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A Shock-Capturing Algorithm for the Differential Equations of Dilation and Erosion

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Abstract

Dilation and erosion are the fundamental operations in morphological image processing. Algorithms that exploit the formulation of these processes in terms of partial differential equations offer advantages for non-digitally scalable structuring elements and allow sub-pixel accuracy. However, the widely-used schemes from the literature suffer from significant blurring at discontinuities. We address this problem by developing a novel, flux corrected transport (FCT) type algorithm for morphological dilation/erosion with a flat disc. It uses the viscosity form of an upwind scheme in order to quantify the undesired diffusive effects. In a subsequent corrector step we compensate for these artifacts by means of a stabilised inverse diffusion process that requires a specific nonlinear multidimensional formulation. We prove a discrete maximum–minimum principle in this multidimensional framework. Our experiments show that the method gives a very sharp resolution of moving fronts, and it approximates rotation invariance very well.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Saarland University, 66041, Saarbrücken, Germany

    Michael Breuß & Joachim Weickert

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  1. Michael Breuß
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  2. Joachim Weickert
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Correspondence to Michael Breuß.

Additional information

Michael Breuß received his diploma and Ph.D. degrees in mathematics from the University of Hamburg in 1998 and 2001, respectively. He worked as an assistant professor at the University of Hamburg, as post-doctoral researcher at the University of Bordeaux 1 and as a researcher at the Technical University Braunschweig. Since April 2006, he is an assistant professor at the Mathematical Image Analysis group at the Saarland University. His research interests are centered around numerical methods for partial differential equations, hyperbolic conservation laws and visual computing.

Joachim Weickert received his diploma and Ph.D. degrees in mathematics from the University of Kaiserslautern (Germany) in 1991 and 1996, respectively. In 2001 he obtained a habilitation degree in computer science from the University of Mannheim (Germany). He worked as research assistant at the University of Kaiserslautern, as post-doctoral researcher at the universities of Utrecht (The Netherlands) and Copenhagen (Denmark), and as assistant professor at the University of Mannheim. Currently he is full professor of mathematics and computer science at Saarland University in Saarbrücken (Germany), where he heads the Mathematical Image Analysis Group. His research interests include image processing, computer vision, and scientific computing.

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Breuß, M., Weickert, J. A Shock-Capturing Algorithm for the Differential Equations of Dilation and Erosion. J Math Imaging Vis 25, 187–201 (2006). https://doi.org/10.1007/s10851-006-9696-7

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