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An optimal control problem in image processing
Author(s) -
Bredies Kristian,
Lorenz Dirk A.,
Maass Peter
Publication year - 2006
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200610409
Subject(s) - frank–wolfe algorithm , generalization , degenerate energy levels , regular polygon , image (mathematics) , shrinkage , mathematical optimization , mathematics , iterative method , inverse , computer science , inverse problem , algorithm , point (geometry) , image processing , convex optimization , artificial intelligence , mathematical analysis , convex set , physics , geometry , statistics , quantum mechanics
As a starting point, we present a control problem in mammographic image processing which leads to non‐standard penalty terms and involves a degenerate parabolic PDE which has to be controlled in the coefficients. We then discuss the classical conditional gradient method from constrained optimization and propose a generalization for non‐convex functionals which covers the conditional gradient method as well as the recently proposed iterative shrinkage method of Daubechies, Defrise and De Mol for the solution of linear inverse problems with sparsity promoting penalty terms. We prove that this new algorithm converges. This also gives a deeper understanding of the iterative shrinkage method. Further, we show an application to the above‐mentioned control problem in image processing. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)