = f(logP) dependence.

Interval of fractalincrement values

(0.975; 1.014) (1.034; 1.140) (1.123; 1.213) (1.205; 1.297) (1.283; 1.392) (1.392; 1.747)

Pairs of values ofthe slit islands PPerimeter (μm)and area (μm2)

(10.00; 5.62)· · ·

(17.15; 11.55)

(22.07; 27.38)· · ·

(74.99; 103.7)

(62.64; 237.1)· · ·

(154.0; 421.7)

(143.3; 930.6)· · ·

(316.2; 2458)

(283.9; 4068)· · ·

(649.4; 4371)

(649.; 23714)· · ·

(4698;83536)

Number ofrepresentativepoints in Figure 1[1–3]

3 8 10 8 6 6

Percentage ofrepresentativepoints

7.318% 19.512% 24.390% 19.512% 14.634% 14.634%

slope si = 2c2(logP)i + c1 of the multifractal scaling logA = c2(logP)2 + c1 logP + c0 and

the local fractal dimensional increment iFi = 2/si were evaluated, the obtained results being

synthesized by Table 2. The synthesis of these clusters features as well as the corresponding

fractal dimensions (or increments) corresponding to each cluster (as a specific representative

of the fractal components of the multifractal set of fracture surfaces slit islands) is presented

by Table 2.

One finds that the small values of the fractal dimension correspond to slit islands of

relatively small dimensions (perimeters of the magnitude order of μm), corresponding to

fracture surfaces not too curly, and even involving some surface breaks (which could explain

eventually the seldom values little less than 2 of the fractal dimension corresponding to some

small parts of the fracture surface).

6. Study of the Fractal Dimensional Increment ofthe Fracture Surfaces Produced by Impact onthe Temperature of the Steels Heat Treatment

Unlike the fracture surfaces produced by uniaxial tensile loading, whose characteristic

parameters were reported for the 300-grade maraging steel by Figure 1 [29], the last part

of this work (Figure 3 [29]) reports the main features of the fracture surfaces produced by

impact.

The evaluation of the slope s and intercept i of the regression line Eimp(J) = s · theat + i

describing the impact energy to fracture in terms of the temperature of the studied steels

heat treatment led us to the results: s ∼= −1.069 J/◦C, i ∼= 494.21 J with a correlation coefficient

r ∼= −0.9563 and a square mean relative error of 10.05%.

Similarly, the evaluation of the slope s′ and of the intercept i′ of the regression line

iF = s′ · theat + i′ describing the fractal dimensional increment of the fracture surface produced

by impact in terms of the temperature of the studied steel heat treatment leads to the results

s′ ∼= 1.25 · 10−3 ( ◦C)−1, i′ ∼= −0.260, with a correlation coefficient r ′ ∼= 0.9243 and a square mean

relative error of 10.971%.

One finds that, as it was expected, (a) the impact energy to fracture decreases (approx-

imately linearly, up to 450◦C) with the temperature of the studied steels heat treatment and


(b) the fracture surface deformation (from its ideal planar shape), measured by its fractal

dimensional increment, increases with the temperature of the heat treatment.

It was possible to obtain also the parameters of a more exact (than that performed in

the frame of Figure 3 [26–29]) regression line Eimp(J) = s′′ · iF + i′′ describing the dependence

of the impact energy to fracture on the corresponding fracture surface deformation (fractal

dimensional increment) s′′ ∼= −781.47 J, i′′ ∼= 258.40 J, correlation coefficient r ′′ ∼= −0.9442, and

square mean relative error 13,31%, but we consider these last results as less important than

the above-indicated ones, referring to the Eimp and iF = f(theat) dependencies.

7. Investigations on the Compatibility with the Experimental Data ofthe Fractal/Multifractal Descriptions of the Fracture Parameters

Taking into account the errors affecting practically all experimental data, the decision about

the compatibility (or incompatibility) of a certain hypothesis (e.g., the fractal character of the

fracture surfaces) has to be established using some statistical tests [45–47]. Unfortunately,

neither [29] nor [30, 32–38] studied statistically the compatibility of the investigated

hypothesis relative to the experimental data, and even these works did not indicate the errors

corresponding to the used experimental data.

In order to evaluate the error risk at the rejection of the compatibility of a certain

representative point relative to the studied correlation Yi = f(X), it is possible to use both

global (for the entire correlation) or local test, respectively. For example, the error risk can be

evaluated by means of the expression (see [44–48])

qk = exp

{− 1

2(1 − r2

k

)[(Yik − Yi,tk

s(Yik)

)2

+(Xk −Xtk

s(Xk)

)2

− 2rk

(Yik − Yi,tk

s(Yik)

)(Xk −Xtk

s(Xk)

)]},

(7.1)

where Yik and Xk are the impact energy and the fractal dimension corresponding to the

representative point (state) k(= 1, 2, . . .N), Yi,tk and Xtk are the impact energy and the fractal

dimension corresponding to the tangency point of the confidence ellipse centered in (Yik, Xk)with the studied correlation plot: Yi = f(X), while rk, s(Yik), and s(Xk) are the correlation

coefficient and the square mean errors corresponding to the individual values Yik and Xk.

Because these errors are not indicated by the studied work [29], we will try evaluate them

from other studies about the fracture energy.

The studies [31, 48] of the published works concerning the (multi)fractal correlations

of some mechanical (fracture) parameters with the specimen size points out the magnitude

orders of the errors corresponding to the fracture energy. The corresponding relative errors

are indicated in Table 3. One finds that for concrete specimens, the average relative errors

affecting the fracture energy is of (approximately) 7%.

Assuming that the relative errors affecting the values of the fractal dimension

are considerably less than those corresponding to the impact energy (approx. 10%), the

expression (7.1) leads to error risks somewhat larger than 2% associated to the rejection of

the compatibility hypothesis of the fractal/multifractal descriptions with the experimental

data. It results that the compatibility hypothesis cannot be rejected, but a more sure decision

needs imperatively the knowledge of the corresponding measurement errors.


Table 3: Relative errors corresponding to the experimental data concerning the fracture energy GF for dif-ferent concrete and rocks specimens.

Material [reference] Concrete [49] Dry concrete [50] Wet concrete [50] Red felsersandstone [50]

Limits of relative errors 4.68 · · · 9.76% 4.028 · · · 16.217% 2.502 · · · 11.585% 3.125 · · · 35.424%

Average relative error 6.062% 8.131% 6.473% 16.236%

8. Conclusions

The accomplished study of the numerical data involved by [29] points out the following main

original findings.

(1) The decision concerning the fractal (or multifractal) character of the fracture sur-

faces of metals needs a previous rigorous study by means of the numerical analysis

procedures.

(2) In this aim, both the errors corresponding to the geometrical parameters (perime-

ters and areas of the slit islands) and to the specific mechanical parameters (impact

energies), respectively, are necessary.

(3) Taking into account the considerable differences between the values of the fractal

dimension resulting from Figure 1 [29], or indicated in the caption of Figure 1 [29],or in Figure 3 [29], we consider that the correct calculation of the fractal dimension

corresponds to the interpretation from work [40], which considers that only the

perimeters of the slit islands present a fractal character: P ∝ RD−1, while the areas of

these slit islands present the usual second degree dependence on their radii A ∝ R2;

we have found that this interpretation [40] leads also to an agreement between the

data from Figure 1 [29] and the values of the fractal dimension indicated by this

work [29].

(4) The accomplished study indicates a multifractal nature of the fracture surfaces of

metals, the size distribution of the fractals (involved by this multifractal structure)being also evaluated by this work.

(5) The influence of the temperature of the studied maraging steels heat-treatment

on the (a) impact energy to fracture and (b) the fracture surface deformation,

measured by its fractal dimensional increment, were also studied, finding the

increase of the fracture surface deformation with the heat-treatment temperature,

particularly.

(6) Using the evaluated errors affecting the fracture energies of some concrete speci-

mens, we have found that the compatibility hypothesis of the fractal/multifractal

descriptions with the experimental data cannot be rejected, but a more sure decision

always needs an accurate knowledge of the corresponding measurement errors.

References

[1] Santa Fe Institute for Studies in the Sciences of Complexity: (i) Bulletin, 2-3 issues/year, 1987-present,(ii) Lectures, 1989-present, (iii) Proceedings, 1986-present.

[2] R. Dobrescu and D. A. Iordache, Complexity Modeling, Politehnica Press, Bucharest, Romania, 2007.[3] R. Dobrescu and D. A. Iordache, Complexity and Information, Romanian Academy Printing House,

Bucharest, Romania, 2010.


[4] P. W. Anderson, “More is different,” Science, vol. 177, no. 4047, pp. 393–396, 1972.

[5] P. W. Anderson, “Physics: the opening to complexity,” Proceedings of the National Academy of Sciencesof the United States of America, vol. 92, no. 15, pp. 6653–6654, 1995.

[6] K. G. Wilson, “Renormalization group and critical phenomena,” Physical Review B, vol. 4, no. 9, pp.3174–3183, 1971.

[7] I. Prigogine and G. Nicolis, Self-Organization in Non-Equilibrium Systems: From Dissipative Structures toOrder through Fluctuations, Wiley, New York, NY, USA, 1977.

[8] S. Solomon and E. Shir, “Complexity; a science at 30,” Europhysics News, vol. 34, no. 2, pp. 54–57, 2003.

[9] S. Solomon, “Evolving uniform and nonuniform cellular automata networks ,” in Annual Reviews ofComputational Physics, D. Stauffer, Ed., pp. 243–294, World Scientific, River Edge, NJ, USA, 1995.

[10] A. L. Barabasi and R. Albert, “Emergence of scaling in random networks,” Science, vol. 286, no. 5439,pp. 509–512, 1999.

[11] R. Albert, H. Jeong, and A. L. Barabasi, “Diameter of the world-wide web,” Nature, vol. 401, no. 6749,pp. 130–131, 1999.

[12] A. L. Barabasi, “Complex networks: from the cell to human diseases,” in Proceedings of the InternationalWorkshop on Quantitative Biology, Bucharest, Romania, May 2007.

