Relaxing Tight Frame Condition in Parallel Proximal Methods for Signal Restoration

TL;DR

This paper extends parallel proximal methods for signal restoration by relaxing the tight frame condition, enabling the use of more flexible non-tight frame representations in various noise scenarios.

Contribution

It introduces a method to relax the tight frame assumption in parallel proximal algorithms, broadening their applicability to non-tight frame representations in signal deconvolution.

Findings

Effective in deconvolving Poisson and Laplacian noise.

Utilizes non-tight dual-tree wavelet and filter bank structures.

Applicable to both frame analysis and synthesis problems.

Abstract

A fruitful approach for solving signal deconvolution problems consists of resorting to a frame-based convex variational formulation. In this context, parallel proximal algorithms and related alternating direction methods of multipliers have become popular optimization techniques to approximate iteratively the desired solution. Until now, in most of these methods, either Lipschitz differentiability properties or tight frame representations were assumed. In this paper, it is shown that it is possible to relax these assumptions by considering a class of non necessarily tight frame representations, thus offering the possibility of addressing a broader class of signal restoration problems. In particular, it is possible to use non necessarily maximally decimated filter banks with perfect reconstruction, which are common tools in digital signal processing. The proposed approach allows us to solve both frame analysis and frame synthesis problems for various noise distributions. In our simulations, it is applied to the deconvolution of data corrupted with Poisson noise or Laplacian noise by using (non-tight) discrete dual-tree wavelet representations and filter bank structures.