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February  2007, 1(1): 29-46. doi: 10.3934/ipi.2007.1.29

Iteratively solving linear inverse problems under general convex constraints

Ingrid Daubechies 1, , Gerd Teschke 2, and  Luminita Vese 3,

1. 

Princeton University, PACM, Washington Road, Princeton, NJ 08544-1000, United States
2. 

Konrad--Zuse--Institute Berlin, Takustr. 7, D-14195 Berlin-Dahlem, Germany
3. 

Department of Mathematics, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, CA 90095-1555, United States

Received  August 2006 Revised  September 2006 Published  January 2007

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: Landweber iteration, Generalized shrinkage., Linear inverse problems, Besov- and BV restoration.
Mathematics Subject Classification: Primary: 35, 41, 49; Secondary: 6.
Citation: Ingrid Daubechies, Gerd Teschke, Luminita Vese. Iteratively solving linear inverse problems under general convex constraints. Inverse Problems and Imaging, 2007, 1 (1) : 29-46. doi: 10.3934/ipi.2007.1.29
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