Research On Contourlets-Based Statistical Modeling For Image And Its Applications

Geometrical structures such as edges and textures are key features in natural images. The traditional wavelet transform only represents point-singularities efficiently, but is less efficient for line-singularities and curve-singularities that always exist in images. Moreover, the separable two-dimensional wavelet cannot capture contour information of images accurately due to the very limited directionality. These limitations of wavelet have led to the development of multiscale geometric analysis (MGA) theory, which includes a serial of MGA tools. The goal of MGA theory is to find the optimal representation for multidimensional signals to capture and process the intrinsic geometrical structures such as contours, edges and textures of images efficiently. Among these tools, contourlet transform and the nonsubsampled contourlet transform (NSCT) distinguish themselves with their excellent efficiencies and flexible structures. In this thesis, both contourlet transform and NSCT are referred as contourlets transforms. The contourlets transforms are directly defined in discrete-domain and use rectangular-shapes for their basis elements to follow the contours, thus forming "true" two-dimensional sparse representations. Contourlets transforms not only possess the features of multiscale and time-frequency localization, but also offer a high degree of directionality and anisotropy, which have proven to be very promising in various image processing applications.The contourlets coefficients of images show strong dependencies across scales, directions and locations (or space). However, the current statistical models of contourlets coefficients are mostly simple extension of those of wavelet coefficients, disregarding the difference in multi-directionality and anisotropy. To take advantage of captured directional information of contourlets transforms, exploring how to integrate the geometrical information into the multiscale statistical model to statistically characterize contourlets coefficients is greatly in need. Therefore, the thesis focuses on the contourlets-based statistical modeling for images. By investigating the statistical correlation of contourlets coefficients, the thesis studies the directional multiscale statistical modeling based on contourlets transforms and develops some new denosing and segmentation method based on the proposed statistical model. In addition, the proposed statistical model can be further extended to image fusion, image reconstruction and other image processing areas, which provides important theoretical significance and wide application prospects.The main contributions of the thesis are as follows:1. The limitation of wavelet transform is first discussed from the views of image sparse representation and receptive characteristics of human visual system. Then the capability of contourlets transforms in capturing the directional information of the natural images is analyzed. Moreover, the current progresses of image processing based on the contourlets transforms are summarized.2. The contourlets representations for images are studied.After reviewing the contourlet transform theory, including its basic principle, filter bank structure and basic characteristics, the thesis discusses the sparsity of contourlet transform by analyzing the shapes of basis functions and their nonlinear approximation rate, and sets up the directional multiresolution frame by functional analysis. To discuss the shift-invariant property, the NSCT transform especially the design of the nonsubsampled filter is studied.3. The statistics of the contourlets coefficients is discussed in detail and described by a newly local contextual hidden Markov model (LCHMM).The thesis adopts the generalized Gaussian distributed (GGD) model and Gaussian mixture model (GMM) respectively to approximate the highly non-Gaussian marginal distributions of contourlets subbands. By joint distribution histogram, the strong interlocation, interscale and interdirection dependencies of contourlets coefficients are analyzed qualitatively. However, the current statistical models of contourlets coefficients do not consider all three dependencies. Moreover, those methods cannot fit the nonstationary properties of images due to the lack of spatial adaptability. To overcome these problems, a new LCHMM framework that statistically characterizes contourlets coefficients is proposed. In the LCHMM model, the Gaussian mixture field (GMF) where each coefficient follows a local Gaussian mixture distribution determined by their neighborhoods is introduced to approximate the non-Gaussian density. Compared to the GMM, the GMF is more suitable for the non-stationary properties of images. To integrate all interscale, interdirection and interlocation dependencies of these corresponding coefficients, a context variable is associated with each coefficient. Based on the GMF, the HMM model combined by context, which called LCHMM model, may describe the complex dependencies between contourlets coefficients effectively and completely. The LCHMM model, which reveals the properties of persistence across scales, multi-directional selectivity within scales and energy clustering in the subbands of contourlets transforms to the full, provides a new