Structured And Element-wise Sparsity Prior With Applications To Image Deblurring

Image is one of the important media that cannot be separated from science,industry,and human daily life.However,during the process of acquisition,transmission,and storing,due to the technical limits of imaging equipment,dust or smoke in the imaging environment,and other factors,the observed image may be corrupted by different noise.In addition,many other factors,such as the camera shake and the movement of the photographed object,may cause the undesired distortions,which will aggravate the loss of significant details of the original image.Image deblurring is one of the ill-posed inverse problems aiming to restore clear images from the degraded ones,which can be regarded as a reverse process of image degradation.To cope with the ill-conditioned blurring operator and the interference of noise,many scholars have considered this image deblurring problem from the perspective of maximum a posteriori(MAP)estimation,so that the recovered image is more likely to be the original one.This means that it is necessary to take the reasonable statistical model of noise and the meaningful image prior for the image deblurring problem.In this thesis,we discuss the problem of recovering clear images from the degraded observation.These images are subject to various blurring interferences(such as motion blur,out-of-focus disk blur,uniform blur,etc.)and are further contaminated by one of Gaussian noise,Poisson noise,or Cauchy noise.In order to estimate the latent sharp image,a series of regularization methods are proposed,providing the simultaneous promotion of the pixel-wise gradient sparsity and structured sparsity inherent to the natural image,and their optimization problem is solved,where the difficulties stem from the non-convexity and overlapping structure of the regularizers.The main contribution of this thesis is as follows:1.For image deblurring under Gaussian noise,we propose a novel regularization method depicting the overlapping group sparsity on hyper-Laplacian(OGS-HL)prior of natural image gradient,based on the analysis of the previous works.According to the statistical characteristics of noise,we build a restoration model.However,due to the non-convex and non-differentiable regularization term,and the complicated overlapping structure,its optimization is still troublesome.Therefore,the alternating direction method of multipliers(ADMM)is adopted to solve the problem.The sub-problem related to the overlapping structure is solved in the framework of the majorization minimization(MM)algorithm,in which the sophisticatedly derived quadratic majorizer is applied to make the intractable optimization more amenable.We prove the convergence of the optimization algorithm,and through some simulation studies,we further verify that the proposed natural image prior can effectively restore the sharp images.2.The problem of image deblurring under Poisson noise is studied.Poisson noise is one of the well-known noises,caused by the particle-like nature of light and the uncertainty of photon detection,which often appears in medical imaging and other fields.In order to remedy the deficiency of traditional methods,a regularization method combining the first-order total variation overlap group sparse prior and nonconvex secondorder total variation prior is proposed,which can alleviate the staircase artifacts while preserving the original sharp edges.Simulation studies show the effectiveness of the proposed method for Poissonian image restoration including denoising and deblurring.3.With the motivation that emphasizing the different contributions of each pixel in the group can promote the more accurate restoration,the proposed OGS-HL regularization method is further improved by introducing the weighting scheme,resulting in the weighted hyper-Laplacian prior with overlapping group sparsity(OGS-WHL).We deal with the image deblurring problem under Cauchy noise,which usually appears in underwater acoustic engineering,synthetic aperture radar systems,and others.The statistical model for Cauchy deblurring is built,however,the OGS-WHL regularizer and the likelihood related to noise distribution are non-convex and non-differentiable.Therefore,on the premise of updating the majorizer of MM algorithm,we further propose a novel optimization method with global convergence.The effectiveness of the proposed prior and its optimization method is verified through simulation studies.