[13] K. Bhattacharya, G. Mukherjee, and S. S. Manna, “The international trade network,” in Econo-Physicsof Markets and Business Networks, A. Chatterjee and B. K. Chakrabarti, Eds., New Economic WindowsSeries, p. 139, Springer, New York, NY, USA, 2008.

[14] S. Y. Chen, Wei Huang, and C. Cattani, “Traffic dynamics on complex networks: a survey,”Mathematical Problems in Engineering, vol. 2012, Article ID 732698, 23 pages, 2012.

[15] E. G. Backhoum and C. Toma, “Dynamical aspects of macroscopic andquantum transitions due tocoherence function and time series events,” Mathematical Problems in Engineering, vol. 2010, Article ID428903, 13 pages, 2010.

[16] E. G. Backhoum and C. Toma, “Specific mathematical aspects of dynamics generated by coherencefunctions,” Mathematical Problems in Engineering, vol. 2011, Article ID 436198, 10 pages, 2011.

[17] C. Cattani and P. Sterian, “Modelling the hyperboloid elastic shell,” Scientific Bulletin of University“Politehnica” of Bucharest, Series A, vol. 71, no. 4, pp. 37–44, 2009.

[18] A. A. Gukhman, Introduction to the Theory of Similarity, Academic Press, New York, NY, USA, 1965.

[19] G. I. Barenblatt, Dimensional Analysis, Gordon and Breach, New York, NY, USA, 1987.

[20] G. I. Barenblatt, Scaling, Self-Similarity and Intermediate Asymptotics, Cambridge Texts in AppliedMathematics, Cambridge University Press, Cambridge, UK, 1996.

[21] C. Shannon, “The mathematical theory of communication,” Bell System Technical Journal, vol. 27, no.3, pp. 379–423, 1948.

[22] C. Shannon, “The mathematical theory of communication,” Bell System Technical Journal, vol. 27, no.4, pp. 623–656, 1948.

[23] C. Shannon, “Prediction and entropy of printed English,” Bell System Technical Journal, vol. 30, no. 1,pp. 50–64, 1951.

[24] A. Carpinteri and B. Chiaia, “Multifractal nature of concrete fracture surfaces and size effects onnominal fracture energy,” Materials and Structures, vol. 28, no. 8, pp. 435–443, 1995.

[25] A. Carpinteri and B. Chiaia, “Multifractal scaling laws in the breaking behaviour of disorderedmaterials,” Chaos, Solitons and Fractals, vol. 8, no. 2, pp. 135–150, 1997.

[26] D. A. Iordache, P. P. Delsanto, St. Pusca, and V. Iordache, “Complexity, similitude, and fractals inphysics,” in Proceedings of the 2nd International Symposium on “Interdisciplinary Applications of FractalAnalysis” (IAFA ’05), vol. 3, pp. 7–12, Politehnica Press Publishing House, Bucharest, Romania, May2005.

[27] P. P. Delsanto, D. A. Iordache, and St. Pusca, “Study of the correlations between different effectivefractal dimensions used for fracture parameters descriptions,” in Interdisciplinary Applications of Fractaland Chaos, R. Dobrescu and C. Vasilescu, Eds., pp. 136–153, Romanian Academy Printing House,Bucharest, Romania, 2004.

[28] P. P. Delsanto, A. S. Gliozzi, C. L. E. Bruno, N. Pugno, and A. Carpinteri, “Scaling laws and fractalityin the framework of a phenomenological approach,” Chaos, Solitons and Fractals, vol. 41, no. 5, pp.2782–2786, 2009.

[29] B. B. Mandelbrot, D. E. Passoja, and A. J. Paullay, “Fractal character of fracture surfaces of metals,”Nature, vol. 308, no. 5961, pp. 721–722, 1984.

[30] Y. Ma and T. G. Langdon, “Observations on the use of a fractal model to predict superplastic ductility,”Scripta Metallurgica et Materiala, vol. 28, no. 2, pp. 241–246, 1993.


[31] D. A. Iordache, St. Pusca, G. Toma, V. Paun, A. Sterian, and C. Morarescu, “Analysis of compatibilitywith experimental data of Fractal descriptions of the fracture parameters,” Lecture Notes in ComputerScience, vol. 3980, pp. 804–813, 2006.

[32] E. E. Underwood and K. Banerji, “Fractals in fractography,” Materials Science and Engineering, vol. 80,no. 1, pp. 1–14, 1986.

[33] C. S. Pande, L. E. Richards, N. Louat, B. D. Dempsey, and A. J. Schwoeble, “Fractal characterizationof fractured surfaces,” Acta Metallurgica, vol. 35, no. 7, pp. 1633–1637, 1987.

[34] C. W. Lung and Z. Q. Mu, “Fractal dimension measured with perimeter-area relation and toughnessof materials,” Physical Review B, vol. 38, no. 16, pp. 11781–11784, 1988.

[35] Z. H. Huang, J. F. Tian, and Z. G. Wang, “Study of the slit island analysis as a method for measuringfractal dimension of fractured surface,” Scripta Metallurgica et Materialia, vol. 24, no. 6, pp. 962–972,1990.

[36] R. E. Williford, “Multifractal fracture,” Scripta Metallurgica, vol. 22, no. 11, pp. 1749–1754, 1988.

[37] Z.Q. Mu and C. W. Lung, “Studies on the fractal dimension and fracture toughness of steel,” Journalof Physics D, vol. 21, no. 5, p. 848, 1988.

[38] H. Su, Y. Zhang, and Z. Yan, Acta Metallurgica Sinica, vol. 3, p. 226, 1990.

[39] St. Pusca, V. Solcan, and D. A. Iordache, “Procedure of extraction of numerical data from experimentalplots. Application to the numerical analysis of some fractal studies,” in Proceedings of the 5th GeneralConference of the Balkan Physical Union, pp. 259–264, Vrnjacka Banja, Serbia, August 2003.

[40] J. F. Gouyet, Physique et Structures Fractales, Masson, Paris, France, 1992.

[41] D. L. Davidson, “Fracture surface roughness as a gauge of fracture toughness: aluminium-particulateSiC composites,” Journal of Materials Science, vol. 24, no. 2, pp. 681–687, 1989.

[42] K. Levenberg, “A method for the solution of certain problems in least squares,” Quarterly of AppliedMathematics, vol. 2, pp. 164–168, 1944.

[43] D. W. Marquardt, “An algorithm fot least-squares estimation of nonlinear parameters,” Journal of theSociety for Industrial and Applied Mathematics, vol. 11, no. 2, pp. 431–441, 1963.

[44] E. Bodegom, D. W. McClure, P. P. Delsanto et al., Computational Physics Guide, Politehnica Press,Bucharest, Romania, 2009.

[45] W.T. Eadie, D. Drijard, F. E. James, M. Roos, and B. Sadoulet, Statistical Methods in Experimental Physics,North-Holland, Amsterdam, The Netherlands, 1982.

[46] W. Ledermann, Ed., Handbook of Applied Mathematics, vol. 6 of Statistics, John Wiley & Sons, New York,NY, USA, 1984.

[47] P. W. M. John, Statistical Methods in Engineering and Quality Assurance, John Wiley & Sons, New York,NY, USA, 1990.

[48] D. A. Iordache and V. Iordache, “Compatibility of multi-fractal and similitude descriptions of thefracture parameters relative to the experimental data for concrete specimens,” in Proceedings of the 1stSouth-East European Symposium on Interdisciplinary Approaches in Fractal Analysis, pp. 55–60, Bucharest,Romania, May 2003.

[49] A. Carpinteri and G. Ferro, “Scaling behaviour amd dual renormalization of experimental tensilesoftening responses,” Materials and Structures, vol. 31, no. 5, pp. 303–309, 1998.

[50] M. R. A. van Vliet, Size effects in tensile fracture of concrete and rock, Ph.D. thesis, University of Delft,2000.


Research ArticleMultidimensional Wave Field Signal Theory:Transfer Function Relationships

Natalie Baddour

Department of Mechanical Engineering, University of Ottawa, Ottawa, ON, Canada K1N 6N5

Correspondence should be addressed to Natalie Baddour, [email protected]

Received 29 August 2011; Accepted 20 September 2011


Copyright q 2012 Natalie Baddour. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The transmission of information by propagating or diffusive waves is common to many fieldsof engineering and physics. Such physical phenomena are governed by a Helmholtz (real wave-number) or pseudo-Helmholtz (complex wavenumber) equation. Since these equations are linear,it would be useful to be able to use tools from signal theory in solving related problems. The aim ofthis paper is to derive multidimensional input/output transfer function relationships in the spatialdomain for these equations in order to permit such a signal theoretic approach to problem solving.This paper presents such transfer function relationships for the spatial (not Fourier) domain withinappropriate coordinate systems. It is shown that the relationships assume particularly simpleand computationally useful forms once the appropriate curvilinear version of a multidimensionalspatial Fourier transform is used. These results are shown for both real and complex wavenumbers.Fourier inversion of these formulas would have applications for tomographic problems in variousmodalities. In the case of real wavenumbers, these inversion formulas are presented in closed form,whereby an input can be calculated from a given or measured wavefield.

1. Introduction

The transmission of information over space and time is often governed by the theory

of waves. Many important physical phenomena are described by a Helmholtz equation,

for example, in the fields of electromagnetism, acoustics, and optics [1–4]. Other physical

phenomena are well described by the use of damped or diffusion waves, such as the pro-

pagation of heat or photonic waves. For example, ultrasound has long been used for medical

imaging [3] while the use of optical radiation via diffuse photon density waves for imaging

inhomogeneities in turbid media is a newer development [5–8]. Similarly, photothermal

tomographic imaging methods have also been used for nondestructive evaluation [9–12],and more recently for biomedical imaging [13, 14]. Furthermore, a combination of ultrasonic


and optical techniques known as photoacoustics has also gained prominence in recent years

[4, 15].Modelled as a linear process, the propagation and scattering of waves obey the prin-

ciples of superposition and homogeneity. One common method of studying linear processes

is to view them as linear systems with an input and output. The input/output relationships

of the system can then be characterized via the system transfer function, which implies

interpreting the problem as a signal theory problem and thus enabling the application of

powerful system theoretic results and concepts. This type of approach applied to propagating

or diffusion wave problems has already yielded promising results [16–18] and the goal of this

paper is to expand the scope of this approach.

