Study On Variational Models And Sparse Optimization With Application For Improving The Quality Of Image

The variation-based digital image processing model estabilished in the early1990s and the sparse optimization theory and method based on wavelet analysis in recent years are two kinds of important mathematic tools in the field of digital image processing. The related research and application have already become one of hotspot directions in mathematical imaging. This thesis uses total variation(TV) combined with sparse optimization theory to overcome the shortcomings of the previous models and methods, and systematically studies effective models and methods for digital image restoration problem. The main research work and innovation are embodied as follows.In this thesis, we propose a TV-based image restoration model with dual method. For the limitations and disadvantages of the original model, a redescribed form of the primal-dual hybrid variation model is derived by introducing first-order finite difference matrix and separating dual variable in the vertical and horizontal directions. On this basis, the corresponding fast algorithm for primal-dual hybrid variation model is designed. Then a unified iterative formula for different algorithms such as the primal-dual hybrid gradient descent(PDHGD) algorithm, the improved version and the semi-implict gradient descent algorithm(SGD) is derived.In this thesis, we propose a fast proximity point algorithm and apply it to TV-based image restoration. The novel method is derived from the idea of establishing a general proximity point operator framework based on which new first-order schemes for TV-based image restoration have been proposed. Many current algorithms for TV-based image restoration, such as Chambolle’s projection algorithm, the split Bregman algorithm, the Bermúdez-Moreno algorithm, the Jia-Zhao denoising algorithm, and the fixed point algorithm, can be viewed as special cases of the new first-order schemes. Moreover, the convergence of the new algorithm has been analyzed at length. Numerical experiments illustrate the accuracy and efficiency of the proposed algorithms.In this thesis, based on Robust Principal Component Analysis(RPCA), we propose a method of recovering a low-rank matrix with an unknown fraction of its entries being arbitrarily corrupted. This problem, under some conditions, can be exactly solved via convex optimization that minimizes a combination of the nuclear norm and the l1norm. This thesis proposes an algorithm based on the Douglas-Rachford splitting method for solving the RPCA problem. First, we solve the convex optimization problem by canceling the constraint of the variable, and then compute the proximity operators of objective function alternately. The new algorithm can exactly recover the low-rank and sparse components simultaneously and it is proved to be convergent. Numerical simulations demonstrate the accuracy and efficiency of our algorithm.This thesis establishes a new first-order optimality condition for the basis pursuit problem. Such a condition provides us with a new approach to choose penalty parameters adaptively for fixed point iteration algorithm. Meanwhile, we extend the result to matrix completion. Numerical experiments of sparse vector recovery and low-rank matrix completion illustrate the accuracy of our theoretic results and efficiency of the algorithm.This thesis proposes the compressed sensing(CS) algorithms for direction-of-arrival(DOA) estimation with nonuniform linear arrays(NLA). By designing the sparse representation matrix and measurement matrix, corresponding to representing process and measuring process respectively in the CS method, we propose a novel model and several CS algorithms for DOA estimation with NLA. The results of experiments demonstrate the good efficiency of the proposed algorithms, especially of the CS-MUSIC and CS-RMUSIC algorithms.