Der Fluss zur Totalvariation : Nichtlinearität und andere Hindernisse

Der Fluss zur Totalvariation wird für viele Algorithmen aus der modernen Bildverarbeitung verwendet. Vor Kurzem ist nun die Herleitung eines Existenzresultates für das Hindernisproblem zu diesem Fluss gelungen, unter Verwendung von klassischen Konzepten aus der geometrischen Maßtheorie, die über 40 Jahre alt sind.

We describe a new result on the existence of solutions for an obstacle problem associated to the total variation flow. The total variation flow is famous for its applications in image reconstruction. The treatment of the obstacle problem, with possibly thin obstacles, requires utilization of concepts that were developed by the ingenious mathematician Ennio De Giorgi more than 40 years ago. We briefly describe these concepts and explain wow they can be applied to the above-mentioned obstacle problem for the total variation flow. In fact, the usual formulation of this problem might not have a solution if the prescribed obstacle contains “thin” parts of lower dimension. The strategy is to replace the original formulation of the obstacle problem with a relaxed version, using the concept of the De Giorgi measure. Together with his co-authors Prof. Dr. Verena Bögelein from Salzburg and Prof. Dr. Frank Duzaar from Erlangen, the author was able to derive an existence result for this relaxed problem by combining modern techniques for time-dependent partial differential equations with ideas from classical mathematical theory.

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