A Hybrid Watermarking Technique Using Singular Value Decomposition

Daniela Stanescu, Daniel Borca, Voicu Groza, Mircea Stratulat

Computer and Software Engineering Department, University “Politehnica”, Faculty of Automation and Computers, 2 Parvan st, Timisoara, ROMANIA, Phone: (40) 256-403271, Fax: (40) 256-403214, E-mail: daniela.stanescu@cs.upt.ro,

dborca@bitdefender.com, groza@site.uottawa.ca, smircea@cs.utt.ro

Abstract

Digital watermarking has been identified as a major

technology to achieve copy control and ownership protection, due to improvements in imaging technologies and the ease with which digital multimedia can be created and manipulated. In this paper we present a hybrid steganographic technique for watermarking, using SVD transform and watermark quantization. 1. Introduction

There are many practical reasons for favoring digital media, such as: widespread usage of computers, smartphones, printers and high-rate digital networks; storage devices are inexpensive and compression technology makes them very appealing to digital information. The popularity of the World Wide Web has demonstrated the commercial potential of the digital multimedia.

However, malicious parties sometimes exploit this accessibility of digital data, creating copies without permission from the rightful authors. The idea of digital watermarking is to detect and trace copyright violations and has stimulated great interest among artists and publishers.

Many of the current techniques for embedding marks in digital images have been inspired by methods of image coding and compression. Although digital watermarking can be applied to any digital signal, we will focus on digital imagery.

Since the watermarked images should have, as far as possible, the same invariance properties as the images they are intended to protect, we will deal with robustness by making use of the Singular Value Decomposition. On one hand, the singular values (SVs) of an image have very good stability. On the other hand, the watermark is quantized before embedding, to increase bit-rate. Therefore, our hybrid method uses the SVs to store main data of quantization process, and the left-singular vectors to store secondary data.

Also, our method is not limited to signing, but can be used for other steganographic purposes.

2. Related works

Liu and Tan [1] first proposed a watermarking technique that utilizes the SVD, for the purpose of protecting owner’s copyright. Their method seems to be robust, but it is somewhat focused on signing images.

Many others have improved techniques based on SVD [2, 3], most notably Bergman and Davidson [4] who developed a method to hide a fixed amount of bits in the left singular vectors of the SVD. The aspect of robustness has concerned many authors, for example, Stanescu et al [5] proposed the Karhunen-Loève Transform. In this paper, we address the special problem of image watermarking with decent resistance to attacks, without disturbing the visual appeal of the original image. 3. Steganography – Data hiding Steganography is the art and science of data hiding for the purpose of covert communication. It has a rich and interesting history, can take many forms and has many applications. We can speak of modern steganography since 1985 with the advent of computers applied to classical steganography problems. Development following that was slow, but has since taken off. It is in contrast to cryptography, where the existence of the message itself is not disguised, but the content is obscured. Data hiding can be used for clandestine transmissions, closed captioning, indexing, or watermarking.

Steganography has three main characteristics: - detectability - robustness - bit rate

Detectability is the primary concern for clandestine transmissions. Robustness against all kinds of transformations is of concern to the watermarking. Bit rate is the maximum amount of data that can be transmitted without serious degradation of the signal.

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A typical data hiding process will begin with a signal, X, and a message, M with an option of using a key, K for encryption of the message. After inserting M into X, the resultant signal Z is then transmitted. During this transmission it may be subject to different attacks ranging from a noisy channel to intentional attempts to remove the message. Using the received signal, Z’, the receiver attempts to recover the original message M, or at least detect its presence. To extract the message, the original signal X, and a key K, may or may not be required. First, the requirement to use the original signal to recover or detect the message is limited to the watermarking community where authenticity or proof of ownership is required; however, a key is used in almost all applications. Due to commercial interest, most data hiding publications are concerned with watermarking; however, many of the concepts remain the same in all applications of this technology. The general encoding process is exemplified in Fig. 1 below.

