%0 Book Section
%A Burgeth, Bernhard
%A Kleefeld, Andreas
%T A Unified Approach to PDE-Driven Morphology for Fields of Orthogonal and Generalized Doubly-Stochastic Matrices
%V 10225
%C Cham
%I Springer International Publishing
%M FZJ-2017-03728
%@ 978-3-319-57239-0 (print)
%B Lecture Notes in Computer Science
%P 284 - 295
%D 2017
%< 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
%X 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.
%B 13th International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing
%C 15 May 2017 - 17 May 2017, Fontainebleau (France)
Y2 15 May 2017 - 17 May 2017
M2 Fontainebleau, France
%F PUB:(DE-HGF)8 ; PUB:(DE-HGF)7
%9 Contribution to a conference proceedingsContribution to a book
%U <Go to ISI:>//WOS:000425356700023
%R 10.1007/978-3-319-57240-6_23
%U https://juser.fz-juelich.de/record/830151