Augmented Lagrangian method for total variation restoration with non-quadratic fidelity
Chunlin Wu - Division of Mathematical Sciences, School of Physical & Mathematical Sciences, Nanyang Technological University, Singapore (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.
Received: December 2009; Revised: September 2010; Available Online: February 2011.
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