Reconstruction Of Structurally-incomplete Matrices And Its Image Processing Applications

Most matrix reconstruction methods impose a low-rank prior or its variants to well pose the problem,which can reconstruct randomly-missing entries in a matrix efficiently.However,in practical applications,these missing entries are not distributed randomly,but represent a trend just like structural missing,which cannot be handled by the rank minimization prior individually.To remedy this,this paper introduces new matrix reconstruction models and algorithms using double priors on the latent matrix.The main contents of this thesis are summarized as follows:1.This paper proposed a structurally-incomplete matrix completion model(MCReLaSP)based on reweighted low-rank and sparsity priors,complementing the classic matrix reconstruction models that handle random missing only.In the proposed model,the matrix is regularized by a low-rank prior to exploit the inter-column and inter-row correlations,and its columns/rows are regularized by a sparsity prior to exploit intracolumn/row correlations.Both the low-rank and sparse priors are reweighted on the fly to promote low-rankness and sparsity,respectively.2.This paper proposed two variant models,i.e.MR-ReLaSP1 model for Gaussian noise and MR-ReLaSP2 model for impulse noise,by considering both structural missing and noise in observed entries based on the MC-ReLaSP model,which enhances the robustness of our proposed models in practical applications.3.Numerical algorithms to solve our models are derived via the alternating direction method under the augmented Lagrangian multiplier(ALM-ADM)framework.The proposed algorithm has a low computational complexity benefiting from ADM.4.We evaluate the recoverability of our proposed models and classic matrix reconstruction models,and apply our models to various image processing applications.Results on synthetic data show that the proposed models outperform the classic ones.Our models are quite effective in image processing applications,such as image inpainting,image denoising and image rain-streak removing.