Abstract AN OPTIMAL WATERMARKING SCHEME BASED ON SINGULAR VALUE DECOMPOSITION

AN OPTIMAL WATERMARKING SCHEME BASED ON SINGULAR VALUE DECOMPOSITION

Emir Ganic, Nasir Zubair and Ahmet M. Eskicioglu*

Department of Computer and Information Science

CUNY Brooklyn College, 2900 Bedford Avenue, Brooklyn, NY 11210, USA

Abstract

Watermarking, the process of embedding data into a multimedia element, can be used primarily for copyright protection and other purposes. The schemes that have recently been proposed modify the pixel values or transform domain coefficients. The Singular Value Decomposition (SVD) is a practical numerical tool with applications in a number of signal processing fields including image compression. In an SVD-based watermarking scheme, the singular values of the cover image are modified to embed the watermark data. We propose an optimal SVD-based watermarking scheme that embeds the watermark twice. In the first layer, the cover image is divided into smaller blocks and a piece of the watermark is embedded in each block. In the second layer, the cover image is used as a single block to embed the whole watermark. Layer 1 allows flexibility in data capacity, and Layer 2 provides additional robustness to attacks.

Key words: watermarking, copyright protection, visual watermark, singular value decomposition.

1.Introduction

Watermarking (data hiding) is the process of embedding data into a multimedia element such as image, audio or video. This embedded data can later be extracted from, or detected in, the multimedia for several purposes including copyright protection, access control and broadcast monitoring. A watermarking algorithm consists of the watermark structure, an embedding algorithm and an extraction, or a detection, algorithm. Watermarks can be embedded in the pixel domain or the transform domain such as the DCT or wavelet. In most multimedia applications, three desired attributes for a watermarking scheme are invisibility, robustness and high capacity. Invisibility refers to the degree of distortion introduced by the watermark and its affect on the viewers or listeners. Robustness is the resistance of an embedded watermark against intentional attacks, and normal audio/visual processes such as noise, filtering, resampling, scaling, *Email address of the corresponding author: eskicioglu@ rotation, cropping and lossy compression. Capacity is the amount of data that can be represented by the embedded watermark.

The SVD for square matrices was discovered independently by Beltrami in 1873 and Jordan in 1874, and extended to rectangular matrices by Eckart and Young in the 1930s. It was not used as a computational tool until the 1960s, however, because of the need for sophisticated numerical techniques. In later years, Gene Golub demonstrated its usefulness and feasibility as a tool in a variety of applications [1].

Every real matrix A can be decomposed into a product of 3 matrices A = UΣV T, where U and V are orthogonal matrices, U T U = I, V T V = I,and Σ = diag (λ1, λ2, ...). The diagonal entries of Σ are called the singular values (SVs) of A, 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

T

i V i U

r

i i

A∑

=

=

1

λ

where r is the rank of matrix A. It is important to note that each SV specifies the luminance of an image layer while the corresponding pair of singular vectors specifies the geometry of the image.

SVD is one of the most useful tools of linear algebra with several applications in image compression [2,3,4,5,6,7], watermarking [8,9,10] and other signal processing fields [11,12,13,14].

2.Watermarking in the SVD domain

We surveyed the relevant literature, and found several approaches for SVD-based watermarking, which are summarized in Table 1. They are either block-based or global schemes with a watermark chosen to be a random sequence or a digital image (i.e., a visual watermark).