Inverse Problems and Imaging (IPI)

Augmented Lagrangian method for total variation restoration with non-quadratic fidelity

Pages: 237 - 261, Volume 5, Issue 1, February 2011      doi:10.3934/ipi.2011.5.237

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Chunlin Wu - Division of Mathematical Sciences, School of Physical & Mathematical Sciences, Nanyang Technological University, Singapore (email)
Juyong Zhang - Division of Computer Communications, School of Computer Engineering, Nanyang Technological University, Singapore (email)
Xue-Cheng Tai - University of Bergen, University of Bergen Bergen, Norway (email)

Abstract: Recently augmented Lagrangian method has been successfully applied to image restoration. We extend the method to total variation (TV) restoration models with non-quadratic fidelities. We will first introduce the method and present an iterative algorithm for TV restoration with a quite general fidelity. In each iteration, three sub-problems need to be solved, two of which can be very efficiently solved via Fast Fourier Transform (FFT) implementation or closed form solution. In general the third sub-problem need iterative solvers. We then apply our method to TV restoration with $L^1$ and Kullback-Leibler (KL) fidelities, two common and important data terms for deblurring images corrupted by impulsive noise and Poisson noise, respectively. For these typical fidelities, we show that the third sub-problem also has closed form solution and thus can be efficiently solved. In addition, convergence analysis of these algorithms are given. Numerical experiments demonstrate the efficiency of our method.

Keywords:  Augmented Lagrangian method, total variation, impulsive noise, Poisson noise, TV-$L^1$, TV-KL, convergence.
Mathematics Subject Classification:  80M30, 80M50, 68U10.

Received: December 2009;      Revised: September 2010;      Available Online: February 2011.