Multi-level thresholding using quantum inspired meta-heuristics Sandip Dey a,1 , Indrajit Saha b,c,⇑,1 , Siddhartha Bhattacharyya d, * , Ujjwal Maulik b a Department of Information Technology, Camellia Institute of Technology, Madhyamgram, Kolkata 700129, India b Department of Computer Science & Engineering, Jadavpur University, Kolkata 700032, India c Institute of Informatics, University of Wroclaw, Wroclaw 50383, Poland d Department of Information Technology, RCC Institute of Information Technology, Beliaghata, Kolkata 700015, India article info Article history: Received 9 October 2013 Received in revised form 3 March 2014 Accepted 6 April 2014 Available online 14 May 2014 Keywords: Image segmentation Multilevel thresholding Otsu’s method Quantum computing Statistical test abstract Image thresholding is well accepted and one of the most imperative practices to accomplish image seg- mentation. This has been widely studied over the past few decades. However, as the multi-level thres- holding computationally takes more time when level increases, hence, in this article, quantum mechanism is used to propose six different quantum inspired meta-heuristic methods for performing multi-level thresholding faster. The proposed methods are Quantum Inspired Genetic Algorithm, Quan- tum Inspired Particle Swarm Optimization, Quantum Inspired Differential Evolution, Quantum Inspired Ant Colony Optimization, Quantum Inspired Simulated Annealing and Quantum Inspired Tabu Search. As a sequel to the proposed methods, we have also conducted experiments with the two-Stage multi- threshold Otsu method, maximum tsallis entropy thresholding, the modified bacterial foraging algo- rithm, the classical particle swarm optimization and the classical genetic algorithm. The effectiveness of the proposed methods is demonstrated on fifteen images at the different level of thresholds quantita- tively and visually. Thereafter, the results of six quantum meta-heuristic methods are considered to cre- ate consensus results. Finally, statistical test, called Friedman test, is conducted to judge the superiority of a method among them. Quantum Inspired Particle Swarm Optimization is found to be superior among the proposed six quantum meta-heuristic methods and the other five methods are used for comparison. A Friedman test again conducted between the Quantum Inspired Particle Swarm Optimization and all the other methods to justify the statistical superiority. Finally, the computational complexities of the pro- posed methods have been elucidated for the sake of finding out the time efficiency of the proposed methods. Ó 2014 Elsevier B.V. All rights reserved. 1. Introduction Image thresholding can be recognized as the easiest and most efficient method that is widely used for image segmentation. This is used as an effective tool to bifurcate images into object and back- ground [1]. For its first kind, the pixel intensities of the image are grouped into two classes, called bi-level image thresholding. While the number of groups exceed two, it is recognized as multi-level thresholding [2]. Both of them can be identified by acclimatizing parametric or nonparametric approaches [3,4]. Moreover, there exists different algorithms for bi-level image thresholding that can also be extended to their corresponding multi-level versions, if necessary [2,5]. When the level increases in multi-level thres- holding, the number of computations increase as well. This could add significant difficulties specially when higher level threshold values are evaluated. Many algorithms have been proposed so far that can handle this situation efficiently, where some of them are developed for a specific purpose. These algorithms have their own advantages and disadvantages. However, in this paper, six new quantum meta-heuristic methods for multi-level thresholding are presented that can be used efficiently for general purpose. Generally a wave function, jwi which exists in Hilbert space, is employed for describing a quantum system. The Schrödinger equa- tion (SE) is assumed to be accountable for overseeing the inherent dynamism of quantum computing (QC). A quantum bit or qubit is considered as the smallest unit for a two-state quantum machine. The qubit may be in ‘‘0’’ state or in ‘‘1’’ state or even in superposi- tion between these two states where, j0i¼ 1 0 and j1i¼ 0 1 . The superposition of the two state vectors are symbolized by the http://dx.doi.org/10.1016/j.knosys.2014.04.006 0950-7051/Ó 2014 Elsevier B.V. All rights reserved. ⇑ Corresponding authors. Address: Department of Computer Science & Engineer- ing, Jadavpur University, Kolkata 700032, India (I. Saha). E-mail addresses: [email protected](I. Saha), dr.siddhartha.bhattacharyya@ gmail.com (S. Bhattacharyya). 1 Both the authors are joint first authors and contributed equally. Knowledge-Based Systems 67 (2014) 373–400 Contents lists available at ScienceDirect Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys
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1 Both the authors are joint first authors and contributed equally.
Sandip Dey a,1, Indrajit Saha b,c,⇑,1, Siddhartha Bhattacharyya d,*, Ujjwal Maulik b
a Department of Information Technology, Camellia Institute of Technology, Madhyamgram, Kolkata 700129, Indiab Department of Computer Science & Engineering, Jadavpur University, Kolkata 700032, Indiac Institute of Informatics, University of Wroclaw, Wroclaw 50383, Polandd Department of Information Technology, RCC Institute of Information Technology, Beliaghata, Kolkata 700015, India
a r t i c l e i n f o
Article history:Received 9 October 2013Received in revised form 3 March 2014Accepted 6 April 2014Available online 14 May 2014
Keywords:Image segmentationMultilevel thresholdingOtsu’s methodQuantum computingStatistical test
a b s t r a c t
Image thresholding is well accepted and one of the most imperative practices to accomplish image seg-mentation. This has been widely studied over the past few decades. However, as the multi-level thres-holding computationally takes more time when level increases, hence, in this article, quantummechanism is used to propose six different quantum inspired meta-heuristic methods for performingmulti-level thresholding faster. The proposed methods are Quantum Inspired Genetic Algorithm, Quan-tum Inspired Particle Swarm Optimization, Quantum Inspired Differential Evolution, Quantum InspiredAnt Colony Optimization, Quantum Inspired Simulated Annealing and Quantum Inspired Tabu Search.As a sequel to the proposed methods, we have also conducted experiments with the two-Stage multi-threshold Otsu method, maximum tsallis entropy thresholding, the modified bacterial foraging algo-rithm, the classical particle swarm optimization and the classical genetic algorithm. The effectivenessof the proposed methods is demonstrated on fifteen images at the different level of thresholds quantita-tively and visually. Thereafter, the results of six quantum meta-heuristic methods are considered to cre-ate consensus results. Finally, statistical test, called Friedman test, is conducted to judge the superiority ofa method among them. Quantum Inspired Particle Swarm Optimization is found to be superior amongthe proposed six quantum meta-heuristic methods and the other five methods are used for comparison.A Friedman test again conducted between the Quantum Inspired Particle Swarm Optimization and all theother methods to justify the statistical superiority. Finally, the computational complexities of the pro-posed methods have been elucidated for the sake of finding out the time efficiency of the proposedmethods.
� 2014 Elsevier B.V. All rights reserved.
1. Introduction
Image thresholding can be recognized as the easiest and mostefficient method that is widely used for image segmentation. Thisis used as an effective tool to bifurcate images into object and back-ground [1]. For its first kind, the pixel intensities of the image aregrouped into two classes, called bi-level image thresholding. Whilethe number of groups exceed two, it is recognized as multi-levelthresholding [2]. Both of them can be identified by acclimatizingparametric or nonparametric approaches [3,4]. Moreover, thereexists different algorithms for bi-level image thresholding thatcan also be extended to their corresponding multi-level versions,
if necessary [2,5]. When the level increases in multi-level thres-holding, the number of computations increase as well. This couldadd significant difficulties specially when higher level thresholdvalues are evaluated. Many algorithms have been proposed so farthat can handle this situation efficiently, where some of them aredeveloped for a specific purpose. These algorithms have theirown advantages and disadvantages. However, in this paper, sixnew quantum meta-heuristic methods for multi-level thresholdingare presented that can be used efficiently for general purpose.
Generally a wave function, jwi which exists in Hilbert space, isemployed for describing a quantum system. The Schrödinger equa-tion (SE) is assumed to be accountable for overseeing the inherentdynamism of quantum computing (QC). A quantum bit or qubit isconsidered as the smallest unit for a two-state quantum machine.The qubit may be in ‘‘0’’ state or in ‘‘1’’ state or even in superposi-
tion between these two states where, j0i ¼ 10
� �and j1i ¼ 0
1
� �. The
superposition of the two state vectors are symbolized by the
374 S. Dey et al. / Knowledge-Based Systems 67 (2014) 373–400
equation jwi ¼ aj0i þ bj1i where, a; b are complex numbers satis-
fying the equation jaj2 þ jbj2 ¼ 1. Coherence in QC exists whenthe states are in superposed form maintaining a constant phaserelationship between them. When the coherence is forced to bedestroyed, decoherence occurs. For decoherence, the requisite
probability for collapsing to the state j1i and j0i are jaj2 and jbj2,respectively. Quantum entanglement is a fascinating feature inquantum system that can be employed to describe the correlationsbetween the diversified qubits [6,7]. Quantum entanglement of aquantum state can be demonstrated using the density matrix[6,7]. Entanglement can be analyzed, distorted, and even washedout, if required, [6,8]. Quantum interference is an another interest-ing feature of a quantum system. The advancement of newresearch era advocates the researchers to entrench the algorithmicconstruction of different quantum-inspired evolutionary algo-rithms (QIEA) by incorporating the philosophy of quantum system[9]. So far, a number of combinatorial optimization problems havebeen resolved using QIEA, where the concept of wave interferencewas introduced [8]. Han et al. designed an QIEA where the qubitwith some probability constraints, linear superposition betweenstates and various Q-gates for population assortment have beenused [9].
