Diffusion, Convection and Erosion on SE(3)/({0} \times SO(2)) and their Application to the Enhancement of Crossing Fibers

Abstract

In this article we study both left-invariant (convection-)diffusions and left-invariant Hamilton-Jacobi equations (erosions) on the space R 3 S 2 of 3D-positions and orientations naturally embedded in the group SE(3) of 3Drigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossingpreserving fiber enhancement on probability densities defined on the space of positions and orientations.The linear left-invariant (convection-)diffusions are forward Kolmogorov equations of Brownian motions on R 3 S 2 and can be solved by R 3 S 2 -convolution with the corresponding Green's functions or by a finite difference scheme. The left-invariant Hamilton-Jacobi equations are Bellman equations of cost processes on R 3 S 2 and they are solved by a morphological R 3 S 2 -convolution with the corresponding Green's functions. We will reveal the remarkable analogy between these erosions/dilations and diffusions. Furthermore, we consider pseudo-linear scale spaces on the space of positions and orientations that combines dilation and diffusion in a single evolution.In our design and analysis for appropriate linear, non-linear, morphological and pseudo-linear scale spaces on R 3 S 2 we employ the underlying differential geometry on SE(3), where the frame of left-invariant vector fields serves as a moving frame of reference. Furthermore, we will present new and simpler finite difference schemes for our diffusions, which are clear improvements of our previous finite difference schemes.We apply our theory to the enhancement of fibres in magnetic resonance imaging (MRI) techniques (HARDI and DTI) for imaging water diffusion processes in fibrous tissues such as brain white matter and muscles. We provide experiments of our crossing-preserving (non-linear) left-invariant evolutions on neural images of a human brain containing crossing fibers.

Left-Invariant Diffusion on the Motion Group in terms of the Irreducible Representations of SO(3)

“…Nevertheless, we want to rely on these crude approximations for the sake practicability. Of course, there are several more sophisticated approximation schemes ( [33,34,35]), but due to the inherent slow-down of the computation speed, in particular in 3D, we only consider the most simple scheme. And we will see in the experiments that it is enough to obtain reasonable good results.…”

“…If we want to implement this with a discretized sphere the operator H has to approximated by interpolation. We just used spherical harmonics to do this interpolation, in the same way like in [3,35] the spherical Laplace-Beltrami operator was approximated. The operator T 2 0 is discretized very similar to our spherical harmonics implementation with finite differences (comparable to [35], Appendix F).…”

“…We just used spherical harmonics to do this interpolation, in the same way like in [3,35] the spherical Laplace-Beltrami operator was approximated. The operator T 2 0 is discretized very similar to our spherical harmonics implementation with finite differences (comparable to [35], Appendix F). Throughout this experiment the sphere was discretized with 512 direction, which were determined such that the energy of the configuration of 512 electrons on the surface of a sphere is minimized, where the electrons repel each other with a force given by Coulomb's law.…”