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

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