Nowadays, meta-heuristic approaches are widely used invarious domains of engineering and science. Many authors haveutilized different meta-heuristic approaches for image threshold-ing. Some distinctive applications of meta-heuristic are given in[10–14]. Mostly, the traditional approaches of multi-level thres-holding for gray scale images use binary encoding scheme, whereeach pixel is represented by 8 bits. Thus, the length of the stringincreases in multiply of 8 for higher levels. As this article is con-fined on gray scale images, this fact motivated us to propose analternative technique based on quantum inspired meta-heuristicmethods for multi-level thresholding using the notion of qubits,where real value encoding scheme is used to determine the activepixel. For this purpose,
ffiffiffiLp
number of random pixels are selected,where, L represents the maximum pixel intensity value of the grayscale test image. With this encoding scheme, Quantum InspiredGenetic Algorithm (QIGA), Quantum Inspired Particle Swarm Opti-mization (QIPSO), Quantum Inspired Differential Evolution (QIDE),Quantum Inspired Ant Colony Optimization (QIACO), QuantumInspired Simulated Annealing (QISA) and Quantum Inspired TabuSearch (QITS) for multilevel thresholding are proposed. The effec-tiveness of these methods is demonstrated on fifteen images atthe different level of thresholds in terms of different quantitativemeasures. Thereafter, consensus results of these six methods arealso computed. Finally, statistical test, called Friedman test[15,16], is conducted to judge the superiority of a method amongthem.
As a part of comparative study to adjudge the efficacy of theproposed quantum inspired methods, we have resorted to fiveclassical algorithms viz., Two-Stage Multithreshold Otsu method(TSMO) [17], Maximum Tsallis entropy Thresholding (MTT)2 [18],Modified Bacterial Foraging (MBF) algorithm [19], classical ParticleSwarm Optimization for multi-level thresholding [20] and classicalGenetic Algorithm for multi-level thresholding [21]. The compara-tive study reveals that the Quantum Inspired Particle Swarm Optimi-zation outperforms the proposed five quantum meta-heuristicmethods and the other five classical algorithms used for comparison.
2. Background
The field of quantum computing became popular since thenotion of quantum mechanical system was anticipated at the early
2 As abbreviated in [18].
1980s [22]. The aforesaid quantum mechanical machine is able tosolve some particular computational problems awfully efficiently[23]. In [24], the author has recognized that classical computerfaces lack of ability while simulating quantum mechanical system.The author has presented a structural framework to build quantumcomputer. Alfares et al. analyzed how the notion of quantum algo-rithms can be applied to solve some typical engineering optimiza-tion problems [25]. According to their perception, some problemsmay arise when the features of QC are applied. These problemscan be avoided by using certain kind of algorithms. Hogg has pre-sented a framework for structured quantum search where Groversalgorithm was applied to correlate the cost with the gate’s behav-ior [26]. In [27], the authors have extended the work and proposeda new quantum version of combinatorial optimization. Rylanderet al. presented a quantum version of genetic algorithm wherethe quantum principles like superposition and entanglement wereemployed on modified genetic algorithm. In [28], Moore et al. pro-posed a framework for general quantum-inspired algorithms.Later, Han et al. [9] developed an evolution algorithm which wasapplied for solving knapsack problem. In their paper, basic quan-tum principles like qubits and rotation quantum gate were used.Afterward, in [9], the authors have designed another version ofquantum inspired evolutionary algorithm by Han et al. where theperformance was evaluated according to the angles of the rotationgates and later, in [29], a new improved version of this algorithmwas presented. They divided the evolution stage into two differentphases and proposed alteration to the quantum gate adding a ter-mination criterion. The improved version of the work presented in[9] was proposed by Zhang et al. where they applied a differentapproach to get the best solution [30]. Narayan et al. presented agenetic algorithm where quantum mechanics was used for modifi-cation of crossover scheme [31]. Moreover, Li et al. developed amodified genetic algorithm using quantum probability representa-tion. They adjusted crossover and mutation processes for attainingthe quantum representation [32]. In [33], the authors presented aquantum-inspired neural network algorithm where also the basicquantum principles were employed to symbolize the problemvariables.
The instinctive compilation of information science with thequantum mechanics resort to construct the concept of quantumcomputing. Quantum evolutionary algorithm (QEA) was admiredas a probability based optimization technique. It uses qubitsencoded strings for its quantum computation paradigm. The intrin-sic principles of QEA help to facilitate for maintaining the equilib-rium between exploitation and exploration. In recent years, someresearchers have presented some QEAs to solve particular combi-natorial optimization problems. A typical example of QEA is Filterdesign by Zhang et al. [34]. The researches are still going on tocreate purposeful and scalable quantum computers.
Meta-heuristic optimization techniques are employed heuristi-cally in searching algorithms. They use iterative approach to havebetter solution by fleeing from local optima. So they coerce somebasic heuristic to compensate from local optima. There are somerenowned Meta-heuristic techniques namely, GA, PSO, DE, ACO,SA and TS which are applied for optimization with different man-ners. Holland has proposed genetic algorithms (GAs) which imper-sonate the belief of some natural fruition. GAs can be appliedefficiently in data mining for classification. In 2006, Jiao et al. pre-sented organizational coevolutionary algorithm for classification(OCEC) [35]. In OCEC, bottom-up searching technique has beenadopted and enthused from the coevolutionary model that can effi-ciently knob multi-class learning. Kennedy and Eberhart first pro-posed Particle Swarm Optimization (PSO) in 1995 inspiring fromthe synchronized movement in flocks of birds [36]. In PSO, thepopulation of particle is the particle swarm. In 2004, Sousa et al.projected PSO in data mining [37]. PSO can be skilfully used in
S. Dey et al. / Knowledge-Based Systems 67 (2014) 373–400 375
rule based searching process. In recent years, many researchershave tried to improve the performance of PSO and proposed vari-ous alternatives of PSO. In [38], the authors presented algorithmon parameters settings. Some authors have furnished their effortsfor the betterment and variation of PSO by combining diversetechniques [39–41].
The actions of ant colonies are governed by heuristic algorithmsuch as Ant colony optimization (ACO) [62] . In 2002, Parpinelliet al. proposed the concepts of Ant-Miner based ACO for generatingthe classification rules [42]. Subsequently, Sousa et al. presented aPSO/ACO algorithm, hybrid in nature, to discover classificationrules [37]. The working principle of PSO/ACO is divided in twophases. In its first part, ACO determines a rule surrounding onlythe ostensible attributes and in later part, PSO determines the rulepotentially comprehensive with incessant attributes. In [43], somefacts on quantum dots (QDs) have been reported.
Stron and Price presented the differential algorithm (DE) in1997. This is a population-based stochastic meta-heuristic optimi-zation technique. DE has proved its excellence in float-point searchspace. There are numerous modified versions of DE algorithmsinvented so far; most of them are built to solve continuous optimi-zation problems. But it has low efficacy to solve discrete optimiza-tion problems and faces several problems such as engineeringrelated problem especially for routing or scheduling or even com-binational type problems. Binary DE, AMDE [44] presented in 2006by Pampara et al. that can solve numerous numerical optimization.
In [45], a coalesce form of simulated annealing and geneticalgorithm (SA–GA) has been proposed by Fengjie et al. The authorshave employed 2D Otsu algorithm in low contrast images. Theiralgorithm can differentiate the ice-covered cables from its back-grounds. Luo et al. have also presented an combined (SA–GA) algo-rithm for colony image segmentation in [46]. According to Luoet al., the collective gives a better result because the individualshortcomings have been eliminated in the proposed algorithm. In[47], SA and improved Snake model based algorithm were pre-sented by Tang et al. in image segmentation. In [48], Nakib et al.presented a research work on non-supervised image segmentationbased on multi-objective optimization. In addition to this theyhave developed another alternative of SA that can able to resolvethe histogram Gaussian curve fitting problem. The details are givenin [48]. Yufei et al. have worked on image segmentation. They havecombined image entropy and SA together and presented a segmen-tation algorithm. According to their research work, the segmenta-tion process can speed up the contours evaluation. Apart from that,the combination of this meta-heuristic approach can cause forterminating the contours evolution automatically to appositeboundaries [49]. In another paper, Garrido et al. have presentedSA based segmentation approach based on MRI for left ventricle.The details were presented in [50]. Zhijun et al. used a combinedapproach of GA and SA in image segmentation to find thresholdvalue for image segmentation [34].
In 1997, tabu search was first proposed by Fred Glover to allowhill climbing to surmount from local optima [51]. The man behinddrafting the basic ideas for tabu search is Hansen [2]. Later, supple-mentary research has been carried out about this meta-heuristicapproach that was reported in Glover [52,53]. There are manycomputational experiments that establish the completeness andefficacy as a meta-heuristic approach. Faigle et al. and Fox alsoinvestigated the theoretical facets of tabu search that have beenreported in [54,55], respectively. Huang et al. have developed analgorithm called Two-Stage Multithreshold Otsu method (TSMO)to improve the effectiveness of Otsu’s method [17]. Zhang et al.presented a Maximum Tsallis entropy Thresholding method(MTT) for image segmentation. They have employed TsallisEntropy using Artificial Bee Colony Approach for optimization[18]. In 2011, Sathya et al. have proposed another multi-level
thresholding algorithm for image segmentation based on histo-gram called Modified Bacterial Foraging (MBF) algorithm [19].
2.1. Brief highlights of image thresholding methods
Sezgin and Sankur presented an extensive and comprehensivestudy on image thresholding. The details have been presented intheir survey paper [56]. They have classified these methods intosix groups as described below.
