Researches And Applications Of Nonconvex Optimization Modeling And Algorithm In Image Restoration

Image restoration is a crucial inverse problem,whose goal is to solve the real image from the observed degraded image inversely.Since the inverse problem is usually ill-posed in the solving process,it is difficult for the algorithm to find the stable solution of the inverse problem.Regularization-based optimization modeling is an effective and stable method to solve ill-posed inverse problems.In image restoration,the regularization terms generally come from the priors of images and have been extensively studied,such as the sparse prior to sparse noise.The dissertation is to construct a reasonable convex or nonconvex optimization model?mainly focus on exploring the nonconvex optimization model?by analyzing the prior properties of the image restoration problems,then design an efficient algorithm for solving the corresponding models.The main contents of this dissertation are as follows:1.A modified Tikhonov regularization is proposed by analyzing the properties of the singular values of the degraded matrix to solve the signal deblurring and denoising problem.The modified model can suppress the noise more effectively and recover the real signal more accurately.Moreover,a preconditioner based on Lanczos bidiagonalization is designed to accelerate the convergence rate of the conjugate gradient least squares algorithm.2.For removing the mixed noise?such as the mixed Gaussian and sparse noise?of remote sensing images,through analyzing the prior properties of spatial domain and spectral domain,such as the similarity among adjacent bands,the sparse tensor-based optimization model based on gradient-domain is proposed.Also,an efficient and stable algorithm is designed under the framework of the alternating direction method of multi-pliers?ADMM?to solve the proposed sparse model.There is an essential relation for the modeling and the algorithm between the con-vex and the nonconvex problems.Although algorithms of the convex optimization model have extensive convergence properties theoretically,the convex optimization model is usually not as accurate as of the nonconvex model in depicting the underlying properties of the images.Besides,the convergence analysis of the nonconvex algorithm is limited,which is a challenging issue.It is also important and difficult in this dissertation.The nonconvex optimization modelings and the corresponding efficient algorithms are gen-erally based on the classical framework of nonconvex algorithms.Thus the theoretical conclusions of convex problems could favor analyzing the theoretical properties of non-convex problems.3.Considering the directional property of stripe noise,the?₀ quasi-norm is utilized to formulate a unidirectional total variation sparse optimization model,aiming to remove the stripe noise of remote sensing images.The mathematical program with equilibrium constraints?MPEC?is used to equivalently transform the proposed nonconvex sparse model to a nonconvex optimization model that can be solved effectively.An efficient algorithm for solving the equivalent nonconvex optimization model is designed by using the proximal ADMM.In particular,the accumulation point of the sequence generated by the developed algorithm is proved theoretically satisfying the Karush-Kuhn-Tucker?KKT?condition of the equivalent nonconvex optimization model.4.An efficient algorithm is designed by using the framework of proximal alternating minimization to solve an existing coupled variable-joint nonconvex model of semi-blind image restoration.The designed algorithm is meaningful and complementary for solv-ing the semi-blind restoration problem,and it is theoretically proved that the sequence generated by the algorithm can converge to the critical point of the nonconvex optimiza-tion model.Compared with the existing algorithm that only guarantees the convergence of the subsequence,the designed algorithm has a more reliable convergence property of sequence.5.The reproducing kernel Hilbert space?RKHS?has a powerful representation for functions,and one image can be regarded as a discrete version of a function.Thus,an im-age could be represented linearly by the reproducing kernels in RKHS.An?₀ quasi-norm based nonconvex optimization model is proposed for the application of image deblur-ring and denoising with the RKHS.Similar to 3,the given model can be equivalently transformed into a nonconvex model based on MPEC.Under the proximal alternating linearized minimization,a stable and efficient algorithm is developed to solve the equiv-alent nonconvex model.Moreover,the sequence generated by the designed algorithm can converge to the critical point of the nonconvex optimization model in the theoretical analysis.