Analysis and Synthesis Prior Greedy Algorithms for Non-linear Sparse Recovery

TL;DR

This paper introduces non-linear greedy algorithms for sparse recovery in inverse problems, demonstrating improved performance in speckle denoising over existing methods.

Contribution

It proposes novel non-linear variants of OMP, CoSamp, and GAP algorithms for both synthesis and analysis prior sparse recovery problems.

Findings

Outperforms state-of-the-art speckle denoising methods

Effective on exponential and logarithmic functions

Shows improved success rates in empirical tests

Abstract

In this work we address the problem of recovering sparse solutions to non linear inverse problems. We look at two variants of the basic problem, the synthesis prior problem when the solution is sparse and the analysis prior problem where the solution is cosparse in some linear basis. For the first problem, we propose non linear variants of the Orthogonal Matching Pursuit (OMP) and CoSamp algorithms; for the second problem we propose a non linear variant of the Greedy Analysis Pursuit (GAP) algorithm. We empirically test the success rates of our algorithms on exponential and logarithmic functions. We model speckle denoising as a non linear sparse recovery problem and apply our technique to solve it. Results show that our method outperforms state of the art methods in ultrasound speckle denoising.