2014 IEEE Region 10 Symposium
Maximum Likelihood Watermark Detection In Absolute Domain Using Weibull Model Luan Dong
Qin Yan*, Meng Liu, Yangxu Pan Hohai University, Nanjing, China
[email protected], {liumeng.hhu, pyangxu}@gmail.com
Hohai University, Nanjing, China Xinjiang Agriculture University, Urumqi, China
[email protected]
Recently, Cui [9] has developed a ML detection scheme by modeling the DWT coefficients with the Bessel K form distribution, which presents an alternative model beyond the GGD. Given the specific distribution, the decision statistic and threshold can be derived from Neyman-Pearson criterion based on Bayes’ decision theory. By comparing these two values, whether a watermark is presented in the image or not can be verified.
Abstract—Maximum Likelihood (ML) detection scheme is regarded as one of key components of many blind image watermarking algorithms in various transform domains. In ML detection, a proper Probability Distribution Function (PDF) such as the Generalized Gaussian Distribution (GGD) is usually required to model the statistical characteristics of the transform coefficients of the watermarked images. However in some cases, the GGD is not the most suitable model due to its limitation in modeling the pulse-shape distribution. In this paper, we propose a novel ML detection scheme. By performing ML detection in the absolute domain, we utilize the Weibull distribution, a special case of the Generalized Gamma distribution, to model the absolute transform coefficients. The experimental results demonstrate that the proposed detection scheme outperforms the conventional ones in both DWT and CT domain for natural images. Furthermore it improves the watermark detection rates averagely by 75.03% for Computer Graphic (CG) images compared with the conventional algorithm. Keywords—watermarking; maximum absolute value; Weibull distribution
likelihood detection;
I. INTRODUCTION Watermarking is an information hiding technique [1]. By embedding specific information in the cover data, such as text, audio, images, video or 3D models, watermarking makes the authenticity and integrity verification of these cover data possible in the digital age. By considering the cover data as “noise” and the watermark as “signal” [2], many Maximum Likelihood (ML) detection schemes have been developed for image watermarking in a variety of transform domains [3-9]. These schemes usually need to select a proper distribution to model the transform coefficients. Barni [3] applies the Weibull distribution to model the magnitudes of a set of full-frame DFT coefficients while Juan [4] and Khalil [5] utilize the Generalized Gaussian Distribution (GGD) to model the Discrete Cosine Transform (DCT) and the Discrete Wavelet Transform (DWT) coefficients respectively. Haifeng [6] and Mohammad [7] also used the GGD to model the Contourlet Transform (CT) coefficients, because of the Gaussian and Laplacian distributions [8] are regarded as the special cases of the GGD.
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In recent years the research on ML detection scheme has shifted its focus on taking the noise [7] into account. However little attention has been paid on the distribution modeling of the transform coefficients. During our experiments we note that the performance of conventional ML detection schemes in CG images is worse than that in natural images. This is largely due to the transform coefficients from CG images exhibits a shaper pulse-shape distribution, which is unable to be modeled by the GGD properly. In this paper, we reform the watermark detection procedure by applying the Weibull distribution in the absolute domain to solve this problem. Two sets of experiments are conducted to evaluate the performance of the proposed scheme. The selected baseline schemes are [5] in the DWT domain and [6] in the CT domain. Both schemes model the transform coefficients with the GGD. The experimental results indicate that the proposed scheme outperforms the conventional ones in both DWT and CT domain. The advantage of the proposed scheme in modeling pulse-shape distribution also sheds light in the robust watermarking scheme for CG images. The rest of the paper is organized as follow. Section II describes the embedding rules in ML detection. The details of the proposed detection scheme are elaborated in Section III. Experiments and discussion are presented in Section IV and Section V concludes the whole paper. II. WATERMARK E MBEDDING In the ML detection scheme, the watermark is embedded in the transform domain using the multiplicative rule, which is defined as
yi
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xi 1 Oi wi
(1)
2014 IEEE Region 10 Symposium
where X = [x1 x2 ··· xN]T and Y = [y1 y2 ··· yN]T are N-vectors representing the transform coefficients of an original image and the associated watermarked image respectively, i = 1,2,…,N. A watermark W = [w1 w2 ··· wN]T is embedded into X giving Y. λ i is a positive scalar, representing the embedding strength. It is tuned to provide a tradeoff between robustness and transparency of the watermarking scheme. To get the watermarked image, the original transform coefficients are replaced by the watermarked coefficients and the inverse transform is performed in succession. Since the process of watermark embedding is not the focus of this paper, we will not brief here and more details can be found in [5,6]. III.
