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Entropy 2019, 21(1), 38; https://doi.org/10.3390/e21010038

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
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Received: 31 October 2018 / Revised: 10 December 2018 / Accepted: 18 December 2018 / Published: 8 January 2019
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Abstract

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 S E ( 2 ) . Here, we extend these approaches to 3D using Fourier transform on the Lie group S E ( 3 ) of rigid body motions. More precisely, we define the homogeneous space of 3D positions and orientations R 3 S 2 : = S E ( 3 ) / ( { 0 } × S O ( 2 ) ) as the quotient in S E ( 3 ) . 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 α -stable Lévy processes on R 3 S 2 . This reduces classical analysis computations and provides an explicit algebraic spectral decomposition of the solutions. We compare the exact probability kernel for α = 1 (the diffusion kernel) to the kernel for α = 1 2 (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. View Full-Text
Keywords: fourier transform; rigid body motions; partial differential equations; Lévy processes; Lie Groups; homogeneous spaces; stochastic differential equations fourier transform; rigid body motions; partial differential equations; Lévy processes; Lie Groups; homogeneous spaces; stochastic differential equations
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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.

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