A Secret Image Sharing Method Using Integer to Integer Wavelet Transform

A SECRET IMAGE SHARING METHOD USING

INTEGER TO INTEGER WAVELET TRANSFORM

PHASE-I

By GANESAMOORTHY.B

A Thesis submitted to the

FACULTY OF INFORMATION AND COMMUNICATION ENGINEERING

in partial fulfillment of the requirements

for the award of the degree

of

MASTER OF ENGINEERING

in

APPLIED ELECTRONICS

COLLEGE OF ENGINEERING, GUINDY

ANNA UNIVERSITY: CHENNAI 600 025

OCTOBER 2007

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ACKNOWLEDGEMENT

I have a great pleasure in expressing my sincere gratitude and heartily thanks

to our Prof. and H.O.D.Dr.N.Kumaravel for providing me an opportunity

to work on this project. I would like to sincerely thank our Prof.Dr.J.Raja

Paul Perinbam, for providing enough time and encouragement.

I express my sincere thanks to my guide Mr.M.Manikandan ,Lecturer,

Department of Electronics and Communication ,Anna University for his

Valuable guidance , keen suggestions ,innovative ideas ,inspirations

discussions , helpful criticisms and kind encouragements in entire phase

of this project work. It had been indeed a great pleasure to work under their

guidance.

I also express my gratitude to all faculty members for their help and support

during entire phase of this project work.

Finally, I express my deep sense gratitude to my members, friends and all

others who directly or indirectly involved in this project, for their valuable

help and consideration towards me.

Place:

Date:

GANESAMOORTHY.B

2

BONAFIDE CERTIFICATE

Certified that this thesis report “A Secret image sharing method using

Integer to Integer wavelet transform” is the Bonafide work of

“Mr.B.GANESAMOORTHY, (200631625)” who carried out the project

work under my supervision. Certified further, that to the best of my

knowledge the work reported herein does not form part of any other thesis or

dissertation on the basis of which a degree or award was concerned on an

earlier occasion on this or any other candidate.

Dr.N.KUMARAVEL, Mr.M.MANIKANDAN

Professor& Head of the Department, Lecturer,Department of Electronics and Department of Electronics and Communication Engineering Communication EngineeringCollege of Engineering, College of Engineering,Anna University Anna UniversityChennai -600 025 Chennai -600 025.

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TABLE OF CONTENTS

CHAPTER NO. TITLE PAGE NO.

ABSTRACT (TAMIL) iii

ABSTRACT (ENGLISH) xvi

LIST OF FIGURES xviii

LIST OF SYMBOLS xxvii

1. INTRODUCTION

1.1LITERATURE SURVEY

2 2.1 OBJECTIVE

. 2.2. OVERVIEW

2.2.1 BLOCK DIAGRAM

3. DESCRIPTION

3.1 WAVELETS

3.2 WAVELET VS FOURIER TRANSFORMS

3.2.1 Similarities

3.2.2 Dissimilarities

3.3 BIO-ORTHOGONAL WAVELETS:

3.4 WAVELET TYPES

3.4.1 Continuous wavelet transform

3.4.2 Discrete wavelet transform

3.5 INTEGER WAVELET TRANSFORMS

3.5.1Advantages:

3.5.2 Daubechies’ 5/3 Wavelet Transform

3.6 DOWNSAMPLING

3.7 SHAMIR’S THRESHOLD SCHEME:

4. MATLAB SIMULATION RESULTS.

4

ABSTRACT (TAMIL)

5

LIST OF FIGURES

FIGURE NO TITLE PAGE N0

Block diagram

6

LIST OF ABBREVIATIONS

ITI Integer to integer wavelet transforms

SPIHT Set hierarchial partitioning of trees

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ABSTRACT

A new image sharing method, based on the reversible

integer-to-integer (ITI) wavelet transform and Shamir’s (r, m) threshold

scheme is presented, that provides highly compact shadows for real time

progressive transmission. This method, working in the wavelet domain,

processes the transform coefficients in each sub band, divides each of the

resulting combination coefficients into m shadows and allows recovery of

the complete secret image by using any r or more shadows (r≤m). By taking

advantages of properties of the wavelet transform multiresolution

representation, such as coefficient magnitude decay and excellent energy

compaction, to design combination procedures for the transform coefficients

and processing sequences in wavelet sub bands such that small shadows for

real time progressive transmission are obtained.