In the first category, some histogram’s property like peaks, val-leys and curvatures are taken into consideration. In the second cat-egory, the gray-level pixels are grouped and clustered into twodivisions; one as foreground and other as background. In [56],the authors have described different cluster based methods by dif-ferent researchers in details. In entropy-based methods, entropy ofthe background and foreground areas is calculated. In some othermethods of this category, cross-entropy of the original image andits corresponding binary version is determined and later opti-mized. For object attribute-based methods, similarities like fuzzyshape similarity, edge coincidence and other various parametersof gray level image and its binary version are investigated. In thenext category, different statistical measures like probability distri-bution of higher-order and the correlation between pixels are ana-lyzed. Finally, local image characteristics are acclimatized todetermine its optimum threshold value. The details of variousmethods of each category are elaborately discussed by Sezginet al. in [56].
3. Otsu’s method for image thresholding
In multi-level thresholding, the original image is segregatedinto K number of classes. There are K � 1 thresholds namely,fh1; h2; . . . ; hK�1g, which are employed as separators between theconsecutive classes. The classes are assumed to beC1;C2; . . . ;Ck; . . . ;CK and the ranges of threshold values in theseclasses are ½0;1; . . . ; h1�; ½h1 þ 1; . . . ; h2�; . . . ; ½hk�1 þ 1; . . . ; hk�; . . . ;
½hK�1 þ 1; . . . ; L� 1� where, L is the maximum pixel intensity valueof the gray scale image.
In this research paper, one cluster based thresholding methodhas been exercised as an objective function. This method wasproposed by Otsu [57]. This is one of the most eminentthresholding method that is frequently used for image segmenta-tion [57]. The method finds a set of optimal thresholdsfh1; h2; . . . ; hK�1g that maximizes the between-class variance Ugiven by [57]
U ¼ #2fðh1; h2; . . . ; hK�1g ¼XK
i¼1
xiðli � lÞ ð1Þ
where K denotes the number of thresholds to be determined fromthe class C ¼ fC1;C2; . . . ;CKg and 0 6 h1 6 � � � 6 hK�1 < L. Here,
xi ¼Xj2Ci
pj; pj ¼ nj=N and li ¼Xj2Ci
jpj=xi ð2Þ
where nj signifies the number of pixels of an image measured at thejth intensity level and N is the total number of pixels intensities ofthe corresponding image. The probability of jth pixel of the image isdenoted by pj. xi is the probability of class Ci whereas, li represents
376 S. Dey et al. / Knowledge-Based Systems 67 (2014) 373–400
the mean of this class. l is known as the mean of class C. Note that,the number of optimum thresholds are determined by maximizingU [57].
4. Basics of quantum computing
The basic physical principles of quantum physics are used inquantum computer (QC) for information processing [7]. Schröding-er’s equation has been incorporated to rule over the mechanism ofquantum system. The state of any quantum mechanical system,jwi, which subsists in Hilbert space, can be considered as a vectordefined in a complex vector space. Let us assume that there existsset of vectors fj/ig; / ¼ 0;1; . . . ;Q� 1 where, the maximum valueof Q may be 1, which form an orthonormal basis for the vectorspace defined above, then jwi can be expressed as
jwi ¼X
/
c/j/i ð3Þ
where, c/ are complex coefficients satisfyingP
/jc/j2 ¼ 1. The‘‘bra-ket’’ notation is very popular and important as well in quan-tum mechanics. The ‘‘ket’’ vector, j�i is equivalent to a column vectorwhereas, its Hermitian conjugate or sometimes called complex con-jugate transpose, ‘‘bra’’ is denoted by h�j. The ‘‘bracket’’ is appearedto be h�j�i [7], which is formed by uniting the above two vectors.
In QC, the basic element of any two-state quantum system iscalled qubit. qubit is generally represented either by the ‘‘groundstate’’, j0i or by ‘‘excited state’’, j1i, which are basically normalizedand mutually orthogonal to each other. It can also be representedby the linear superposition of the basic states as jwi ¼ caaþ cbb
Fig. 1. Quantum inspired meta-heuristic m
with the unitary condition jcaj2 þ jcbj2 ¼ 1 where, ca; cb 2 C. InQC, coherence is described as a linear superposition of the basisstates of jwi as given in Eq. (3). If the above defined linear superpo-sition is forced to be destroyed, it is called decoherence.
The wave function w is crumpled into the basic state wk withprobability ck. When an inimitable correlation exists in QC, it iscalled entanglement. The quantum gates are basically hardwaredevices which are used to update qubits individuals using a prede-fined unitary operator. The quantum gates act over a fixed timeperiod. Formally, quantum gates hold the relation Uþ ¼ U�1 andUUþ ¼ UþU ¼ I where, U stands for unitary operator. A typicalexample of quantum rotation gate, which is used to update jthqubit value ðaj; bjÞ, can be depicted as
a0jb0j
" #¼
cosðhjÞ � sinðhjÞsinðhjÞ cosðhjÞ
� � aj
bj
" #ð4Þ
where, hj is a rotation angle of each qubit, which is generallydesigned compliant with specific problem.
For measurement purpose, we introduce the von Neumannmeasurement strategy which establishes one of a set of basisparticipating states as output. To continue this process, we firstselect a basis at random and ensure the basis states the systemexists in. QC initiates some probabilistic measurement proce-dure that transforms the states in superposed form to a specificstate required. To accomplish multi-qubit measurement, a succes-sion of single-qubit measurements in the defined basis areperformed. Suppose, at initial, the system is in the state jwi, thenthe probability of finding the state / will be jc/j2.
ethods for multi-level thresholding.
S. Dey et al. / Knowledge-Based Systems 67 (2014) 373–400 377
In this section, six different quantum inspired meta-heuristicmethods are presented (see Fig. 1). Each method is described elab-orately in the subsections given below.
5.1. Quantum Inspired Genetic Algorithm
In this subsection, the authors have tied up the genetic algo-rithm, a meta-heuristic optimization technique with quantumprinciples to develop a new Quantum Inspired Genetic Algorithmfor multi-level thresholding (QIGA). The proposed QIGA is depictedin Algorithm 1. The details of QIGA is described in different subsec-tions given below.
5.1.1. Population generationThis is the initial part of the proposed QIGA. Here, a number of
qubit encoded chromosomes are employed to generate an initialpopulation. Initially, a n� L matrix is formed using two superposedquantum states, as given below [8]
Algorithm 1. Steps of QIGA for multi-level thresholding
Input: Number of Generation: MxGenSize of the Population: VCrossover probability: Pcr
Mutation probability: Pm
No. of thresholds: K
Output: Optimal threshold values: h
1: Selection of pixel values randomly to generate V numberof initial chromosomes (POP) where length of eachchromosome is L ¼
ffiffiffiLp
, where, L represents maximumgray value of a selected image.
2: Using the concept of qubits to assign real value between(0,1) to each pixel encoded in POP. Let it produces POP0.
3: Using quantum rotation gate as given in Eq. (4), updatePOP0.
4: POP0 undergoes quantum orthogonality to generate POP00.5: Finding K number of thresholds as pixel values from each
chromosome in POP. It should satisfy the conditiongreater than ð>Þ randð0;1Þwith its corresponding value inPOP00. Let it gives POP�.
6: Compute fitness of each of chromosome in POP� using Eq. (1).7: Record the best chromosome b 2 POP00 and its threshold
values in TB 2 POP�.8: Apply tournament selection strategy to replenish the
chromosome pool.9: repeat
10: Select two chromosomes, k and m at random from ½1;V�where k – m.
11: if (randð0;1Þ < Pcr) then12: Select a random position, pos 2 ½1;L�.13: Perform crossover operation between two
chromosomes, k and m at the position pos.
14: end if15: until the pool of chromosomes are filled up16: for all k 2 POP00 do17: for For all position in k do18: if (randð0;1Þ < Pm) then19: Flip the corresponding position with random real
number.20: end if21: end for22: end for23: Repeat steps 3, 4 and 5.24: Save the best chromosome in c 2 POP00 and its
corresponding threshold values in TB 2 POP00.25: Compare the fitness value of the chromosomes of
b and c.26: Store the best chromosome in b 2 POP00 and its
corresponding threshold values in TB 2 POP00 (elitism).27: Repeat steps 8 to 26 for fixed number of generations,
MxGen.28: Report the optimal threshold values, h ¼ TB.
Here, jaijj2 and jbijj2 are the probabilities to find the state jw1i
and jw2i, respectively where, i ¼ 1;2; . . . ;n; j ¼ 1;2; . . . ; L and Lrepresents the maximum pixel intensity value of the input grayscale image. Each row in Eq. (5) signifies qubit representation ofa single chromosome. This is the possible scheme for encoding par-ticipating chromosomes for required number of solution usingsuperposition principle. jw1i and jw2i represent ‘‘0’’ state and ‘‘1’’state, respectively where jw1i þ jw2i is the superposition of thesetwo states for a two state quantum computing.
5.1.2. Quantum orthogonalityThe second step of the proposed QIGA entails an quantum
orthogonality of the population of chromosomes as shown inEq. (5). This step preserves the basic constraint of the qubit individ-ual in Eq. (5) as given by
jaijj2 þ jbijj2 ¼ 1 ð6Þ
where, i ¼ 1;2; . . . ; n and j ¼ 1;2; . . . ; L.Each qubit experiences a quantum orthogonality to validate
Eq. (6). A typical example of quantum orthogonality is shown inFig. 2 where, ð0:2395Þ2 þ ð0:9708Þ2 ¼ 1; ð�0:1523Þ2 þ ð0:9883Þ2 ¼1, etc. hold.
5.1.3. Determination of threshold values in population andmeasurement of fitness
In quantum measurement, a special treatment in measurementpolicy is espoused that leads to find optimal solution. Since, theproposed methods are executed on conventional computers, itnecessitates to destruct the superposed quantum states to havecoherent solution and the states are conserved for the iterationof next generation. Depending on the level of computation inmulti-level thresholding, the number of states having exactly sin-gle qubit are determined. This phase resorts to quantum measure-ment for determining optimal thresholds. Firstly, V number ofpixel intensity values are randomly selected from the test imageto form a population, POP. The length of each participating chromo-some in POP is taken as L ¼
ffiffiffiLp
, where L signifies the maximumpixel intensity value of the selected image. Afterward, the conceptof qubits is applied to assign a real number between (0,1) at ran-dom to each pixel encoded in POP, which produces POP0. Next,
Fig. 2. Quantum orthogonality.
