On sub-Riemannian geodesics in $SE(3)$ whose spatial projections do not have cusps

Duits, Remco; Ghosh, Arpan; Haije, Tom Dela; Mashtakov, Alexey

doi:10.48550/arxiv.1305.6061

Cited by 1 publication

(1 citation statement)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Future work will include extensive analysis of the Monte-Carlo simulations, briefly described in Appendix E. Furthermore, we will put connections of hypo-elliptic diffusion (and Brownian bridges) on SE(3) and recently derived sub-Riemannian geodesics in SE(3) [44].…”

Section: Discussionmentioning

confidence: 99%

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

Portegies

Duits

2017

Differential Geometry and its Applications

View full text Add to dashboard Cite

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.

show abstract

Section: Discussionmentioning

confidence: 99%