On Level-Set Type Methods for Recovering Piecewise Constant Solutions of Ill-Posed Problems

Authors:
Adriano Decezaro;Antonio Leitão;Xue-Cheng Tai
Affiliations:
Institute of Mathematics Statistics and Physics, Federal University of Rio Grande, Rio Grande, Brazil 96201-900;Department of Mathematics, Federal University of St. Catarina, Florianópolis, Brazil 88040-900;Division of Mathematical Science, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore and Department of Mathematics, University of Bergen, Bergen, Norway N-50 ...
Venue:
SSVM '09 Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision
Year:
2009

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Abstract

We propose a regularization method for solving ill-posed problems, under the assumption that the solutions are piecewise constant functions with unknown level sets and unknown level values. A level set framework is established for the inverse problem and a Tikhonov regularization approach is proposed. Existence of generalized minimizers for the Tikhonov functional is proven. Moreover, we establish convergence and stability results, characterizing our Tikhonov approach as a regularization method. Based on the necessary conditions of optimality for the Tikhonov functional, a level-set type method is derived and implemented numerically for solving an inverse source problem. This allow us to test the quality of the proposed algorithm.