378 S. Dey et al. / Knowledge-Based Systems 67 (2014) 373–400
POP0 passes through an quantum orthogonality to generate POP00.In QIGA, a predefined number of threshold values as pixelintensities are obtained using a probability criteria. Next,another population matrix, POP� is created by applying thecondition randð0;1Þ > jbij
2; i ¼ 1;2; . . . ; L in POP. Finally, each
chromosome in POP� is evaluated to derive the fitness usingEq. (1). The entire process is repeated for a predefined numberof generations and the global best solution is updated andreported.
A typical example is presented below to demonstrate themeasurement strategy. Let, at a particular generation, theaforementioned equation is satisfied at the positions ð15;3Þ;ð16;8Þ and ð17;9Þ, respectively in population, POP. A solutionmatrix, SOL is introduced of identical size as POP. The abovethree positions in SOL are set to 1 where the remaining posi-tions are set to 0. Therefore, the optimum threshold valuesare determined as ð15� 3Þ ¼ 45, ð16� 8Þ ¼ 128 andð17� 9Þ ¼ 153 (in case of multi-level thresholding) (shown bycircle) for that generation. This phenomenon is depicted inFig. 3.
Fig. 4. Quantum
Fig. 3. The solut
5.1.4. SelectionThis phase leads to an elitist selection mechanism for each chro-
mosome in POP that possesses the best fitness value. At each gen-eration, tournament selection strategy is invoked to fill the pool ofchromosomes in population, POP and the best chromosome isrecorded (elitism). The best chromosome is merged at the bottomof the pool in POP at the prior of initiation of the next generation.
5.1.5. Quantum crossoverIn this phase, a population diversity may occur as the effect of
quantum crossover. Based on predefined crossover probability,Pcr , two random chromosomes are selected for crossover at an ran-dom position. After the occurrence of quantum crossover at eachgeneration, a new pool of chromosomes is created in POP. Fig. 4shows an distinctive example of quantum crossover.
5.1.6. Quantum mutationLike quantum crossover, another population diversity may
crops up in quantum mutation. In this phase, each position inthe participating chromosome in POP may be muted with another
crossover.
ion matrix.
S. Dey et al. / Knowledge-Based Systems 67 (2014) 373–400 379
real random number based on a predefined mutation probability,Pm. Fig. 5 presents a typical example of quantum mutation.
5.1.7. Complexity analysisThe worst case time complexity of the proposed QIGA is ana-
lyzed by describing the following steps.
1. The time complexity to generate the initial chromosomes inpopulation, POP in QIGA becomes OðV � LÞ, where, V representsthe population size in QIGA. Note that, the length of each chro-mosome is L ¼
ffiffiffiLp
where, L is the utmost pixel intensity valuein a gray scale image.
2. For assigning real value to each pixel encoded in the populationof chromosomes in POP, the time complexity of QIGA turns intoOðV � LÞ.
3. The time complexity for updating POP0 using quantum rotationgate in QIGA becomes OðV � LÞ.
4. The time complexity for quantum orthogonality in QIGA isOðV � LÞ.
5. The time complexity to create POP� is OðV � LÞ.6. QIGA employs Otsu’s method as an evaluation function to com-
pute the fitness of each chromosome in POP. The time complex-ity to evaluate the fitness of each chromosome in POP isOðV � KÞ.
7. The time complexity for performing selection using tournamentselection strategy in POP is OðVÞ.
8. Again, the time complexities for mutation and crossover areOðV � LÞ each.
9. So, to execute for a predefined number of generations, the timecomplexity of QIGA becomes OðV � L �MxGenÞ. Here, MxGenstands for number of generations.
Therefore, the overall worst case time complexity (by summingup all above steps) for the proposed QIGA for multi-level thres-holding becomes OðV � L �MxGenÞ.
5.2. Quantum Inspired Particle Swarm Optimization
The congregation of bird’s behavior has been examined andimposed in Particle Swarm Optimization (PSO). This meta-heuristicoptimization technique was first presented by Kennedy and Eber-hart in 1995 [36]. The panorama of particle swarm is that birdsalways try to discover some dedicated search space to fly. Theirtendency is to pursue some paths which have been visited beforewith high efficacy [2,36]. The influence of quantum mechanismover PSO facilitated to construct a new quantum version of meta-heuristic method namely, Quantum Inspired Particle Swarm Opti-mization for multi-level thresholding (QIPSO).
The first step of QIPSO is to produce a population, POP having Vnumber initial particles by picking up pixel intensities at random.The length of each particle in POP is L ¼
ffiffiffiLp
, where L represents themaximum pixel intensity value of a selected image. Afterward,using the theory of qubits, a random real number between (0,1)is selected and assigned to each pixel encoded in POP to createPOP0. Then POP0 endures an quantum orthogonality to producePOP00. A predefined number of threshold values as pixel intensitiesare derived based on some probability criteria. The particles ofQIPSO are considered as the participating points in a D dimensionalspace. The jth particle in the swarm is represented asS ¼ ðsj1; sj2; . . . ; sjDÞ. The best prior position of each particle is docu-mented, which can be symbolized as Pk ¼ ðpk1; pk2; . . . ; pkDÞ
T . b rep-resents the index of the best particle in the swarm. Here, at t-thgeneration, v t
kðDÞ is regarded as the current velocity whereas,st
kðDÞ represents the position at the search space of the k-th particleof dimension D.
rand represents a random real number where0 6 randð0;1Þ 6 1. c1 and c2 are called positive acceleration con-stants. Lastly, x is refers to inertia weight. The outline of the pro-posed QIPSO is expressed in Algorithm 2.
Algorithm 2. Steps of QIPSO for multi-level thresholding
Input: Number of Generation: MxGenSize of the population: VAcceleration coefficients: c1 and c2
Inertia weight: xNo. of thresholds: K
Output: Optimal threshold values: h
1: Select pixel values randomly for generating V number ofinitial particles, POP, where length of each particle isL ¼
ffiffiffiLp
, where, L represents the maximum intensityvalues of the selected image.
2: Use the notion of qubits for allocating real value between(0,1) to each pixel encoded in POP. Let it creates POP0.
3: Update POP0 by using quantum rotation gate as given inEq. (4).
4: Each particle in POP0 experiences an quantumorthogonality to generate POP00.
5: Find K number of thresholds as pixel values from eachparticle in POP satisfying corresponding value inPOP00 > randð0;1Þ. Let it produces POP�.
6: Work out fitness of each particle in POP� using Eq. (1).7: Record the best particle b 2 POP00 and its threshold values
in TB 2 POP�.8: for a predefined number of generations, MxGen do9: for all k 2 POP00 do
10: The best prior position of each particle and the indexof the best particle in POP00 are recorded.
11: v tþ1k ðDÞ ¼ x � v t
kðDÞ þ c1 � randð0;1Þ � ðptkðDÞ � st
kðDÞÞþc2 � randð0;1Þ � ðpt
uðDÞ � stkðDÞÞ.
12: stþ1k ðDÞ ¼ st
kðDÞ þ v tþ1k ðDÞ.
13: Repeat steps 3 and 5 to update POP0 and POP�
respectively.14: Evaluate the fitness of particles in POP� using Eq. (1).15: The best particle in c 2 POP0 and its threshold values
in TB 2 POP� are recorded.16: Compare the fitness of b and c. Update the best
particle in b and its corresponding threshold values inTB 2 POP�.
17: end for18: end for19: Report the threshold values h ¼ TB.
5.2.1. Complexity analysis
The following steps are to be followed while computing theworst case time complexity of the proposed QIPSO. The time com-plexity analysis for the first six steps of QIPSO have already beendiscussed in Section 5.1. The rest parts of time complexity for theproposed method are described below.
1. The time complexity for manipulation of swarm at each gener-ation is OðV � LÞ.
2. So, time complexity to execute QIPSO for a predefined numberof generations is OðV � L �MxGenÞ where, MxGen represent thenumber of generations.
Fig. 5. Quantum mutation.
380 S. Dey et al. / Knowledge-Based Systems 67 (2014) 373–400
Therefore, summing up all the steps discussed above, the over-all worst case time complexity for the proposed QIPSO for multi-level thresholding becomes OðV � L �MxGenÞ.
5.3. Quantum Inspired Differential Evolution
In 1995, Storn and Prince proposed an evolutionary optimiza-tion technique, called Differential Evolution (DE) [58]. DE is simpleto use and implement, converges fast and may furnish remarkableresults with almost no or a few parameter alteration. It is stochas-tic in nature and very effective for optimizing unimodal or multi-modal search spaces [59,60]. In the recent years, number ofresearch works have been carried out that points to the superiorityof DE over other meta-heuristics [61]. In this paper, a new quan-tum version of conventional Differential Evolution for multi-levelthresholding namely, Quantum Inspired Differential Evolution(QIDE) has been proposed.
