New Exact and Numerical Solutions of the (Convection-)Diffusion Kernels on SE(3)

TL;DR

This paper derives exact and numerical solutions for hypo-elliptic diffusion and convection-diffusion kernels on the Lie group SE(3), extending methods from SE(2) and introducing new analytical and computational techniques.

Contribution

It extends the derivation of convolution kernels to SE(3) using eigenfunction expansions and Fourier transforms, and introduces a numerical method and an analytic approximation for these kernels.

Findings

Exact formulas for kernels in terms of spheroidal wave functions.

Extended numerical procedure from SE(2) to SE(3).

Comparison of analytic approximation with exact kernels.

Abstract

We consider hypo-elliptic diffusion and convection-diffusion on $\mathbb{R}^3 \rtimes S^2$, the quotient of the Lie group of rigid body motions SE(3) in which group elements are equivalent if they are equal up to a rotation around the reference axis. We show that we can derive expressions for the convolution kernels in terms of eigenfunctions of the PDE, by extending the approach for the SE(2) case. This goes via application of the Fourier transform of the PDE in the spatial variables, yielding a second order differential operator. We show that the eigenfunctions of this operator can be expressed as (generalized) spheroidal wave functions. The same exact formulas are derived via the Fourier transform on SE(3). We solve both the evolution itself, as well as the time-integrated process that corresponds to the resolvent operator. Furthermore, we have extended a standard numerical procedure from SE(2) to SE(3) for the computation of the solution kernels that is directly related to the exact solutions. Finally, we provide a novel analytic approximation of the kernels that we briefly compare to the exact kernels.