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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
1
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.
3
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
7
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.
8
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.
9
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.
10
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
11
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.
12
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
13
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
14
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.
15
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
: :
16
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
17
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.
18
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
19
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
20
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
21
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)]
22
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
23
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
24
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.
25
6 LEVEL WAVELET DECOMPOSITION
26
AFTER COMBINATION
27
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.
30