Fourier transform on the homogeneous space of 3D positions and orientations for exact solutions to linear PDEs

Abstract

Fokker-Planck PDEs (including diffusions) for stable Lévy processes (includingWiener 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 SE(2). Here, we extend these approaches to 3D using Fourier transform on the Lie group SE(3) of rigid body motions. More precisely, we define the homogeneous space of 3D positions and orientations ℝ3 ⋊ S2 := SE(3)/0 × SO(2)) as the quotient in SE(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 a-stable Lévy processes on ℝ3 ⋊ S2. 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.