A Watermarking Algorithm Based on Contourlet Transform and Nonnegative Matrix Factorization

Silja M S Computational Engineering and Networking

Amrita Vishwa Vidyapeetham Coimbatore, India.

siljasudhakar@gmail.com

Soman K P Computational Engineering and Networking

Amrita Vishwa Vidyapeetham Coimbatore, India.

kp_soman@ettimadai.amrita.edu

Abstract -This paper present a robust digital image watermarking algorithm based on NMF and SVD in Contourlet transform. Firstly, the orgianl image is transformed into directional subband coefficients using Discrete Contourlet Transform(CT).Then apply NMF and SVD to factorize the directional subband coefficients. After that ,embed grayscale watermark image into the SVD coefficients. The experiments based on this scheme shows that this method is very robust and it gives very good Peak Signal to Noise Ratio(PSNR). Keywords — Watermarking, Contourlet Transform,Nonnegative Matrix Factorization

I. INTRODUCTION

The main idea in digital watermarking is to insert machine readable information within the content of a digital data (image, audio, or video) for the purpose of copy right protection ,content authentication, tamper proofing etc. The embedding information is called watermark ,it can be a pseudo-random signal, binary logo or digital signal.Now a days internet is commonly used to transmit digital multimedia signals. So digital watermarking has more important due to increasing growth of internet usage.. In recent times, for data analysis we are using Nonnegative Matrix Factorization (NMF). In NMF we factorize a nonnegative matrix into two physically meaningful nonnegative matrices, and has been successfully applied in many signal processing analysis. Till now only few watermarking algorithms are developed based on NMF (Ghaderpanah and Hamza 2006,Wie lu et al 2008,Ouhasain and Hamza 2009). In this paper ,we are introducing an image watermarking algorithm based on NMF and SVD in CT domain. Contourlet transform is more powerful than Wavelet transform because it can captures smooth contours and fine details

in image, so it is very useful for images like maps. This paper is organized as follows: In section 2 Contourlet transform methods are introduced, in Section 3 NMF methods are introduced, in section 4 SVD methods are introduced, in section 5 watermarking procedure are presented, in section 6 some experimental results, in section 7 conclusions are presented.

II. DISCRETE CONTOURLET TRANSFORM

The Contourlet transform gives a flexible multiscale ,local and multidirectional decomposition of an image using a combination of a Laplacian pyramid (LP) and a directional filter bank (DFB),so it is called Pyramidal Directional Filter Bank(PDFB). It means that it can be capture smooth contours and fine details in an image. [1] Figure 1 shows contourlet filter bank ,it is used to obtain sparse expansion for images having smooth contours. As shown in this figure 1 first stage is LP decomposition and second stage is DFB decomposition. After LP decomposition we are getting low pass image and band pass image. The band pass image are directly fed into DFB so we can capture directional information. In each pyramidal level we can get only one band pass image, this is the unique feature of LP. This band pass image is decomposed by l-level DFB into 2l directional sub band. This process can be iterated on downsampled low pass images [2].

Figure.1. Contourlet filter bank.

2009 International Conference on Advances in Recent Technologies in Communication and Computing

978-0-7695-3845-7/09 $25.00 © 2009 IEEE

DOI 10.1109/ARTCom.2009.198

279

2009 International Conference on Advances in Recent Technologies in Communication and Computing

978-0-7695-3845-7/09 $26.00 © 2009 IEEE

DOI 10.1109/ARTCom.2009.198

279

2009 International Conference on Advances in Recent Technologies in Communication and Computing

978-0-7695-3845-7/09 $26.00 © 2009 IEEE

DOI 10.1109/ARTCom.2009.198

279

III. NONNEGATIVE MATRIX FACTORIZATION

Nonnegative matrix factorization (NMF) is a matrix factorization technique, which decomposes nonnegative matrices into two nonnegative matrices, and has been used for signal analysis. After applying NMF we are getting a reduced representation of original data. A description of nonnegative matrix factorization can be described as follows. Given a nonnegative matrix V of size MxN and a positive integer P< min(M,N). The aim of NMF is to find two nonnegative matrix factors, W of size MxP and H of size PxN, to approximately represent the original matrix V such that V ≈ WH, where W and H are called the base matrix and the coefficient matrix (or encoding matrix), respectively. In this work, Lee and seung’s ‘’multiplicative iteration algorithm’’ is used to find the nonnegative matrix factors W and H.[3]. In multiplicative iterative algorithm ,first randomly initialize W and H with positive numbers. For each basic vector W ,update corresponding encoding vector H by using equations (1) and (2)

1 , , ,, ,

1 ,

/( )Nn p n m n m n

m p m p Nn p n

H V WHW W

H=

=

Σ←

Σ (1)

1 , , ,, ,

1 ,

/( )Mm m p m n m n

p n p n Mm m p

W V WHH H

W=

=

Σ←

Σ (2)

where m = 1, 2 ….M , n = 1, 2…….N, p =1, 2……P. Repeat this process until (3) is convergence.

,1 1 , , ,

,

[ log ( ) ]( )

mnM Nm n mn mn mn

mn

VF V V WH

WH= ==Σ Σ − + (3)

which is known as the generalized Kullback-Leibler (KL) divergence.

