Wavelet-based hybrid natural image modeling using generalized Gaussian and α-stable distributions☆
Introduction
Statistical analysis of natural images is of both theoretical and practical importance to image/video processing tasks which assume priori knowledge or probability models over images. Since this area of research appeared in the mid-1950s, related studies have drawn increasing attention and enthusiasm. Natural image statistics is an important premise for many image processing applications such as denoising, compression, content classification, texture analysis, quality assessment, etc. In general, many of these applications need to learn some image properties even without access of specific data.
State of the art literatures have revealed that there are two striking properties in natural image statistics: (1) scale invariance, which is manifested by the empirical evidence that the power spectrum of natural scenes obeys the power-law pattern in the spatial domain [1]; (2) non-Gaussian performance of image statistics, represented by leptokurtic and heavy-tailed distributions both in spatial and transform domains. Statistics of natural images originates from the investigation of pure natural scenes such as forest environments. In [2], Ruderman et al. studied the statistics of an ensemble of images taken in the woods and reported that distributions of local quantities (e.g., contrast, contrast gradients, local variance) exhibited scale invariance and exponential tails which were very non-Gaussian. In his further research [3], Ruderman indicated that natural images were collages of regions corresponding to statistically independent fractions and provided evidences for their power-law distribution of sizes within images. In [4], Luo et al. proposed a Bayesian estimation framework to provide a unified, simultaneous modeling of both spatial and intensity distribution of the wavelet transform coefficients, where the spatially localized characteristics of a given sub-band are modeled by Gibbs random fields. Along with the insight of images’ stochastic structures, Markov Random Field (MRF) has been used to derive more efficacious image models. Roth and Black [5] extended traditional MRF model and established a Fields of Experts (FOE) model for learning image priors. In [6], Yousefi et al. introduced bilateral Markov mesh random field to overcome shortcomings of conventional MRF method consisting of computational intractability in image probability modeling and the resulted non-symmetrical image models. Besides, Zhai et al. [7] proposed a Minimum Description Length (MDL) principle based image context model, which was highly competitive in lossless predictive coding and denoising. Very recently, Zoran and Weiss [8] showed the superiorities of Gaussian Mixture Model (GMM) in learning image statistics, which fused components including covariance structure, contrast variation and intricate textures. These MRF or learning based algorithms have shown their impressive performances by modeling image statistics, whereas accompanied with the expense of higher computation complexity.
Image decomposition techniques transform the image data into more uncorrelated coefficients, which facilitate the statistical analysis. In [9], [10], coefficients of discrete cosine transform (DCT) bases have been modeled by Laplacian distribution and generalized Gaussian density (GGD) respectively. As for a specific application, Zoran and Weiss [11] demonstrated the scale invariance for the kurtosis of marginal filter (DCT basis) response distributions and suggested that the noise present in the image would alter the kurtosis through scales, according to which they proposed a novel noise variance estimation algorithm. Along with the exploration of human visual system (HVS) and the development of multi-resolution analysis, wavelet domain methods have been investigated in order to capture image features in different scales. In [12], Mallat pointed out that the coefficients of multi-scale, orthonormal wavelet sub-bands of images could be described by a GGD model. Subsequently, Zhu and Mumford [13] extended the scale invariance of images in wavelet domain, and learned the priors with a smart Gibbs-diffusion. In [14], Simoncelli described a natural image model based on multi-scale wavelet decomposition. Both the marginal and pairwise joint histograms of wavelet coefficients at adjacent spatial locations, orientations, and spatial scales were examined for the statistical modeling. Besides, sparse coding of natural images was also highlighted because of its consistency with HVS. In [15], Liu et al. presented multiview Hessian discriminative sparse coding (mHDSC) which seamlessly integrates Hessian regularization with discriminative sparse coding for multiview learning problems. Recently, relevant studies showed that the heavy-tailed behavior of image wavelet coefficients was better modeled by α-stable distributions, which gave birth to pertinent statistical image processing algorithms based on wavelet decomposition [16], [17], [18], [19].
In this paper, we analyze the non-Gaussian properties of natural images in wavelet domain, and indicate the limitations of modeling wavelet coefficients using GGD or α-stable density. Aimed for a more accurate description of the transform domain image statistics, we propose a hybrid model that fuses GGD and α-stable densities by linear weighted averaging, while determine the weights by an objective function guided convex optimization. We evaluate the hybrid model on three databases, and the Kullback–Leibler Divergence (KLD) is employed as the similarity index between the estimated model and true probability distribution. Furthermore, we derive the close-formed KLD of the hybrid model for image comparison application using model parameters. Comprehensive experiments show the reliability and superiority of our model.
The rest of the paper is laid out as follows: In Section 2, we start with the perspective on non-Gaussian behavior of natural images’ wavelet coefficients in SubSection 2.1, while performances of GGD and α-stable modeling are analyzed in SubSection 2.2 that followed. Given the demonstrated limitations of single modeling, we introduce the hybrid model of natural images in Section 3, with the mathematical description and detailed model realization procedures elaborated in SubSections 3.1 Mathematical description, 3.2 Model realization. Experiments and evaluations are presented in Section 4. Moreover, application of image comparison using the model are illustrated in Section 5. Finally, Section 6 gives concluding remarks.
Section snippets
Statistical modeling of natural images in wavelet domain
As the receiving terminal of images, HVS is characterized by its comprehensiveness and complexity. Basic mechanism of HVS is the sparse representation of natural scenes. Since the multi-scale and multi-resolution analysis in wavelet theory are very effective for representing image contents in a HVS-like way, as well as the favorable statistical characteristics of wavelet coefficients, modeling natural image statistics in wavelet domain has become an attractive research scope. Fig. 1 illustrates
Mathematical description
The probability density function (PDF) of GGD is given as follows:where is the Gamma Function, i.e. . Here ν models the width of the PDF peak (standard deviation), and σ is inversely proportional to the decreasing rate of the peak. In general, ν is referred to as the scale parameter while σ is called the shape parameter. So that the GGD PDF are characterized by these two factors.
α-stable interference modeling is widely used in economics,
Experiments and performance evaluation
According to the algorithm framework shown in Fig. 5, we model the CD wavelet coefficients with GGD/α-stable to achieve preliminary estimation of parameter set , then we optimize each of the parameters by fixing the other five and minimizing Eq. (7) using MATLAB’s fminsearch function. The starting value of is set to 0.5. After that, is estimated using the optimized parameters, and .
Image comparison using the hybrid model
The parametric density of hybrid model makes it attractively for practical image processing tasks. One such application could be quantifying differences between two given images, i.e. the distortion measurement for a test image relative to a reference one. All the natural images are regularized in some kinds of statistics, regardless of content, unless distorted during acquisition, processing, or reproduction. It is the presence of distortion that deteriorate the natural statistical properties,
Conclusion and future work
We proposed a hybrid model to describe statistics of image wavelet coefficients. We investigated the non-Gaussian configuration of the coefficients, then analyzed the modeling performances of prevalent GGD and α-stable densities. We found that the GGD density was preferable in fitting leptokurtosis yet inferior for heavy tails, while the α-stable was just opposite. Accordingly, We formulated the hybrid model integrating the two densities, and devised a realization framework based on an
Acknowledgments
This work was supported by National Nature Science Foundation of China (NSFC) (61025005, 60932006, 61001145, 61102098, 61221001), Science and Technology Commission of Shanghai Municipality (STCSM) (12DZ2272600), 111 Project (B07022).
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This paper has been recommended for acceptance by Prof. Yehoshua Zeevi.