Semi-explicit Solution and Fast Minimization Scheme for an Energy with l1-Fitting and Tikhonov-Like Regularization

Authors:
Mila Nikolova
Affiliations:
CMLA, ENS Cachan, CNRS, PRES UniverSud, Cachan, France 94230
Venue:
Journal of Mathematical Imaging and Vision
Year:
2009

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Abstract

Regularized energies with l1-fitting have attracted a considerable interest in the recent years and numerous aspects of the problem have been studied, mainly to solve various problems arising in image processing. In this paper we focus on a rather simple form where the regularization term is a quadratic functional applied on the first-order differences between neighboring pixels. We derive a semi-explicit expression for the minimizers of this energy which shows that the solution is an affine function in the neighborhood of each data set. We then describe the volumes of data for which the same system of affine equations leads to the minimum of the relevant energy. Our analysis involves an intermediate result on random matrices constructed from truncated neighborhood sets. We also put in evidence some drawbacks due to the l1-fitting. A fast, simple and exact optimization method is proposed. By way of application, we separate impulse noise from Gaussian noise in a degraded image.