Improving SVDbased image watermarking via blockbyblock optimization on singular values
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Abstract
The purpose of this paper is to improve the robustness of traditional image watermarking based on singular value decomposition (SVD) by using optimizationbased quantization on multiple singular values in the wavelet domain. In this work, we divide the middlefrequency parts of discretetime wavelet transform (DWT) into several square blocks and then use multiple singular value quantizations to embed a watermark bit. To minimize the difference between original and watermarked singular values, an optimizedquality formula is proposed. First, the peak signaltonoise ratio (PSNR) is defined as a performance index in a matrix form. Then, an optimizedquality functional that relates the performance index to the quantization technique is obtained. Finally, the Lagrange Principle is utilized to obtain the optimizedquality formula and then the formula is applied to watermarking. Experimental results show that the watermarked image can keep a high PSNR and achieve better biterror rate (BER) even when the number of coefficients for embedding a watermark bit increases.
Keywords
Singular Value Decomposition Watermark Image Watermark Scheme Host Image JPEG Compression1 Introduction
With the rapid development of activity on the internet, much digital information is widely spread. Digital watermarking was developed to hide digital information and protect the copyright of multimedia signals, like audio, images, etc. Due to the fact that discretetime wavelet transform (DWT) provides a useful platform, numerous DWTbased algorithms for digital watermarking have been proposed in recent years.
Watermarking in the spatial domain [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] is usually more vulnerable than watermarking in the frequency domain [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29] with the same embedding capacity due to the fact that spatialdomain methods are generally fragile to imageprocessing operations and other attacks [23, 24, 25]. The spatialdomain singular value decomposition (SVD) for image watermarking was first introduced by Liu et al. [8]. In this paper, the authors used a spreadspectrum technique to embed a watermark by modifying the singular values of the host image in the spatial domain. Some authors embedded watermark to U and V components to increase embedding capacity [9, 10] while Ghazy et al. [11] presented a blockbyblock SVDbased imagewatermarking scheme to increase embedding capacity. However, the robustness of SVDbased image watermarking in the spatial domain is low. In recent years, many imagewatermarking techniques combine DWT and SVD to achieve better transparency and robustness [17, 18, 24, 25]. Bao et al. [17] proposed a novel, yet simple, imageadaptive watermarking scheme for image authentication by applying a simple quantizationindexmodulation process on each single singular value of the blocks in the wavelet domain. Their watermarking scheme is blind and is robust against JPEG compression but extremely sensitive to malicious manipulation such as filtering and random noising. Ganic et al. [18] applied SVD to all details, approximating part of the DWT and watermark image to increase embedding capacity. Gaurav and Balasubramanian [24] embedded a watermark into the reference image by modifying the singular value of the reference image using the singular values of the watermark. The robustness is slightly enhanced. However, the computation is significantly increased. Lai and Tsai [25] reduced the computation in [24] by directly embedding the watermark into the singular values in the wavelet domain.
In this work, we first divide the DWT middlefrequency parts LH3 and HL3 into several square blocks to have high embedding capacity. Unlike the traditional spreadspectrum technique on single singular values [24, 25], we use multiple singular value quantizations to embed a watermark bit. It does not only keep a high embedding capacity but also achieves strong robustness against median filtering. On the other hand, an optimizedquality formula is proposed by minimizing the difference between original and watermarked singular values. First, the peak signaltonoise ratio (PSNR) is defined as a performance index in matrix form. Then, an optimizedquality functional that relates the performance index to the quantization technique is obtained. Finally, the Lagrange Principle is utilized to obtain the optimizedquality formula; then, the formula is applied to watermarking. Experimental results show that the watermarked image can keep a high PSNR and achieve a better biterror rate (BER) even when the number of coefficients for embedding a watermark bit increases. In particular, the robustness against median filtering is significantly improved.
This paper is organized as follows. In Section II, we review some mathematical preliminaries. Section III introduces the proposed watermark embedding and extraction. In Section IV, we rewrite PSNR as a performance index. An optimizedquality equation that relates the performance index to the quantization constraint is proposed, and the Lagrange Principle is used to solve the optimizedquality problem. The solution is utilized to embed the watermark, and we discover a very good result; the watermark is extracted without the original image. In Section V, we present some experiments to test the performance of the proposed scheme. Finally, conclusions are drawn in Section VI.
2 Preliminaries
In this section, some related steps for the proposed imagewatermarking scheme are reviewed.
2.1 Discretetime wavelet transform (DWT)
2.2 Singular value decomposition (SVD)
2.3 Optimization solver
To find the extreme of the matrix function, some optimization methods are summarized in [29, 30, 31]. The operations of the matrix function are first shown as follows.
In order to apply the Lagrange Principle, we have to introduce the gradient of a matrix function \( f\left(\overline{\mathbf{\mathsf{C}}}\right) \) as follows.
In order to solve (9), we apply the Lagrange Principle as follows.
3 Proposed optimizationbased DWTSVD watermarking scheme
The proposed watermarking scheme is introduced in this section. The watermark is extracted without the original image.
3.1 Watermark embedding
 (1)
Use threelevel DWT to decompose the original image A into four subbands (i.e., LL3, LH3, HL3, and HH3).
 (2)
Divide LH3 and HL3 into nonoverlapping blocks A ^{ k }, k = 1, 2, ⋅ ⋅⋅, N.
 (3)
Apply SVD to each block, i.e.,
 (4)
Watermark B = {β _{ j }} randomly generated using a binary PN sequence is embedded by modifying singular values λ _{ i } ^{ k }, i = 1, ⋅ ⋅⋅, r of the matrix A ^{ k } as follows: Let

