Inverse Problems and Imaging (IPI)

Iteratively solving linear inverse problems under general convex constraints

Pages: 29 - 46, Volume 1, Issue 1, February 2007      doi:10.3934/ipi.2007.1.29

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Ingrid Daubechies - Princeton University, PACM, Washington Road, Princeton, NJ 08544-1000, United States (email)
Gerd Teschke - Konrad--Zuse--Institute Berlin, Takustr. 7, D-14195 Berlin-Dahlem, Germany (email)
Luminita Vese - Department of Mathematics, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, CA 90095-1555, United States (email)

Abstract: We consider linear inverse problems where the solution is assumed to fulfill some general homogeneous convex constraint. We develop an algorithm that amounts to a projected Landweber iteration and that provides and iterative approach to the solution of this inverse problem. For relatively moderate assumptions on the constraint we can always prove weak convergence of the iterative scheme. In certain cases, i.e. for special families of convex constraints, weak convergence implies norm convergence. The presented approach covers a wide range of problems, e.g. Besov-- or BV--restoration for which we present also numerical experiments in the context of image processing.

Keywords:  Linear inverse problems, Landweber iteration, Besov- and BV restoration, Generalized shrinkage.
Mathematics Subject Classification:  Primary: 35, 41, 49; Secondary: 65.

Received: August 2006;      Revised: September 2006;      Available Online: January 2007.