Adaptive Parameter Selection for Block Wavelet-Thresholding Deconvolution

Abstract In this paper, we propose a data-driven block thresholding procedure for wavelet-based non-blind deconvolution. The approach consists in appropriately writing the problem in the wavelet domain and then selecting both the block size and threshold parameter at each resolution level by minimizing Stein's unbiased risk estimate. The resulting algorithm is simple to implement and fast. Numerical illustrations are provided to assess the performance of the estimator.

[1]  Ja-Yong Koo,et al.  Wavelet deconvolution , 2002, IEEE Trans. Inf. Theory.

[2]  Jean-Luc Starck,et al.  Stein block thresholding for wavelet-based image deconvolution , 2010 .

[3]  B. Vidakovic,et al.  Adaptive wavelet estimator for nonparametric density deconvolution , 1999 .

[4]  Mohamed-Jalal Fadili,et al.  Adaptive structured block sparsity via dyadic partitioning , 2011, 2011 19th European Signal Processing Conference.

[5]  L. Cavalier.  Nonparametric statistical inverse problems , 2008 .

[6]  D. Donoho.  Nonlinear Solution of Linear Inverse Problems by Wavelet–Vaguelette Decomposition , 1995 .

[7]  A. Tsybakov,et al.  Sharp adaptation for inverse problems with random noise , 2002 .

[8]  Amel Benazza-Benyahia,et al.  A SURE Approach for Digital Signal/Image Deconvolution Problems , 2008, IEEE Transactions on Signal Processing.

[9]  Charles Dossal,et al.  Local Behavior of Sparse Analysis Regularization: Applications to Risk Estimation , 2012, 1204.3212.

[10]  Christophe Chesneau.  Wavelet Estimation Via Block Thresholding : A Minimax Study Under The $L^p$ Risk , 2006 .

[11]  J. Johannes.  DECONVOLUTION WITH UNKNOWN ERROR DISTRIBUTION , 2007, 0705.3482.

[12]  Y. Meyer.  Wavelets and Operators , 1993 .

[13]  Harrison H. Zhou,et al.  A data-driven block thresholding approach to wavelet estimation , 2009, 0903.5147.

[14]  Thierry Blu,et al.  A New SURE Approach to Image Denoising: Interscale Orthonormal Wavelet Thresholding , 2007, IEEE Transactions on Image Processing.

[15]  A. Tsybakov,et al.  Block Thresholding and Sharp Adaptive Estimation in Severely Ill-Posed Inverse Problems , 2004 .

[16]  I. Johnstone.  WAVELET SHRINKAGE FOR CORRELATED DATA AND INVERSE PROBLEMS: ADAPTIVITY RESULTS , 1999 .

[17]  T. Tony Cai,et al.  On adaptive wavelet estimation of a derivative and other related linear inverse problems , 2002 .