The seeds of such an approach to the solution of tomographic-type problems can be

seen in [1, 8, 10, 19, 20]; however, in those papers there is no clear reference to any such

system theoretic concepts. As such, a unifying framework for a signal theoretic approach to

wavefields does not exist. The goal of this work is to propose a clear and cohesive signal

theoretic framework for working with acoustic, thermal, photonic, or other wavefield-related

problems. This would be particularly powerful, enabling the application of a vast array of

results from a signal processing point of view and in particular, recent novel results from

algebraic signal processing [21–27] should find ground-breaking applications.

With these ideas in mind, the goal of this exposition is to derive multidimensional

spatial transfer function relationships for the Helmholtz and pseudo-Helmholtz equation so

that the tools of linear system theory can be used to solve related problems.

We consider the input quantity to be the time- and space-dependent inhomogeneity

term (the “forcing” term in the relevant partial differential equation). The output quantity

is then taken as the resulting field present at any point in space. The goal is to develop

transfer function relationships in spatial multidimensions so that general results of the theory

of signals and systems may be used. It is known that the relationship between the input

(inhomogeneity term) and the output (resulting wavefield) is one of convolution with the

impulse response (Green’s function in classical terms). In the Fourier domain, the equivalent

relationship is a transfer function relationship, that is to say one of multiplication. In this

paper, key transfer function relationships are shown to hold for the spatial (not Fourier)domain with the use of appropriate coordinate systems. That is to say, the relationship

between input and output is shown to be one of multiplication (not convolution) even in the

spatial domain, hence yielding the signal theoretic construction that is the aim of the paper.

Rather than focusing on a specific application area (acoustics, medical imaging,

seismic, etc.), the end goal of this paper is a unifying framework. The backbone of

this framework is Fourier transforms in multidimensions which are also transcribed into

curvilinear coordinate systems. The representations in this paper are embedded in a time-

dependent, spatially one- to three-dimensional description.

The outline of this paper is as follows. Sections 2 to 7 establish the background

preliminaries, definitions, and sign conventions that are important to the developments in

the rest of the paper. The definition of multidimensional Fourier transforms is given in

Section 2. Section 3 presents the mathematical forms of the types of signal to which this

development applies, namely, any signal governed by a Helmholtz or pseudo-Helmholtz

equation. Section 4 introduces the Green’s and transfer functions for the (generalized)Helmholtz equation and points out that the spatial transfer function assumes a particularly

simple form in curvilinear coordinates. Section 5 explains the sign conventions that will be

employed in this paper to ensure consistency and simplicity of results. Sections 6 and 7

develop multidimensional Fourier transforms in curvilinear coordinates. Sections 8, 9, and 10


present the relevant transfer function relationships in curvilinear coordinates in 3D, 2D, and

1D, respectively, and form the core results of this paper. Section 11 summarizes the results

of the previous three sections in a tabular format that is easy to refer to. Sections 12 and 13

present some applications while Section 14 concludes the paper.

2. Fourier Transforms in n-Dimensions

The theory of Fourier transforms can be extended to Rn in a way that is completely analogous

to the treatment of one-dimensional Fourier transforms. Let elements of Rn be denoted by

�x = (x1,x2, . . . ,xn) or more generally �r, and elements of the corresponding Fourier dual space

Ωn be denoted by �ω = (ω1, ω2, . . . , ωn). Under the suitability of integration of f , then the

Fourier transform in Rn is defined for �ω in Ωn as

F( �ω) =∫

Rn

f(�x)e−j �ω·�xd �x. (2.1)

Under suitable conditions, the function f can be recovered from the inverse transform

through

f(�x) =1

(2π)n

∫Ωn

F( �ω)ej �ω·�xd �ω. (2.2)

Various other conventions are possible regarding the location of the positive and negative

signs and also the factors of 2π in (2.1) and (2.2).

3. The Helmholtz and Pseudo-Helmholtz Equation

All wave fields governed by the wave equation (such as acoustic or electromagnetic waves)lead to the Helmholtz equation once a Fourier transform is used to transform the time domain

to the frequency domain:

∇2u(�r, ω) + k2su(�r, ω) = −s(�r, ω), (3.1)

where k2s = ω2/c2

s . Here, s(�r, ω) is the temporal Fourier transform of the inhomogeneous

time- and space-dependent source term for the wave equation. From a signal theory

perspective, this is considered to be the input to the system. The variable u(�r, ω) represents a

physical variable that is governed by the wave equation, for example, acoustic pressure, and

is considered to be the output from a signal theory point of view. Both s(�r, ω) and u(�r, ω) are

functions of position, �r, and (temporal) frequency, ω. The variable cs represents the speed of

the wave, which for an acoustic wave would be the speed of sound and for an electromagnetic

wave would be the speed of light. For wavefields governed by the wave equation, k2s is a real

(and positive) quantity.

Other physical phenomena lead to a “pseudo-” Helmholtz equation upon Fourier

transformation of the time variable to a (temporal) frequency variable. For example, the

equation for a diffuse photon density wave (DPDW) which describes the photon density


u(�r, t) in a solid due to incident energy intensity s(�r, t) (optical source function) is given in

the time domain by [2]

D∇2u(�r, t) − 1

c

∂

∂tu(�r, t) − μau(�r, t) = −s(�r, t). (3.2)

In the above equation, μa is the optical absorption coefficient (m−1), c is the speed of

light in the turbid medium (m/s), and D is the optical diffusion coefficient (m). Taking the

temporal Fourier transform (FT) of (3.2) to transform time to frequency gives the following:

∇2u(�r, ω) + k2pu(�r, ω) = −s(�r, ω)

D, (3.3)

where k2p = −(μa/D) − (iω/cD) is the complex wavenumber. Equation (3.3) is a pseudo-

Helmholtz equation. The term “pseudo” is used because although (3.3) has the form of

a Helmholtz equation similar to (3.1), the wavenumber k2p is a complex variable, with

the imaginary part indicating a decaying or damped wave. In this development the

terms “Helmholtz” and “pseudo-Helmholtz” will generally be used interchangeably unless

otherwise noted. Similarly, the standard heat equation is given by

∇2u(�r, t) − 1

α

∂

∂tu(�r, t) = −s(�r, t), (3.4)

where s(�r, t) is the time- and space-dependent heat source, u(�r, t) describes the temperature

in the material as a function of time and space, and α is the thermal diffusivity of a material.

As done previously, we take the temporal FT of (3.4) to obtain

∇2u(�r, ω) + k2t u(�r, ω) = −s(�r, ω), (3.5)

where k2t = −iω/α is the complex wavenumber. Once again (3.5) has the form of a Helmholtz

equation, although it is a pseudo-Helmholtz equation to be exact since the wavenumber is

complex and indicates a damped wave.

Thus, we see that the general Helmholtz form of (3.1) can be used to describe several

different physical phenomena, from the propagation of light or acoustic waves to the heavily

damped nature of photonic or thermal waves. The exact form of the wavenumber in each case

is the best indicator as to the propagation characteristics of a wave with a real k2 indicating a

propagating wave and a complex k2 indicating a damped wave.

4. Green’s Function and Transfer Function for the Helmholtz Equation

Taking the full spatial Fourier transform of the Helmholtz (or pseudo-Helmholtz) equation

and rearranging yields

U( �ω,ω) = U(ωx, ωy, ωz, ω

)=

1(ω2

x +ω2y +ω2

z − k2)S( �ω,ω), (4.1)


where we have used the shorthand notation ( �ω) = (ωx, ωy, ωz) to denote a point in 3D spatial

Fourier frequency space. The wavenumber may be real or complex. A capital letter is used to

denote the full 3D spatial and temporal Fourier transform of a function, although it should

be clear from the arguments which is being indicated. In cases where it may not be clear, a

tilde (∼) will be used to denote the function in spatial Fourier space. For shorthand notation,

let ω2k= ω2

x +ω2y +ω2

z, so that ωk is the length of the spatial Fourier vector. The Fourier space

has the same spatial frequency dimension as the spatial dimension of the problem. So in 2D

space, a ωz spatial frequency variable would not be required and so forth.

By inverse spatial Fourier transformation of (4.1), the wavefield in n dimensions is

given by

u(�r, ω) =1

(2π)n

∫∞

−∞

S( �ω,ω)(ω2

k− k2

)ei �ω·�rd �ω. (4.2)

Using the definition of the Fourier transform of s(�r, ω), the above equation can be rewritten

as

u(�r, ω) =1

(2π)n

∫∞

−∞

ei �ω·�r(ω2

k− k2

) ∫∞

−∞s(�x, ω)e−i �ω·�xd�x d �ω. (4.3)

Let us define the spatial Green’s function in n-dimensional space as

g(�r | �x, ω) = g(�r − �x, ω) =1

(2π)n

∫∞

−∞

ei �ω·(�r−�x)

ω2k− k2

d �ω. (4.4)

This is the Green’s function for the (pseudo-) Helmholtz equation. By switching the order

of integration, (4.3) can be rewritten such that it can be clearly interpreted as a spatial

convolution of Green’s function with the input source:

u(�r, ω) =∫∞

−∞s(�x, ω)g(�r − �x, ω)d�x = s(�r, ω)∗�rg(�r, ω). (4.5)

The notation ∗�r has been used to denote a (multidimensional) space-only convolution. The

frequency domain equivalent to (4.5) is (4.1), which can be interpreted as a multiplication in

terms of the spatial transfer function as

U( �ω,ω) = G( �ω,ω)S( �ω,ω), (4.6)

where the Fourier transform of the spatial Green’s function is the spatial transfer function:

G( �ω,ω) = F[g(�r, ω)

]=

1(ω2

k− k2

) . (4.7)

The dependence of the spatial Green’s function on temporal frequency is via the wavenumber

k of the wavefield (which may be real or complex) and this is emphasized by writing G( �ω,ω)as a function of spatial and temporal frequencies.