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CHAPTER 1

INTRODUCTION

With the rapid development of computer and

communication networks, Internet has been established worldwide that

brings numerous applications such as commercial services, telemedicine and

military document transmissions. Due to the nature of the network, Internet

is an open system; to transmit secret application data securely is an issue of

great concern. Security could be introduced in many different ways, for

example, by image hiding and watermarking. However, the common weak

point of them is that a secret image is protected in a single information

carrier, and once the carrier is damaged or destroyed the secret is lost. If

many duplicates are used to overcome this deficiency, the danger of security

exposure will also increase. A secret image sharing method provides a viable

solution. To process the received data efficiently is another problem. As

transmission proceeds, the receiver may gradually access images with

increased visual quality. If the received data is of no interest, the

transmission can be terminated immediately to increase efficacy. Therefore,

the functionality of progressive reconstruction is very essential to be built in

the scheme. The goal is to develop an efficient secret image sharing method

with progressive transmission capability.

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LITERATURE SURVEY

In 1979 Adi Shamir proposed “How to share a secret“.

In this paper he show how to divide data D into n pieces in such a way that

D is easily reconstruct able from any k pieces, but even complete knowledge

of k - 1 pieces reveals absolutely no information about D. This technique

enables the construction of robust key management schemes for

cryptographic systems that can function securely and reliably even when

misfortunes destroy half the pieces and security breaches expose all but one

of the remaining pieces.

In 2002 Chih-Ching Thien and Ja-Chen Lin proposed

“Secret image sharing” In this paper they suggested the concept of image

sharing for both lossy as well as lossless image .In this method such that

secret image can be shared by several shadow images. The size of each

shadow image is 1∕ r of the secret images in our method, and this small size

property gives our method certain benefits including easier process for

storage, transmission, and hiding.

In 2005 Shang-Kuan Chen and Ja-chen Lin proposed

“Fault tolerant and progressive transmission of images “.In this paper the

image is divided into n parts with equal importance to avoid worrying about

which part is lost or transmitted first and if the image is a secret image, then

the transmission can use n distinct channels and intercepting up to r1-1

channels by the enemy will not reveal any secret.

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In 2006 Ran-Zan Wang and Chin-Hui Su proposed

“Secret image sharing with smaller shadow images”. In this paper Secret

image sharing is a technique for protecting images that involves the

dispersion of the secret image into many shadow images. This endows the

method with a higher tolerance against data corruption or loss than other

image-protection mechanisms, such as encryption or steganography. In the

method proposed in this study, the difference image of the secret image is

encoded using Huffman coding scheme, and the arithmetic calculations of

the sharing functions are evaluated in a power-of-two Galois Field GF (2t).

The shadow image in this method is about 40% smaller than that of the

method used in Secret image sharing which improves its efficiency in

storage, transmission, and data hiding.

In 1998 HyungJun Kim and C. C. Li proposed

“Lossless and Lossy Image Compression Using Biorthogonal Wavelet

Transforms with Multiplierless Operations”. In this paper they proposed

lossless and lossy image compression algorithms, based on biorthogonal

wavelets, which provide high computational speed and excellent

compression performance.

In 1995 Ahmad Zandi James, D. Allen Edward,

L. Schwartz and Martin Boliek proposed “CREW:

Compression with Reversible Embedded Wavelets”.

Compression with Reversible Embedded Wavelets (CREW) is a unified

lossless and lossy continuous-tone still image compression system. It is

wavelet-based using a “reversible” approximation of one of the best wavelet

filters. Reversible wavelets are linear filters with non-linear rounding which

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implement exact-reconstruction systems with minimal precision integer

arithmetic.

In 1996 A. R. Calderbank, Ingrid daubechies, Wim

sweldens, and Boon-lock yeo proposed “Wavelet Transforms That Map

Integers to Integers”. Invertible wavelet transforms that map integers to

integers have important applications in lossless coding. In this paper we

present two approaches to build integer to integer wavelet transforms. The

first approach is to adapt the precoder of Laroia et al., which is used in

information transmission; combine it with expansion factors for the high and

low pass band in subband filtering. The second approach builds upon the

idea of factoring wavelet transforms into so-called lifting steps. This allows

the construction of an integer version of every wavelet transform.

In 2003 Chih-Ching Thien and Ja-Chen Lin proposed

“An Image-Sharing Method with User-Friendly Shadow Images”. This

study presents a user-friendly image-sharing method for easier management

of the shadow images. The sharing of images among several branches

(distributed disks) using this method has several characteristics: 1) fast

transmission among branches; 2) fault tolerance; 3) a secure storage system;

4) reduced chance of pirating of high-quality images and 5) most

importantly, the provision to each branch manager an easy-to-manage

environment This approach still has the small-size and channel-independent

properties of our the size of each shadow image is only 1/r of that of the

original image, and any shadow images can be used for restoration.

