ncawoct-10

National Conference on Advancements in Wireless and Optical Communication Technologies-2010

SINGULAR VALUE DECOMPOSITION TECHNIQUE FORDIGITAL IMAGE WATERMARKING

Ms. N.Singh1 Mr. M.M.Sharma2

1Asst. Prof., Department of Electronics and Communication EngineeringInstitute of Engineering and Technology, Alwar (India)

2Assoc. Prof., Department of Electronics and Communication EngineeringMalaviya National Institute of Technology, Jaipur (India)

Index terms Digital Image Watermarking,, watermark, Singular Value Decomposition, Singular values, attacks, Bit Correct Rate.

Abstract This paper, aims at embedding an invisible watermark into a digital image using the Singular Value Decomposition (SVD). The technique uses the U matrix of the SVD and modifies two of its value according to the watermark bit. The proposed technique is a modification of a recent scheme[1] proposed by B. Chandra Mohan and S. Srinivas Kumar in terms of Bit Correct Rate and robustness. Superior results are obtained when the watermarked image is attacked by histogram equalization, cropping, setting and resetting LSB plane, linear and non-linear filtering, noise and compression.

I. INTRODUCTION

Over the past few years, the tremendous growth in the computer network, World Wide Web and computer performance has facilitated the distribution of multimedia images with an increased concern for security, copyright and authentication. Digital watermarks have been used to tackle these issues.

Watermarking aims at including imperceptible information into the digital image. It would be then possible to recover the embedded information at any time, even when the image undergoes any intentional or unintentional nondestructive attack. This information can be a text, an image or a digital signal or pattern and is referred to as a watermark [3,8,10].

Image watermarking techniques are generally classified [2] as spatial domain or transform domain techniques. The former [3] changes the intensity of original image or gray levels of its pixels and the latter embeds the watermark into the transformed image. The latter [3] techniques offer higher security and robustness. A variety of image transforms are exploited for watermark embedding. Some of the commonly used transforms are Discrete Cosine Transform, Singular Value Decomposition [6-8] and Discrete Wavelet Transform.

The basic idea behind embedding watermark is to modify the transform coefficients in accordance with the bits of the watermark image.

It is desirable that the watermark is secure, imperceptible, robust and tamper resistant. However, several intentional and unintentional operations with the watermarked image may provide possibility for disabling the watermark. Commonly, these operations (especially the intentional ones) are referred as attacks [4,5,9] against watermarks. These manipulations include compression, geometric operations, signal processing operations and forgery.

II. SINGULAR VALUE DECOMPOSITION

Singular Value Decomposition (SVD) is a mathematical tool to analyze matrices, which decomposes a square matrix into 3 matrices of same size. For example, a square matrix A of size N X N, is decomposed into U, V and D matrix such that A= UDVT where VT is the transpose of matrix V. Here U and V are orthogonal and D is square diagonal. That is, UUT = Irank(A), VVT= Irank(A), U is rank(A) X M, V is rank(A) X N and D is a rank(A) X rank(A) diagonal matrix.

These diagonal entries σi' s are called singular values of A and their number is equal to the rank of A. These singular values satisfy the relation

σ 1 ≥ σ 2 ≥ σ 3......σrank ( A) > 0.

Each singular value specifies [6,7] the luminance of an image layer while the corresponding pair of singular vectors specifies the geometry of the image. For majority


of the attacks, the change in the largest singular value is very small.

The columns of U are called the left singular vectors of A, and the columns of V are called the right singular vectors of A. This decomposition is known as the Singular Value Decomposition (SVD) of A, and can be written as

SVD(A) = [U D V ].SVD (A) = λ1U1V1

T + λ 2U2V2

T + ...... + λ U V

A = UDVT.

Calculating the SVD consists of finding the eigenvalues and eigenvectors of AAT

and ATA. The eigenvectors of ATA make up the columns of V , the eigenvectors of AAT

make up the columns of U. Also, the singular values in S are square roots of eigenvalues from AAT

or ATA. The singular values are the diagonal entries of the S matrix and are arranged in descending order. The singular values are always real numbers. If the matrix A is a real matrix, then U and V are also real.

III. PERFORMANCE ANALYSIS

Some of the early literature considered a binary robustness metric that only allows for two different states (the watermark is either robust or not). However, it makes sense to use a metric that allows for different levels of robustness. The use of the bit-correct ratio (BCR) has become common recently, as it allows for a more detailed scale of values. The bit correct ratio (BCR) is defined as the ratio of correct extracted bits to the total number of embedded bits and can be expresses using the formula:

Where, l is the watermark length, Wn corresponds to the nth

bit of the embedded watermark and W'n corresponds to the nth bit of the recovered watermark.

Imperceptibility of an embedded watermark can be expressed either as fidelity or quality measure. Fidelity represents a measure of similarity between the original and watermarked cover.

The widely used peak signal-to-noise ratio (PSNR) measurement [4,10] which measures the maximum signal to noise ratio found on an image is used as an objective measure for the distortions introduced by the watermarking system. The PSNR is given by [9]:

where MSE is mean square error between the original image Iorg and the watermarked one Iw.

IV. PROPOSED METHOD

The proposed algorithm is presented below. The image used as cover image is a 512 X 512, 8-bit, Lena.bmp image with 256 gray levels, as shown in figure 1 and the watermark image is shown in figure 2. The watermark image is 32 X 128 black and white image. The size of watermark is required to be equal to the number of blocks in which the cover image is divided. If not, it is padded with appropriate number of ones.