4.1. Spatial Transfer Function in Curvilinear Coordinates

Equations (4.4) and (4.7) hold in any n-dimensional space with Green’s function of (4.4)being the n-dimensional spatial inverse Fourier transform of the spatial transfer function

of (4.7). More specifically, in 1D, ω2k

= ω2x, and in 2D ω2

k= ω2

x + ω2y. Transforming to

polar coordinates in the 2D spatial frequency domain via the transformation ωx = ρ cosψ,

ωy = ρ sinψ gives ω2k= ω2

x + ω2y = ρ2 where ρ is the radial frequency variable in polar

coordinates. Similarly, in 3D, ω2k= ω2

x+ω2y+ω

2z becomes ω2

k= ω2

x+ω2y+ω

2z = ρ2 where ρ is the

spherical radial frequency variable in spherical polar coordinates in spatial frequency space

via the coordinate transformation given by ωx = ρ sinψω cos θω, ωy = ρ sinψω sin θω and ωz =ρ cosψω. Thus, the spatial transfer functions assume particularly simple, radially symmetric

forms in polar coordinates for 2D space and spherical polar coordinates for 3D space. These

particularly simple forms for the 2D and 3D transfer functions in polar coordinates motivates

the development of the general 2D and 3D Fourier transforms in curvilinear coordinates

[28, 29] so that this simple form of the transfer function may be exploited.

5. Notation and Sign Conventions

Wavenumbers are defined by the squares of their quantities and arise as a result of taking

the Fourier transform of the corresponding propagation equation (be it acoustic, thermal,

or otherwise), which in turn leads to a Helmholtz or pseudo-Helmholtz equation. These

wavenumbers are defined so that the wave propagation equation transformed to the

temporal frequency domain all have the form of a pseudo-Helmholz equation as given

by (3.1) or (3.5) with a complex or real wavenumber. For the rest of the paper, we will

consider that kt represents a generic complex wavenumber while ks represents a generic real

wavenumber.

Rather than the squared wavenumber, the quantity of interest will prove to be the

wavenumber itself, namely, ks and kt, which are the square roots of the given squared

wavenumber in the Helmholtz equation. Each k can be considered as the sum of a real and an

imaginary part, so that kt = ktr + ikti with ktr denoting the real part of kt and kti denoting the

imaginary part. Since the square root of any k2 can be ±k, we will use the convention that for

a given complex k, the required square root of the corresponding k2 is defined such that the

imaginary part of k is negative. Hence, kt is chosen as the square root of k2t such that kti < 0 and

so forth. If this sign convention is adopted, then it was shown in [36] that the many results

for complex wavenumber use the same mathematical form of travelling wave solution as for

the real wavenumbers.This makes the notation and book-keeping considerably simpler. It is

noted that this sign convention is the opposite of what this author adopts in [30].

5.1. Sommerfeld Radiation Condition

To aid in the selection of a causal solution, the Sommerfeld radiation condition is required.

The Sommerfeld radiation condition states that the sources in the field must be sources not

sinks of energy. Therefore, energy radiated from sources must scatter to infinity and cannot

radiate from infinity into the field. Mathematically, a solution u(x), where x is the spatial

variable, to the Helmholtz equation is considered to be radiating if it satisfies

lim|x|→∞

|x|(n−1)/2

(∂

∂|x| + ik

)u(x) = 0, (5.1)


where n is the dimension of the space and k is the wavenumber in the Helmholtz equation.

This is the radiation equation based on an implied time variation of eiωt which is implicit in our

chosen definition of the Fourier transform, (2.1). For example, with the standard definition of

the Fourier transform, the transform of f ′(t) is iωF(ω), clearly implying the eiωt dependence.

Had a different definition of the Fourier transform been used, the implied time variation

would have been e−iωt, in which case the sign of i in (5.1) would be reversed.

6. 2D Fourier Transforms in Terms of Polar Coordinates

Given the desire to exploit the simple form of the spatial transfer function in curvilinear

coordinates, we consider 2D and 3D Fourier transform in terms of curvilinear coordinates.

Let us first consider the Fourier transforms in 2D in polar coordinates. In order to define this,

some preliminary definitions of Hankel transforms are required first.

6.1. Hankel Transforms

The Hankel transform of order n is defined by the integral [31]

�

Fn

(ρ)= Hn

(f(r)

)=∫∞

0

f(r)Jn(ρr)r dr, (6.1)

where Jn(z) is the nth order Bessel function. If n > −1/2, the transform is self-reciprocating

and the inversion formula is given by

f(r) = H−1n

{�

Fn

(ρ)}

=∫∞

0

�

Fn

(ρ)Jn(ρr)ρ dρ. (6.2)

6.2. Connection between the 2D Fourier Transform and Hankel Transform

To develop Fourier transforms in 2D in polar coordinates, both the function f(�r) and its

Fourier transform F( �ω) are expressed in polar coordinates. In general f(�r) = f(r, θ) is not

radially symmetric and is a function of both r and θ so that the θ dependence can be expanded

into a Fourier series due to the 2π periodicity of the function in θ:

f(r, θ) =∞∑

n=−∞fn(r)ejnθ, (6.3)

where the Fourier coefficients fn(r) can be found from

fn(r) =1

2π

∫2π

0

f(r, θ)e−jnθdθ. (6.4)


Similarly, the 2D Fourier transform F( �ω) = F(ρ, ψ) can also be expanded into its own Fourier

series so that

F(ρ, ψ

)=

∞∑n=−∞

Fn

(ρ)ejnψ, (6.5)

and where those Fourier coefficients are similarly found from

Fn

(ρ)=

1

2π

∫2π

0

F(ρ, ψ

)e−jnψdψ, (6.6)

The relationship between the Fourier coordinates in normal space fn(r) in (6.3) and

the spatial frequency space Fourier coordinates Fn(ρ) in (6.5) is desired. The details of this

are given in [28], and the results are summarized here. It is emphasized that the relationship

between fn(r) and Fn(ρ) is not a Fourier transform. The relationship between them is given

by

Fn

(ρ)= 2πi−n

∫∞

0

fn(r)Jn(ρr)r dr = 2πi−nHn

{fn(r)

}, (6.7)

where Hn is the nth order Hankel transform. The reverse relationship is given by

fn(r) =in

2π

∫∞

0

Fn

(ρ)Jn(ρr)ρ dρ =

in

2πHn

{Fn

(ρ)}

. (6.8)

The nth term in the Fourier series for the original function will Hankel transform into the

nth term of the Fourier series of the Fourier transform. However, it is an nth order Hankel

transform for the nth term, namely, all the terms are not equivalently transformed. The

mapping from fn(r) to Fn(ρ) is one of nth-order Hankel transform, which in general is not a

2D Fourier transform.

The operation of taking the 2D Fourier transform of a function is thus equivalent to

(1) first finding its Fourier series expansion in the angular variable and (2) then finding the

nth-order Hankel transform (of the radial variable to the spatial radial variable) of the nth

coefficient in the Fourier series. Since each of these operations involves integration over one

variable only with the others being considered parameters vis-a-vis the integration, the order

in which these operations are performed is interchangeable.

7. Spherical Hankel Transform

We introduce the spherical Hankel transform, which will form part of the 3D Fourier trans-

form in spherical coordinates.


7.1. Definition of the Spherical Hankel Transform

The spherical Hankel transform can then be defined as [32, 33]

�

Fn

(ρ)= Sn

{ff(r)

}=∫∞

0

f(r)jn(ρr)r2dr. (7.1)

Sn is used to specifically denote the spherical Hankel transform of order n. The inverse

transform is given by [29]

f(r) =2

π

∫∞

0

�

Fn

(ρ)jn(ρr)ρ2dρ. (7.2)

The spherical Hankel transform is particularly useful for problems involving spherical

symmetry.

7.2. Spherical Harmonics

The spherical harmonics are the solution to the angular portion of Laplace’s equation in

spherical polar coordinates and can be shown to be orthogonal. These spherical harmonics

are given by [32]

Yml

(ψ, θ

)=

√(2l + 1)(l −m)!

4π(l +m)!Pml

(cosψ

)eimθ, (7.3)

where Yml

is called a spherical harmonic function of degree l and order m, Pml

is an associated

Legendre function, 0 ≤ ψ ≤ π represents the colatitude and 0 ≤ θ ≤ 2π represents the

longitude. With the normalization of the spherical harmonics as given in (7.3), the spherical

harmonics are orthonormal so that∫2π

0

∫π

0

Yml Ym′

l′ sinψ dψ dθ = δll′δmm′ . (7.4)

Here δij is the kronecker delta and the overbar indicates the complex conjugate. It is

important to note that there are several different normalizations of the spherical harmonics

that are possible, that will differ from (7.3) and thus lead to a different version of (7.4).Several of these are nicely catalogued in [34], including which disciplines tend to use which

normalization. This is important to note since any result that uses orthogonality will differ

slightly depending on the choice of normalization.

The spherical harmonics form a complete set of orthonormal functions and thus form

a vector space. When restricted to the surface of a sphere, functions may be expanded on

the sphere into a series approximation much like a Fourier series. This is in fact a spherical

harmonic series. Any square-integrable function may be expanded as

f(ψ, θ

)=

∞∑l=0

l∑m=−l

fml Ym

l

(ψ, θ

), (7.5)


where

fml =

∫2π

0

∫π

0

f(ψ, θ

)Yml

(ψ, θ

)sinψ dψ dθ. (7.6)

The coefficients fml

are sometimes referred to as the spherical Fourier transform of f(ψ, θ) [35].