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In 2003 Michael D. Adams and Rabab Kreidieh Ward

proposed “Symmetric-Extension Compatible Reversible Integer-To-

Integer Wavelet Transforms”. In this paper they proposed Symmetric

extension is explored by means for constructing nonexpansive reversible

integer-to-integer (ITI) wavelet transforms for finite-length signals. Two

families of reversible ITI wavelet transforms are introduced, and their

constituent transforms are shown to be compatible with symmetric

extension.

In 1996 Amir Said and William A.

Pearlman, proposed “A New, Fast, and Efficient Image

Codec Based on Set Partitioning in Hierarchical

Trees”. In this paper Embedded zero tree wavelet (EZW)

coding is a very effective and computationally simple

technique for image compression. These principles are

partial ordering by magnitude with a set partitioning sorting

algorithm, ordered bit plane transmission, and exploitation

of self-similarity across different scales of an image wavelet

transform. Moreover a new and different implementation

based on set partitioning in hierarchical trees (SPIHT), which

provides even better performance than our previously

reported extension of EZW that surpassed the performance

of the original EZW. The image coding results, calculated

from actual file sizes and images reconstructed by the

decoding algorithm, are either comparable to or surpass

previous results obtained through much more sophisticated

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and computationally complex methods. In addition, the new

coding and decoding procedures are extremely fast, and

they can be made even faster, with only small loss in

performance, by omitting entropy coding of the bit stream

by arithmetic code.

CHAPTER 2

2.1 OBJECTIVE:

To present a new image sharing method based on the

integer-to-integer (ITI) wavelet transform and Shamir’s (r, m) threshold

scheme that provide highly compact shadows for real time progressive

transmission

2.2 OVERVIEW OF THE PROJECT:

An integer-to-integer reversible

wavelet transform maps an integer-valued image to integer-

valued transform coefficients and provides the exact

(lossless) reconstruction of the original image. Its

multiresolution representation is the same as usual, but can

be fast computed with only integer addition and bit-shift

operations. Most of the signal energy is concentrated in the

low frequency bands and the transform coefficients therein

are expected to be better magnitude-ordered as moving

downward in the multi-resolution pyramid in the same

spatial orientation. These properties are very important for

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the development of an image sharing method with real time

progressive transmission. Instead of using permutation to

de-correlate pixels prior to applying the (r, m) threshold

scheme as in, first apply ITI wavelet transform and then

process transform coefficients in a preprocessing stage to

de-correlate pixels (coefficients) and increase security. The

preprocessing stage is performed on sub band basis and the

resulting coefficients in each sub band are processed in a

zigzag sequence from the smooth sub band to detail sub

bands. The most important information of the smooth sub

band will be processed first and then the detail bands so that

the progressive transmission can be obtained. In SPIHT, the

progressive transmission is achieved by checking several

times the transform coefficients. In this method, the

progressive transmission is enabled by ordering the importance

of the sub band information and checking the coefficients

only one time to speed up the processing. This method,

based on the ITI wavelet transform, provides small shadows,

lossless secret image reconstruction, and more importantly

the capability of real time progressive transmission. In our

proposed method described below, we take a0, a1, a2,…ar -1 as

values of r processed transform coefficients to generate m

shadows. A secret image is ITI- wavelet transformed down to

a selected scale level j to form its multiresolution

hierarchical representation.

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Combination procedures for transform coefficients in

individual subbands are developed first based on the strong

intra-band correlation and small absolute values of the

coefficients in the detail bands. Thus, we expect

to have small values of differences between neighboring

coefficients in the smooth subband and small coefficients in

the detail subbands. These are

used in the combination processes in the respective

subbands to produce combination coefficients for use in the

(r, m) threshold scheme. The sequence of the combination

process starts from the smooth subband and follows a zigzag

path to the detail subbands in a hierarchical tree [8] such

that the progressive transmission can be readily achieved.

BLOCK DIAGRAM:

1

2

: :

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INPUT IMAGE

INTEGER WAVELET

TRANSFORM

PRE PROCESSINGSTAGE

SHARING

m (shadows)

1

2

:

:

m

CHAPTER 3

3.1 WAVELETS:

The very name wavelet comes from the requirement

that they should integrate to zero, “waving" above and below the x-axis. The

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REVEALPOST

PROCESSING STAGE

RECONSTRUCTED WAVELET

COEFFICIENTS

diminutive connotation of wavelet suggests the function has to be well

localized. Other requirements are technical and needed mostly to insure

quick and easy calculation of the direct and inverse wavelet transform.

