Sharpening fibers in diffusion weighted MRI via erosion

In this chapter erosion is generalized to the space of diffusion weighted MRI data. This is done effectively by solving a Hamilton-Jacobi-Bellman (HJB) system (erosion) on the coupled space of three dimensional positions and orientations, embedded as a quotient in the group of three dimensional rigid body motions. The solution to the HJB equations is given by a well-posed morphological convolution. We present two numerical approaches to solve the HJB equations: analytical kernels, and finite differences. Proof of concept is given by showing improved visibility of major fiber bundles in both artificial and human data. Furthermore, the method is shown to significantly improve the output of a probabilistic tractography algorithm used to extract the optic radiation.