L₁-norm Minimization Method Based On Bregman Iteration And Its Applications

With the development of regularization Bregman iteration and its theory. The theory on the sparse representation, which is the core of the signal and image processing field has an important application value, and becomes many scholars’ focus of study. This dissertation is mainly based the optimization theory about the l₁-norm and Bregman iteration, studies the algorithms of the basis pursuit problem and the applications in image and signal processing fields. The main research results and innovations consist of three parts:Firstly, making use of the matrix theory, we give out an equivalent form of the basis pursuit problem, then solve the problem of the equivalent form, and deduce the +A linearized Bregman iterative.Secondly, We propose simple and extremely efficient methods for solving the basis pursuit problem1min{ u: Au= g, u? Rn},which is used in compressed sensing. We prove that the linearized Bregman with some steps is equivalent to the dual problem and provide a convergence analysis for the linearized Bregman with some steps. We can find one way to accelerate the+A linearized Bregman method for solving the basis and related sparse optimization problems. We show that the new method can reduce this iteration complexity to 1. Numerical results are presented that for sparse signal recovery problems, the new algorithm can reconstruct signal exactly from the observational signal. Meanwhile. The new method can be faster, less iterative steps, specially, reduce the stagnation of iterative procedure efficiently compared with the +A linearized Bregman iterative procedure.At last, because of some faults about l₂ norm constraint in image and signal processing fields, we change the l₂-norm constraint into l₁-norm constraint. Simultaneously, a new model of the split Bregman iterative algorithm is deduced. The numerical results show that the new model can reconstruct the origin signal from the noise signal in both image and signal processing. At the same time, the number of iteration and iterative time are significantly reduced.