Simulation-based Exploration of SVD-based

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robustness of the steganographic channel distorted by JPEG compression while .... In [9] a format for SVD-based lossy compression of digital images was.
Simulation-based Exploration of SVD-based Technique for Hidden Communication by Image Steganography Channel Vladimir Gorodetsky, Vladimir Samoilov St. Petersburg Institute for Informatics and Automation 39, 14-th Liniya, St.Petersburg, 199178, Russia {gor, samovl}@mail.iias.spb.su

Abstract. The paper presents an empirical study of the properties of a new image-based information hiding technique that are critical for hidden communication. The technique is based on the Singular Value Decomposition (SVD) transform of a digital image and uses embedding a bit of data through slight modifications of a linear combination of singular values of a small block of the segmented cover image. The primary objective of the study is to establish the dependence between the capacity rate of the invisibly embedded data and robustness of the steganographic channel distorted by JPEG compression while varying embedding procedure attributes. The second objective of the empirical study is to establish practical recommendations concerning adjusting of the controllable attributes of the SVD-based data embedding procedure given the message size and the threshold for Bit Error Rate. The results of the study provide evidence that the developed information hiding approach is promising for hidden communication.

1 Introduction Hiding information in digital images is a rapidly developing area providing an effective way for information assurance in covert communications, for watermarking of digital objects and for other applications. Unlike cryptography that aims at hiding the message content, the goal of information hiding is the concealment of the very fact that a message exists. The common requirement to information hiding techniques is providing invisibility of a message. Given invisibility, quality of a hiding technique is determined by rate of the hidden data and robustness to common and specific types of intentional distortions. These properties are always contradictive and particular applications favor one of them [1]. To date a number of techniques for hiding information in digital images have been developed. Most of them are of value because theory and practice proved that there is no one superior technique. Each technique has its own advantages and flaws and the overall evaluation of a technique is application dependent. The overviews of hiding information techniques are given in [7], [8], [12], [13], [14], [21], [27], etc.

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Vladimir Gorodetsky, Vladimir Samoilov

Embedding techniques that insert data into the spatial image domain by the use of a selected subset of the image pixels and/or a bit-wise approach are described in [3], [4], [6], [16], [20], [22], [28], [29], etc. Transform-based techniques, that operate images represented by a finite set of orthogonal or bi-orthogonal “basis functions” are presented in [5], [10], [15], [19], [23], [24], [26], etc. Examples of such transforms are Discrete Cosine, Wavelet transform, Singular Value Decomposition, etc. A specific technique is provided by fractal-based approach that constructs a "fractal code" encoding both the cover image and hidden message [25]. It should be noticed that most developed techniques are watermarking-oriented which major requirement is robustness to a wide range of distortions, but not of the high rate of the embedded data. However, many military and industrial applications like hidden communication (HC), in particular, hidden transmission of digital images call for both invisibility of high volumes of data and survivability of the transmitted information. Examples are transmission of industrial secrets, plans of covert operations [12], etc. Unfortunately, most existing techniques capable of providing the ratio of transparently embedded data, required for HC, are highly sensitive to many distortions, in particular, to lossy compressions like JPEG. In contrast, paper [19] presents a technique oriented to HC application. It proposes an approach called Spread Spectrum Image Steganography). It is a blind scheme in which one does not need to use the cover image in order to extract the hidden message. The central point of this approach is to embed the message in the form of a signal provided that the signal has the same characteristics as noise inherent to this image. The last property is provided by a special modulation of the message data. HC application is the focus of paper [10], which presents a technique primarily destined for meeting the requirements of the covert communication. The technique uses Singular Value Decomposition (SVD) transform of digital images. According to the properties of SVD of a digital image, each singular value (SV) specifies the luminance (energy) of an SVD image layer, whereas the respective pair of singular vectors specifies the image geometry. Accordingly, slight variations of SVs cannot affect the visual perception of the quality variations of the cover image. The robustness of the approach is provided by the fact that it embeds of data through slight modifications of the largest SVs of a small block of the segmented covers, i.e. it embeds a message into low frequency of a cover image. Simulation has proved this anticipation. This paper presents the results of the detailed simulation-based study of the HC critical properties of SVD-based data hiding. Primarily, the paper studies interdependencies of both the capacity of invisibly embedded data and robustness of its transmission through communication channel distorted by JPEG compression, on the one hand, and values of attributes determining the embedding procedure, on the other hand. The simulated situation assumes that a message is embedded into digital image (cover) represented in the bmp format and the resulting image is transmitted in the JPEG format. The received image with embedded data is again transformed to the JPEG format. Thus, the cover image can be considered as a noisy channel distorted by the JPEG compression. In addition, it is assumed that the channel can also be intentionally distorted by applying of a JPEG compression to the transmitted cover with embedded data. As the robustness metric the value of Bit Error Rate (BER) is used.

Simulation-based Exploration of SVD-based Technique

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The rest of the paper is organized as follows. In Section 2 the concept of the SVDbased approach to and algorithm for hiding data in digital images is briefly explained and one of the previously developed algorithms is outlined. Section 3 describes a distortion model as applied to a message transmitted through a steganographic channel. Section 4 discuss the results of simulation-based study of the influence of attributes specifying the SVD-based embedding procedure on both the capacity of the invisibly embedded data and robustness of its transmission through steganographic communication channel distorted by JPEG compression of different quality. The Conclusion summarizes the paper results and future work.

2 SVD of Digital Images and Information Hiding Technique Assume that a digital image in the bitmap format is specified by a m n matrix, A {a i , j } m , n . If an image is represented in RGB format then it is specified by three 

such matrices, AR , AG and A B . An arbitrary matrix A of size m n can be represented by its SVD [8] in the form

A X YT =

i r







i



i 1

X i Yi

(1)

T





where X, Y are orthogonal m m and n n matrices respectively, X 1 , X 2 ,..., X m and

Y1 , Y2 ,...,Yn are their columns, is a diagonal matrix with non-negative elements, and r min{ m , n } is the rank of the matrix A. Diagonal terms 1 , 2 ,..., r of 









are called singular values (SV) of the matrix A and r is the total number of matrix non-zero singular values. Columns of the matrices X, Y are known as left and right singular vectors of matrix A respectively. There are several ways to calculate SVs 1 , 2 ,..., r . The most simple way is to 





calculate them as $$

T

$

, or

T



Yi

i

i=1,2,…,r, where

i

is the i-th eigenvalue of the matrix

. The left singular vector X i , i

eigenvector of the matrix vector





i

$



T

$$

corresponding to 

i





 

    U

, is equal to the

. Similarly, the right singular

, i=1, 2,.., r , is equal to the eigenvector of the matrix

T

$

$

that

corresponds to its eigenvalue i . If an image is given in RGB format then it can be represented by three such SVDs in the form (1). Thus, SVD of an image is 

decomposition of each its matrix, AR , 

2

X 2 Y 2T ,...,

mode: if 

i 



1

X 1Y1T ,

s

X r Y rT . As a rule, singular values are enumerated in descending

j

then i