We consider morphological and linear scale spaces on the space R3¿S 2 of 3D positions and orientations naturally embedded in the group SE(3) of 3D rigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossing-preserving fiber enhancement on probability densities defined on the space of positions and orientations. The strength of these enhancements is that they are expressed in a moving frame of reference attached to fiber fragments, allowing us to diffuse along the fibers and to erode orthogonal to them. The linear scale spaces are described by forward Kolmogorov equations of Brownian motions on R3¿S 2 and can be solved by convolution with the corresponding Green’s functions. The morphological scale spaces are Bellman equations of cost processes on R3¿S 2 and we show that their viscosity solutions are given by a morphological convolution with the corresponding morphological Green’s function. For theoretical underpinning of our scale spaces on R3¿S 2 we introduce Lagrangians and Hamiltonians on R3¿S 2 indexed by a parameter ¿¿[1,8). The Hamiltonian induces a Hamilton-Jacobi-Bellman system that coincides with our morphological scale spaces on R3¿S 2. By means of the logarithm on SE(3) we provide tangible estimates for both the linear- and the morphological Green’s functions. We also discuss numerical finite difference upwind schemes for morphological scale spaces (erosions) of Diffusion-Weighted Magnetic Resonance Imaging (DW-MRI), which allow extensions to data-adaptive erosions of DW-MRI. We apply our theory to the enhancement of (crossing) fibres in DW-MRI for imaging water diffusion processes in brain white matter.
Duits, R., Dela Haije, T. C. J., Creusen, E. J., & Ghosh, A. (2013). Morphological and linear scale spaces for fiber enhancement in DW-MRI. Journal of Mathematical Imaging and Vision, 46(3), 326-368. https://doi.org/10.1007/s10851-012-0387-2