A fast Iterative Shrinkage-Thresholding Algorithm with application to wavelet-based image deblurring

We consider the class of Iterative Shrinkage-Thresholding Algorithms (ISTA) for solving linear inverse problems arising in signal/image processing. This class of methods is attractive due to its simplicity, however, they are also known to converge quite slowly. In this paper we present a Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) which preserves the computational simplicity of ISTA, but with a global rate of convergence which is proven to be significantly better, both theoretically and practically. Initial promising numerical results for wavelet-based image deblurring demonstrate the capabilities of FISTA.

[1]  Y. Nesterov.  A method for solving the convex programming problem with convergence rate O(1/k^2) , 1983 .

[2]  I. Johnstone,et al.  Adapting to Unknown Smoothness via Wavelet Shrinkage , 1995 .

[3]  Antonin Chambolle,et al.  Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage , 1998, IEEE Trans. Image Process..

[4]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[5]  Robert D. Nowak,et al.  An EM algorithm for wavelet-based image restoration , 2003, IEEE Trans. Image Process..

[6]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

[7]  E. Candès,et al.  Astronomical image representation by the curvelet transform , 2003, Astronomy & Astrophysics.

[8]  Patrick L. Combettes,et al.  Signal Recovery by Proximal Forward-Backward Splitting , 2005, Multiscale Model. Simul..

[9]  José M. Bioucas-Dias,et al.  A New TwIST: Two-Step Iterative Shrinkage/Thresholding Algorithms for Image Restoration , 2007, IEEE Transactions on Image Processing.

[10]  Y. Nesterov.  Gradient methods for minimizing composite objective function , 2007 .

[11]  Mário A. T. Figueiredo,et al.  Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems , 2007, IEEE Journal of Selected Topics in Signal Processing.

[12]  Michael Elad,et al.  Coordinate and subspace optimization methods for linear least squares with non-quadratic regularization , 2007 .

[13]  Michael Unser,et al.  A fast iterative thresholding algorithm for wavelet-regularized deconvolution , 2007, SPIE Optical Engineering + Applications.

[14]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..