In QIDE, pixel intensities are randomly picked to produce a pop-ulation POP having V number initial vectors. Each vector in POPhave a length of L ¼
ffiffiffiLp
, where L denotes the maximum pixelintensity value of the selected gray scale image. A real numberbetween (0,1) is randomly chosen and assigned to each pixelencoded in POP to produce POP0 using the notion of qubits. POP0
goes for an quantum orthogonality to create POP00. QIDE providesan user defined number of thresholds as pixel intensities basedon a probability condition. Three basic operators namely, mutation,crossover and selection are consecutively applied in each genera-tion. The applications of mutation and crossover are different inQIDE from those exercised in QIGA. For quantum mutation, threevectors namely, r1; r2 and r3 are randomly selected from POP00 sat-isfying r1; r2 and r3 2 ½1;V�; j – r2 – r3 – r1 where, V and j repre-sent the number of vectors and a vector individual in POP00. Theweighted difference between two population vectors are beingscaled by u and then added to the third vector to get the newmutant vector solution in POP00 where, u is called scaling factor.Afterward, POP00 undergoes crossover operation to have anothernew vector solution POP00. A random integer, ct is generated from½1;V� and mutant vector in POP00 goes for crossover based on thecondition given by j – ct and randð0;1Þ > Pc where, Pc representsa predefined crossover probability and j represents a particularposition in the selected mutant vector.
The tournament selection mechanism is applied to get the pop-ulation of vector for the next generation. A particular vector solu-tion j is substituted with the new vector solution having betterfitness value in POP00. The entire process is repeated for a prede-fined number of generation, MxGen. In QIDE, the values of u andPc are taken in the range 0 < u 6 1:2 and ½0;1� respectively. Thedetails of QIDE is delineated in Algorithm 3.
Algorithm 3. Steps of QIDE for multi-level thresholding
Input: Number of Generation: MxGenSize of the population: VScaling factor: uCrossover probability: Pc
No. of thresholds: K
Output: Optimal threshold values: h
1: Choose the pixel values randomly to produce V number ofinitial vectors, POP, where length of each vector is L ¼
ffiffiffiLp
,where, L is the maximum pixel intensity value of an image.
2: Using the conception of qubits to allocate real valuebetween (0,1) to each pixel encoded in POP. Let it makesPOP0.
3: Update POP0 by using quantum rotation gate as depictedin Eq. (4).
4: POP0 goes for an quantum orthogonality to produce POP00.5: Locate K number of thresholds as pixel values from each
vector in POP satisfying corresponding value inPOP00 > randð0;1Þ. Let it makes POP�.
6: Calculate fitness of each vector in POP� using Eq. (1).7: Save the best vector b from POP� and its threshold values
in TB 2 POP�.8: BKPOP ¼ POP00.9: for all k 2 POP00 do
10: for all jth position in k do11: Select three random integers r1; r2 and r3 from ½1::V�
satisfying r1 – r2 – r3 – j.12: POP00ðjÞ ¼ BKPOPðr1Þ þuðBKPOPðr2Þ � BKPOPðr3ÞÞ13: end for14: end for15: for all k 2 POP00 do16: for all jth position in k do17: Generate a random integer ct from ½1::V�.18: if (j – ct and randð0;1Þ > Pc) then19: POP00kj ¼ BKPOPkj.20: end if21: end for22: end for23: Follow the steps 5 and 6 to evaluate the fitness of each
vector in POP00 using Eq. (1).24: The best vector in c 2 POP00 and its threshold values in
TB 2 POP� are recorded.25: Compare the fitness of b and c.26: Update the best vector in b and its threshold values in
TB 2 POP�.
S. Dey et al. / Knowledge-Based Systems 67 (2014) 373–400 381
27: for all k 2 POP00 do28: if (Fitness of BKPOPk is better than the fitness of POP00k)
then29: POP00ðkÞ ¼ BKPOPk.30: end if31: end for32: POP00 undergoes an quantum orthogonality.33: Repeat steps 5 to 31 for fixed number of generation,
MxGen.34: The optimal threshold values, h ¼ TB are reported.
5.3.1. Complexity analysis
The outline of worst case time complexity analysis of the pro-posed QIDE is described below. The first six steps of time complex-ity analysis for QIDE have already been illustrated in Section 5.1.The worst case time complexity analysis for the next parts of QIDEare given below.
1. Time complexities for mutation, crossover and the selectionparts entail OðV � LÞ.
2. So, to execute QIDE for a predefined number of generations, thetime complexity becomes OðV � L �MxGenÞ. Here, MxGen sig-nifies the number of generations.
Therefore, aggregating the overall discussion as stated above, itcan be concluded that the worst case time complexity for the pro-posed QIDE for multi-level thresholding is OðV � L �MxGenÞ.
5.4. Quantum Inspired Ant Colony Optimization
In 1996, Dorigo et al. presented a population-based optimiza-tion technique, called Ant Colony Optimization (ACO) [62]. ACOimitates the basic behaviors of real ants to accomplish the solu-tions of various optimization problem. In real life, ants strugglefor food to sustain their existence. They traverse different pathsfor food and squirt a chemical called pheromone from their body.This chemical helps them to exchange information among them-selves and to locate the shortest path to be followed between theirnest and a food source. It is be observed that a particular pathwhich contains more amount of pheromone, is outlined by morenumber of ants. Many scholars have been motivated by the com-munal behavior of real ants and established numerous algorithmsfor solving combinatorial optimization problems [42].
In the proposed Quantum Inspired Ant Colony Optimization(QIACO), pixel intensities are randomly selected to create a popu-lation POP having V number initial strings. The length of eachstring in POP is L ¼
ffiffiffiLp
, where L is the utmost pixel intensity valuein a gray scale image. Using the concept of qubits, a real randomnumber between (0,1) is generated and allocated to each pixelencoded in POP to create POP0. Then POP0 undergoes an quantumorthogonality to produce POP00. Based on a probability condition,the method resorts to an user defined number of thresholds aspixel intensity values. At each generation, the method exploresthe best search path. At the outset, a pheromone matrix, sj is gen-erated for each ant, j. For each individual j 2 POP00, the maximumpheromone integration is deduced as threshold value in the grayscale image, if POP00ðjÞ > q0. Here, q0 is the priory defined numberwhere, 0 6 q0 6 1. This leads to POP00ðkjÞ ¼ arg maxskj. IfPOP00ðjÞ 6 q0; POP00ðkjÞ ¼ randð0;1Þ. The pheromone trial matrix isupdated at the end of each generation using skj ¼ qskj þ ð1� qÞbwhere, b represents the best string of each generation and k repre-sents the corresponding position in a particular string, j. In QIACO,MxGen represents the number of generations to be executed,K and V are the user defined number of thresholds and population
size, respectively. q is known as persistence of trials, q 2 ½0;1�. Thedetails of the proposed QIACO is illustrated in Algorithm 4.
5.4.1. Complexity analysisThe worst case time complexity of the proposed QIACO is ana-
lyzed in this section. The time complexities of the first six steps ofQIGA and QIACO are identical, which are already discussed in theSection 5.1. The time complexity analysis for the remaining partsof proposed QIACO are given below.
1. The time complexity to construct the pheromone matrix, s isOðV � LÞ.
2. The time complexity to determine POP�� from s at each gener-ation is OðV � LÞ.
3. The pheromone matrix is needed to be updated at each gener-ation. The time complexity for this computation is OðLÞ
4. Again, time complexity to execute for a predefined number ofgenerations for QIACO is OðV � L �MxGenÞ, where, MxGen isthe number of generations.
Algorithm 4. Steps of QIACO for multi-level thresholding
Input: Number of Generation: MxGenSize of the population: VNo. of thresholds: KPersistence of trials: qPriory defined number: q0
Output: Optimal threshold values: h
1: The pixel values are randomly selected to generate Vnumber of initial strings, POP, where length of each stringis L ¼
ffiffiffiLp
, where, L signifies the greatest pixel value of thegray scale image.
2: The thought of qubits is employed to assign real value between(0,1) to each pixel encoded in POP. Let it produces POP0.
3: Quantum rotation gate is employed to update POP0 usingEq. (4).
4: POP0 endures an quantum orthogonality to create POP00.5: Finding K number of thresholds as pixel values from each
string in POP satisfying corresponding value inPOP00 > randð0;1Þ. Let it creates POP�.
6: Evaluate fitness of each string in POP� using Eq. (1).7: Save the best string b from POP�.8: Repeat step (5) to produce POP��.9: Construct the pheromone matrix, s.
10: for a fixed number of generations (MxGen) do11: for all j 2 POP00 do12: for all kth location in j do13: if (randð0;1Þ > q0) then14: POP00jk ¼ arg maxsjk.15: else16: POP00jk ¼ randð0;1Þ.17: end if18: end for19: end for20: Use Eq. (1) to calculate the fitness of POP00.21: Save the best string c from POP��.22: The fitness of b and c is compared.23: Use POP�� to update the string with best string along of
its corresponding threshold values in TBS.24: Save the best string of step (22) in b and its
corresponding thresholds TBS 2 POP��.25: for all j 2 POP00 do
(continued on next page)
382 S. Dey et al. / Knowledge-Based Systems 67 (2014) 373–400
26: for all kth location in j do27: sjk ¼ qsjk þ ð1� qÞb.28: end for29: end for30: end for31: Report the threshold values h ¼ TBS.
Therefore, summing up all the steps discussed above, the over-all worst case time complexity for the proposed QIACO for multi-level thresholding becomes OðV � L �MxGenÞ.
5.5. Quantum Inspired Simulated Annealing
In 1983, Kirkpatrick et al. first used Simulated annealing (SA) asa search algorithm for optimization [63]. The basic principle of thismeta-heuristic optimization technique is to heat a physical bodywith a very high temperature and then cool it gradually to forma sturdy chemical structure. This chemical phenomena is calledcrystallization. A strategy is introduced in SA that ensures avoidingto stuck at its local minima. In this subsection, a variant QuantumInspired Simulated Annealing for multi-level thresholding (QISA) ispresented.