If μ _{ j } ^{ k } mode 2 = β _{ j }, the singular values are modified to

If μ _{ j } ^{ k } mode 2 ≠ β _{ j } and \( {\mu_j}^k\left\lfloor \raisebox{1ex}{${\displaystyle {\sum}_{i=\mathsf{1}}^r}{\lambda}_i^k$}\!\left/ \!\raisebox{1ex}{$q$}\right.\right\rfloor =\mathsf{0} \), the singular values are modified to

If μ _{ j } ^{ k } mode 2 ≠ β _{ i } and \( {\mu}_j^k\left\lfloor \raisebox{1ex}{${\displaystyle {\sum}_{i=\mathsf{1}}^r}{\lambda}_i^k$}\!\left/ \!\raisebox{1ex}{$q$}\right.\right\rfloor \ne \mathsf{0} \), the singular values are modified to
3.2 Watermark extraction
4 Optimization of PSNR on singular values
Generally, the quality of a watermarked image is evaluated by the peak signaltonoise ratio (PSNR). Since a tradeoff exists between image quality measured by PSNR and robustness measured by BER, a scalar parameter ξ is applied to connect the PSNR and the quantization equation to optimize the tradeoff in this section. The details are in the following:
5 Experimental results
Embedding capacity under fixed PSNR
Method  Image genre  Parameters  Embedding capacity (bits) 

Reference [24]  Lena  α = 28  256 
Jet  α = 25  256  
Peppers  α = 28  256  
Cameraman  α = 27  256  
Reference [25]  Lena  α = 28  256 
Jet  α = 25  256  
Peppers  α = 28  256  
Cameraman  α = 27  256  
Reference [27]  Lena  α = 28  256 
Jet  α = 25  256  
Peppers  α = 28  256  
Cameraman  α = 27  256  
The proposed method  Lena  r = 2, q = 27  512 
r = 4, q = 55  256  
r = 8, q = 135  64  
Jet  r = 2, q = 26  512  
r = 4, q = 44  256  
r = 8, q = 129  64  
Peppers  r = 2, q = 28  512  
r = 4, q = 51  256  
r = 8, q = 131  64  
Cameraman  r = 2, q = 28  512  
r = 4, q = 50  256  
r = 8, q = 140  64 
 (1)JPEG compression and JPEG2000 compression are the most popular compression methods. They are widely used to reduce the sizes of images. Usually, an image is compressed before it is transmitted over the Internet. Table 2 and Table 3 concern the compression of the 40 watermarked images by JPEG compression and JPEG2000 compression with different quality factors. The average BER of the proposed method is much lower than the other methods in cases k = r = 4 and k = r = 8. At the same time, the average BER of the proposed method decreases as the parameter k increases.Table 2
JPEG compression
 (2)Table 4 shows the robustness against Gaussian noise with different means and variances. By testing the 40 watermarked images, the average BER of the proposed method is lower than other methods. As the parameter k increases, the average BER of the proposed method decreases.
 (3)Table 5 shows the robustness against median filtering with different radii in pixels. By testing the 40 watermarked images, the average BER of the proposed method is still much lower than other methods in cases k = r = 2 and k = r = 4. As the parameter k increases, the average BER of the proposed method also increases a little.
 (4)Table 6 shows the performance against rotation attack with different degrees. By testing the 40 watermarked images, the average BER of the proposed method is slightly higher than other methods.
From the above discussion, the proposed method performs better than the method in [25] except for the rotation attack. And as the parameter k increases, the BER decreases as well except for the JPEG 2000 compression. To conclude, the proposed method is acceptable for its better performance except for the rotation attack.
6 Conclusions
This study improved the robustness of traditional SVDbased image watermarking by using optimizationbased quantization on multiple singular values in the wavelet domain. Experimental results show that the watermarked image can keep a high PSNR and achieve a better BER even when the number of coefficients for embedding a watermark bit increases. In particular, the robustness against JPEG compression, Gaussian noise, and median filtering is significantly improved. The future work is the consideration of improving robustness against rotation.
Notes
Acknowledgments
This work is partially supported under the grand MOST 1032115029003.
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