Some Regularization Models And Algorithms For Image Restoration

Image restoration is a key problem in image processing and low-level vision, and is a foundation of subsequent pattern recognition and high-level understanding. In the decades, this technology has gone deeply into the various application fields, such as remote sensing image, medical imaging, military target recognition, etc. Therefor, the research of image restoration has important theoretical significance and application value.Image restoration is a ill-posed inverse problem in mathematical nature. Effective way adopted to stabilize the inversion of ill-posed problem is called regularization. This thesis focuses on image modeling for edge-preservation, contrast-preservation, texture-preservation, and structure-preservation. Several image restoration algorithms are put for-ward for multiplicative noise removal, image deblurring, image super-resolution, and color demosaicking. Followings are the primary works and achievements of the dissertation.(1) Most existing methods for multiplicative noise removal consider little the influ-ence of human vision psychology. A non-convex variational model for removing multi-plicative Gamma noise is proposed under the MAP framework. The data fidelity term is constructed according to the distribution characteristic of the multiplicative Gamma noise. The regularizer based on Weberized TV is designed on the basis of the Weber’s Law in HVS. Moreover, we study the issues of existence and uniqueness of a minimizer for this variational model, and adaptively compute the regularization parameter according to the variance of the recovered noise which matches that of our prior knowledge.In order to solve the proposed optimization problem, two numerical algorithms are put forward. One is a simple alternative minimization algorithm by using the well-known variable splitting and quadratic penalty techniques in optimization. The other is a lin-earized gradient method to solve the associated Euler-Lagrange equation via a fixed-point iteration. Our experimental results show that our algorithm is effective and efficient to filter out multiplicative noise while well preserving the feature details.(2) In order to improve the retention capacity of important structures (e.g. edges and textures) in restored image, this thesis proposes a image restoration model combining the Poisson singular integral (PSI) regularization and Curvelet sparse representation. As a new regularization tool, PSI can be used to depict the regularity of a texture image. Curvelet can optimally represent smooth and edge components of image with sparsity.To solve the variational model, an efficient algorithm is put forward by using the operator splitting technique in convex analysis. The basic idea of the algorithm is that the model can be divided into some simple subproblems that can be solved iteratively. Only fast Fourier transformation or simple Curvelet shrinkage is used in each iteration of the algorithm. Experimental results demonstrate that our compound regularization method is not only able to restrain the noise and blurring effect, but also able to preserve important image features, such as edges and textures.(3) Two fast numerical algorithms are proposed for image super-resolution Based on the TV regularization. The first is a fast decoupling algorithm, which exploit the alternating minimization approach based on the variable splitting and quadratic penalty techniques. This method takes full advantage of the structural properties of the degenerative operators (i.e., geometrical motion matrices, blur matrix, and the first-order finite-difference ma-trix all have circulant structure under periodic boundary condition) in the image degraded model. As such, the treatment is separated into measurements fusion, deblurring and de-noising. The fusion part is shown to be a very simple non-iterative algorithm. The linear equation systems for deblurring part can be effectively solved by Fourier transformation. The denoising part can be solved by the subspace projection method. The second method is proposed to solve the optimization problem with TV constraint. The Douglas-Rachford splitting technique is applied to the dual problem. In this way, the problem is decomposed into three simple sub-problems, and each sub-problem has a closed-form solution. More-over, to speed up convergence, we provide an accelerated scheme based on preconditioned design of initial guess and forward-backward operator splitting technique.(4) Currently, most of the color demosaicking (CDM) algorithm using only a local spatial and spectral correlation, easily lead CDM restored picture to blur edges and loss of fine structure. When cyclical small structures exist in the image, these local methods are prone to the distortion of the zipper effect, the raster effect, false color, etc. To solve these problems, unifying dictionary learning and sparse coding into a variational framework, a nonlocal adaptive sparse representation model is proposed through nonlocal similarity clustering and adaptive dictionary online learning. Using the local and nonlocal redun-dancy information, sparse coding constraints forced sparse coding close to its nonlocal means to reduce coding errors. Moreover, the fidelity term is characterized by l1-norm to suppress the heavy-tailed visual artifacts. Finally, the joint alternating minimization method and operator splitting techniques are utilized to effectively solve the model. Exper-imental results show that the proposed method not only improves the peak signal-to-noise ratio, and reduces the jagged effect and zipper effect ratio (ZER), but also sharpens the image edge and texture, and greatly improves the visual quality of the image.