Reconstituting The Sparse Signal Based On Truncated ?₁ Minimization

Rapidly developing society generates huge amounts of information,how to restore the original data with less observations has become a huge research challenge.In recent years,the emergence of compressed sensing has greatly helped solve this problem,and has been widely applied to statistics,image processing,information theory,bio-sensing and many other fields.As the research progresses,we can find that the noise pollution has a great impact on us,so it is very important to study the noise problem of compressed sensing.Combined with compressed sensing theory,this work studies the signal reconstruction problem affected by noise under the truncated model.The main contents are as follows:The first chapter introduces the background of compressed sensing,describes the research history and current situation of traditional compressed sensing and block compressed sensing,and finally gives the structure of the whole article.The second chapter elaborates theory of compressed sensing and block compression sens-ing with structural properties.Then one introduced what is truncation.Finally,the alter-nating direction multiplier method framework is introduced.In the third chapter,based on the compressed sensing,we consider the complete pertur-bation problem and propose a truncated?₁minimization model for complete perturbation.This completely perturbed model extends previous work by including a multiplicative noise term in addition to the usual additive noise term.We derived the RIP for perturbed matrix A?.Our theoretical result showed that the stability of the truncated?₁solution of the com-plectly perturbed scenario was limited by the total noise and the best k-term approximation.In addition,the numerical simulations are carried out to demonstrate the validation of our results.The fourth chapter is based on block compressed sensing,by using the generalized?_p-norm noise constraint for 2?p<?to replace the popular?₂-norm,we put forward a truncated?₁model for recovering block-sparse signal.A theoretical analysis is first presented to guarantee the validity of proposed method.If the measurement matrix satisfies an extend-ed block restricted isometry property,the reconstruction error is bounded in the optimization.Moreover,in order to solve the induced optimization problem effectively,we present an alter-nating direction method of multipliers via embedding Karush-Kuhn-Tucker system of?_p-norm functions into the frame structure of augmented Lagrangian methods.When compared with some of the state-of-the-art methods,the proposed method becomes more competitive.The last chapter summarizes the work of this paper,and it has a corresponding outlook.