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A Comparative Performance Evaluation of SVD and Schur Decompositions for Image Watermarking

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A Comparative Performance Evaluation of SVD and Schur Decompositions for Image Watermarking International Conference on VLSI, Communication & Instrumentation (ICVCI) 2011 Proceedings published by International Journal of Computer Applications® (IJCA) A Comparative performance evaluation of SVD and Schur Decompositions for Image Watermarking B.Chandra Mohan K.Veera Swamy S.Srinivas Kumar Professor, ECE dept Professor, ECE dept Professor, ECE dept Bapatla Engineering College QIS College of Engg JNTU College of Engg Bapatla-522101 Ongole Kakinada ABSTRACT image. There might be no or little perceptible difference In this paper, the performance of SVD and Schur between the host image and watermarked image. One major decomposition is evaluated and compared for image copyright application of digital image watermarking is copyright protection applications. The watermark image is embedded in protection. After embedding the watermark, the watermarked the cover image by using Quantization Index Modulus image is sent to the receiver via the Internet or transmission Modulation (QIMM) and Quantization Index Modulation channel. (QIM). Watermark image is embedded in the D matrix of Schur The watermark image is to be sustained against various attacks, decomposition and Singular Value Decomposition (SVD). viz., filtering, compression, cropping, and rotation, etc., on Watermarking in SVD domain is highly flexible . This is due to watermarked image. The algorithms proposed so far can be the availability of three matrices for watermarking. Singular classified according to the embedding domain of the cover values in SVD and Schur decomposition are highly stable. image. They are, spatial, transform, and hybrid domain Compared to Singular Value Decomposition (SVD), Schur watermarking algorithms. decomposition is computationally faster and robust to image Three major requirements of a digital watermarking system are attacks. The proposed algorithms based on SVD and Schur imperceptibility, robustness, and capacity. Imperceptibility is decompositions are more secure and robust to various attacks, defined as “perceptual similarity between the original and the viz., rotation, low pass filtering, median filtering, resizing, salt watermarked versions of the cover work”. Robustness is the & pepper noise. Superior experimental results are observed with “ability to detect the watermark after common signal processing the proposed algorithm over a recent scheme proposed by operations”. Capacity describes amount of data that should be Chung et al. in terms of Normalized Cross correlation (NCC) embedded as a watermark to successfully detect during and Peak Signal to Noise Ratio (PSNR). extraction. General Terms Basically there are two main types of watermarks that can be Image Processing, Image Watermarking Algorithms embedded within an image, viz., pseudo random gaussian sequence and logo (binary or grey) image watermarks. Based on Keywords: Digital Image Watermarking, Schur the type of watermark embedded, an appropriate decoder is to be Decomposition, SVD, PSNR and NCC designed to detect the presence of watermark. 1. INTRODUCTION Watermarking in transform domain is more secure and robust. Several transforms like Discrete Cosine Transform (DCT) Distribution of multimedia data such as images, video and audio [2,10], Discrete Hadamard Transform (DHT) [5], Discrete over the internet requires secure computer networks. Multimedia Wavelet Transform (DWT) [3,4], Contourlet Transform (CT) data can be duplicated and distributed with out the owner’s [9], and Singular Value Decomposition (SVD) [6,7,8] are consent. Digital watermarking technique is a viable solution common transforms used in image watermarking. Each of these proposed to tackle this complex issue. Digital watermarking is a transforms has its own characteristics and represents the image branch of information hiding which is used to hide proprietary in different ways. information (company logo) in digital media like digital images, digital music, or digital video. Image watermarking has attracted SVD and Schur decompositions are two mathematical tools a lot of attention in the research community compared to video used to analyze matrices. When some perturbation occurs in the watermarking and audio watermarking. This is due to the watermarked image, the extraction of the watermark is not availability of various types of images and amount of redundant affected much. SVD is computationally expensive. information present in images. Digital image watermarking also Watermarking using Schur decomposition is faster compared to called watermark insertion or watermark embedding represents SVD decomposition. In this work, the performance of SVD and the scheme that inserts the hidden