Theoretical foundations for spatially discrete 1-D shock filtering

Authors:
Martin Welk;Joachim Weickert;Irena Galić
Affiliations:
Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Saarland University, Building E11, 66041 Saarbrücken, Germany;Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Saarland University, Building E11, 66041 Saarbrücken, Germany;Mathematical Image Analysis Group, Faculty of Mathematics and Computer Science, Saarland University, Building E11, 66041 Saarbrücken, Germany
Venue:
Image and Vision Computing
Year:
2007

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Abstract

While shock filters are popular morphological image enhancement methods, no well-posedness theory is available for their corresponding partial differential equations (PDEs). By analysing the dynamical system of ordinary differential equations that results from a space discretisation of a PDE for 1-D shock filtering, we derive an analytical solution and prove well-posedness. We show that the results carry over to the fully discrete case when an explicit time discretisation is applied. Finally we establish an equivalence result between discrete shock filtering and local mode filtering.