7.3. 3D Fourier Transforms in Spherical Polar Coordinates

To find 3D Fourier transforms in spherical polar coordinates, both the function and its

3D Fourier transform are written in terms of polar coordinates in the spatial and spatial

frequency domains. That is, a function in 3D space is expressed as f(�r) = f(r, ψr , θr) as a

function of spherical polar coordinates and its 3D Fourier transform is written as F( �ω) =F(ρ, ψω, θω) in frequency spherical polar coordinates. Certain relationships can be shown to

hold. The relationship between the function and its transform are summarized here, with the

relevant details omitted for brevity. The function itself can be expanded as a series in terms

of the spherical harmonics as

f(�r) = f(r, ψr , θr

)=

∞∑l=0

l∑k=−l

fkl (r)Y

kl

(ψr, θr

), (7.7)

where Ykl(ψr, θr) are the spherical harmonics and the Fourier coefficients are given by

fkl (r) =

∫2π

0

∫π

0

f(r, ψr , θr

)Ykl

(ψr, θr

)sinψrdψrdθr. (7.8)

The 3D Fourier transform of f can also be written in Fourier space in spherical polar coor-

dinates as

F( �ω) = F(ρ, ψω, θω

)=

∞∑l=0

l∑k=−l

Fkl

(ρ)Ykl

(ψω, θω

), (7.9)

where

Fkl

(ρ)=∫2π

0

∫π

0

F(ρ, ψω, θω

)Ykl

(ψω, θω

)sinψωdψωdθω. (7.10)

We emphasize again that the relationship between fkl(r) and Fk

l(ρ) is not that of a Fourier

transform. In fact, this relationship is given by [29]

Fkl

(ρ)= 4π(−i)l

∫∞

0

fkl (r)jl

(ρr)r2dr = 4π(−i)lSl

{fkl (r)

}, (7.11)


where Sl denotes a spherical Hankel transform of order l. The inverse relationship is given by

fkl (r) =

1

4π(i)l

2

π

∫∞

0

Fkl

(ρ)jl(ρr)ρ2dρ =

1

4π(i)lS−1

l

{Fkl

(ρ)}

. (7.12)

8. Spatial Transfer Function Relationship in 3D

With this long set of preliminaries, definitions, and conventions established, we can now

begin the derivation of the 3D spatial transfer function relationship between input s(�r, ω)and output u(�r, ω). The foregoing section makes use of several key results in [36] which

will be presented here without proof. The results in [36] address general cases, not all of

which are relevant to the work here. The versions of the theorems required herein are stated

based on the sign conventions presented in Section 5 with regards to the sign conventions

for the complex wavenumbers and also for the chosen definition of the Fourier transform.

These affect the implied time dependence, the Sommerfeld radiation condition, and thus

the physically correct solution that is chosen from several mathematical possibilities. The

versions of the theorems presented here ensure that the chosen waves are outwardly

propagating (Sommerfeld radiation condition) and also decay to zero at infinity for a damped

wavefield, thus ensuring bounded and physically meaningful solutions.

Theorem 8.1. It is shown in [36] that the following result holds true:

I =∫∞

0

φ(x)x2 − k2

jn(xr)x2dx = −πik2h(2)n (kr)φ(k). (8.1)

Here, φ is any analytic function defined on the positive real line that remains bounded as x goes toinfinity, jn(x) is a spherical Bessel function of order n, h(2)

n (x) is a spherical Hankel function of ordern, and k is a wavenumber which may be real or complex, chosen such that the imaginary part of kis negative. Given the definition of the Fourier transform that is being currently used, the presentedresult satisfies the Sommerfeld radiation condition, ensuring an outwardly propagating wave.

8.1. 3D Transfer Function Relationship in Spherical Polar Coordinates

From (4.7) and using the conversion to spherical polar coordinates in 3D, the transfer function

for the Helmholtz equation in the Fourier domain can be written as

G( �ω,ω) =1(

ρ2 − k2) , (8.2)

which depends on the frequency spherical radial variable ρ only.

In general, as the imaginary part of the wavenumber k gets smaller, the transfer

function becomes more frequency selective, in the sense of strongly passing a smaller

bandwidth. As the imaginary part of the wavenumber k gets larger, the transfer function

becomes more low-pass in nature, passing lower frequencies more strongly than the higher

frequencies, with a corresponding larger bandwidth of frequencies being passed. This is

physically meaningful as the imaginary part of k represents the damping inherent in this


wave modality. Wavenumbers with larger imaginary parts are more heavily damped and

that damping tends to affect the higher frequencies more—that is, the higher frequencies are

attenuated and the lower ones are passed through, implying a low-pass nature to the physical

system.

Recall that ρ is the magnitude of the spatial frequency variables. The case for the

imaginary part of k equal to zero (in other words, a real wavenumber) is easy to visualize.

For the case of a real k, the function is discontinuous at ρ = k, meaning that only those

spatial frequencies where exactly ρ = k are passed and the rest are attenuated. In other

words, the resulting wave must have ρ2 = ω2x + ω2

y + ω2z = k2. While these statements are

fairly mathematical in nature, they explain the nature of the resolution of various imaging

modalities. For example, it is far more difficult to achieve the resolution of acoustic imaging

with thermal imaging and one of the explanations for this is the “low-pass” blurring nature

of the complex wavenumber of a thermal wave versus the highly selective band-pass nature

of the acoustic wavenumber. This is discussed further in other papers, for example, in [18].

8.2. 3D Green Function Coefficients

Let us define a set of functions that will be referred to as the Green function coefficients.

The actual Green function for the system is the full 3D inverse Fourier transform of the

transfer function, which for a spherically symmetric function is equivalent to a spherical

Hankel transform of order zero only [33]. With the help of Theorem 8.1, we define these Green

function coefficients as

g3Dn (r, k) = S−1

n

{1

ρ2 − k2

}=

2

π

∫∞

0

1

ρ2 − k2jn(ρr)ρ2dρ = −ikh(2)

n (kr), (8.3)

where g3Dn (r, k) has been used to denote the nth order Green function coefficient, with

the subscript indicating the order of the coefficient and the wavenumber k included as a

parameter of the function.

8.3. Transfer Function Relationship in Spherical Polar Coordinates

The 3D Fourier transform of the input (source) function is written in polar coordinates and

expanded in terms of a spherical harmonic series as

S( �ω,ω) =∞∑l=0

l∑m=−l

Sml

(ρ,ω

)Yml

(ψω, θω

). (8.4)

Hence, (4.1) for the output wavefield becomes

U( �ω,ω) =∞∑l=0

l∑m=−l

Sml

(ρ,ω

)ρ2 − k2

Yml

(ψω, θω

). (8.5)


The symbol k is used to denote the wavenumber. The spatial inverse Fourier transform and

thus the output in spatial coordinates is given by

u(r, ψr , θr , ω

)=

∞∑l=0

l∑m=−l

(i)l

4π

{2

π

∫∞

0

Sml

(ρ,ω

)jl(ρr)

ρ2 − k2ρ2dρ

}Yml

(ψr, θr

). (8.6)

The temporal frequency ω is a constant as far as the spatial inverse Fourier transformation

is concerned. We recognize that the quantity within the curly brackets can be evaluated with

the help of Theorem 8.1 and using the definition of the Green function coefficients in (8.3) to

obtain

2

π

∫∞

0

Sml

(ρ,ω

)jl(ρr)

ρ2 − k2ρ2dρ = g3D

l (r, k)Sml (k,ω). (8.7)

It now follows that the wavefield expression becomes

u(�r, ω) =∞∑l=0

l∑m=−l

il

4πg3Dl (r, k)Sm

l (k,ω)Yml

(ψr, θr

). (8.8)

If the measured wavefield itself, namely, the left-hand side of (8.8), is expanded as in a

spherical harmonic series so that

u(�r, ω) = u(r, ψr , θr , ω

)=

∞∑l=0

l∑m=−l

uml (r, ω)Ym

l

(ψr, θr

), (8.9)

then (8.8) gives us the simple input-output transfer function relationship we seek:

uml (r, ω) =

il

4πg3Dl (r, k)Sm

l (k,ω). (8.10)

Note how this relationship is in the spatial domain (not the frequency spatial domain) and

gives a multiplicative transfer function relationship between the input coefficients Sml(k,ω)

and the resulting wavefield uml(r, ω).

Using the relationships proposed between the coefficients of the function in the spatial

domain and the coefficients in the Fourier domain as given in (7.11), the general relationship

between Hankel and Fourier transforms is given by

Sml

(ρ,ω

)= 4π(−i)mSl

{sml (r, ω)

}= 4π(−i)l

∫∞

0

sml (r, ω)jl(ρr)r2dr. (8.11)

Hence (8.10) can be written as

uml (r, ω) = g3D

l (r, k)∫∞

0

sml (x, ω)jl(kx)x2dx, (8.12)


so that the value of the measured wavefield is related to the spherical Hankel transform of

the input function, evaluated at the wavenumber of the wavefield in question. Using the

definition of the spherical Hankel transforms, this becomes a very compact yet powerful

expression:

uml (r, ω) = g3D

l (r, k)Sl

{sml (k,ω)

}. (8.13)

The value of defining the Green function coefficients now becomes apparent in that it permits

the relationship between uml(r, ω) and sm

l(k,ω) to have a simple multiplicative transfer

function relationship. In other words, the relationship between output wave coefficients and

input is one of multiplication with the transfer function coefficients instead of a complicated

convolution-type relationship.

If we also assume that the input function is separable in time and space (a fairly

general assumption) so that s(�r, ω) = q(�r)η(ω). This would be the case for a spatial

inhomogeneity and a temporal input. In this case, then sml(r, ω) = qm

l(r)η(ω) and the formula

for the forward problem is given by

uml (r, ω) = g3D

l (r, k)η(ω)∫∞

0

qml (r)jl(kr)r2dr = g3D

l (r, k)η(ω)Sl

{qml (k)

}. (8.14)

8.4. Discussion and Relationship with the Fourier Diffraction Theorem

Equation (8.13) is the key input-output relationship that is sought. It gives the relationship

between input and output and the relationship is a transfer function (multiplication) type

of relationship, even in the spatial domain where the normal relationship to be expected is

one of convolution. This relationship is for the spherical harmonic expansion coefficients, not

between the full functions themselves. However, because it is a direct, proportional type of

relationship between the (l,m)th term of the input and the (l,m)th term of the output, it is

particularly useful and simple to apply.

In particular, (8.13) states that the wavefield (output) coefficients are directly

proportional to the transfer function coefficients and the proportionality factor between

them is the spherical Hankel transform of the input coefficients, evaluated on the sphere

ρ = k. Note that (l,m)th order output coefficients are related to (l,m)th order transfer

function coefficients in proportion to the lth order spherical Hankel transform of the (l,m)thorder input coefficient. This is in fact a generalization of the Fourier diffraction theorem of

tomography [3] which loosely states that the output wavefield is proportional to the Fourier

transform of the input inhomogeneity evaluated somewhere on the ρ = k sphere in Fourier

space. This is still true but we have expressed a more precise version of this theorem in the

sense that we have removed any ambiguity regarding the “somewhere” on the ρ = k sphere

and replaced it with a relationship (8.14) that does not depend on angular location. This was

enabled by the use of the Fourier transform in curvilinear coordinates so that a full Fourier

transform requires an (l,m) set of Fourier coefficients and spherical Hankel transforms.

