A Hybrid Iterative Thresholding Method for Solving Sparsity-Regularized Linear Inverse Problems and Its Application in Seismic Sparse-Spike Deconvolution

Abstract

Sparsity-regularized linear inverse problem has been served as the base in many disciplines, such as remote sensing imaging, image processing and analysis, seismic deconvolution, compressed sensing, medical imaging, and so forth. Iterative hard thresholding algorithm (IHTA) and iterative soft thresholding algorithm (ISTA) are two frequently used methods to solve sparsity-regularized linear inverse problems. They are also the basic unit of other more complex methods. IHTA and ISTA are derived under steepest descent method. I.e., iteratively perform gradient descent and thresholding shrinkage. Steepest descent method is a first order algorithm, which is a powerful way to solve optimization due to its relatively simple implementation. However, a known issue of first order method is possible poor convergence rate. Fast iterative thresholding-like algorithms have been proposed to overcome this issue in the existing literatures. In history, another alternative way is second order algorithms or quasi-second order algorithms. In this paper, we include a quasi-Newton’s method, i.e., DFP (Davidon-Fletcher-Powell) formulations in the framework of iterative thresholding-like algorithms to replace gradient descent to form a hybrid method to further increase convergence rate. The proposed method has been performed on two numerical examples and a real-life application in sparse-spike seismic deconvolution. The numerical examples and real-life application showed that it provides an effective alternative method to solve sparsity-regularized linear inverse problems.