Image And Signal Processing And Its Application Based On Sparse Optimization

With the development of the society and technology,the big data time is coming.The big data is pluralism including video,image and signal.Facing the abundant,high-dimensional,high-speed and diversified data,sparse optimization made it possible for us to reconstruct the high dimensional data with few samples.Sparse optimization becomes more and more important since it can save memory and use few sensors.It has been a significant work for many researchers to analyse and handle the video,image and signal data.Most of image and signal processing problems belong to the ill-posed problem.We consider using the sparse regularization in image processing problem?including image decomposition,image restoration and image compressive sensing?.We propose the new mathematical models and efficient algorithms to solve these image processing problems.We also consider the signal reconstruction problem,and propose a weighted joint sparsity approach.In particular,we apply the weighted joint sparsity approach to solve the multi-task feature learning problem.The main contributions of this thesis are reported in the following five parts:1.In cartoon-texture image decomposition,cartoon part is often characterized by total variation?TV?regularization due to its edge-preserving feature.For texture part,it is oscillating and low-rank.There are many advances using the nuclear norm to charac-terize the low-rank property of texture part,which is suboptimal since each eigenvalue has its physical meaning and should be treated differently.However,the nuclear norm equally punishes each eigenvalue.To remedy the defect,in our image decomposition model,cartoon and texture parts are characterized simultaneously by TV regularization and log det function,where log det function treats eigenvalues with varying degrees to facilitate better characterization of texture part.2.In compressive sensing?CS?image recovery,there are several methods that can be used to efficiently solve the?₁ regularization based model,but they only achieve sub-optimal performance due to their relaxation of the?₀ norm.To achieve better results,we use the smoothed rank function?SRF?as a low-rank regularization to characterize image nonlocal self-similarity.We employ alternative minimization method?AMM?to decouple variables,and respectively solve each subproblem.Numerical experiments demonstrate that the proposed approach achieves the clear restored images under low sampling rate and high noise level situation.3.Although SRF based CS image recovery approach achieved good numerical per-formances,the theoretical analysis of this approach was missing.To guarantee the theory of the proposed CS approach,we build a new CS model based on SRF by adding two regularizations.We provide the corresponding theoretical analysis of the new model.We develop an iterative descent algorithm for the computation of a stationary point of this new model and prove its convergence.4.Based on joint sparse structure of signals,we propose a truncated joint sparsity model.It uses a threshold based weighted method to detect support information of signal.The model obtains an accurate solution by gradually updating the support information in iterations.The convergence and a sufficient recovery condition of the proposed model are guaranteed.We develop a multi-stage convex relaxation algorithm to solve the model.The proposed method not only efficiently reconstructs the joint sparse signals,but also solves collaborative spectrum sensing problem and multi-task feature learning problem.Many experiments demonstrate the superiority of the proposed method.5.In multi-task feature learning problem,we adopt an adaptive threshold selection rule to improve a multi-stage multi-task feature learning algorithm?MSMTFL?which used a fixed threshold.We show the convergence and the uniqueness of solution of the improved method.Compared with other efficient multi-task feature learning approaches and MSMTFL,the improved method obtains the outstanding performance.