TY  - CHAP
AU  - Burgeth, Bernhard
AU  - Kleefeld, Andreas
TI  - A Unified Approach to PDE-Driven Morphology for Fields of Orthogonal and Generalized Doubly-Stochastic Matrices
VL  - 10225
CY  - Cham
PB  - Springer International Publishing
M1  - FZJ-2017-03728
SN  - 978-3-319-57239-0 (print)
T2  - Lecture Notes in Computer Science
SP  - 284 - 295
PY  - 2017
AB  - 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.
T2  - 13th International Symposium on Mathematical Morphology and Its Applications to Signal and Image Processing
CY  - 15 May 2017 - 17 May 2017, Fontainebleau (France)
Y2  - 15 May 2017 - 17 May 2017
M2  - Fontainebleau, France
LB  - PUB:(DE-HGF)8 ; PUB:(DE-HGF)7
UR  - <Go to ISI:>//WOS:000425356700023
DO  - DOI:10.1007/978-3-319-57240-6_23
UR  - https://juser.fz-juelich.de/record/830151
ER  -