information into an image Schur decomposition in image watermarking application is known as host or cover image [1]. The hidden information may compared. Three scalar quantization schemes are adopted for be the serial number, the random number sequence, copyright the watermark embedding. messages, logos or any ownership identifiers called the This paper is organized as follows. SVD and Schur watermark. After inserting or embedding the watermark by decomposition are discussed in section 2. Scalar quantization using specific algorithms, the cover image will be slightly and proposed algorithm is presented in sections 3 and 4 modified and the modified image is called the watermarked 25 International Conference on VLSI, Communication & Instrumentation (ICVCI) 2011 Proceedings published by International Journal of Computer Applications® (IJCA) respectively. Experimental results are presented in section 5. • Flexible for watermark embedding. Conclusions are given in section 6. • Highly stable singular values 2. SVD AND SCHUR DECOMPOSITION However, one major drawback of SVD is, it’s computational SVD is a mathematical tool used to analyze matrices. In SVD, a complexity. The computational complexity of Schur square matrix is decomposed into three matrices of same size. A decomposition is less compared to SVD and hence it is useful real matrix g of size N × N is decomposed into three for real time applications. Schur decomposition can be applied to any real matrix. There are two versions of this decomposition: matrices (U, D, and V ) of same size. U and V are the complex Schur decomposition and the real Schur T T orthogonal matrices, i.e., U U = I and V V = I . Here, decomposition. In complex schur decomposition, Decomposition (complex version): = , where is a I is an identity matrix. Superscript T indicates transpose g UTU' U operation. D is a square diagonal matrix given by unitary matrix. U' is the conjugate transpose of U , and T D = diag(λ ,λ ...λ ) , where, the diagonal entries is an upper triangular matrix called the complex Schur form 1 2 r which has the eigen values of g along its diagonal. Schur λ λ λ 1 , 2 ... r are known as singular values of g . Here, r is the Decomposition (real version) is given by rank of the matrix g .The columns of U are called left g = VSV' singular vectors of g and the columns of V are called the right where g,V , S and V ' are matrices that contain real numbers singular vectors of g . This decomposition is known as Singular only. In this case, V is an orthogonal matrix, V ' is the Value Decomposition and can be represented as transpose of V , and S is a block upper triangular called the 8 3 real Schur form. Schur decomposition requires about N SVD(g) = [U D V ] (2.1) 3 3 T T T SVD(g) = λ U V + λ U V + ... + λ U V (2.2) flops. SVD computation requires 11N flops. Eigen values in 1 1 1 2 2 2 r r r the Schur decomposition are also highly stable. = T gˆ UDV (2.3) 3. SCALAR QUANTIZATION There are many scalar quantization techniques that are available in the literature for image watermarking applications. An where, U and V are real N × N unitary matrices with small extensive theoretical study on the scalar quantization techniques values. D is a diagonal matrix of N × N size with large for image watermarking applications was carried by Chen and singular values. gˆ is the reconstructed matrix after applying Wornell [11]. Quantization Index Modulation (QIM), Quantization Index Modulus Modulation (QIMM), and Dither inverse SVD transformation. The singular values satisfy the Modulation are variants of scalar quantization schemes. relation λ ≥ λ ... ≥ λ ≥ 0 . Each singular value specifies 1 2 r Two methods in QIMM and one method in QIM are the luminance of an image layer while the corresponding pair of = singular vectors specifies the geometry of the image. SVD presented here. Host signal is X {x1 , x2 ,...xn } The matrix of an image has good stability. Singular values have three original watermark W = {w , w ,...w },w ∈ }1,0{ is important properties: 1 2 n i the binary message signal. The extracted watermark is • Singular values of the image are stable, i.e., when a = ∈ small perturbation is added to an image, variance of its denoted by W' {w1 ,' w2 ',...wn },' wi ' }1,0{ singular values does not occur. Quantization Index Modulus Modulation: • Singular values exhibit the algebraic and geometric invariance to some extent. • Singular values represent the algebraic attributions of Method I (QIMM1): an image which are intrinsic and not visual. • These three properties of stability, algebraic and geometric In QIMM, the host signal = invariance of the singular values are utilized to embed the X {x1 , x2 ,...xn } is first divided by the watermark image in the largest singular values of the D matrix ∆ of the cover image. Image attacks tend to modify the singular quantization step size . It is rounded to the values of
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