Furthermore, the idea of proportionality between Fourier transform of the inhomogeneity

and output wavefield is also made more precise by stating that the proportionality between

them is actually the coefficients of the Green function itself. In essence, the output wavefield

can be seen as being the Green function coefficients, with each term weighted by the spherical

Hankel transform of the input inhomogeneity.


9. 2D Transfer Function Relationship in 2D Polar Coordinates

We proceed to develop the transfer function expressions for the 2D case in polar coordinates.

As for the 3D case, we begin with a necessary theorem which is stated without proof. As

previously mentioned, the results in [36] address general cases, and the versions of these

theorems required herein are stated here based on the sign conventions presented in Section 5

for the complex wavenumbers and also for the Fourier transform which affects the implied

time dependence and thus the Sommerfeld radiation condition.

Theorem 9.1. It is shown in [36] that the following result holds true:

I =∫∞

0

φ(ρ)

ρ2 − k2Jn(ρr)ρ dρ = −πi1

2H

(2)n (kr)φ(k). (9.1)

Here, φ is an analytic function defined on the positive real line that remains bounded as x goes toinfinity, Jn(x) is a Bessel function of order n, H(2)

n (x) is a Hankel function of order n, and k is awavenumber which may be real or complex, chosen such that the imaginary part of k is negative.Given the definition of the Fourier transform that is being currently used, the presented result satisfiesthe Sommerfeld radiation condition, ensuring an outwardly propagating wave.

9.1. 2D Green Function Coefficients

The overall system transfer function in 2D is given by

G( �ω,ω) =1(

ρ2 − k2) , (9.2)

which depends on the frequency radial variable only.

As for the 3D case, we define the 2D Green function coefficients as the nth-order

inverse Hankel transform of the overall system spatial transfer function. These will be shown

to be the required transfer functions for working in the 2D polar formulation. From the

definition of the inverse Hankel transform, these are given by

g2Dn (r, k) = H−1

n

{1

ρ2 − k2

}=∫∞

0

1

ρ2 − k2Jn(ρr)ρ dρ = −πi

2H

(2)n (kr). (9.3)

Recall that the actual (full) Green function for the system is the full 2D inverse Fourier

transform of the transfer function, which is an inverse Hankel transform of order zero only.

In (9.3), g2Dn (r, k) has been used to denote the nth order Green function coefficients with the

subscript indicating the order.


9.2. Fourier Theorem for 2D Fourier Transforms in Polar Coordinates

The 2D Fourier transform of the input function is written in polar coordinates as

S(ρ, ψ,ω

)=

∞∑n=−∞

Sn

(ρ,ω

)ejnψ, (9.4)

where Sn(ρ,ω) can be found from

Sn

(ρ,ω

)=

1

2π

∫2π

0

S(ρ, ψ,ω

)e−jnψdψ. (9.5)

The output wavefield u(�r, ω) is similarly expanded as a series so that in the spatial domain

we can write

u(�r, ω) = u(r, θ, ω) =∞∑

n=−∞un(r)ejnθ, (9.6)

while in the spatial Fourier domain this is

U( �ω,ω) = U(ρ, ψ,ω

)=

∞∑n=−∞

Un

(ρ,ω

)ejnψ. (9.7)

The output wavefield is given by U( �ω,ω) = G( �ω,ω)S( �ω,ω) and since G( �ω,ω) is radially

symmetric and does not require a full series, the output wavefield in the Fourier domain is

given by

U(ρ, ψ,ω

)=

∞∑n=−∞

Un

(ρ,ω

)ejnψ =

∞∑n=−∞

Sn

(ρ,ω

)(ρ2 − k2

)ejnψ, (9.8)

or equivalently as

Un

(ρ,ω

)=

Sn

(ρ,ω

)(ρ2 − k2

) . (9.9)

The equivalent expression to (9.8) in the spatial domain is given by

u(�r, ω) =∞∑

n=−∞un(r)ejnθ =

∞∑n=−∞

in

2π

{∫∞

0

Sn

(ρ,ω

)(ρ2 − k2

)Jn(ρr)ρ dρ}ejnθ, (9.10)

or

un(r) =in

2π

∫∞

0

Sn

(ρ,ω

)(ρ2 − k2

)Jn(ρr)ρ dρ. (9.11)


The temporal frequency ω is a constant as far as the spatial inverse Fourier transformation

is concerned. The quantity within the curly brackets can be evaluated with the help of

Theorem 9.1 as

∫∞

0

Sn

(ρ,ω

)(ρ2 − k2

)Jn(ρr)ρ dρ = −πi2H

(2)n (kr)Sn(k,ω) = g2D

n (r, k)Sn(k,ω). (9.12)

Hence, the output wavefield expression is given by

u(�r, ω) =∞∑


∞∑n=−∞

in

2πg2Dn (r, k)Sn(k,ω)ejnθ. (9.13)

The definition of forward and inverse Hankel transforms, the general relationship between

Hankel and Fourier transforms is given by

Sn

(ρ)= 2πi−nHn(sn(r)) = 2πi−n

∫∞

0

sn(r)Jn(ρr)r dr, (9.14)

and may be used to simplify (9.13) to yield

u(�r, ω) =∞∑


∞∑n=−∞

g2Dn (r, k)

∫∞

0

sn(x, ω)Jn(kx)x dxejnθ, (9.15)

or more compactly as

un(r, ω) = g2Dn (r, k)

∫∞

0

sn(x, ω)Jn(kx)x dx. (9.16)

It is in (9.15) that the interpretation of the evaluation of the 2D Fourier transform becomes

apparent, namely through the evaluation of the Bessel function at the (real or complex)wavenumbers. Using the definition of the Hankel transform, (9.16) can be written in the

compact and powerful formulation of

un(r, ω) = g2Dn (r, k)Hn(sn(k,ω)). (9.17)

If it is also further assumed that the inhomogeneity function is separable in time and space

(a fairly general assumption) so that s(�r, ω) = q(�r)η(ω), and

∞∑n=−∞

sn(r, ω)ejnθ = η(ω)∞∑

n=−∞qn(r)ejnθ, (9.18)

then (9.16) can be further reduced to

un(r, ω) = g2Dn (r, k)η(ω)Hn

{qn(k)

}. (9.19)


It is noted that the value of the measured wavefield at any position r is related to the Hankel

transform of the heterogeneity function evaluated at the wavenumber of the wavefield in

question, with the Green function coefficient acting as the proportionality term.

Equation (9.19) is the 2D equivalent of (8.14). The same comments can be made

as those made in the discussion after the 3D version of this problem, as the relationship

is identical, save for translating 3D tools such as the spherical harmonic expansions and

spherical Hankel transforms into 2D tools involving Fourier series and Hankel transforms.

10. Transfer Function Relationship in 1D

We proceed to develop the equivalent relationship for the 1D case. We have seen from (4.7)that the 1D transfer function in the Fourier domain is given by

G(ωx, ω) =1(

ω2x − k2

) , (10.1)

which depends on only a single spatial frequency. Equation (4.2) in 1D then becomes

u(r, ω) =1

(2π)

∫∞

−∞

S(ωx, ω)(ω2

x − k2)eiωxrdωx. (10.2)

The temporal frequency ω is a constant as far as the spatial inverse Fourier transformation is

concerned. We are thus interested in calculating integrals of the form

I =1

2π

∫∞

0

φ(x)x2 − k2

eixrdx, (10.3)

where φ is an analytic function defined on the positive real line that approaches zero as x goes

to infinity. The required theorem is given in [36] and the relevant result is presented below.

Theorem 10.1. It is shown in [36] that the following is true

1

2π

∫∞

−∞

φ(ρ)

ρ2 − k2eiρrdρ =

⎧⎪⎪⎨⎪⎪⎩1

2ikφ(−k)e−ikr , r > 0,

1

2ikφ(k)eikr , r < 0.

(10.4)

Here, φ is an analytic function defined on the positive real line that remains bounded as x goes toinfinity and k is a wavenumber which may be real or complex, chosen such that the imaginary part ofk is negative. Given the definition of the Fourier transform that is being currently used, the presentedresult satisfies the Sommerfeld radiation condition, ensuring an outwardly propagating wave.


10.1. 1D Green’s Functions via Inverse Fourier Transformation of the SpatialTransfer Functions

The Green’s function for the Helmholtz equation is 1D and is given by inverse Fourier

transform of the spatial transfer function as

g1D(r, k) =1

2π

∫∞

−∞

eiωxr

ω2x − k2

dωx, (10.5)

and can be evaluated with the help of Theorem 10.1 to give

g1D(r, k) =

⎧⎪⎪⎨⎪⎪⎩1

2ike−ikr r > 0

1

2ikeikr r < 0

⎫⎪⎪⎬⎪⎪⎭ =1

2ike−ik|r|. (10.6)

10.2. Transfer Function Relationship in 1D

Equation (10.2) can now be evaluated with the help of Theorem 10.1 as well as the result in

(10.6) as

u(r, ω) =1

(2π)

∫∞

−∞

S(ωx, ω)(ω2

x − k2)eiωxrdωx =

⎧⎪⎨⎪⎩1

2ikS(−k,ω)e−ikr r > 0

1

2ikS(k,ω)eikr r < 0

= g1D(r, k)S(− sgn(r)k,ω

),

(10.7)

where g1D(r, k) = (1/2ik)e−ik|r|. This is the 1D version of the transfer function relationship,

with the wavefield being directly proportional to the Fourier transform of the object function

evaluated at k. This is similar to the results for the 2D and 3D cases. For a real wavenumber,

a “sphere” in 1D becomes the two points on the real line at ±k and the proportionality term

is the Green’s function for the space in question.

As before, we assume that the inhomogeneity function is separable in time and space

(a fairly general assumption) so that s(r, ω) = q(r)η(ω) → S(ωx, ω) = Q(ωx)η(ω), where

Q(ωx) is the 1D Fourier transform of q(r). The relevant result now reads

u(r, ω) = g1D(r, k)η(ω)Q(− sgn(r)k

). (10.8)

This result is directly applicable for computational purposes.