Figure 1 Steganographic encoding

Data hiding in images and video is usually accomplished with imperceptible modifications to the digital data. In a general sense, data hiding can be segmented into two major divisions, those in the spatial domain and those in the frequency domain. In the spatial domain, the variation of a few pixels in which the location is only known by the sender and intended receiver is one technique. In the frequency domain, the transform (e.g., FFT, DCT) of the image is taken, and again some or all of the coefficients are altered. 4. Proposed method The Singular Value Decomposition (SVD) is a widely used technique to decompose a matrix into several component matrices, exposing many of the useful and interesting properties of the original matrix. Using the SVD, we can determine the rank of matrix, quantify the sensitivity of a linear system to numerical error, or obtain an optimal lower-rank approximation to the matrix. Every real matrix A can be decomposed into a product of 3 matrices: TUSVA = , where U and V are orthogonal matrices and S is a diagonal matrix. The diagonal entries of S are called the singular values 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. It is important to note that each singular value specifies the luminance of an image

layer while the corresponding pair of singular vectors specifies the geometry of the image layer. Our purpose is to hide a digital image into another digital image, and we base our method on the method proposed by Bergman and Davidson[4]. As such, the same strengths and weaknesses apply to our method as well. However, we will show that it is possible to greatly enhance bit-rate of a hidden image by quantizing it first, and also exploiting the properties of the singular values, not only left-singular vectors. Because the SVD is not unique, we will use the canonical SVD. In this form, the SVs are in non-increasing order, and the left-singular vectors are lexicographically positive[4]. Also, Bergman and Davidson[4] showed that there is a possibility that we cannot compute the canonical SVD. In our tests, we found that, with carefully chosen images, the aforemoentioned possibility is rather low. A digital image can be regarded as a matrix with non-negative scalar elements. Let such an image be designated I. This is the carrier image, and we will assume, for the sake of simplicity, that it is a square, grayscale image. Start by dividing the carrier image I in 16x8 blocks. This block is further divided into two 8x8 subblocks. leftA is the left-hand subblock and rightA is the right-hand subblock. Then, we compute the canonical SVD of each subblock:

Tleftleftleftleft VSUA =

Trightrightrightright VSUA =

Bergman and Davidson proved that in U matrix of SVD of a given 8x8 subblock, we find room for 15 bits. This means that for a 16x8 block, we will have 2x15 bits.

Figure 2 Embeddable bits shown in green

We cannot touch the first two columns, because that would distort the stego image too much. Also, we cannot touch the first row, because we need the column vectors to remain lexicographically positive. We will use the mid-upper triangle to hide 15 bits, the sign of each

value reflecting the value of the bit to be hidden. The lower-right trapezoid needs to be modified in order to keep the U matrix orthogonal.

Now assume that the watermark W is divided in 4x4 blocks. Each block of the watermark is compressed using DXT [9]. DXT is a lossy compression algorithm which quantizes values, based on the spatial coherency. First, of all 16 values of the watermark block, we find the two extrema. Each value in the block can be expressed as a linear combination of these two extrema:

highlowc *)1(* αα −+= , 10 ≤≤ α We now quantize the values of α to only 4 possible values: As a result of this, the 4x4 block can be compressed to only two values and 4x4x2 = 32 bits. We try to embed the 4x4 watermark block in a 16x8 image block. As shown earlier, 15 bits can be embedded into leftU , another 15 bits can be embedded

into rightU . What we need now is room for another 2 bits and the two extrema. Recall that: ],[ ,...21 leftleftleft diagS σσ=

,...],[ 21 rightrightright diagS σσ=

with ...21 ≥≥ σσ Modify S matrices like this:

],_*[ ,...21 leftleftleft extremumhighdiagS σλσ +=

],_*[ ,...2'

1 rightrightright extremumlowdiagS σλσ +=

with 'λλ = and the signs of the two lambdas reflecting

the two remaining bits; λ is the same for all blocks. It is a weighting factor which must be chosen statistically, to minimize the distortion effects on the final image. A typical value for λ is 0.05.

After modifying leftU , rightU , leftS and rightS , the inverse SVD is computed for each 8x8 subblock and the 16x8 block is written back to the image.

