A Directional Rouy-Tourin Scheme for Adaptive Matrix-Valued Morphology

In order to describe anisotropy in image processing models or physical measurements, matrix fields are a suitable choice. In diffusion tensor magnetic resonance imaging (DT-MRI), for example, information about the diffusive properties of water molecules is captured in symmetric positive definite matrices. The corresponding matrix field reflects the structure of the tissue under examination. Recently, morphological partial differential equations (PDEs) for dilation and erosion known for grey scale images have been extended to matrix-valued data. In this article we consider an adaptive, PDE-driven dilation process for matrix fields. The anisotropic morphological evolution is steered with a matrix constructed from a structure tensor for matrix valued data. An important novel ingredient is a directional variant of the matrix-valued Rouy-Tourin scheme that enables our method to complete or enhance anisotropic structures effectively. Experiments with synthetic and real-world data substantiate the gap-closing and line-completing properties of the proposed method.