Diffusion on the 3D Euclidean Motion Group for Enhancement of HARDI Data

In previous work we studied linear and nonlinear left-invariant diffusion equations on the 2D Euclidean motion group SE(2), for the purpose of crossing-preserving coherence-enhancing diffusion on 2D images. In this paper we study left-invariant diffusion on the 3D Euclidean motion group SE(3), which is useful for processing three-dimensional data. In particular, it is useful for the processing of High Angular Resolution Diffusion Imaging (HARDI) data, since these data can be considered as orientation scores directly, without the need to transform the HARDI data to a different form. In principle, all theory of the 2D case can be mapped to the 3D case. However, one of the complicating factors is that all practical 3D orientation scores are not functions on the entire group SE(3), but rather on a coset space of the group. We will show how we can still conceptually apply processing on the entire group by requiring the operations to preserve the introduced notion of alpha-right-invariance of such functions on SE(3). We introduce left-invariant derivatives and describe how to estimate tangent vectors that locally fit best to the elongated structures in the 3D orientation score. We propose generally applicable techniques for smoothing and enhancing functions on SE(3) using left-invariant diffusion on the group. Finally, we will discuss implementational issues and show a number of results for linear diffusion on artificial HARDI data.