Initially, an initial configuration, P is created by choosing thepixel intensities randomly of length L ¼
ffiffiffiLp
, where, L representsthe highest pixel intensity value in a gray scale image. The theoryof qubits is employed to a newly introduced encoded scheme toallocate real random value between (0,1) to each pixel encodedin P and it is named as P0. Afterward, P0 goes for an quantumorthogonality to create P00. QISA finds an user defined numberof thresholds as pixel intensities based on a defined probabilitycondition. Firstly, a very high temperature, T max is assigned to anewly invoked temperature parameter, T . QISA is allowed to exe-cute for I number of iterations for each temperature. Thereafter,T is reduced by a reduction factor, t. The execution continuesuntil T reaches at the predefined final temperature, T min. At eachiteration of I , a better configuration, ðSÞ may expected becausethe old configuration is perturbed randomly at multiple points.Subsequently, a new configuration, S� is generated from S byenduring the similar process which was acclimatized before tocreate P� from P. The acceptance of the configurations S and S�
depends on the condition given by FðS�Þ > FðP�Þ, by revisingthe previous configurations P and P�, respectively; otherwise,the newly generated configuration may be admitted with a prob-ability expð�ðFðP�Þ � FðS�ÞÞÞ=T . In general, the probability isdetermined by the Boltzmann distribution. t is chosen withinthe range of ½0:5;0:99� whereas, K signifies user defined numberof thresholds. h represents the output as thresholds. The detailsof the proposed QIGA for multi-level thresholding is describedin Algorithm 5.
5.5.1. Complexity analysisThe following steps describes the worst case time complexity
for the proposed QISA for multi-level thresholding.
1. For initial configuration in QISA, the time complexity is OðLÞ,where length of configuration is L ¼
ffiffiffiLp
where, L signifies themaximum intensity value of a gray scale image.
2. For assignment of real value to each pixel encoded in the pop-ulation of configuration, the time complexity is OðLÞ.
3. The time complexity for updating P0 turns into OðLÞ.4. The time complexity to perform quantum orthogonality is OðLÞ.5. The time complexity to create P� is OðLÞ.6. The time complexity for fitness computation using Eq. (1) turns
into OðKÞ.
7. In a similar way, for the fitness computation of the configura-tion after perturbation, the time complexity is OðKÞ.
8. Let the outer loop and inner loop are executed MxIn and I timesrespectively in QISA. Therefore, the time complexity to executethis step in QISA happens to be I �MxIn.
Algorithm 5. Steps of QISA for multi-level thresholding
Input: Initial temperature: T max
Final temperature: T min
Reduction factor: tNo. of thresholds: KNumber of Iterations: I
Output: Optimal threshold values: h
1: Randomly select pixel intensities to create an initialconfiguration, P, where length of the configuration isdenoted by L ¼
ffiffiffiLp
, where, L is taken as the maximumintensity value of a gray scale image.
2: Apply the thought of qubits to assign real value between(0,1) to each pixel encoded in P. Let it generates P0.
3: Quantum rotation gate is used to update P0 using Eq. (4).4: The configuration in P0 passes through an quantum
orthogonality to generate P00.5: Discover K number of thresholds as pixel intensities from
the configuration in P. It should hold corresponding valuein P00 > randð0;1Þ. Let it generates P�.
6: Evaluate fitness of the configuration in P� using Eq. (1). Letit be symbolized by FðP�Þ.
7: T ¼ T max.8: repeat9: for j ¼ 1 to I do
10: Perturb P. Let it create S.11: Repeat step (2), (3) and (4) to create S�.12: Use Eq. (1) to evaluate fitness EðS�; TÞ of the
configuration S�.13: if (FðS�Þ � FðP�Þ > 0) then14: Set P ¼ S; P� ¼ S� and FðP�Þ ¼ FðS�Þ.15: else16: Set P ¼ S; P� ¼ S� and FðP�Þ ¼ FðS�Þ with
probability expð�ðFðP�Þ � FðS�ÞÞÞ=T .17: end if18: end for19: T ¼ T � t.20: until T >¼ T min
21: Report the optimal threshold values, h ¼ P�.
Therefore, aggregating the steps discussed above, the proposedQISA for multi-level thresholding possesses the worst case timecomplexity OðL � I �MxInÞ.
5.6. Quantum Inspired Tabu Search
The Tabu Search (TS) is a popular meta-heuristic technique thatwas first proposed by Glover and Laguna in 1997 [52]. The search-ing mechanism facilitates TS to trounce local optima. So, it explorethe solution space to let it go for hill climbing. It incorporates theconcept of adaptive memory strategy, popularly known as tabumemory to record the intermediate tabu restrictions. Basically, thiskind of memory stores the information regarding solution attri-butes that may require while transforming from one solution moveto another. The functionality of TS may be described as the
Fig. 6. Original synthetic images with (a) comlex backgrounds and (b) low contrast.
S. Dey et al. / Knowledge-Based Systems 67 (2014) 373–400 383
combination of the set of three activities. Firstly, generating set ofmoves to get the initial trial solutions and afterward add in tabumemory for prohibiting some transitional moves followed by liber-ating some outlawed moves on the basis of some aspiration crite-ria. In its way, TS compares the newly produced trial solution withthe existing solution in tabu and accepts the solution if it is notpresent in tabu memory. TS continues executing until it does notreach to the assigned terminating condition.
In this article, a Quantum Inspired Tabu Search for multi-levelthresholding namely, QITS is proposed. Like QISA, at beginning,pixel intensities are randomly selected to produce an string, P.The length of P is taken as L ¼
ffiffiffiLp
, where, L represents the maxi-mum gray scale intensity value of an image. A new encodedscheme have been introduced for assigning real random valuebetween (0,1) to each pixel encoded in P to produce P0. For thispurpose, the concept of qubits was applied along with the encodedtechnique. P0 again passes through an quantum orthogonality toproduce P00. An user defined number of thresholds as pixel intensi-ties is computed based on a defined probability condition. The tabumemory, mem is introduced and assigned to null at the prior of itsexecution. QITS starts with a single vector, vbest and stop executingwhen it reaches the predefined number of iterations, I . For eachiteration of I , a new set of vectors, VðBSÞ is created in the neighbor-hood areas of vbest . For each vector in VðBSÞ, if it is not in mem andpossesses more fitness value than vbest then vbest is updated withthe new vector. The vector is pushed into mem. When the tabumemory is full, it follows FIFO to eliminate a vector from the list.The outline of QITS is elaborately described in Algorithm 6.
5.6.1. Complexity analysisThe worst case time complexity of the proposed QITS is ana-
lyzed in this subsection. The worst case time complexity for thefirst six steps have already been discussed in the Section 5.5. Thetime complexity analysis for the remaining parts of QITS are dis-cussed below.
1. For each generation, the time complexity to create a set ofneighbors of the best vector turns into OðWÞ, where, W is thenumber of neighbors.
2. For assigning the best vector at each generation, the time com-plexity happens to be OðW � LÞ, where length of string isL ¼
ffiffiffiLp
where, L signifies the maximum pixel intensity valueof a gray scale image.
3. Hence, the time complexity to execute for a predefined numberof generations for QITS happens to be OðW � L�MxGenÞ,where, MxGen is the number of generations.
Therefore, summing up all the above steps, it can be concludethat the overall worst case time complexity for the proposed QITSfor multi-level thresholding is OðW � L �MxGenÞ.
Algorithm 6. Steps of QITS for multi-level thresholding
Input: Number of Generation: MxGenNumber of thresholds: K
Output: Optimal threshold values: h
1: Select the pixel values randomly to create an initial string,P, where length of the string is L ¼
, where, L isthe greatest pixel intensity value of a gray scale image.
2: Apply the conception of qubits to assign real valuebetween (0,1) to each pixel encoded in P. Let it creates P0.
3: Quantum rotation gate is utilized to update P0 using Eq.(4).
4: The string in P0 goes for an quantum orthogonality to createP00.
5: Finding K number of thresholds as pixel values fromthe string in P satisfying corresponding value inP00 > randð0;1Þ. Let it creates P�.
6: Compute fitness of the string in P� using Eq. (1).7: Record the best string b from P�.8: Initialize the tabu memory, mem ¼ /.9: for j ¼ 1 to MxGen do
10: vbest ¼ b.11: Create a set of neighbors, VðBSÞ of vector vbest
12: for each v 2 VðBSÞ do13: if v R mem and ðFitnessðvÞ > FitnessðvbestÞ) then14: vbest ¼ v .15: end if16: end for17: Set c ¼ vbest .18: mem ¼ mem [ vbest .19: Compare the fitness of b and c.20: Store the best individual in b 2 POP00 and its
corresponding threshold values in TB 2 POP�.21: end for22: Report the optimal threshold values, h ¼ TB.
6. Experimental results
In this article, six different quantum inspired meta-heuristicmethods for multi-level thresholding are presented. The proposedquantum inspired methods employ Otsu’s method [57] as a evalu-ation function to determine a predefined number of optimalthreshold values of gray level images. The method proposed byOtsu [57], maximizes the between-class variance ðUÞ as given inEq. (1). The optimal threshold values are reported for ten gray scaleimages and five synthetic images, each of size 256� 256 at differ-ent levels. The original synthetic images having complex back-ground and low contrast are depicted in Fig. 6(a) and (b),respectively. The original gray scale images are Lena, B2, Barbara,Boat, Cameraman, Jetplane, Pinecone, Car, Woman, and House areshown in Fig. 8(a)–(j). Four new synthetic images have beendesigned by adjusting different noise and contrast level on the syn-thetic image of Fig. 6(b), which are portrayed in Fig. 7(a)–(d). Thesenoisy images are named as ImageN30C30 (noise-30, contrast-30),ImageN30C80 (noise-30, contrast-80), ImageN80C30 (noise-80,contrast-30) and ImageN80C80 (noise-80, contrast-80). The firstsynthetic image as shown in Fig. 6(a) and these four modified ver-sions of synthetic images have been used as the test images.
