Sparsity Signal Inversion By Multi-parameter Regularization And Optimal Parameter Choices

In this paper, we discuss multi-parameter regularization, the combination of traditional smooth penalty l2 and sparsity penalty l1 for the recovery of sparsity signal. Typically,sparse regularization method is appropriate for recovering sparse signal. However, the relative research are often devoted to compressive sensing problem, where random matrices are well-conditioned. For ill-conditioned problems, for example image inpainting, image deblurring and parameter identification, etc., these methods are often unstable. Then the traditional sparse regularization method cannot effectively recovery the approximate solution with ill-conditioned problems. Inspired by multi-regularization theory, a smooth l2 term is added to original functional of regularization which can improve the stability. This method admits sparsity inverse problems have large conditional number, which will admit more extensive application. For numerical realization, multi-parameter regularization iterative threshold algorithm is proposed. Several experiments involving compressed sensing and image inpainting are presented showing that our proposed approaches are robust and efficient.