ADDITIVE WATERMARK DETECTORS BASED ON A NEW HIERARCHICAL SPATIALLY ADAPTIVE IMAGE MODEL Antonis Mairgiotis1, Nikolaos Galatsanos1, and Yongi Yang2 1
2
Dept. of Computer Science University of Ioannina Ioannina, Greece 45110 {mairgiot,
[email protected]} ABSTRACT In this paper we propose a new family of watermark detectors for additive watermarks in digital images. These detectors are based on a recently proposed two-level, hierarchical image model, which was found to be beneficial for image recovery problems. The top level of this model is defined to exploit the spatially-varying local statistics of the image, while the bottom level is used to characterize the image variations along two principal directions. Based on this model we derive a class of detectors for the additive watermark detection problem, including the generalized likelihood ratio test (GLRT) and Rao detectors. Index Terms— watermarking, statistical methods, source modeling 1. INTRODUCTION Additive watermark detection can be formulated as a hypothesis testing problem, where one needs to determine the presence or absence of a known watermark in an image. Within such a formulation, the watermark is treated as the known signal and the image is treated as the corrupting noise [1]. To derive a test statistic for this problem, such as the likelihood ratio test detector, a statistical model for the image has to be defined. In this paper we propose the use of a hierarchical, locally adaptive image model for watermark detection. The top level of this model is defined to exploit the spatiallyvarying local statistics of the image. This model can be viewed as a generalization of the concept of line process used in the context of compound Markov random fields [6]. The difference is that a continuous model, rather than binary edges, is used for characterizing the local discontinuities in the image. Using this image model we will derive detectors for additive watermarking, which include the generalized likelihood ratio test (GLRT) and Rao detectors. We note that the hierarchical image model used in this study was recently developed for image restoration in [5]. It is interesting to note that the development of image models
1-4244-1437-7/07/$20.00 ©2007 IEEE
Dept. of Electrical and Computer Engineering Illinois Institute of Technology Chicago, IL 60616, USA {
[email protected]} has also been very important for the classical image denoising and restoration problems, in which a statistical image model is essential for various estimation methodologies, e.g., maximum a posteriori estimation. The rest of this paper is organized as follows. In Section 2 we introduce the hierarchical image model and formulate its use for additive watermark detection. In Section 3 the GLRT detector is derived and methods to estimate the necessary parameters of the model are given. Rao test based detector is derived in Section 4. In Section 5 numerical experiments are given to demonstrate the proposed detectors. 2. IMAGE MODELS AND ADDITIVE WATERMARKS The image model we propose to use in this paper is based on the first order differences of the image along the two principal directions. Specifically, consider an image f , whose pixels are denoted by f i, j . At pixel location
i, j ,
we define the image directional differences (IDD)
along the horizontal and vertical directions, respectively, as follows: H1 (i, j ) f (i, j ) f (i, j 1), (1) H 2 (i, j ) f (i, j ) f (i 1, j ) We assume that these IDDs obey a Gaussian probability density function (pdf), given by
1
H k (i, j ) ~ N 0, ak (i, j ) ,
(2)
1
where ak (i, j ) is the variance parameter. Since for each spatial location different variances are assumed this model is very flexible to model the fluctuations of the IDDs in both smooth and edge image areas. For notational convenience, in the rest of the paper we will denote the IDDs using a single index as
İk
V - 461
>H k (1), H k (2),..., H k ( N )@
T
, k=1,2, where N is the total
ICIP 2007
number of image pixels. In addition, let İ
T
ª¬İ1T , İ 2T º¼ , a
vector consisting of IDDs in both directions. Assuming independence of the IDDs, we can write the joint pdf 2 N ª 1/ 2 2 ·º § 1 (3) p İ; a v « ak (i ) exp ¨ ak (i ) H k (i ) ¸ » © 2 ¹¼ k 1 i 1 ¬ ª¬a1T ,a 2T º¼ , a k > ak (1), ak (2),..., ak ( N ) @ , which denotes the corresponding variance parameters. The pdf in (3) allows the local variance to vary from pixel to pixel. This is desirable for modeling the spatially non-stationary properties of the image (e.g., edges). Unfortunately, it includes as many variance parameters ak (i ) as the number of image pixels. To avoid the problem of over-fitting, it is necessary to impose additional constraint on the model to limit its degrees of freedom. For this purpose we model ak (i ) as random variables, and define a hyper-prior on them. In this work we use a Gamma pdf for the hyper-prior, which is of the form T
where a
p(ak (i ); m, l ) v ak
l 2 2
T
(i ) exp ^ m(l 2)ak (i )` , k
1, 2 ,
E
> ak i @
l 2m l 2 , Var ¬ª ak i ¼º
l 2m l 2 2
2 1
Assuming that ak (i ) are independent and identically distributed, then we have 2 N § l 2 · p(a ; m, l ) C ¨ ak 2 (i ) exp ^m(l 2)ak (i )` ¸ (5) k 1 i 1 © ¹ where C is a normalization constant. In the additive watermark detection problem one has to decide between the following two hypotheses H0 :y f (6) H1 : y f J w where y and f are the observed and the original images, respectively, and w is the watermark signal and J is its strength. Applying the directional difference operators Q k , k 1, 2 , to the observed image in (6), we obtain H 0 : y 'k H1 : y
' k
İk İ k J w ck
İ k w cck
2· 2 N ½ § 1 2 N C ® ak (i )1/ 2 ¾ exp ¨ ¦¦ ak i ykc i ¸ © 2k 1i1 ¹ ¯k 1 i 1 ¿ p(y c; a , H1 )
(7)
(8)
2· 2 N ½ §1 2 N C ® ak (i )1/ 2 ¾ exp ¨ ¦¦ ak i ykc i wkcc i ¸ ©2 k 1 i 1 ¹ ¯k 1 i 1 ¿ T
where y c
ª¬ y1cT , y c2T º¼ . In what follows these pdfs will be used to derive the GLRT and Rao test detectors.
3. GENERALIZED LIKELIHOOD RATIO DETECTOR The GLRT is given by H
1 ° p(y c; aˆ / H , H1 ) ½° > 1 log ® ¾ 0, ˆ °¯ p(y c; a / H 0 , H 0 ) °¿ H
