Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization

Juan Carlos De los Reyes; Carola-Bibiane Schönlieb

doi:10.3934/ipi.2013.7.1183

Article Contents

2013, Volume 7, Issue 4: 1183-1214. Doi: 10.3934/ipi.2013.7.1183

This issue Previous Article Identification of nonlinearities in transport-diffusion models of crowded motion Next Article Hybrid regularization for MRI reconstruction with static field inhomogeneity correction

Image denoising: Learning the noise model via nonsmooth PDE-constrained optimization

Juan Carlos De los Reyes^1, and
Carola-Bibiane Schönlieb^2,

1.
Research Center on Mathematical Modelling (MODEMAT), Escuela Politécnica Nacional de Quito, Ladrón de Guevara E11-253, Quito, 170109,
2.
Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA

Received: June 30, 2012

Revised: August 31, 2013

Published: October 2013

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Abstract

We propose a nonsmooth PDE-constrained optimization approach for the determination of the correct noise model in total variation (TV) image denoising. An optimization problem for the determination of the weights corresponding to different types of noise distributions is stated and existence of an optimal solution is proved. A tailored regularization approach for the approximation of the optimal parameter values is proposed thereafter and its consistency studied. Additionally, the differentiability of the solution operator is proved and an optimality system characterizing the optimal solutions of each regularized problem is derived. The optimal parameter values are numerically computed by using a quasi-Newton method, together with semismooth Newton type algorithms for the solution of the TV-subproblems.

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Mathematics Subject Classification: Primary: 49J20, 49J40, 49K20, 68U10; Secondary: 65K10.

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