Sparse Representations Of Images And Its Application To Inverse Problems In Image Processing

The sparse and overcomplete representations of images are a new image model, which can represent images in a compact and efficient way. Most atom coefficients are zero, only few coefficients are big, and the nonzero coefficients can reveal the intrinsic structures and essential properties of images. Besides, redundant systems are also robust to noise and error. For these reasons, sparse representations are beneficial to subsequent image processing applications. At the same time, sparse representation model can effectively match the sparse coding strategy in the primary visual cortex of mammal. Sparse representation theory has already attracted large numbers of international and domestic scholars. At present it is a research hotspot and difficult problem. This paper mainly revolves around the three aspects of sparse representation theory, which include the design of overcomplete dictionary, sparse decomposition (approximation) algorithms and the application of overcomplete and sparse representation model to inverse problems in image processing. The obtained achievements include:(1) In terms of geometric properties of the image structures and the perception characters of human visual system, two dimensional Gabor function is adopted as the generating function of the dictionary and a multi-component Gabor perception dictionary matching various image structures is constructed, which includes smooth, edge and texture sub-dictionaries. Meanwhile, discretization sampling densities of all free parameters in the Gabor function are allocated according to the receptive field properties of neurons in visual cortex and geometric characters of the image structures.Thus, the number of atoms in the dictionary is reduced dramatically.Furthermore, an effective algorithm based on the matching pursuit method is proposed to obtain sparse decomposition of images with our dictionary. The experimental results indicate that the Gabor multi-component perception dictionary can adaptively provide a precise and complete characterization of local geometry structures, such as plain, edge and texture structures in images. In comparison with the anisotropic refinement-Gaussian (AR-Gauss) mixed dictionary, our dictionary has a much sparser representation of images.(2) A structure adaptive matching pursuit subspace search algorithm is proposed to obtain effective sparse representations of images. Firstly, images are adaptively segmented into quad-tree blocks in terms of geometrical structure character. Then each block is classified as one of smooth, edge or texture structure types. When seeking for sparse decomposition of every quad-tree block, it is only to search in subspace of single component sub-dictionary with the same structure type as current block. Due to the reduction of dimension of image and complexity of search in the dictionary, our algorithm for sparse decomposition is effective and fast. (3) Adopting Bayesian-MAP estimation framework and using the sparse representations of the underlying image in an overcomplete dictionary, a sparsity regularized convex functional model is proposed to deconvolve (denosie) Poisson noisy image. The negative-log Poisson likelihood functional is used for data fidelity term and non-smooth regularization term constrains the sparse image representation over the dictionary. An additional term is also added in the functional to ensure the non-negative of the restored image. Inspired form the Split Bergman iteration method, a multi-step fast iterative algorithm is proposed to the model above numerically. By introducing an intermediate variable and Bergman distance, the original problem is transformed into solving two simple sub-problems iteratively, thus decreases the computational complexity rapidly. Experimental results demonstrate the effectiveness of our recovery model and numerical iteration algorithm.(4) Making use of the prior knowledge of image sparse representation, a general variational model is proposed for image recovery with sparsity regularization. The objective functional can be formulated as minimizing the sum of two lower semicontinuous convex functions (not necessarily differentiable) in a real Hilbert space. According to the difference of sparsity regularization term, the general model can be classified as decomposition prior or synthesis prior types. Further, a peceman-rachford operator splitting method is proposed to solve this general recovery model numerically. With regard to the synthesis type, the variable is the sparse representation coefficients and it is usually separable in the sparsity regularization term. The primal peceman-rachford operator splitting method is adopted to solve it directly and conjugate cradient method is employed to rapidly solve the subproblem in the iteratioa With respect to the decomposition type, the problem variable is image-self and it is usually unseparable in the non-smooth sparisty regularization term, thus it is necessary to employ dual peceman-rachford operator splitting method. Dual method can decouple the original recovery model. At the same time, FFT transform method is used to fast solve the subporlem in the iteration, which can simplify the solving of problem and improve operation efficiency.(5) A convex variational model is proposed for multi-frame image super-resolution with sparse representation regularization. The regularization term of the model constrains the underlying image to have a sparse representation in a proper dictionary. The fidelity term of the model restricts the consistency with the measured image in terms of the data degradation model. The existence, uniqueness and character of solution to the model are analyzed. Furthermore, forward and backward operator splitting method and linearized bregman iteration are adopted to numerically solve the above model respectively. Both the algorithms can decompose each iteration into the forward (explicit) gradient sub-step only for the fidelity term and the backward (implicit) sub-step only for regularization term, thus decreasing the computational complexity rapidly. The advantages and disadvantages of the two algorithms are analysed and compared. Numerical results demonstrate that our super-resolution model can efficiently keep the edge and contour structures in the reconstructed image.(6) Two incoherent geometry and texture sub-dictionaries are constructed, which can provide sparse representation of cartoon and texture structures respectively, thus constructs an image multi-morphology sparse representation model. Furthermore, a convex variational model is proposed for multi-frame image super-resolution with multi-morphology sparsity regularization. The regularization term of the model constrains the underlying image to have a sparse representation in a multi-component dictionary. The fidelity term of the model restricts the consistency with the measured image in terms of the data degradation model. An alternate minimization iteration algorithm is proposed to solve it numerically and adopts proximal forward-backward operator splitting method for each sub-problem. Our super-resolution model can efficiently keep the edge, contour and texture structures in the reconstructed image synchronously.