WATERMARK DETECTION
Different from the conventional ML detection schemes, the proposed scheme performs watermark detection in the absolute domain. It takes the absolute value of the transform coefficients and then models the absolute coefficients with the Weibull distribution. Fig.1 (b) and (d) show the normalized histogram of the absolute coefficients and the Weibull models. This contrasts the GGD models in the original transform domain as shown in Fig.1 (a) and (c). By taking absolute values on both sides of (1), the following equation can be obtained
yi
xi 1 Oi wi
xi 1 Oi wi
Fig. 1. Comparison of the normalized histograms of the transform coefficients and the corresponding models. (a) The histogram of DWT coefficients of a subband in “Bridge” image and its GGD model; (b) The histogram of absolute DWT coefficients of a subband in “Bridge” image and its Weibull model. (c) The histogram of CT coefficients of a subband in a test CG image and its GGD model; (d) The histogram of the absolute CT coefficients of a subband in a test CG image and its Weibull model.
(2)
The above equation implies that embedding the watermark in the original coefficients followed by taking the absolute value is equivalent to embedding the watermark directly in the absolute value. Furthermore the watermark can be considered as hidden in the absolute value of Y. Therefore detecting the watermark in the absolute domain makes sense. The equation holds as long as (1+λi wi ) ≥ 0, which can be satisfied by choosing appropriate λi and wi . The PDF of the Weibull distribution is defined as
f X x
U§x· ¨ ¸ D ©D ¹
U 1
ª § x ·U º exp « ¨ ¸ » ¬« © D ¹ ¼»
Fig. 2. Standard test images.
(3)
where α and ρ are the positive constants controlling the mean, variance and shape of the distribution. The special cases of the Weibull distribution are the Exponential and Rayleigh distributions, which can be obtained by letting ρ = 1 and ρ = 2 respectively. When ρ < 1, fX(x) approaches infinity while x approaches zero. It can be seen in Fig.1(c)(d) that the Weibull model fits the coefficients histogram better than the GGD model at both the origin and the tail. The GGD model assumes that the distribution of coefficients is symmetrical and the mean is zero, which is not always the case. The Weibull model discards these assumptions, and is capable of capturing the characteristic when the distribution has a pulse-shape as shown in Fig.1 (d). The transform coefficients is mapped into absolute domain before modeled by the Weibull distribution. Given the distribution, the decision statistic and threshold can be derived
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from Neyman-Pearson criterion based on Bayes’ decision theory. The details can be found in [3]. IV. EXPERIMENTS AND DISCUSSION In this section two sets of experiments are conducted to evaluate the proposed scheme against the conventional ones in both DWT and CT domain for both natural and CG images. A. Comparison in the DWT and CT domain To demonstrate the advantage of the proposed detection scheme in different transform domains, we compared the detection performance of our scheme with the ML detection scheme [5] in the DWT domain and [6] in the CT domain. Both baseline schemes model the transform coefficients with the GGD. The detection schemes are tested on a set of 512 × 512 standard images selected from the USC-SIPI database [10], which are shown in Fig.2. The Weibull and the GGD
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distribution parameters are estimated from the transform coefficients of the watermarked image using ML estimation respectively. The decision statistic and threshold are calculated given the distribution parameters and probability of false alarm PFA. In the experiments PFA equals to 10-9. By choosing a PN sequence as the watermark in which the entries are in {-1, 1} and letting the embedding strength λ = 0.2, the requirement of (2) can be satisfied. Each time a watermark is selected from 100 randomly generated watermarks and embedded into a test image. A detection is said to be correct only if the decision threshold is exceeded for the embedded watermark but not for any other watermarks. The correct detection rate is obtained after 10000 times repetition of above process. In the DWT domain, the experimental setup is similar to [5]. Each image is transformed by the DWT using the db4 filter to obtain a three level pyramid decomposition. The watermark is then embedded in the highest energy sub-band of the 3rd level high resolution sub-bands. The results are shown in Fig.3. It can be seen that our detection scheme outperforms the scheme in [5] in almost every circumstances. The correct detection rate improves by 21.85% on average. In particular, compared with the baseline scheme, the watermark detection rates under jpeg compression and Gaussian blurring of the proposed scheme are very promising with 49.15% and 28.34% up on average respectively. In the CT domain, as described in [6], the test images are decomposed by CT using the “9-7” pyramid filter and the “PKVA” directional filter to obtain a two level decomposition. The watermark is then embedded in the most energetic subband of the 2nd level directional sub-bands.
Fig.4 shows the experimental results. Our scheme performs better almost in every standard watermarking attacks except some cases for Gaussian and Salt & Pepper noising attack. Overall the correct detection rate of our scheme is 47.14% higher than that of [6]. The above results imply that it may be more appropriate using the Weibull distribution than the GGD to model the DWT or CT coefficients in the absolute domain, which leads to a great potentiality of the proposed scheme to be a substitute for the conventional ones. B. Watermark detection in CG images The advantage of the proposed detection scheme lies in the ability to model the transform coefficients with a pulse-shape distribution, which is very suitable for the CT coefficients of computer graphic (CG) images as shown in Fig.1(c). Actually due to the sparsity of the CT coefficients, any image that has less details like the CG images tends to have a pulse-shape distribution. Due to no standard test dataset for CG image, 24 CG images are randomly collected from the internet as shown in Fig.5. The experimental setup is similar to the CT domain experiment in the first part of section IV. The proposed detection scheme is tested at different levels of probability of false alarm subject to various kinds of attacks. The average correct detection rates are depicted in Fig.6. As a comparison, the average correct detection rates of [6] are also plotted, marked with cross in Fig.6.