In Section 5, the proposed quantum inspired meta-heuristicmethods are described elaborately. The selection of best combina-tion of parameter in each method can accelerate its performance.The parameter set for the proposed methods are listed in Table 2.
Fig. 8. Original test images (a) Lena, (b) B2, (c) Barbara, (d) Boat, (e) Cameraman, (f) Jetplane, (g) Pinecone, (h) Car, (i) Woman and (j) House.
Table 1List of worst case time complexity of the proposed methods for multi-levelthresholding.
Proposed method Time complexity
QIGA OðV � L �MxGenÞQIPSO OðV � L �MxGenÞQIDE OðV � L �MxGenÞQIACO OðV � L �MxGenÞQISA OðL � I �MxInÞQITS OðW � L �MxGenÞ
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The worst case time complexity for each method is analyzed indetails in the respective section. The list of worst case timecomplexities of each method is reported in Table 1. It should benoted that the worst case time complexities for QIGA, QIPSO, QIDEand QIACO are same. These four methods possess lower time
Table 2Parameter sets of the proposed QIGA, QIPSO, QIDE, QIACO, QISA and QITS for multi-level
QIGA QIPSO
Population size: V ¼ 50 Population size: V ¼ 5Number of generation: MxGen ¼ 1000 Number of generationCrossover probability: Pcr ¼ 0:9 Inertia weight: x ¼ 0Mutation probability: Pm ¼ 0:1 Acceleration coefficieNo. of thresholds: K ¼ 2;3;4;5 No. of thresholds: K ¼
QIACO QISA
Population size: V ¼ 50 Initial temperature: TNumber of generation: MxGen ¼ 1000 Final temperature: TPriori defined number: q0 ¼ 0:5 Number of Iterations:Persistence of trials: q ¼ 0:2 Reduction factor: t ¼No. of thresholds: K ¼ 2;3;4;5 No. of thresholds: K ¼
complexities than QITS if V <W. In case of QISA, the worst casetime complexity depends on the parameters selected for execution.
Each method has been executed 20 times for first four levels ofthresholds. These methods may discover varying number of opti-mal threshold values depending on the level chosen for optimiza-tion. The results of the proposed methods are reported in Tables 3and 4, respectively. These tables encompass the number of thresh-old ðKÞ, optimum threshold ðhÞ at each level, the corresponding fit-ness value ðUbestÞ, and also the execution time ðtÞ (in seconds).Later, the mean fitness ðUavgÞ and standard deviation ðrÞ of themean fitness over 20 individual runs are reported in Tables 5 and6, respectively.
Each method sounds good result for K ¼ 2 and 3. For K ¼ 2,each method reports the same mean fitness ðUavgÞ value as wellas same standard deviation ðrÞ for all test images. It has beennoticed that, for K ¼ 3, the average fitness ðUavgÞ values for all testimages remain equal for QIGA, QIPSO and QIDE whereas, a very
thresholding.
QIDE
0 Population size: V ¼ 50: MxGen ¼ 1000 Number of generation: MxGen ¼ 1000:4 Scaling factor: F ¼ 0:5nts: c1; c2 ¼ 0:5 Crossover constant: Pc ¼ 0:9
2;3;4;5 No. of thresholds: K ¼ 2;3;4;5
QITS
max ¼ 100 Number of generation: MxGen ¼ 1000
min ¼ 0:5 No. of thresholds: K ¼ 2;3;4;5I ¼ 500:92;3;4;5
Table 3Best results of QIGA, QIPSO, QIDE, QIACO, QISA and QITS for multi-level thresholding for Image1, ImageN30C30, ImageN30C80, ImageN80C30 and ImageN80C80.
K QIGA QIPSO QIDE QIACO QISA QITS
h Ubest t h Ubest t h Ubest t h Ubest t h Ubest t h Ubest t
Table 5Average values of fitness Uavg and standard deviation ðrÞ of QIGA, QIPSO, QIDE, QIACO, QISA and QITS for multi-level thresholding of Image1, ImageN30C30, ImageN30C80,ImageN80C30 and ImageN80C80.
Table 6Average values of fitness Uavg and standard deviation ðrÞ of QIGA, QIPSO, QIDE, QIACO, QISA and QITS for multi-level thresholding of Lena, B2, Barbara, Boat, Cameraman, Jetplane,Pinecone, Car, Woman and House.
Table 7Data sets for QIGA, QIPSO, QIDE, QIACO, QISA and QITS, are used in Friedman test for K = 4 and 5, respectively. The rank of each method is shown in the parentheses after statisticaltest.
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small deviation found in ðUavgÞ for all respective images for theother three methods. Similar to mean fitness values, the standarddeviations ðrÞ also variate for QIACO, QISA and QITS at littleamount for this level of computation. Each method reports unequalvalues of optimal thresholds, average fitness values and the stan-dard deviations for all test images for K ¼ 4 and 5.
For K ¼ 4 and 5, the Friedman test [15,16] was conducted forthe proposed methods using 15 different data set (both for realand synthetic test images) as given in Table 7. In general, Friedmantest is introduced to compare the performances of multiple meth-ods using multiple data set. This test exposes the average rank ofeach individual method as the output. In this statistical test, the
Table 8Consensus results of multi-level thresholding for Image1, ImageN30C30, ImageN30C80, ImageN80C30 and ImageN80C80
a – Signifies that only QIGA reports, b – signifies that only QIPSO reports, c – signifies that only QIDE reports, d – signifies that only QIACO reports, e – signifies that only QISAreports, f – signifies that only QITS reports, and g – signifies that all methods report.
0 100 200 300 400 500 600 700 800 9001910
1910.5
1911
1911.5
1912
1912.5
1913
1913.5
1914
Number of generation
Fitn
ess
valu
e (U
)
Convergence curves for the proposed algorithms for K=4
QIGAQIPSOQIDEQIACOQISAQITS
(n)/Temperature (T)
(a)
0 100 200 300 400 500 600 700 800 9001967
1967.5
1968
1968.5
1969
1969.5
1970
1970.5Convergence curves for the proposed algorithms for K=5
Fitn
ess
valu
e (U
)
QIGAQIPSOQIDEQIACOQISAQITS
(T)Number of generation /Temperature(n)
(b)
Fig. 9. Convergence curves of the proposed methods for Lena: (a) for K ¼ 4 and (b) for K ¼ 5.
S. Dey et al. / Knowledge-Based Systems 67 (2014) 373–400 389
Table 9Consensus results of multi-level thresholding for Lena, B2, Barbara, Boat, Cameraman, Jetplane, Pinecone, Car, Woman and House.
a – Signifies that only QIGA reports, b – signifies that only QIPSO reports, c – signifies that only QIDE reports, d – signifies that only QIACO reports, e – signifies that only QISAreports, f – signifies that only QITS reports, and g – signifies that all methods report.
3 Supplementary is available at http://sysbio.icm.pl/indra/supplementary.pdf.
S. Dey et al. / Knowledge-Based Systems 67 (2014) 373–400 391
null hypothesis, H0 affirms the equal behavior of the participatingmethods. Hence, under H0, each method possesses equal rank,which confirm that each method is equally efficient among others.The alternative hypothesis, H1 endorses the difference in perfor-mances among the partaken methods. Since each method reportsequal fitness values for all test images for K ¼ 2 and 3, the Fried-man test [15,16] finds equal average rank for all of them. Hence,the null hypothesis is accepted, which proves that the proposedmethods behave equally for K ¼ 2 and 3. Table 7 shows that aver-age ranks for QIGA, QIPSO, QIDE, QIACO, QISA and QITS are foundto be 2.46, 1.73, 2.86, 3.23, 5.16 and 5.53, respectively for K ¼ 4whereas, for K ¼ 5, the corresponding average ranks are 2.73,1.30, 2.86, 3.50, 5.20 and 5.40, respectively. Note that, each methodacquires unequal rank for K ¼ 4 and 5. Moreover, Friedman testfinds the chi-square ðX2Þ value 48.854 for K ¼ 4 and 50.344 forK ¼ 5. This statistical test also determines the p-values forK ¼ 4 and 5 as 2:1� 10�4 and 3:5� 10�4, respectively. From chi-square ðX2Þ distribution table, we find that the critical value forð6� 1Þ ¼ 5 degree of freedom with 0.05 significance level is11.070. Since both of these chi-square values for K ¼ 4 and 5 aregreater than the critical value, H0 is rejected and H1 is accepted.Furthermore, the p-values for K ¼ 4 and 5 are found to be verysmall (very closed to zero), which confirms the rejection of H0
and find some significant difference in behavior among the pro-posed methods. Beside that, for K ¼ 4 and 5, QIPSO possesses low-est rank and QITS has the highest rank among six differentmethods. Hence, QIPSO can be declared as the best performingmethod whereas, QITS is worst performing method among others.
We have performed our experiments on Toshiba Intel (R) Core(TM) i3, 2.53 GHz PC with 2 GB RAM. The computational time atdifferent level for each of the proposed method and for each testimage is reported in Tables 3 and 4, respectively. It can be noticedthat QIPSO takes least time for computation among all the pro-posed methods. As QIPSO outperforms the other methods andthe paper size is the main concern, only QIPSO is selected for
reporting for the visual representation of results. The images ofQIPSO after thresholding are given for synthetic test images inFig. 10 and that of for real test images in Figs. 11 and 12, respec-tively for each different level. Since the threshold values of allthe proposed methods are identical or very close to each other,only the set of images for QIPSO after thresholding are presentedin Figs. 10–12 for each level. As there are no significant differencesfound in results for K ¼ 2;3 among the proposed methods, the con-vergence curves for K ¼ 4;5 are only reported. The convergencecurves of the proposed methods for K ¼ 4;5 for lena are presentedin Fig. 9 (a) and (b). Moreover, the other convergence curves areenlisted in supplementary.3 It is clearly perceptible from Fig. 9(a)and (b) that QIPSO is the fastest converging method among others.
