An Iterative Re-Weighting Algorithm for Element to Solve Sparsity-Regularized Linear Inverse Problem and an Application in Sparse-Spike Deconvolution of Seismic Data

The sparsity-regularized linear inverse problem has been widely used in many fields, such as remote sensing imaging, image processing and analysis, seismic deconvolution, compressed sensing, medical imaging, etc. This method offers greater flexibility in solving real-world problems. There are various types of sparse regularization methods available in the literature, such as L0-norm, Lp-norm(0<p<1) regularization, L1- norm, weighted L1-norm regularization, Cauchy regularization, modified Cauchy regularization, t regularization, and so on. Among these, L1-norm regularization is the most well-known and widely used due to its convexity and linearity. It can be efficiently solved by the iterative soft thresholding algorithm (ISTA) or its faster variants. The soft thresholding operator works on each element individually and is highly efficient. In contrast, all other sparse regularization methods are nonlinear, making them more challenging to solve. However, they can address the limitations of L1-norm regularization, such as degeneracy. An effective approach to solving sparsity-regularized problems is the iterative re-weighting least squares algorithm (IRLS). However, IRLS is computationally intensive and may not be suitable for massive data applications. In this paper, we propose a unified algorithmic framework for solving sparsity-regularized linear inverse problems. Our method combines the principles of ISTA and IRLS to create an element-wise iterative re-weighting algorithm. We have applied this algorithm to the sparse spike deconvolution of real seismic data and demonstrated its effectiveness as an alternative method for solving sparsityregularized linear inverse problems.