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URN: urn:nbn:de:bsz:291-scidok-46281
URL: http://scidok.sulb.uni-saarland.de/volltexte/2012/4628/


Mathematical morphology on tensor data using the Loewner ordering

Burgeth, Bernhard ; Feddern, Christian ; Welk, Martin ; Weickert, Joachim

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Freie Schlagwörter (Englisch): dilation , erosion , matrix-valued images , positive definite matrix
Institut: Fachrichtung 6.1 - Mathematik
DDC-Sachgruppe: Mathematik
Dokumentart: Preprint (Vorabdruck)
Schriftenreihe: Preprint / Fachrichtung Mathematik, Universit├Ąt des Saarlandes
Bandnummer: 160
Sprache: Englisch
Erstellungsjahr: 2005
Publikationsdatum: 05.03.2012
Kurzfassung auf Englisch: 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.
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