11. Summary of Results

The relationships in the previous sections are summarized in Table 1.


Table 1

Dimension TransformsPseudo-Green’s (gn)

functionInput/output relationship

3Spherical Hankel and

spherical harmonicg3Dn (r, k) = −ikh(2)

n (kr)ukl(r, ω) =

g3Dl

(r, k)η(ω)Sl{qkl (k)}2 Hankel and Fourier Series g2D

n (r, k) = −πi2H

(2)n (kr) un(r, ω) =

g2Dn (r, k)η(ω)Hn{qn(k)}

1 1D Fourier g1D(x) =−i2k

e−ik|x| u(x, ω) =g1D(x)η(ω)Q(− sgn(x)k)

12. Applications of the Transfer Function Relationships tothe Wave Equation

The transfer function relationships can be used to find time domain Green’s functions where

the temporal Fourier integral can be easily inverted. In particular, we consider the case where

the input to the standard wave equation is a Dirac-delta function at the origin

∇2u(�r, t) − 1

c2

∂2

∂t2u(�r, t) = −δ(�r)δ(t). (12.1)

This corresponds to the Helmholtz equation above with k = ω/c. The temporal and

spatial Fourier transform of δ(�r)δ(t) is 1, which is spherically symmetric so that the transfer

function relationships given above only need the zeroth-order component and simplify

considerably. In 3D, the transfer function relationship is given by

u(r, ω) = g3D0 (r, k) · 1 = −ikh(2)

0 (kr). (12.2)

In 2D, the relationship is given by

u(r, ω) = g2D0 (r, k) · 1 = −πi

2H0(kr). (12.3)

In 1D, the relationship is

u(r, ω) = g1D(r, k) · 1 =1

2ike−ik|r|. (12.4)

These three relationships can now be inverse Fourier transformed in time.

For the 3D case, we use the fact that h(2)0 (kr) = j0(kr)− iy0(kr) = i exp(−ikr)/kr so that

the inverse Fourier transform of (12.2) gives

u(r, t) =1

2π

∫∞

−∞

1

re−iωr/ceiωtdω

=1

rδ

(t − r

c

).

(12.5)


The 1D time response can be found from the inverse Fourier transform of (12.4)

u(r, t) =1

2π

∫∞

−∞

c

2iωe−iω|r|/ceiωtdω

=c

4

[2H

(t − |r|

c

)− 1

]=

c

4sgn

(t − |r|

c

),

(12.6)

where H(x) is the Heaviside unit step function.

The 2D case is a little more complicated but can nevertheless be evaluated in closed

form by an inverse Fourier transform of (12.3). To do this, we write the zeroth-order Hankel

function in its integral form as [37]

H(2)0 (x) = J0(x) − iY0(x) =

2i

π

∫∞

1

e−ixτ√τ2 − 1

dτ. (12.7)

Hence from (12.7) and (12.3), the inverse Fourier transform of (12.3) gives

u(r, t) =1

2π

∫∞

−∞

∫∞

1

e−iωrτ/c

√τ2 − 1

dτeiωtdω. (12.8)

Changing the order of integration gives

u(r, t) =∫∞

1

1√τ2 − 1

1

2π

∫∞

−∞eiω[t−rτ/c]dωdτ. (12.9)

But

1

2π

∫∞

−∞eiω[t−rτ/c]dω = δ

(t − rτ

c

), (12.10)

so that (12.9) becomes

u(r, t) =∫∞

1

1√τ2 − 1

δ

(t − rτ

c

)dτ =

∫∞

−∞

1√τ2 − 1

H(τ − 1)δ(t − rτ

c

)dτ. (12.11)

Changing variables so that x = rτ/c gives

u(r, t) =c

r

∫∞

−∞H

(cx

r− 1

)1√

(cx/r)2 − 1

δ(t − x)dx = H

(t − r

c

)1√

t2 − r2/c2, (12.12)

which yields the desired result in closed form.


13. Application: Inversion Formulas for Tomographic Applicationswith Real Wavenumbers

In the case of a real wavenumber k, an immediate application of the above formulas is

for closed-form inversion formulas which are useful for tomographic applications. These

inversion formulas are applicable to applications where the wavefield (e.g., acoustic field)is measured and the goal is to reconstruct the source (input) that led to that measured

wavefield (output). As we have seen, formulas above give the output (wavefield) in terms

of the transform of the input evaluated at k. Thus, if the output wavefield is measured, this

implies knowledge of the transform of the input. An inverse transform then leads to the input

itself.

13.1. Inversion Formula in 3D

Equation (8.14) admits an inversion useful for tomographic applications. It can be written as

iukl(r0, ω)

ksh(2)l(ksr0)η(ω)

=∫∞

0

φkl (x)jl(ksx)x

2dx, (13.1)

where x has been used as a dummy integration variable in order to avoid possible confusion

and r0 has been used as the radial variable and indicates the position where a measurement

of the wavefield is made. Multiplying both sides by jl(ksr)k2s and integrating over ks gives

∫∞

0

iukl(r0, ω)

ksh(2)l(ksr0)η(ω)

jl(ksr)k2sdks =

∫∞

0

∫∞

0

φkl (x)jl(ksx)x

2dxjl(ksr)k2sdks

=∫∞

0

∫∞

0

φkl (x)jl(ksx)jl(ksr)k

2sdksx

2dx.

(13.2)

Using the orthogonality of the spherical Bessel functions, it follows that (13.2) becomes

∫∞

0

iukl(r0, ω)

ksh(2)l(ksr0)η(ω)

jl(ksr)k2sdks =

∫∞

0

φkl (x)

π

2x2δ(x − r)x2dx =

π

2φkl (r). (13.3)

Recalling that ks = ω/cs, the inversion formula for the spatial source becomes

φkl (r) =

2i

πc2s

∫∞

0

ukl(r0, ω)

h(2)l(ωr0/cs)η(ω)

jl

(ωr

cs

)ωdω. (13.4)

If the measured wavefield is also separable in space and time so that ukl(r0, ω) = χk

l(r0)T(ω),

then (13.4) can be further simplified to

φkl (r) =

2iχkl(r0)

πc2s

∫∞

0

T(ω)

h(2)l(ωr0/cs)η(ω)

jl

(ωr

cs

)ωdω. (13.5)


Equation (13.5) gives a simple form for finding the spatial source function at any position

r once having measured the temporal response of the wavefield at a position r0. However,

since the spherical harmonic transform ukl(r0, ω) of the wavefield is required, this implies

that sufficient angular information about the wavefield u must be obtained in order to permit

this spherical harmonic calculation.


Equation (9.19) admits an inversion leading to the input function. We assume that the source

function is separable in time and space (a fairly general assumption) so that s(�r, ω) =φ(�r)η(ω). In this case, then sk

l(r, ω) = φk

l(r)η(ω) and (8.12) becomes

2iun(r0, ω)

πH(2)n (ksr0)η(ω)

=∫∞

0

φn(x)Jn(ksx)x dx, (13.6)

where x has been used as a dummy integration variable in order to avoid possible confusion

and r0 indicates the radial position where the wavefield is measured. Multiplying both sides

of (13.6) by Jn(ksr)ks and integrating over ks gives

∫∞

0

2iun(r0, ω)


Jn(ksr)ksdks =∫∞

0

∫∞

0

φn(x)Jn(ksx)x dxJn(ksr)ksdks,

=∫∞

0

φn(x)∫∞

0

Jn(ksx)Jn(ksr)ksdks x dx.

(13.7)

Using the orthogonality of the spherical Bessel functions, it follows that (13.7) becomes

∫∞

0

2iun(r0, ω)


Jn(ksr)ksdks =∫∞

0

φn(x)1

xδ(x − r)x dx = φn(r). (13.8)

Recalling that ks = ω/cs, the inversion formula for the inhomogeneity becomes

φn(r) =2i

πc2s

∫∞

0

un(r0, ω)

H(2)n (ωr0/cs)η(ω)

Jn

(ωr

cs

)ωdω. (13.9)

If the measured wavefield is also separable in space and time so that un(r0, ω) = χn(r0)T(ω),then (13.4) can be further simplified to

φn(r) =2iχn(r0)

πc2s

∫∞

0

T(ω)

H(2)n (ωr0/cs)η(ω)

Jn

(ωr

cs

)ωdω. (13.10)

Equation (13.10) gives a simple form for finding the spatial source function at any position

r once having measured the wavefield response at a radial position r0. Comparing (13.10)and (13.5), it is noted that they have identical forms with the exception of the replacement


of Hankel and Bessel functions for the 2D case with spherical Hankel and Bessel functions

for the 3D case.


We can also further assume that the wavefield is also separable in time and space so that

u(r, ω) = χ(r)T(ω), which leads to to

φ(−ks) =χ(r)T(ω)η(ω)

2ikseiksr =

∫∞

−∞φ(y)eiyksdy r > 0,

φ(ks) =χ(r)T(ω)η(ω)

2ikse−iksr =

∫∞

−∞φ(y)e−iyksdy r < 0.

(13.11)

Using the orthogonality of the Fourier kernel, both sides are multiplied by e±ixks and

integrated over all ks to give

χ(r)∫∞

−∞

T(ω)η(ω)

2ikseiksre−ixksdks

=∫∞

−∞φ(y) ∫∞

−∞eiykse−ixksdksdy = 2π

∫∞

−∞φ(y)δ(y − x

)dy r > 0

=⇒ φ(x) =iχ(r)

πc2s

∫∞

−∞

T(ω)η(ω)

eiω((r−x)/cs)ωdω r > 0,

(13.12)

and similarly

χ(r)∫∞

−∞

T(ω)η(ω)

2ikse−iksreixksdks

=∫∞

−∞φ(y) ∫∞

−∞e−iykseixksdksdy = 2π

∫∞

−∞φ(y)δ(y − x

)dy r < 0

=⇒ φ(x) =iχ(r)

πc2s

∫∞

−∞

T(ω)η(ω)

eiω((x−r)/cs)ωdω r < 0.