theoretical idea for the MGA-domain statistical modeling. The thesis gives the mathematical description of LCHMM model in detail and develops the fusion method of three dependencies. To obtain a LCHMM from an image, typically a training procedure based on the iterative expectation maximization algorithm is utilized. The robustness of training process is achieved by providing a good initial parameter setting.4. A spatially adaptive BayesShrink thresholding algorithm with elliptic directional windows'estimation in the NSCT domain is proposed for image denoising.The anisotropic energy clusters in NSCT subbands reveal that the distribution of correlated coefficients in the neighborhood is not isotropic, but exhibits anisotropic and specific directional features. Therefore, instead of the traditional square windows, the elliptic directional windows, which match the direction of the energy clusters' distribution in each subband, are used to select the neighboring coefficients with strong dependency for signal estimation. Reasonably, based on Bayesian estiamtion, the spatially adaptive BayesShrink thresholding algorithm with elliptic directional windows' estimation in the NSCT domain is proposed for image denoising. The proposed threshold that explores the anisotropic and directional features of NSCT deeply obtains the adaptability in anisotropic neighborhood as well as in scale and direction. The thesis discusses the definition of elliptic directional windows, the signal estimation method based on elliptic directional windows and noise variance estimation in detail. Experimental results show that, compared with some current outstanding denoising algorithms based on the contourelts transforms, the proposed algorithm gains the better denoising performance in both the value of peak signal-to-noise ratio (PSNR) and the visual quality.5. An image denoising algorithm based on the proposed LCHMM model in the NSCT domain is developed, which is particularly used to remove the noise with high level.The NSCT coefficients of an image display strong dependencies not only across scales, but also across directions and space. By contrast, the NSCT transform of zero-mean white Gaussian noise is still zero-mean white Gaussian noise of the same power. Thus, the statistical correlation difference between NSCT coefficients of signal and those of noise can be used to remove noise, especially for the noise with high level. To take full use of three correlation information, the weight combination for integrating three dependencies of NSCT coefficients is studied by using mutual information as a measuring tool. Then the mapping function from the correlation information to context variable is determined for the denoising application. Based on the prior information captured by LCHMM model, the denoised image is obtained by the Bayesian estimation. The thesis presents the fusion method of correlation information, the context construction method and the denoising steps based on the LCHMM model in detail. Simulation experiments show that the proposed algorithm outperforms the denoising algorithms based on both the wavelet-domain contextual hidden Markov model and the contourlet-domain hidden Markov tree model (HMT) in terms of PSNR value and visual appearance. The proposed algorithm reduces the Gibbs artifacts obviously and preserves edges and texture information of original images effectively.6. A new texture image segmentation algorithm based on the contourlet-domain HMT model with a modified context scheme is proposed.The thesis studies the multiscale image segmentation based on Bayesian criterion, especially the image segmentation based on the wavelet-domain HMT. However, the blurry edges and singular diffusion often occur in wavelet-domain HMT segmentation results. By analyzed, the reason lies in that (1) the wavelet can not represents contours and edges efficiently; and (2) the context model mainly describe interscale dependencies and encourage the formation of large uniformly classified regions with less consideration on directional and texture characters. To solve these problems, the context scheme is modified for the contourlet-domain HMT and applied to the texture image segmentation. Firstly, the likelihoods of all dyadic squares of the image given the different texture classes are obtained during the model training of contourlet-domain HMT. According to the maximum likelihood classification, direct block-by-block comparison of the likelihoods yields the multiscale raw segmentations. Then, to capture the interscale and neighboring dependencies of class labels, the new contextual design is proposed by voting the correlated class labels. Using the contextual prior, the interscale fusions of multiscale raw segmentations proceed in a multiscale, coarse-to-fine manner and achieve the final pixel-level segmentation. Since the proposed contextual scheme strengthens the boundary dependency, it can improve the accuracy of segmentation around boundaries, as well as provides the reliable outline segmentations. The thesis discusses the context construction and gives the detail segmentation steps. Experiments show that the proposed algorithm outperforms the wavelet-domain HMT segmentation and produces an accurate segmentation of texture images,by improving misclassification and boundary localization.