The wavelet transform has the advantage -over

Fourier-based transform that it has both time (space) and frequency

resolution instead of frequency resolution only. The wavelet transform cuts

the input signal into several parts and each part is analyzed separately. They

are given by

Where a is the scale parameter and b is the translation parameter.

3.2 Wavelet vs Fourier Transforms 3.2.1 Similarities

The fast Fourier transform (FFT) and the discrete

wavelet transform (DWT) are both linear operations that generate a data

structure that contains segments of various lengths, usually filling and

transforming it into a different data vector of length .

The mathematical properties of the matrices involved in the transforms are

similar as well. The inverse transform matrix for both the FFT and the DWT

is the transpose of the original. As a result, both transforms can be viewed as

a rotation in function space to a different domain. For the FFT, this new

domain contains basis functions that are sines and cosines. For the wavelet

transform, this new domain contains more complicated basis functions

called wavelets, mother wavelets, or analyzing wavelets.

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Both transforms have another similarity. The basis

functions are localized in frequency, making mathematical tools such as

power spectra (how much power is contained in a frequency interval) and

scalegrams (to be defined later) useful at picking out frequencies and

calculating power distributions.

3.2.2 Dissimilarities

The most interesting dissimilarity between these two kinds

of transforms is that individual wavelet functions are localized in space.

Fourier sine and cosine functions are not. This localization feature, along

with wavelets' localization of frequency, makes many functions and

operators using wavelets "sparse" when transformed into the wavelet

domain. This sparseness, in turn, results in a number of useful applications

such as data compression, detecting features in images, and removing noise

from time series.

One way to see the time-frequency resolution differences

between the Fourier transform and the wavelet transform is to look at the

basis function coverage of the time-frequency plane

3.3 BIO-ORTHOGONAL WAVELETS:

A biorthogonal wavelet is a wavelet where the

associated wavelet transform is invertible but not necessarily orthogonal.

Designing biorthogonal wavelets allows more degrees of freedoms than

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orthogonal wavelets. One additional degree of freedom is the possibility to

construct symmetric wavelet functions.

In the biorthogonal case, there are two scaling functions , which may

generate different multiresolution analyses, and accordingly two different

wavelet functions . So the numbers M, N of coefficients in the scaling

sequences may differ. The scaling sequences must satisfy the following

biorthogonality condition

.

Then the wavelet sequences can be determined as ,

n=0,...,M-1 and , n=0,....,N-1.

Types:3.4

3.4.1

3.4.2

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2) DISCRETE WAVELET TRANSFORM:

The DWT of a signal x is calculated by passing it

through a series of filters. First the samples are passed through a low pass

filter with impulse response g resulting in a convolution of the two:

The signal is also decomposed simultaneously using a high-pass filter h. The

outputs giving the detail coefficients (from the high-pass filter) and

approximation coefficients (from the low-pass). It is important that the two

filters are related to each other and they are known as a quadrature mirror

filter.

However, since half the frequencies of the signal have now been removed,

half the samples can be discarded according to Nyquist’s rule. The filter

outputs are then down sampled by 2:

3.5 INTEGER WAVELET TRANSFORMS

3.5.1Advantages:

A few characteristics of reversible ITI wavelet

transforms that make them well suited for signal coding applications. In

order to efficiently handle lossless coding in a transform-based coding

system, we require transforms that are invertible. If the transform employed

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is not invertible, the transformation process will typically result in some

information loss. In order to allow lossless reconstruction of the original

signal, this lost information must also be coded along with the transform

data. Determining this additional information to code, however, is usually

very costly in terms of computation and memory requirements. Moreover,

coding this additional information can adversely affect compression

efficiency. Thus, invertible transforms are desired. Often the invertibility of

a transform depends on the fact that the transform is calculated using exact

arithmetic. In practice, however, finite-precision arithmetic is usually

employed, and such arithmetic is inherently inexact due to rounding error.

Consequently, we need transforms that are reversible (i.e., invertible in

finite-precision arithmetic). Reversible ITI wavelet transforms approximate

the behavior of their parent linear transforms, and in so doing inherit many

of the desirable properties of their parent transforms. For example, linear

wavelet transforms are known to be extremely effective for decor relation

and also have useful multiresolution properties. For all of the reasons

described above, reversible ITI wavelet transforms are an extremely useful

tool for signal coding applications. Such transforms can be employed in

lossless coding systems, hybrid lossy/lossless coding systems, and even

strictly lossy coding systems as well.

