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

Ramesh Neelamani, Hyeokho Choi, Richard Baraniuk

(IEEE Transactions on Signal Processing, Vol. 52, No. 2, pgs 418--433, Feb. 2004)

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, while 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 upon WVD's performance.

Support: This work was supported by the NSF grant CCR--99--73188, AFOSR grant F49620--01--1--0378, ONR grant R13820, DARPA grant R13360, and Texas Instruments.