Convex Denoising using Non-Convex Tight Frame Regularization

TL;DR

This paper introduces a convex optimization approach for signal denoising that employs a non-convex regularizer within a tight-frame analysis framework, ensuring global convergence and improved estimation of non-zero signal values.

Contribution

It proposes a novel non-convex regularizer constrained to maintain convexity, enabling effective denoising with guaranteed convergence using ADMM.

Findings

Enhanced accuracy in estimating non-zero signal components

Convexity of the objective function is maintained despite non-convex regularization

ADMM converges to the global optimum in the proposed framework

Abstract

This paper considers the problem of signal denoising using a sparse tight-frame analysis prior. The L1 norm has been extensively used as a regularizer to promote sparsity; however, it tends to under-estimate non-zero values of the underlying signal. To more accurately estimate non-zero values, we propose the use of a non-convex regularizer, chosen so as to ensure convexity of the objective function. The convexity of the objective function is ensured by constraining the parameter of the non-convex penalty. We use ADMM to obtain a solution and show how to guarantee that ADMM converges to the global optimum of the objective function. We illustrate the proposed method for 1D and 2D signal denoising.