Research On Shearlet-based Statistical Model For Images And Its Applications

Images are often corrupted by noise and blur due to the undesired conditions for im-age acquisition, processing and transmission. The noise and blur in images have severelydegraded image quality and affected the subsequent image processing tasks. Thus noise andblur reduction has been a very important pre-processing step for improving the quality ofimages. In the past decade, the wavelet transform has been successfully used in image de-noising and deblurring due to its multiresolution capability. However, despite its remarkablesuccess in dealing with pointwise singularities, the standard separable wavelet transformfails to provide an optimal sparse representation for images that contain other types of singu-larities. Shearlets, a new directional multiresolution transform, can efficiently represent thedirectional information of images.This thesis mainly investigates the properties of the shearlet transform, especially thestatistical characteristic for shearlet transform coefficients of natural images. Then new im-age denoising and deblurring methods are derived using maximum a posteriori (MAP) esti-mation theory with the proposed statistical model. The main achievements of this thesis areas follows:Firstly, after discussing the implementation of the discrete shearlet transform (DST),the computational complexity, redundancy, reconstruction accuracy and sparsity of the DSTare analyzed. Since multiscale and multidirectional decomposition of images is the mostimportant characteristic of the DST, we can implement the DST by performing the multi-scale and multidirectional decomposition steps separately as done in contourlets. One wayof achieving a multiscale and multidirectional decomposition is to use Laplacian pyramidfollowed by fast pseudo-polar Fourier transform.Secondly, estimation methods, coefficients statistical models and correlation betweencoefficients are key problems for image denoising and deblurring based on the shearlet trans-form. In this thesis, a comprehensive study of these three problems is given. In essence,image denoising and deblurring can be regarded as a standard estimation problem. We useMAP estimation theory to estimate the shearlet coefficients. In order to model the shearletcoefficients, we propose a probability model, generalized spherically contoured exponential(GSCE) model, which is used as prior information for MAP. Theoretical analysis is pre-sented to prove that GSCE is heavy-tailed. Furthermore, correlation between coefficients isanalyzed by using the mutual information as quantitative measure. The quantitative resultsare used to develop a image denoising model. Thirdly, three shearlet-based denoising methods are proposed for image noise reduction.According to the correlation characterization, GSCE is exploited to establish a trivariatemaximum a posteriori model. In the case of variates having the same standard deviation, asimple closed-form solution (Method I) is derived from this trivariate model. For variateswith different standard deviations, an iterative denoising method (Method II) is presented,and the convergence of the iterative algorithm is proved. Related parameters estimationmethods are given. Then we propose another denoising method (Method III) based on thegeneralized model. Experimental results demonstrate the efficiency of the proposed methods.Finally, a shearlet-based deblurring method using the proximal iteration is presentedfor image deblurring. Using the coefficients model and MAP, an image deblurring model isderived. Since this deblurring model is equivalent to an optimization problem, the proximaliterative algorithm is used to solve it. The parameters estimation methods, the computationalcomplexity and the convergence of the proximal iteration are presented. Experimental re-sults show that the proposed deblurring method obtains better performance than the existingwavelet-based and curvelet-based methods.