(13.13)

Several points need to be made regarding (13.12) and (13.13). First, they both give the source

function as a function of position x along the real line. The interpretation of the variable r

is that of the position at which the measurement is made and is considered to be a fixed

quantity. Both (13.12) and (13.13) are inverse Fourier transforms of (T(ω)/η(ω))ω, evaluated

at (x − r)/cs or (r − x)/cs, depending on whether measurements are made in transmission or

reflection. Clearly |x − r|/cs is the time taken for a wave to travel the distance |x − r|.The 2D and 3D equivalent to (13.12) and (13.13) are (13.5) and (13.10) which also

similarly involve an inverse transform of (T(ω)/η(ω))ω but have a different kernel for the

integration. The reason for the differences in the nature of the kernels from the 2D/3D cases

to the 1D case is that in the 2D and 3D cases the kernels used for the Fourier transforms are

the Bessel and spherical Bessel functions, respectively, which are the standing wave solutions


in those dimensions. However, Green’s functions in those dimensions are the Hankel and

spherical Hankel functions which are the travelling wave solutions. We note that the kernels

for the inversion formulas of (13.5) and (13.10) involve inverse Hankel and spherical Hankel

functions, which represent the inverse of Green’s function. In contrast, the travelling wave

solutions in 1D are the complex exponential and those are also the kernel of 1D Fourier

transform. The ratio of the two complex exponentials (one represents the Fourier kernel, the

other represents the inverse of 1D Green’s function) finally give another complex exponential

which represents the shift |x − r|/cs.

14. Summary and Conclusions

This paper presents the derivation of spatial transfer function relationships for the Helmholtz

equation. The focus has been on deriving forward transfer function relationships so that

once given a particular input function, the resulting output wavefield can be calculated

at any point in space. These are termed “transfer function” relationships because the

relationship between the input and output quantities is one of multiplication and no

convolution is involved, even in the spatial domain where normally a convolution would

be required. Interestingly, instead of Green’s function itself, Green’s function coefficients

for the space are required. These Green function coefficients form the kernel of the output

response. Each element of the kernel is then weighted (multiplicatively) by the relevant

Fourier/Hankel/Spherical Hankel transform of the input, evaluated on the n-dimensional

sphere of radius k, where k is the wavenumber and n is the dimension of the space.

Combined together, these form the final output response. This is true for dimensions n = 1,

2, 3. These simple but powerful results serve to cast the entire problem as an input-output

problem with a transfer function relating input to output. In this view of the problem, the

input (inhomogeneity) and output (resulting wavefield) in space are related by a simple

multiplicative transfer function relationship and not via a convolution. Some applications of

these results were shown in the manuscript.

References

[1] A. Mandelis, “Theory of photothermal-wave diffraction and interference in condensed media,”Journal of the Optical Society of America A, vol. 6, no. 2, pp. 298–308, 1989.

[2] A. Mandelis, Diffusion-Wave Fields. Mathematical methods and Green Function, Springer-Verlag, NewYork, NY, USA, 2001.

[3] M. Slaney and A. C. Kak, Principles of Computerized Tomographic Imaging, Society for Industrial andApplied Mathematics, Philadelphia, Pa, USA, 1988.

[4] M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Review of Scientific Instruments, vol.77, no. 4, pp. 041101.1–041101.22, 2006.

[5] A. Mandelis and C. Feng, “Frequency-domain theory of laser infrared photothermal radiometricdetection of thermal waves generated by diffuse-photon-density wave fields in turbid media,”Physical Review E—Statistical Nonlinear, and Soft Matter Physics, vol. 65, no. 2, pp. 021909/1–021909/19,2002.

[6] A. Mandelis, Y. Fan, G. Spirou, and A. Vitkin, “Development of a photothermoacoustic frequencyswept system: theory and experiment,” Journal De Physique. IV : JP, vol. 125, pp. 643–647, 2005.

[7] A. Mandelis, R. J. Jeon, S. Telenkov, Y. Fan, and A. Matvienko, “Trends in biothermophotonics andbioacoustophotonics of tissues,” in Proceedings of the International Society for Optics and Photonics (SPIE’05), vol. 5953, pp. 1–15, The International Society for Optical Engineering, Warsaw, Poland, 2005.

[8] C. L. Matson, “A diffraction tomographic model of the forward problem using diffuse photon densitywaves,” Optics Express, vol. 1, no. 1, pp. 6–11, 1997.


[9] L. Nicolaides and A. Mandelis, “Image-enhanced thermal-wave slice diffraction tomography withnumerically simulated reconstructions,” Inverse Problems, vol. 13, no. 5, pp. 1393–1412, 1997.

[10] L. Nicolaides, M. Munidasa, and A. Mandelis, “Thermal-wave infrared radiometric slice diffractiontomography with back-scattering and transmission reconstructions: experimental,” Inverse Problems,vol. 13, no. 5, pp. 1413–1425, 1997.

[11] O. Pade and A. Mandelis, “Computational thermal-wave slice tomography with backpropagationand transmission reconstructions,” Review of Scientific Instruments , vol. 64, no. 12, pp. 3548–3562,1993.

[12] O. Pade and A. Mandelis, “Thermal-wave slice tomography using wave field reconstruction,” inProceedings of the 8th International Topical Meeting on Photoacoustic and Photothermal Phenomena, vol.4, 1994.

[13] S. A. Telenkov, G. Vargas, J. S. Nelson, and T. E. Milner, “Coherent thermal wave imaging of sub-surface chromophores in biological materials,” Physics in Medicine and Biology, vol. 47, no. 4, pp. 657–671, 2002.

[14] S. A. Telenkov, J. S. Nelson, and T. E. Milner, “Tomographic reconstruction of tissue chromophoresusing thermal wave imaging,” Laser Tissue Interaction XIII, vol. 4617, pp. 40–46, 2002.

[15] S. A. Telenkov and A. Mandelis, “Fourier-domain biophotoacoustic subsurface depth selectiveamplitude and phase imaging of turbid phantoms and biological tissue,” Journal of Biomedical Optics,vol. 11, no. 4, 2006.

[16] N. Baddour, “Theory and analysis of frequency-domain photoacoustic tomography,” Journal of theAcoustical Society of America, vol. 123, no. 5, pp. 2577–2590, 2008.

[17] N. Baddour, “A multidimensional transfer function approach to photo-acoustic signal analysis,”Journal of the Franklin Institute, vol. 345, no. 7, pp. 792–818, 2008.

[18] N. Baddour, “Fourier diffraction theorem for diffusion-based thermal tomography,” Journal of PhysicsA, vol. 39, no. 46, pp. 14379–14395, 2006.

[19] A. Mandelis, “Theory of photothermal wave diffraction tomography via spatial Laplace spectraldecomposition,” Journal of Physics A, vol. 24, no. 11, pp. 2485–2505, 1991.

[20] A. Mandelis, “Green’s functions in thermal-wave physics: cartesian coordinate representations,”Journal of Applied Physics, vol. 78, no. 2, pp. 647–655, 1995.

[21] M. Puschel and J. M. F. Moura, “Algebraic signal processing theory: cooley-Tukey type algorithms forDCTs and DSTs,” IEEE Transactions on Signal Processing, vol. 56, no. 4, pp. 1502–1521, 2008.

[22] M. Puschel, “Algebraic signal processing theory: An overview,” in Proceedings of the IEEE 12th DigitalSignal Processing Workshop and 4th IEEE Signal Processing Education Workshop, pp. 386–391, Moose,Wyo, USA, 2006.

[23] M. Puschel and J. Moura, “Algebraic signal processing theory: foundation and 1-D time,” IEEETransactions on Signal Processing, vol. 56, no. 8, pp. 3572–3585, 2008.

[24] M. Puschel and J. M. F. Moura, “Algebraic signal processing theory: 1-D space,” IEEE Transactions onSignal Processing, vol. 56, no. 8, pp. 3586–3599, 2008.

[25] M. Puschel and J. M. F. Moura, “The algebraic structure in signal processing: time and space,” inProceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP ’06),vol. 5, 2006.

[26] M. Puschel and M. Rotteler, “Algebraic signal processing theory: 2-D spatial hexagonal lattice,” IEEETransactions on Image Processing, vol. 16, no. 6, pp. 1506–1521, 2007.

[27] M. Puschel and M. Rotteler, “Algebraic signal processing theory: cooley-Tukey type algorithms on the2-D hexagonal spatial lattice,” Applicable Algebra in Engineering, Communication and Computing, vol. 19,no. 3, pp. 259–292, 2008.

[28] N. Baddour, “Operational and convolution properties of two-dimensional Fourier transforms in polarcoordinates,” Journal of the Optical Society of America A, vol. 26, no. 8, pp. 1767–1778, 2009.

[29] N. Baddour, “Operational and convolution properties of three-dimensional Fourier transforms inspherical polar coordinates,” Journal of the Optical Society of America, vol. 27, no. 10, pp. 2144–2155,2010.

[30] N. Baddour, “Theory and analysis of frequency-domain photoacoustic tomography,” Journal of theAcoustical Society of America, vol. 123, no. 5, pp. 2577–2590, 2008.

[31] G. S. Chirikjian and A. B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, CRCPress, Boca Raton, Fla, USA, 2000.

[32] G. S. Chirikjian and A. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis: WithEmphasis on Rotation and Motion Groups, Academic Press, New York, NY, USA, 2001.


[33] R. Piessens, “The Hankel transform,” in The Transforms and Applications Handbook, vol. 2, pp. 9.1–9.30,CRC Press, Boca Raton, Fla, USA, 2000.

[34] “Spherical harmonics—wikipedia, the free encyclopedia,” http://en.wikipedia.org/wiki/Sphericalharmonics, Accessed: 16-Nov-2009.

[35] J. R. Driscoll and D. M. Healy, “Computing Fourier transforms and convolutions on the 2-sphere,”Advances in Applied Mathematics, vol. 15, no. 2, pp. 202–250, 1994.

[36] N. Baddour, “Multidimensional wave field signal theory: mathematical foundations,” AIP Advances,vol. 1, no. 2, p. 022120, 2011.

[37] M. Abramowitz and I. Stegun, Stegun, Handbook of Mathematical Functions with Formulas, Graphs, andMathematical Tables, Dover Publications, New York, NY, USA, 1964.

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