Generalised diffusion based regularization for inverse problem in image processing

Innocent Souopgui 1 Arthur Vidard 1 François-Xavier Le Dimet 1
1 MOISE - Modelling, Observations, Identification for Environmental Sciences
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
Abstract : Due to the ill-posedness of inverse problems, it is important to make use of most of the \textit{a priori} informations while solving such a problem. These informations are generally used as constraints to get the appropriate solution. In usual cases, constrains are turned into penalization of some characteristics of the solution. A common constraint is the regularity of the solution leading to regularization techniques for inverse problems. Regularization by penalization is affected by two principal problems: - as the cost function is composite, the convergence rate of minimization algorithms decreases - when adequate regularization functions are defined, one has to define weighting parameters between regularization functions and the objective function to minimize. It is very difficult to get optimal weighting parameters since they are strongly dependant on the observed data and the truth solution of the problem. There is a third problem that affects regularization based on the penalization of spatial variation. Although the penalization of spatial variation is known to give best results (gradient penalization and second order regularization), there is no physical underlying foundation. Penalization of spatial variations lead to smooth solution that is an equilibrium between good and bad characteristics. Here, we introduce a new approach for regularization of ill-posed inverse problems. Penalization of spatial variations is weighted by an observation based trust function. The result is a generalized diffusion operator that turns regularization into pseudo covariance operators. All the regularization informations are then embedded into a preconditioning operator. On one hand, this method do not need any extra terms in the cost function, and of course is affected neither by the ill-convergence due to composite cost function, nor by the choice of weighting parameters. On the other hand, The trust function introduced here allows to take into account the observation based a priori knowledges on the problem. We suggest a simple definition of the trust function for inverse problems in image processing. Preliminary results show a promising method for regularization of inverse problems.
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Innocent Souopgui, Arthur Vidard, François-Xavier Le Dimet. Generalised diffusion based regularization for inverse problem in image processing. EngOpt 2010 - 2nd International Conference on Engineering Optimization, Technical University of Lisbon and Institute of Mechanical Engineering, Sep 2010, Lisbon, Portugal. ⟨inria-00519330⟩

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