Embedding:Input: cover image, watermark (w(i)).Output: watermarked image.Symbols: a = constant

d = absolute difference

1. Read the cover image and the watermark.2. Reshape the watermark into vector3. Set a = 0.2 and block size = 8.4. Divide the cover image into blocks.5. For each block i,

a. Find SVD.b. Obtain the U matrix and find difference d = |U(1,1)| – |U(2,1)|.c. For corresponding watermark bit If (w(i) = 1 & d > a) or (w(i) = 0 & d < a)

U(2,1) = -||U(2,1)| + (a - d)/2|.U(1,1) = -||U(1,1)| - (a + d)/2|

if (w(i) = 1 & d < a) or (w(i) = 0 & d > a)U(2,1) = -||U(2,1)| - (a - d)/2|.U(1,1) = -||U(1,1)| + (a + d)/2|

d. Take inverse SVD.

Recovery:Input: watermarked image.Output: retrieved watermark (wm).

1. Read the watermarked image.2. Define block size.3. Divide the watermarked image into blocks.4. For each block i,

a. Find its SVD.b. Obtain the U matrix.c. if (U(1,1) > U(2,1))

Set wm(i) = 1. else

Set wm(i) = 0.

V. EXPERIMENTS AND RESULTS

The watermark is embedded with block size = 8 and the value of constant varies from 0 to 0.1. The resultant watermarked images for constant value of 0.01 and 0.1 are shown in figure 3(a) and (b) respectively. The watermark is imperceptible in figure 3(a) with a PSNR of 5.9151e+004, but when the value of constant is increased the PSNR drops greatly to 11.8113 and the watermarked image is highly corrupted. But, on the other hand an


increase in the value of constant, results in better performance of the technique at detection end.

This is measured with the performance metric of BCR. A higher BCR implies that the retrieved watermark is closer to the original one. The variations of PSNR and BCR with the constant value ‘a’ are shown in figure 4. The two curves intersect at ‘a’ = 0.01 which is then chosen as the constant for experimentation.

The watermark is retrieved with a BCR of 89.5996 by the proposed algorithm, at constant = 0.01. The result is compared with the algorithm described in [1] which gives the retrieved watermark with 83.0322% correctness only. The images watermarked with both the algorithms are subjected to different intentional and unintentional attacks to experimentally determine the BCR. Table I summarizes the results obtained for the two algorithms.

The techniques are analyzed for JPEG compression by retaining only 8 of the DCT coefficients for each block of size 8 X 8. The next test, for cropping, is done by setting the first 128 X 128 elements of the image to black. To test the techniques for robustness against noise two types of

noise are taken viz. Gaussian noise and salt and pepper noise. Both the types of noises are added at the density of 0.01. Test for robustness against filtering is carried out for rotationally symmetric low pass Gaussian filter, which is a linear filter, as well as for non-linear median filter. The median filter is taken for size 3 X 3.

Attack AlgorithmProposed in [6]

Proposedalgorithm

JPEG Compression 69.1406 72.4609HistogramEqualization

82.7881 88.6475

Setting LSB plane= 1

83.0322 89.5996

Cropping(1:128,1:128)=0

77.9785 83.2031

Salt and pepperNoise

78.3691 83.9600

Gaussian Noise 55.4932 60.2051Low passGaussian filtering

83.3008 86.8896

Median filtering 51.9531 52.4414

Table I: Table for comparison of results for the two algorithms

VI. DISCUSSIONS AND CONCLUSIONS

SVD thus, proves to be promising domain for watermark embedding. Thus, it is very clear from the table that the proposed algorithm is better than that proposed in [6] in terms of robustness against attacks. The proposed technique is more resistive to the attacks like histogram equalization, compression, cropping, setting LSB plane to 1, noise and median filtering.

References[1] Mohan B. Chandra, Kumar S. Srinivas, “A Robust Image Watermarking Scheme Using Singular Value Decomposition”. Journal of Multimedia, Volume 3, No. 1, (May 2008).[2] Dickman Shawn D., “An overview of Steganography”. James Madison University Infosec Techreport, (July 2007).[3] Genov P. Eugene, “Digital Watermarking of Bitmap Images”. International Conference on Computer Systems and Technologies- CompSysTech ’07, (2007).[4] Kutter M., Petitcolas F.A.P., “Fair Evaluation Methods for Image Watermarking Systems”.[5] Miller Matt L., Cox Ingemar J., Linnartz Jean-Paul M. G., Kalker Ton, “A Review of Watermarking Principles and Practices”. Chapter 18, Digital Signal Processing in Multimedia Systems, Ed. K. K. Parhi and T. Nishitani, Marcell Dekker Inc., 461-485, (1999).[6] Judge James C., “Steganography:Past, Present, Future”, SANS Institute InfoSec Reading Room, (2001).[7] Mohan B. Chandra, Kumar S. Srinivas, Chhatterji B. N., “A Robust Digital Image Watermarking Scheme Using Singular Value Decomposition (SVD), Dither


Quantization and Edge Detection”. ICGST-GVIP Journal, Volume 8, issue 1, (June 2008).[8] Pei Soo-Cheng, Liu Hsin-Hua, “Improved SVD based Watermarking for Digital Images”, Sixth Indian Conference on Computer Vision, Graphics and Image Processing, IEEE Computer Society, (2008).[9] Petitcolas Fabien A., Anderson Ross J., Kuhn Markus G., “Information Hiding- A Survey”, Proceedings of IEEE, Special issue on protection of multimedia content, 1062-1078, (July 1999).[10] Wolfgangand Raymond B., Delp Edward J., “Overview of Image Security Techniques with Applications in Multimedia Systems”. Prudune University.

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