Please use this identifier to cite or link to this item: doi:10.22028/D291-26325
Title: Mathematical morphology for tensor data induced by the Loewner orderingin higher dimensions
Author(s): Burgeth, Bernhard
Papenberg, Nils
Bruhn, Andres
Welk, Martin
Weickert, Joachim
Language: English
Year of Publication: 2005
Free key words: dilation
top hats
DDC notations: 510 Mathematics
Publikation type: Other
Abstract: Positive semidefinite matrix fields are becoming increasingly important in digital imaging. One reason for this tendency consists of the introduction of diffusion tensor magnetic resonance imaging (DTMRI). In order to perform shape analysis, enhancement or segmentation of such tensor fields, appropriate image processing tools must be developed. This paper extends fundamental morphological operations to the matrix-valued setting. We start by presenting novel definitions for the maximum and minimum of a set of matrices since these notions lie at the heart of the morphological operations. In contrast to naive approaches like the component-wise maximum or minimum of the matrix channels, our approach is based on the Loewner ordering for symmetric matrices. The notions of maximum and minimum deduced from this partial ordering satisfy desirable properties such as rotation invariance, preservation of positive semidefiniteness, and continuous dependence on the input data. We introduce erosion, dilation, opening, closing, top hats, morphological derivatives, shock filters, and mid-range filters for positive semidefinite matrix-valued images. These morphological operations incorporate information simultaneously from all matrix channels rather than treating them independently. Experiments on DT-MRI images with ball- and rod-shaped structuring elements illustrate the properties and performance of our morphological operators for matrix-valued data.
Link to this record: urn:nbn:de:bsz:291-scidok-46299
Series name: Preprint / Fachrichtung Mathematik, Universität des Saarlandes
Series volume: 161
Date of registration: 6-Mar-2012
Faculty: MI - Fakultät für Mathematik und Informatik
Department: MI - Mathematik
Collections:SciDok - Der Wissenschaftsserver der Universität des Saarlandes

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