Fig. 3. The comparison of correct detection rate of the proposed detection scheme and [5] under various attacks. (a) JPEG compression with 30% of quality. (b) Scaling 50%. (c) Gaussian noising with σ = 0.01. (d) Gaussian blurring with σ = 1. (e) Histogram equalization. (f) Salt & Pepper with noise density 0.05. (g) Median filtering with window size 3 × 3. Image abbrievation: B: Bridge, M: Man, P: Pepper, Ba: Baboon, L: Lena, F: F16.
Fig. 4. The comparison of correct detection rate of the proposed detection scheme and [6] under various attacks. (a) JPEG compression with 30% of quality. (b) Scaling 50%. (c) Gaussian noising with σ = 0.01. (d) Gaussian blurring with σ = 1. (e) Histogram equalization. (f) Salt & Pepper with noise density 0.05. (g) Median filtering with window size 3 × 3. Image abbrievation: B: Bridge, M: Man, P: Pepper, Ba: Baboon, L: Lena, F: F16.
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Compared with the baseline scheme, the correct detection rate of the proposed scheme increases 35.08% in the attack of histogram equalization, 131.33% in JPEG compression, 39.07% in Gaussian noising and 94.63% in scaling respectively. The average rate of increase is 75.03%. Note that these rates of increase are calculated by the top detection rates of both schemes.
V.
CONCLUSION
In this paper we propose an improved ML detection scheme for image watermark detection by modeling absolute values of the transform coefficients with the Weibull distribution. The Weibull distribution has been utilized in other ML detection scheme [3]. This paper expands its territory to the GGD ruled region by simply taking the absolute value of the transform coefficients. The experimental results show that the proposed scheme improves the watermark detection rate by 21.85% in the DWT domain and 47.14% in the CT domain respectively. The proposed scheme discards the prerequisite of the GGD and better fits coefficients distribution at both the origin and the tail. Thus it exhibits advantages compared with the conventional schemes and is capable of overcome the impact of the pulseshape distribution. Although CG can be stored in vector format, it still appears in pixel format. The proposed scheme provides an approach to make the ML detection on CG-like images effectively. Future work will be focus on the improvement of the proposed detection scheme by modeling the watermark attacks. ACKNOWLEDGMENT We would like to thank for the sponsorship of National Natural Science Foundation of China (No.61170297), startup funding of Hohai University Human Resource (20080604), Qing Lan Project of Jiangsu Province of China. REFERENCES [1]
Fig. 5. The test CG images. [2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
M. Barni, F. Bartolini, A. D. Rosa, A. Piva, "A new decoder for the optimum recovery of nonadditive watermarks," IEEE Transactions on Image Processing, vol. 10, no. 5, pp.755-766, May 2001. J. R. Hernandez, M. Amado, F. Perez-Gonzalez, "DCT-domain watermarking techniques for still images: detector performance analysis and a new structure," IEEE Transactions on Image Processing, vol. 9, no. 1, pp.55-68, Jan. 2000. K. Zebbiche, F. Khelifi, A. Bouridane, "Maximum-likelihood watermarking detection on fingerprint images," in Proc. of Bio-inspired, Learning, and Intelligent Systems for Security, 2007, pp.15-18. H. F. Li, W. W. Song, S. Wang, "A novel blind watermarking algorithm in contourlet domain," in Proc. of 18th International Conference on Pattern Recognition, 2006, pp.639-642. M. A. Akhaee, S. M. E. Sahraeian, F. Marvasti, "Contourlet-based image watermarking using optimum detector in a noisy environment," IEEE Transactions on Image Processing, vol. 19, no. 4, pp.967-980, April 2010. T. M. Ng, H .K. Garg, "Maximum-likelihood detection in DWT domain image watermarking using laplacian modeling," IEEE Signal Processing Letters, vol. 12, no. 4, pp.285-288, April 2005.
L. H. Cui, L N Zhao, W. G Li. "A novel optimal decoding algorithm of multibit multiplicative watermarks," in Proc. of Wavelet Analysis and Pattern Recognition Int. Conf., 2011, pp:73-76. [10] SIPI Image Database, University of Southern California, [online] 2013, http://sipi.usc.edu/database/ (Accessed: 24 October 2013)
Fig. 6. The average correct detection rate of the proposed detection scheme and the baseline scheme [6] at various PFA subject to four types of attacks. (a) Histogram equalization. (b) JPEG compression with 30% of quality. (c) Gaussian noising with σ = 0.01. (d) Scaling 50%.
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A. A. J. Altaay, S. B. Sahib, M. Zamani, "An Introduction to Image Steganography Techniques," in Proc. of Advanced Computer Science Applications and Technologies Int. Conf., 2012, pp.122-129. J. Miller, J. B. Thomas, "Detectors for discrete-time signals in nonGaussian noise," IEEE Transactions on Information Theory, vol. 18, no. 2, pp.241-250, Mar. 1972.
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