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An Inexact Alternating Directions Algorithm for Constrained Total Variation Regularized Compressive Sensing Problems

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Abstract

Recently, the efficient solvers for compressive sensing (CS) problems with Total Variation (TV) regularization are needed, mainly because of the reconstruction of an image by a single pixel camera, or the recovery of a medical image from its partial Fourier samples. In this paper, we propose an alternating directions scheme algorithm for solving the TV regularized minimization problems with linear constraints. We minimize the corresponding augmented Lagrangian function alternatively at each step. Both of the resulting subproblems admit explicit solutions by applying a linear-time shrinkage. The algorithm is easily performed, in which, only two matrix-vector multiplications and two fast Fourier transforms are involved at per-iteration. The global convergence of the proposed algorithm follows directly in this literature. Numerical comparisons with the sate-of-the-art method TVLA3 illustrate that the proposed method is effective and promising.

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Authors and Affiliations

  1. Institute of Applied Mathematics, Department of Mathematics, Henan University, Kaifeng, 4750001, China

    Yun-Hai Xiao

  2. Department of Mathematics, Henan University, Kaifeng, 4750001, China

    Hui-Na Song

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  1. Yun-Hai Xiao
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  2. Hui-Na Song
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Correspondence to Yun-Hai Xiao.

Additional information

This work is supported by Chinese NSF grant 11001075, and the Natural Science Foundation of Henan Province Eduction Department grant 2010B110004.

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Xiao, YH., Song, HN. An Inexact Alternating Directions Algorithm for Constrained Total Variation Regularized Compressive Sensing Problems. J Math Imaging Vis 44, 114–127 (2012). https://doi.org/10.1007/s10851-011-0314-y

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  • DOI: https://doi.org/10.1007/s10851-011-0314-y

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