# American Institute of Mathematical Sciences

February  2011, 5(1): 237-261. doi: 10.3934/ipi.2011.5.237

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

 1 Division of Mathematical Sciences, School of Physical & Mathematical Sciences, Nanyang Technological University, Singapore 2 Division of Computer Communications, School of Computer Engineering, Nanyang Technological University, Singapore 3 University of Bergen, University of Bergen Bergen, Norway

Received  December 2009 Revised  September 2010 Published  February 2011

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.
Citation: Chunlin Wu, Juyong Zhang, Xue-Cheng Tai. Augmented Lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Problems & Imaging, 2011, 5 (1) : 237-261. doi: 10.3934/ipi.2011.5.237
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##### References:
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