A generic approach to diffusion filtering of matrix-fields

Authors:
B. Burgeth;S. Didas;L. Florack;J. Weickert
Affiliations:
Saarland University, Department of Mathematics and Computer Science, 66041, Saarbrücken, Germany;Saarland University, Department of Mathematics and Computer Science, 66041, Saarbrücken, Germany;Eindhoven University of Technology, Department of Biomedical Engineering, 66041, Eindhoven, The Netherlands;Saarland University, Department of Mathematics and Computer Science, 66041, Saarbrücken, Germany
Venue:
Computing - Special Issue on Industrial Geometry
Year:
2007

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Abstract

Diffusion tensor magnetic resonance imaging, is a image acquisition method, that provides matrix- valued data, so-called matrix fields. Hence image processing tools for the filtering and analysis of these data types are in demand. In this article, we propose a generic framework that allows us to find the matrix-valued counterparts of the Perona–Malik PDEs with various diffusivity functions. To this end we extend the notion of derivatives and associated differential operators to matrix fields of symmetric matrices by adopting an operator-algebraic point of view. In order to solve these novel matrix-valued PDEs successfully we develop truly matrix-valued analogs to numerical solution schemes of the scalar setting. Numerical experiments performed on both synthetic and real world data substantiate the effectiveness of our novel matrix-valued Perona–Malik diffusion filters.