PDE-Driven Adaptive Morphology for Matrix Fields

Matrix fields are important in many applications since they are the adequate means to describe anisotropic behaviour in image processing models and physical measurements. A prominent example is diffusion tensor magnetic resonance imaging (DT-MRI) which is a medical imaging technique useful for analysing the fibre structure in the brain. Recently, morphological partial differential equations (PDEs) for dilation and erosion known for grey scale images have been extended to three dimensional fields of symmetric positive definite matrices. In this article we propose a novel method to incorporate adaptivity into the matrix-valued, PDE-driven dilation process. The approach uses a structure tensor concept for matrix data to steer anisotropic morphological evolution in a way that enhances and completes line-like structures in matrix fields. Numerical experiments performed on synthetic and real-world data confirm the gap-closing and line-completing qualities of the proposed method.