IV. SINGULAR VALUE DECOMPOSITION

SVD is an effective numerical analysis tool used to analyze matrices.. Consider a matrix A, denoted as A ∈ R m×n , where R represents the real number domain, then SVD of A is defined as A=USV T . Where U ∈ R mxm and V ∈ R n×n

are orthogonal matrices, and S ∈ R m×n is a diagonal matrix with r (rank of A matrix) nonzero elements called singular values of A matrix. Singular values comes in decreasing order, this feature is used in SVD based compression methods.

V. WATERMARKING PROCEDURE

In this section we introduce a image watermarking algorithm based on NMF and SVD in Contourlet domain.The detailed algorithm is explained below.

A.Watermark Embedding Algorithm

Our digital watermark embedding process is divided into 6 steps and is briefly described below Step1:Take a gray scale image of size 512x512 and apply Contourlet transform to the image. In this algorithm , after applying CT we are taking the number of pyramidal level is four. At each successive level, number of directional sub bands are 2, 4, 8, and 16 respectively. Last directional sub band is selected for watermark embedding because this sub band possesses maximum energy .In this we are taking first 4096 coefficients of 16th band of fourth level for watermark embedding and is denoted by I. Step 2: Apply NMF to I . After NMF, we get two matrices W and H, that is I≈WH. Take randomly one value p < 64 (let it be 60).Then the size of W is 64x60 and size of H is 60x64. Step 3: Apply SVD to H. H ⇒ USVT and obtain U ,V and S . Step 4: Take a grey scale watermark image (B) of size 60x64.Embedded this watermark into the singular values of H, according to the equation f1=S+α * B, where ’α‘ is a scaling factor. Step 5:Apply SVD to f1 , f1 ⇒ U1S1V1

T , obtain U1 ,S1, and V1, then obtain d1 ⇐US1VT . Step 6:Now watermark is embedded and apply inverse CT to produce watermarked image of size 512x512. B. Watermark Extracting Algorithm Our digital watermark extracting process is divided into 3 steps and is briefly described below. For watermark retrieval we need host image so it is a non-blind algorithm. Step 1: Apply CT to watermarked image. Take the first 4096 coefficients of 16th sub band of 4th pyramidal level and is denoted as I* . Apply NMF and we get two matrices W* and H*,I*=W*H*. Step 2: Apply SVD to the H* ,H*⇒U*S*V*T and obtain U*,S* and V*T.

280280280

Step 3:By using U1 ,V1 and S*,we are getting d1* ⇐ U1S*V1

T . According to the rule, B* ⇐

α1

(d1*−S), we are extracting the watermark B*.

VI. EXPERIMENTAL RESULTS

In this section some experiments are carried out to determine the effectiveness of the proposed algorithm. Matlab 7.5 is used for testing the proposed scheme. To check the robustness ,it is tested under different attacks like salt and pepper, Gaussian noise,etc. In this algorithm ,scaling factor α=.003. We are taking the cover image of size 512x512 and the watermark image of size 60x64. The original , watermarked,watermark and watermark extracted images is shown in figure 2 . There was no visual difference between the original and watermarked images. After retrieving the watermark we calculate the normalized correlation coefficient(C_C) of the retrieved watermark(w’) with the original watermark(w). Table I give the PSNR and normalized correlation coefficients.

Figure. 2 a) Original b)Watermarked c) Watermark d) Watermark Extracted

TABLE I.

PSNR AND NORMALIZED CORRELATION COEFFICENTS

Type PSNR C_C

No attack 55.78 .9919 Salt & pepper 45.04 .9735 Gaussian noise 43.39 .9407 Low pass filter 37.76 .9749

Gamma correction 19.81 .9800 Resizing 38.38 .9100 Cropping 26.43 .9892

Row column blanking 25.53 .9919 Row column copying 29.45 .9914

Rotation 21.25 .8295 Translation 20.84

.9203

JPEG compression 55.82 .9869

.

0

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5 6 7 8 9 10 11 12

C_C

Proposed scheme SVD

Figure 3 .Correlation Coefficent Comparison Results (1)No attack (2)cropping (3)Low pass filtering (4)Gaussian Noise (5)Gamma correction (6)Resizing (7)salt & pepper (8)R_C

blanking (9)R_C Copying (9)Rotation (10)Translation (11)JPEG compression(60%)

We compared our results with ‘An efficient block by block SVD based image watermarking scheme’ by Ghazy et al[4]. The result is shown in figure 3.

VII .CONCLUSION

In this paper , a watermarking algorithm based on CT and NMF is proposed. This method gives robustness against different types of image processing attacks. The experimental results shows that the proposed method is more robust than the block based SVD method.

REFERENCES

[1] Minh N. Do and Martin Vetterli.’’ The contourlet transform: An efficient directional multiresolution image representation.’’ IEEE Trans. on Image Processing, 14:2091–2106, December 2005.

[2] M. N. Do, “ Directional Multiresolution Image Representations Phd Thesis Department of Communication Systems, Swiss Federal Institute of Technology Lausanne, November 2001

[3] Lee DD, Seung HS. Algorithms for non- negative matrix factorization. Adv Neural Inform Process System 2001;13:556–62. [4] R. A. Ghazy , N. A. El-Fishawy, M. M. Hadhoud, M. I.

Dessouky and F. E. Abd El-Samie” An efficient block by block SVD based image watermarking scheme ” IEEE Radio science conference 2007

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