A Unified Approach to PDE-Driven Morphology for Fields of Orthogonal and Generalized Doubly-Stochastic Matrices

Burgeth, Bernhard; Kleefeld, Andreas

doi:10.1007/978-3-319-57240-6_23

Contribution to a conference proceedings/Contribution to a book

FZJ-2017-03728

A Unified Approach to PDE-Driven Morphology for Fields of Orthogonal and Generalized Doubly-Stochastic Matrices

Burgeth, B. (Corresponding author) ; Kleefeld, A.FZJ*

2017
Springer International Publishing Cham
ISBN: 978-3-319-57239-0 (print), 978-3-319-57240-6 (electronic)

Mathematical Morphology and Its Applications to Signal and Image Processing / Angulo, Jesús (Editor) ; Cham : Springer International Publishing, 2017, Chapter 23 ; ISSN: 0302-9743=1611-3349 ; ISBN: 978-3-319-57239-0=978-3-319-57240-6
13th International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing, ISMM 2017, Fontainebleau, France, 15 May 2017 - 17 May 2017 Cham : Springer International Publishing, Lecture Notes in Computer Science 10225, 284 - 295 (2017) [10.1007/978-3-319-57240-6_23]

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Abstract: In continuous morphology two nonlinear partial differential equations (PDEs) together with specialized numerical solution schemes are employed to mimic the fundamental processes of dilation and erosion on a scalar valued image. Some attempts to tackle in a likewise manner the processing of higher order data, such as color images or even matrix valued images, so-called matrix fields, have been made. However, research has been focused almost exclusively on real symmetric matrices. Fields of non-symmetric matrices, for example rotation matrices, defy a unified approach. That is the goal of this article. First, the framework for symmetric matrices is extended to complex-valued Hermitian matrices. The later offer sufficient degrees of freedom within their structures such that, in principle, any class of real matrices may be mapped in a one-to-one manner onto a suitable subset of Hermitian matrices, where image processing may take place. Second, both the linear mapping and its inverse are provided. However, the non-linearity of dilation and erosion processes requires a backprojection onto the original class of matrices. Restricted by visualization shortcomings, the steps of this procedure are applied to the set of 3D-rotation matrices and the set of generalized doubly-stochastic matrices.

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