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. Renaut, Rosemary A.; Vatankhah, Saeed; Ardestani, Vahid E.: Hybrid and iteratively reweighted regularization by unbiased predictive risk and weighted GCV for projected systems (2017)
  2. Ma, Liyan; Zeng, Tieyong: Image deblurring via total variation based structured sparse model selection (2016)
  3. Deng, Liang-Jian; Guo, Huiqing; Huang, Ting-Zhu: A fast image recovery algorithm based on splitting deblurring and denoising (2015)
  4. Kulik, Rafal; Sapatinas, Theofanis; Wishart, Justin Rory: Multichannel deconvolution with long range dependence: upper bounds on the $L^p$-risk $(1 \leq p < \infty)$ (2015)
  5. Starck, Jean-Luc; Murtagh, Fionn; Fadili, Jalal M.: Sparse image and signal processing. Wavelets and related geometric multiscale analysis (2015)
  6. Zhou, Xu; Zhou, Fugen; Bai, Xiangzhi; Xue, Bindang: A boundary condition based deconvolution framework for image deblurring (2014)
  7. Benhaddou, Rida; Pensky, Marianna; Picard, Dominique: Anisotropic de-noising in functional deconvolution model with dimension-free convergence rates (2013)
  8. 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)
  9. Chung, Julianne; Chung, Matthias; O’leary, Dianne P.: Optimal filters from calibration data for image deconvolution with data acquisition error (2012)
  10. Donatelli, Marco: An iterative multigrid regularization method for Toeplitz discrete ill-posed problems (2012)
  11. Xiang, Shiming; Meng, Gaofeng; Wang, Ying; Pan, Chunhong; Zhang, Changshui: Image deblurring with matrix regression and gradient evolution (2012)
  12. Aizenberg, Igor: Complex-valued neural networks with multi-valued neurons. (2011)
  13. Averbuch, Amir Z.; Zheludev, Valery A.; Khazanovsky, Marie: Deconvolution by matching pursuit using spline wavelet packets dictionaries (2011)
  14. Chen, Yang; Yu, Weimin; Li, Yinsheng; Yang, Zhou; Luo, Limin; Chen, Wufan: Bayesian image restoration using a large-scale total patch variation prior (2011)
  15. Chesneau, Christophe: On adaptive wavelet estimation of a quadratic functional from a deconvolution problem (2011)
  16. Chung, Julianne; Easley, Glenn; O’Leary, Dianne P.: Windowed spectral regularization of inverse problems (2011)
  17. Lou, Chun; Li, Wen-Hao; Zhou, Huai-Chun; Salinas, Carlos T.: Experimental investigation on simultaneous measurement of temperature distributions and radiative properties in an oil-fired tunnel furnace by radiation analysis (2011)
  18. Lou, Yifei; Bertozzi, Andrea L.; Soatto, Stefano: Direct sparse deblurring (2011)
  19. Pensky, Marianna; Sapatinas, Theofanis: Multichannel boxcar deconvolution with growing number of channels (2011)
  20. Averbuch, Amir; Zheludev, Valery; Neittaanmäki, Pekka; Koren, Jenny: Block based deconvolution algorithm using spline wavelet packets (2010)

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