Using this method, we can embed 4x4 values of watermark in 16x8 values of cover image, giving a 1:8 ratio.

Detecting the watermark is quite simple as long as we have the original image. Divide the (possibly modified) image I' into 16x8 blocks and perform the canonical SVD on each subblock. leftU and rightU respectively gives us 2x15 bits.

Then, divide the image I into 16x8 blocks and perform the canonical SVD on each subblock. The two

extrema and the missing 2 bits can be obtained by subtracting the major singular values of I and I’. After extracting the watermark information, apply inverse DXT. 5. Experimental results

The algorithm is better visualized using grayscale

images, but we can use color images and hide in only one channel (for example: blue). Also, for the purpose of this paper, we will use a non-encrypted watermark image.

Figure 3 Grayscale carrier

Figure 4 Proposed watermark image (2x)

After embedding, we found that highest pixel difference was 78. This can be reduced by lowering lambda to a smaller value.

The difference between original image and watermarked image is shown below; also, the extracted watermark and histogram of errors. The difference of watermarks will reveal the blocks where we could not obtain the canonical SVD.

The contrast and brightness were greatly enhanced, in order to be able to visualize the difference.

α Bit representation 0 00

1/3 01 2/3 10 1 11

Figure 5 Difference after embedding (hi-contrast)

Figure 6 The extracted watermark (2x)

0

50

100

150

200

250

1 11 21 31 41 51 61 71 81 91

Abs difference

Affected pixels

By looking at the difference of watermarks, we

see that there were three blocks in the carrier image, for which we were unable to obtain the canonical SVD.

Figure 7 Distortion of watermark (2x) (hi-contrast)

Figure 8 Extracted after JPEG compression (left) and

sharpen (right)

The algorithm proved to be semi-fragile. We also tested geometrical transformations, noise and JPEG compression with varying degree of success. 6. Conclusions

We have described a new method of hiding a graphic image into another image using a two level approach: the major characteristics of the watermark are embedded into the singular values of the carrier, while the details of the watermark are embedded into the left singular vectors of the carrier. While the second layer is less resistant to attacks, the first layer is more robust. Also, the DXT transformation allows us to greatly enhance bit-rate. More aspects of the attack-resistant capability are currently under investigation such as: error correcting codes and checksums, custom partitioning of image instead of fixed n x n blocks.

7. References [1] R. Liu and T. Tan, "An SVD-based watermarking scheme

for protecting rightful ownership," IEEE Trans. Multimedia 4(1), pp. 121-128, 2002;

[2] A. Sverdlov, S. Dexter, A. M. Eskicioglu, “Secure DCT-SVD Domain Image Watermarking: Embedding Data in All Frequencies”;

[3] B. C. Mohan, S. Srinivaskumar, B. N. Chatterji, “A Robust Digital Image Watermarking Scheme using Singular Value Decomposition (SVD), Dither Quantization and Edge Detection”;

[4] C. Bergman and J. Davidson, “Unitary Embedding for Data Hiding with the SVD”, Security, Steganography, and Watermarking of Multimedia Contents VII, SPIE Vol. 5681, San Jose, CA, Jan. 2005;

[5] Stanescu, D et al, “Digital Watermarking using Karhunen-Loeve transform”, Applied Computational Intelligence and Informatics, 2007. SACI ‘07. 4th International Symposium on Volume , Issue , Yearly 17 2007-May 18 2007 Page(s):187 – 190;

[6] I. Cox, J. Killian, W. J. Dowling, F. M. Boland, “Watermarking digital images for copyright protection”, IEEE Proceedings on Vision, Image and Signal Processing 143, pp 250-256, August 1996;

[7] Piyu Tsai, Yu-Chen Hu, and Chin-Chen Chang, "A color image watermarking scheme based on color quantization," Signal Processing, vol. 84, pp. 95106, 2004;

[8] Neil F. Johnson and Sushil Jajodia. “Steganography: Seeing the Unseen”, IEEE Computer, February 1998: 26-34;

[9] Pat Brown, EXT texture compression s3tc, http://opengl.org/registry/specs/EXT/texture_compression_s3tc.txt.