New exact and numerical solutions of the (convection–)diffusion kernels on SE(3)

Abstract

We consider hypo-elliptic diffusion and convection-diffusion on R 3 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.

Measure-Valued Variational Models with Applications to Diffusion-Weighted Imaging

“…Furthermore, modifications of our total variation seminorm that take into account the coupling of positions and orientations according to the physical interpretation of ODFs in DW-MRI could close the gap to state-of-the-art approaches such as [28,63].…”

“…This approach has been used to enhance the quality of CSD as a prior in a variational formulation [67] or in a post-processing step [64] that also includes additional angular regularization. Due to the linearity of the underlying linear PDE, convolution-based explicit solution formulas are available [28,63]. Implemented efficiently [55,54], they outperform our more computationally demanding model, which is not tied to the specific application of DW-MRI, but allows arbitrary metric spaces.…”

Total Variation and Mean Curvature PDEs on the Space of Positions and Orientations

“…For {a, b} = {0, 1} we have a Total Variation Flow (TVF) [8]. For {a, b} = {0, 0} we obtain a linear diffusion for which exact smooth solutions exist [27]. Remark 1.…”

Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D

“…We reason that the Folland-Kaplan-Korányi provides an accurate approximation to the fundamental solution on SE(n) as well, as it provides the exact fundamental solution on the Heisenberg type approximation (SE(n)) 0 . As such, we provide an approach to approximating the heat kernel and fundamental solution of the sub-Laplacian on SE(n), as an alternative to the works [22,47,12].…”

“…Also here, the Folland-Kaplan-Korányi-type norm can be used to approximate the fundamental solutions of the sub-Laplacian on SE(3). The norm ||c|| ξ,ζ with ζ = 1 was for example used in [23] approximations of the heat kernel and the fundamental solution on SE(3), of which only recently exact solutions were found in [47]. In the context of this paper, we can approximate the exact solutions of the sub-Laplacian on SE(3) by c 2−Q 1,16 , with homogeneous dimensions Q = 9, as the exact solution of the sub-Laplacian on the approximation group (SE(3)) 0 .…”