ForWaRD: Fourier-Wavelet Regularized Deconvolution for Ill-Conditioned Systems
Baraniuk, Richard G.
deconvolution; restoration; deblurring; wavelets; wavelet-vaguelette
We propose an efficient, hybrid <i>Fourier-Wavelet Regularized Deconvolution</i> (ForWaRD) algorithm that performs noise regularization via scalar shrinkage in both the Fourier and wavelet domains. The Fourier shrinkage exploits the Fourier transform's sparse representation of the colored noise inherent in deconvolution, while the wavelet shrinkage exploits the wavelet domain's sparse 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 sparser wavelet representations require less Fourier shrinkage. ForWaRD is applicable to all ill-conditioned deconvolution problems, unlike the purely wavelet-based <i>Wavelet-Vaguelette Deconvolution</i> (WVD), and its estimate features minimal ringing, unlike purely Fourier-based Wiener deconvolution. We analyze ForWaRD's MSE decay rate as the number of samples increases and demonstrate its improved performance compared to the optimal WVD over a wide range of practical sample-lengths.