Research On Sparse Optimization Models And Algorithms For Data Restoration Problems

Nowadays,in the big data era,many real-world problems have a higher requirement against the accuracy and completeness of the collected data.However,due to the adverse environment conditions or the limitation of acquisition cost,most real data is usually contaminated by the noise or lose detailed information in the process of acquisition and transmission.Such degradation will further affect the subsequent data processing tasks,like object tracking,target detection,instance classification,etc.Therefore,the restoration of the degraded data has become a fundamental and important task,and also drawn much attention from both the industry and academia.The goal of data restoration is to estimate the original clean and complete data from the corrupted or incomplete data.Mathematically,this is an inverse problem with ill-posed property.The key issue to solve the inverse problem is to rationally explore and exploit the intrinsic priors of the clean and complete data,and then enforce the desired solution to approximate the original data.Due to the underlying structures and characteristics,most data often have sparsity or sparse representation under suitable based or proper transformation.Thus,the optimization method with sparse regularization has become an important way to tackle the data restoration problem in recent years.This dissertation mainly focuses on exploring and depicting the sparse priors of complicated data,then constructing rational regularization model for data restoration and developing efficient algorithm to solve the model.The main contents can be summarized as the following four parts:1.We propose a single-image rain streaks removal method based on the directional total variation regularization.The goal of single-image rain streaks removal is to eliminate the rain effects and recover the clean original image from an given corrupted image which is captured in raining conditions.Based on the directionality of rain streaks,we propose to use the directional total variation to characterize the discriminative directional smoothness of rain streaks and clean images.Then we build the sparse optimization model for single-image rain streaks removal,and develop an efficient alternating direction method of multipliers to solve the proposed model.Experiments on both synthetic and real data shows that the proposed method can effectively remove the rain streaks with arbitrary directions,and illustrates the superiority of our method in rain streaks removal task.2.In the single-image deraining problem,despite of the directionality,the local patterns of rain streaks often have the similarity and repeatability.Based on this fact,we propose a tensor-based low-rank model which utilizes the similar and repeated rain patterns for single-image deraining task.Different from the existing matrix-based methods,we first divide the rain streaks layer into small patches and stack them to form a three-order tensor.Then we use the low-rankness of tensors to characterize the similar rain patterns and the dependency of rain patches.At last,we build the corresponding sparse optimization model and design an efficient algorithm to solve the model.Numerical experiments demonstrate that our method can effectively remove the rain effects and recover the details of original clean images.3.We propose a video rain streaks removal method based on the nonlocal selfsimilarity of clean videos.To depict the redundancy of videos,we divide the video into3 D patches,then for each 3D patches,we group its similar 3D patches and rearrange them to form a three-order tensor.We characterize the redundancy of videos by regularizing the similar-patch-formed tensors with their low-rankness,and then build our low-rank regularization model based on the nonlocal self-similarity.We develop an efficient algorithm based on alternating direction method of multipliers framework.The numerical experiments demonstrate the effectiveness and superiority of the proposed method especially when dealing with the videos with remarkable moving objects and dynamic backgrounds.4.we propose a method of high-rank matrix completion with side information.In matrix completion task,we consider the columns of data matrix lie in an union of multiple low-rank subspaces,but not a single one.In this case,the data matrix may be high-rank or even full-rank,thus the methods based on low-rank assumption are not effective for the completion task.To handle this problem,we utilize the self-expressiveness of data matrix,and cast the problem as finding the sparse representation of the data matrix with the dictionary being the data matrix itself.We also incorporate the side information for the matrix completion task.Moreover,we propose a strategy to decouple the partial observed data matrix and the sparse coefficients so that the proposed model is amenable to convex relaxation,and design a linearized alternating direction method to solve the proposed model.Numerical experiments shows that our method can complete the high rank matrix with high accuracy.