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

Ramesh Neelamani, Hyeokho Choi, Richard Baraniuk

Research output: Contribution to journalArticlepeer-review

407 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)418-433
Number of pages16
JournalIEEE Transactions on Signal Processing
Issue number2
StatePublished - Feb 2004


  • Deblurring
  • Deconvolution
  • Restoration
  • Wavelet-vaguelette
  • Wavelets

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Signal Processing


Dive into the research topics of 'ForWaRD: Fourier-wavelet regularized deconvolution for Ill-conditioned systems'. Together they form a unique fingerprint.

Cite this