ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems

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.