6.1. Consensus results of the quantum inspired methods
A consensus of the six quantum inspired methods is describedin this subsection. The optimum set of thresholds has beenreported for K ¼ 2;3;4 and 5 in Table 8 for synthetic test imagesand in Table 9 for real test images. Here, 1� signifies that a partic-ular set of threshold values is reported by a single method at a par-ticular level, 2� signifies that two out of six of the proposedmethods report another set of thresholds for different levels andso on. The name of the methods and the corresponding thresholdvalues are also reported in the above mentioned tables. It is clearlyvisible from Tables 8 and 9 that all methods report same thresholdvalue/values for all test images for K ¼ 2 and 3. The variant resultsmay be found for K P 3. A typical example can be outlined forK ¼ 4 where each method reports 73 and 121 as the optimumthreshold value for barbara image except QITS. In some cases itmay be observed that most of the proposed methods report
Table 10Best results of QIPSO, TSMO, MTT, MBF, PSO and GA for multi-level thresholding for Image1, ImageN30C30, ImageN30C80, ImageN80C30 and ImageN80C80.
K QIPSO TSMO MTT MBF PSO GA
h Ubest t h Ubest t h Ubest t h Ubest t h Ubest t h Ubest t
Table 11Best results of QIPSO, TSMO, MTT, MBF, PSO and GA for Multi-level Thresholding for Lena, B2, Barbara, Boat, Cameraman, Jetplane, Pinecone, Car, Woman and House.
K QIPSO TSMO MTT MBF PSO GA
h Ubest t h Ubest t h Ubest t h Ubest t h best t h Ubest t
Table 13Average values of fitness ðUavgÞ and standard deviation ðrÞ of QIPSO, TSMO, MTT, MBF, PSO and GA for multi-level thresholding of Lena, B2, Barbara, Boat, Cameraman, Jetplane,Pinecone, Car, Woman and House.
Table 12Average values of fitness ðUavgÞ and Standard deviation ðrÞ of QIPSO, TSMO, MTT, MBF, PSO and GA for multi-level thresholding of Image1, ImageN30C30, ImageN30C80,ImageN80C30 and ImageN80C80.
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Table 14Data sets for QIPSO, TSMO, MTT, MBF, PSO and GA, used in Friedman test for K = 4 and 5, respectively. The rank of each method is shown in the parentheses after statistical test.
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maximum number of threshold values for some images for anylevel of thresholding. In another observation it is seen that veryfew methods report maximum number of threshold values foranother set of images at another levels as shown in Tables 8 and9, respectively.
6.2. Comparison of QIPSO with other existing methods
The experimental results prove that QIPSO is the best perform-ing method among the six different proposed methods. Hence,QIPSO is compared with five other methods namely, TSMO, MTT,
Fig. 10. For K ¼ 2;3;4;5, (a)–(d), for Image1, (e)–(h), for ImageN30C30, (i)–(l), for ImageN30C80, (m)–(p), for ImageN80C30, and, (q)–(t), for ImageN80C80, after using QIPSOfor multi-level thresholding.
396 S. Dey et al. / Knowledge-Based Systems 67 (2014) 373–400
MBF, conventional PSO and conventional GA to find the effective-ness of this proposed method. These five other methods have beenexecuted for same number of runs as the proposed methods on thesimilar images. The best results for all comparable methods havebeen reported in Tables 10 and 11, respectively with the same for-mat as used for the proposed methods. Later, the mean fitnessðUavgÞ and standard deviation ðrÞ for different runs have beenreported in Tables 12 and 13, respectively for different level ofthresholding. These five methods find equal threshold value asQIPSO for each image for K ¼ 2. The mean fitness ðUavgÞ and stan-dard deviation ðrÞ obtained for each test image are found to beidentical for lowest level of thresholding. The results variate whenthe upper level of thresholding is considered. There are some sig-
nificant changes found in the mean fitness values and standarddeviations for these five methods for K > 2. Hence, QIPSO outper-forms the comparable methods in terms of accuracy and effective-ness. As the level of thresholding increases, the effectiveness ofQIPSO increases as well. In addition, QIPSO takes least time to exe-cute as compared to others.
For K ¼ 4 and 5, the Friedman test [15,16] was again conductedfor these five methods along with QIPSO using the same data setused as earlier. Since, there are significant changes in mean fitnessvalues and standard deviations for K P 3, the efficiency of QIPSOcan easily be determined by calculating the average rankamong these six comparable methods. Therefore, Friedman testwas conducted among the six methods over same data sets for
Fig. 11. For K ¼ 2;3;4;5, (a)–(d), for Lena, (e)–(h), for B2, (i)–(l), for Barbara, (m)–(p), for Boat, and, (q)–(t), for Cameraman, after using QIPSO for multi-level thresholding.
S. Dey et al. / Knowledge-Based Systems 67 (2014) 373–400 397
K ¼ 4 and 5. The average ranks for the comparable methods arereported in Table 14 for K ¼ 4 and 5. The Friedman test deter-mines the average ranks for QIPSO, TSMO, MTT, MBF, PSO andGA as 1.00, 3.13, 3.73, 2.13, 5.20 and 5.80, respectively for K ¼ 4and that of 1.00, 3.23, 3.60, 2.20, 4.90 and 5.86, respectively forK ¼ 5. For both of K ¼ 4 and 5, QIPSO possesses lowest averagerank among these six methods, which proves its superiority ascompared to others. Furthermore, this test determines the chi-square ðX2Þ value and p-values for K ¼ 4 as 69.847 and a very smallvalue (close to zero), whereas, these two measures for K ¼ 5 are69.847 and another very small number (close to zero). Beside that,at 0.05 significance level and ð6� 1Þ ¼ 5 degree of freedom, thechi-square values are larger than the predetermined critical valuesfor K ¼ 4 and 5. Hence, these two measures validate the rejection
of H0 and confirm the acceptance of H1, which substantiates thebehavioral disparity among the comparable methods. It is evi-dently appreciable from Fig. 13(a) and (b) that QIPSO is the fastestconverging method among all comparable methods. The conver-gence curves for the other images are given in supplementary.Summarizing the facts discussed above, it can be concluded thatQIPSO is the best performing method among all the methods com-pared with.
7. Conclusion
In this paper, six different quantum inspired meta-heuristicmethods, namely Quantum Inspired Genetic Algorithm, Quantum
Fig. 12. For K ¼ 2;3;4;5, (a)–(d), for Jetplane, (e)–(h), for Pinecone, (i)–(l), for Car, (m)–(p), for Woman, and, (q)–(t), for House, after using QIPSO for multi-level thresholding.
398 S. Dey et al. / Knowledge-Based Systems 67 (2014) 373–400
Inspired Particle Swarm Optimization, Quantum Inspired Differen-tial Evolution, Quantum Inspired Ant Colony Optimization, Quan-tum Inspired Simulated Annealing and Quantum Inspired TabuSearch for multi-level thresholding are proposed. The quantumversion of meta-heuristic methods is developed to determine thepredefined number of optimal threshold values from five syntheticimages and ten gray level test images. These proposed methodsused Otsu’s method as an objective function. The effectiveness ofthe proposed methods have been shown at different levels ofthresholding in terms of optimal threshold values with fitnessmeasure, average fitness measure, standard deviation of the fitnessmeasure, computational time and the convergence plot. Finally,Friedman test is conduced to judge the superiority of a methodamong them. It has been noticed that each method performs
almost equally for lower level of thresholding, specially forK ¼ 2;3. While K P 4, the efficiency varies among different meth-ods. As a result, Quantum Inspired Particle Swarm Optimization isfound to be superior to the other methods as far as computationaltime is concerned. Moreover, it has also been noticed that theQuantum Inspired Particle Swarm Optimization is the best per-forming method while the proposed Quantum Inspired TabuSearch is the inferior among them for the higher level ofthresholding.
A comparative study of the proposed methods has been carriedout with the two-stage multithreshold Otsu method, the maxi-mum tsallis entropy thresholding, the modified bacterial foragingalgorithm, the classical particle swarm optimization and the classi-cal genetic algorithm. It has been found from empirical observation
1914Convergence curves for the comparable algorithms for K=4
Number of generation
Fitn
ess
valu
e (U
)
QIPSOTSMOMTTMBFPSOGA
(n) Number of generation (n)
(a)
0 200 400 600 800 1000 1200 1400 1600 1800 2000
1958
1960
1962
1964
1966
1968
1970
Fitn
ess
valu
e (U
)
Convergence curves for the comparable algorithms for K=5
QIPSOTSMOMTTMBFPSOGA
(b)
Fig. 13. Convergence curves of the comparable methods for Lena: (a) for K ¼ 4 and (b) for K ¼ 5.
S. Dey et al. / Knowledge-Based Systems 67 (2014) 373–400 399
that the Quantum Inspired Particle Swarm Optimization outper-forms these methods in terms of average ranking, accuracy androbustness. Apart from this, as a scope of further research, quan-tum inspired multiobjective algorithm [64] can also be designedfor multi-level thresholding.
Acknowledgements
This work was partially supported by UGC sponsored UPE-IIproject grant of Jadavpur University, India and European UnionSeventh Framework Programme (FP7/2007–2013) under the grantagreement number: 246016. Apart from this, we would like tothank the reviewers for their valuable comments.
Appendix A. Supplementary material
Supplementary data associated with this article can be found,in the online version, at http://dx.doi.org/10.1016/j.knosys.2014.04.006.
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