ForWaRD: Fourier-Wavelet Regularized Deconvolution for Ill-Conditioned Systems

TLDR

The authors introduce ForWaRD, a hybrid Fourier‑wavelet regularized deconvolution algorithm that applies scalar shrinkage in both domains to regularize noise. ForWaRD achieves this by combining Fourier shrinkage, which leverages the Fourier transform’s efficient representation of colored noise, with wavelet shrinkage, which exploits the wavelet domain’s efficient representation of piecewise‑smooth signals, and optimizes their balance via an approximate MSE criterion. ForWaRD outperforms existing methods, being applicable to all ill‑conditioned deconvolution problems, producing minimal ringing compared to Wiener deconvolution, achieving MSE decay at the optimal WVD rate as sample size grows, and improving performance over WVD across a wide range of practical sample lengths.

Abstract

We propose an efficient, hybrid Fourier-wavelet regularized deconvolution (ForWaRD) algorithm that performs noise regularization via scalar shrinkage in both the Fourier and wavelet domains. The Fourier shrinkage exploits the Fourier transform's economical representation of the colored noise inherent in deconvolution, whereas the wavelet shrinkage exploits the wavelet domain's economical representation of piecewise smooth signals and images. We derive the optimal balance between the amount of Fourier and wavelet regularization by optimizing an approximate mean-squared error (MSE) metric and find that signals with more economical wavelet representations require less Fourier shrinkage. ForWaRD is applicable to all ill-conditioned deconvolution problems, unlike the purely wavelet-based wavelet-vaguelette deconvolution (WVD); moreover, its estimate features minimal ringing, unlike the purely Fourier-based Wiener deconvolution. Even in problems for which the WVD was designed, we prove that ForWaRD's MSE decays with the optimal WVD rate as the number of samples increases. Further, we demonstrate that over a wide range of practical sample-lengths, ForWaRD improves on WVD's performance.