Please use this identifier to cite or link to this item: doi:10.22028/D291-26324
Title: Mathematical morphology on tensor data using the Loewner ordering
Author(s): Burgeth, Bernhard
Feddern, Christian
Welk, Martin
Weickert, Joachim
Language: English
Year of Publication: 2005
Free key words: dilation
matrix-valued images
positive definite matrix
DDC notations: 510 Mathematics
Publikation type: Other
Abstract: The notions of maximum and minimum are the key to the powerful tools of greyscale morphology. Unfortunately these notions do not carry over directly to tensor-valued data. Based upon the Loewner ordering for symmetric matrices this paper extends the maximum and minimum operation to the tensor-valued setting. This provides the ground to establish matrix-valued analogues of the basic morphological operations ranging from erosion/dilation to top hats. In contrast to former attempts to develop a morphological machinery for matrices, the novel definitions of maximal/minimal matrices depend continuously on the input data, a property crucial for the construction of morphological derivatives such as the Beucher gradient or a morphological Laplacian. These definitions are rotationally invariant and preserve positive semidefiniteness of matrix fields as they are encountered in DT-MRI data. The morphological operations resulting from a component-wise maximum/minimum of the matrix channels disregarding their strong correlation fail to be rotational invariant. Experiments on DT-MRI images as well as on indefinite matrix data illustrate the properties and performance of our morphological operators.
Link to this record: urn:nbn:de:bsz:291-scidok-46281
Series name: Preprint / Fachrichtung Mathematik, Universität des Saarlandes
Series volume: 160
Date of registration: 5-Mar-2012
Faculty: MI - Fakultät für Mathematik und Informatik
Department: MI - Mathematik
Collections:SciDok - Der Wissenschaftsserver der Universität des Saarlandes

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