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.

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  1. Jiang, Jiahua; Chung, Julianne; de Sturler, Eric: Hybrid projection methods with recycling for inverse problems (2021)
  2. Jiang, Jiahua; Chung, Julianne; de Sturler, Eric: Hybrid projection methods with recycling for inverse problems (2021)
  3. Bergounioux, Maïtine; Abraham, Isabelle; Abraham, Romain; Carlier, Guillaume; Le Pennec, Erwan; Trélat, Emmanuel: Variational methods for tomographic reconstruction with few views (2018)
  4. Escande, Paul; Weiss, Pierre: Accelerating (\ell^1)-(\ell^2) deblurring using wavelet expansions of operators (2018)
  5. Lamash, Yechiel: Algorithm and constraints for exact non-blind deconvolution (2018)
  6. Zhao, Xueqing; Huang, Keke; Wang, Xiaoming; Shi, Meihong; Zhu, Xinjuan; Gao, Quanli; Yu, Zhaofei: Reaction-diffusion equation based image restoration (2018)
  7. Jiang, Ming; Bobin, Jérôme; Starck, Jean-Luc: Joint multichannel deconvolution and blind source separation (2017)
  8. Ma, Liyan; Xu, Li; Zeng, Tieyong: Low rank prior and total variation regularization for image deblurring (2017)
  9. Renaut, Rosemary A.; Vatankhah, Saeed; Ardestani, Vahid E.: Hybrid and iteratively reweighted regularization by unbiased predictive risk and weighted GCV for projected systems (2017)
  10. Ma, Liyan; Zeng, Tieyong: Image deblurring via total variation based structured sparse model selection (2016)
  11. Deng, Liang-Jian; Guo, Huiqing; Huang, Ting-Zhu: A fast image recovery algorithm based on splitting deblurring and denoising (2015)
  12. Kulik, Rafal; Sapatinas, Theofanis; Wishart, Justin Rory: Multichannel deconvolution with long range dependence: upper bounds on the (L^p)-risk ((1 \leqp < \infty)) (2015)
  13. Starck, Jean-Luc; Murtagh, Fionn; Fadili, Jalal M.: Sparse image and signal processing. Wavelets and related geometric multiscale analysis (2015)
  14. Zhou, Xu; Zhou, Fugen; Bai, Xiangzhi; Xue, Bindang: A boundary condition based deconvolution framework for image deblurring (2014)
  15. Benhaddou, Rida; Pensky, Marianna; Picard, Dominique: Anisotropic de-noising in functional deconvolution model with dimension-free convergence rates (2013)
  16. Liu, Qiegen; Liang, Dong; Song, Ying; Luo, Jianhua; Zhu, Yuemin; Li, Wenshu: Augmented Lagrangian-based sparse representation method with dictionary updating for image deblurring (2013)
  17. Chung, Julianne; Chung, Matthias; O’leary, Dianne P.: Optimal filters from calibration data for image deconvolution with data acquisition error (2012)
  18. Donatelli, Marco: An iterative multigrid regularization method for Toeplitz discrete ill-posed problems (2012)
  19. Xiang, Shiming; Meng, Gaofeng; Wang, Ying; Pan, Chunhong; Zhang, Changshui: Image deblurring with matrix regression and gradient evolution (2012)
  20. Aizenberg, Igor: Complex-valued neural networks with multi-valued neurons. (2011)

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