3.5.2 DAUBECHIES’ 5/3 WAVELET TRANSFORMS :

For transforming the image I have taken daubechies’ 5/3

bioorthogonal wavelet for decomposition and the equation is given by

d[n]=d0[n]-[1/2(s0[n+1]+s0[n])]

s[n]=s0[n]+[1/4(d[n]+d[n-1]+1/2)]

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where d[n]is the high pass subband signal and s[n] is the low pass subband

signal and s0[n]=x[2n] and d0[n]=x[2n+1]

ADVANTAGES

1) low computational complexity

2) efficient handling of lossless coding

3) minimal memory usage

4) performs best for images with a greater amount of high frequency

content.

3.6 DOWNSAMPLING:

Downsampling is one of the fundamental processes in

multirate systems, and is performed by a processing element known as the

downsampler. The downsampler, takes an input signal x[n] and produces

the output signal

Y (n) = x (Mn)

where M is a sampling matrix. The relationship between the input and

output of the downsampler in the z-domain is given by

Where

and mk is the kth column of M .The frequency domain relation

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3.7 SHAMIR’S THRESHOLD SCHEME:

In Shamir’s (r,m) threshold scheme , the

secret D is divided into m shadows (D1,D2, . . . ,Dm) and any r or more

shadows can be used to reconstruct it. To split D into m pieces, a prime p,

which is bigger than both D and m, is randomly selected and an (r − 1) th

degree polynomial is chosen,

q(x) = (a0 + a1x + · · · + ar−1xr−1)mod p

in, a0 = D, and {a1, a2, . . . , ar−1} are random numbers selected from numbers

0 ~ (p − 1). The pieces are obtained by evaluating

D1 = q(1), . . . ,Di = q(i), . . . ,Dm = q(m).

Note that Di is a shadow. Given any r pairs from these m pairs {(i,Di); i = 1,

2, . . . ,m}, the coefficients a0, a1, a2, . . . , ar−1 can be solved using

Lagrange’s interpolation, and hence the secret data D can be revealed. In

Thien and Lin’s work, they took a0, a1, a2,….. ar−1 as the gray levels of r

pixels in a secret image to generate m shadows.

CHAPTER 4

MATLAB SIMULATION RESULTS

I have taken Lena as my input and its details are given below

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File size:262144

Format:’gif’

Width:512

Height: 512

Bit depth:8

Colortype:’gray scale’

Bits per sample: 8

The input image is ITI wavelet transformed by daubechies ‘5/3 biorthogonal

wavelet, 6 level decomposition.

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6 LEVEL WAVELET DECOMPOSITION

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AFTER COMBINATION

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REFERENCE:

[1] A. Shamir, “How to share a secret,” Communication of

ACM, vol. 22(11), pp. 612-613 1979.

28

[2] C. C. Thien and J. C. Lin, “An Image-sharing method with

user-friendly shadow images”, IEEE Trans. on CSVT, vol.

13(12), 2003, pp. 1161-1169.

[3] C.C.Thien and J.C. Lin, “Secret image sharing,”

Computers & Graphics, vol. 26, pp.765-770, 2002.

[4] S. K. Chen and J.C. Lin, “Fault-tolerant and progressive

transmission of images,” Pattern Recognition,vol. 38, pp.

2466-2471, 2005.

[5] R. Z. Wang and C.-H. Su, “Secret image sharing with

smaller images,” Pattern Recognition Lett., vol. 27, pp. 551-

555, 2006.

[6] H. Kim and C.C. Li, “Lossless and lossy image

compression using biorthogonal wavelet transforms with

multiplierless operations,” IEEE Trans. on Circuit And

Systems-II: Analog And Digital Signal Processing,

vol. 45(8), pp. 1113-1118, 1998.

[7] A. Zandi, J. Allen, E. Schwartz, and M. Boliek, “CREW:

Compression with reversible embedded wavelets,” Proc. 5th

IEEE Data Compression Conf., Snowbird, UT, pp. 212-221,

1995.

[8] A.R. Calderbank, I. Daubechies, W. Sweldens, and

B.L.Yeo, “Wavelet transforms that map integers to integers,”

Applied and Computational Harmonic Analysis, vol. 5, pp.

332-369, 1998.

[9] M.D. Adams, and R.K. Ward, “Symmetric-extension-

compatible reversible integer-to-integer wavelet

29

transforms,” IEEE Trans. on Signal Processing, vol. 51(10),

pp. 2624-2636, 2003.

[10] A. Said and W.A. Pearlman, “A new, fast and efficient

image codec based on set partitioning in hierarchical trees,”

IEEE Trans. on Circuits Syst. Video Technol., vol. 6(3), pp.

243-250, 1996.

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