Fourier Transform on the Homogeneous Space of 3D Positions and Orientations for Exact Solutions to Linear PDEs
Department of Mathematics and Computer Science (CASA), Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
Author to whom correspondence should be addressed.
Received: 31 October 2018 / Revised: 10 December 2018 / Accepted: 18 December 2018 / Published: 8 January 2019
Fokker–Planck PDEs (including diffusions) for stable Lévy processes (including Wiener processes) on the joint space of positions and orientations play a major role in mechanics, robotics, image analysis, directional statistics and probability theory. Exact analytic designs and solutions are known in the 2D case, where they have been obtained using Fourier transform on
. Here, we extend these approaches to 3D using Fourier transform on the Lie group
of rigid body motions. More precisely, we define the homogeneous space of 3D positions and orientations
as the quotient in
. In our construction, two group elements are equivalent if they are equal up to a rotation around the reference axis. On this quotient, we design a specific Fourier transform. We apply this Fourier transform to derive new exact solutions to Fokker–Planck PDEs of
-stable Lévy processes on
. This reduces classical analysis computations and provides an explicit algebraic spectral decomposition of the solutions. We compare the exact probability kernel for
(the diffusion kernel) to the kernel for
(the Poisson kernel). We set up stochastic differential equations (SDEs) for the Lévy processes on the quotient and derive corresponding Monte-Carlo methods. We verified that the exact probability kernels arise as the limit of the Monte-Carlo approximations.
This is an open access article distributed under the Creative Commons Attribution License
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
Share & Cite This Article
MDPI and ACS Style
Duits, R.; Bekkers, E.J.; Mashtakov, A. Fourier Transform on the Homogeneous Space of 3D Positions and Orientations for Exact Solutions to Linear PDEs. Entropy 2019, 21, 38.
Duits R, Bekkers EJ, Mashtakov A. Fourier Transform on the Homogeneous Space of 3D Positions and Orientations for Exact Solutions to Linear PDEs. Entropy. 2019; 21(1):38.
Duits, Remco; Bekkers, Erik J.; Mashtakov, Alexey. 2019. "Fourier Transform on the Homogeneous Space of 3D Positions and Orientations for Exact Solutions to Linear PDEs." Entropy 21, no. 1: 38.
Show more citation formats
Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.
[Return to top]
For more information on the journal statistics, click here
Multiple requests from the same IP address are counted as one view.