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

Remco Duits (Corresponding author), Erik J. Bekkers, Alexey Mashtakov

Research output: Contribution to journalArticleAcademicpeer-review

1 Citation (Scopus)
21 Downloads (Pure)

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.

Original languageEnglish
Article number38
Number of pages38
JournalEntropy
Volume21
Issue number1
DOIs
Publication statusPublished - 8 Jan 2019

Fingerprint

pulse detonation engines
quotients
rigid structures
robotics
image analysis
Monte Carlo method
differential equations
statistics
decomposition
approximation

Bibliographical note

This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century.

Keywords

  • Fourier transform
  • Homogeneous spaces
  • Lie Groups
  • Lévy processes
  • Partial differential equations
  • Rigid body motions
  • Stochastic differential equations

Cite this

@article{143ca2def14e4ff09986c878d6da384a,
title = "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 L{\'e}vy 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 L{\'e}vy 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 L{\'e}vy 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.",
keywords = "Fourier transform, Homogeneous spaces, Lie Groups, L{\'e}vy processes, Partial differential equations, Rigid body motions, Stochastic differential equations",
author = "Remco Duits and Bekkers, {Erik J.} and Alexey Mashtakov",
note = "This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century.",
year = "2019",
month = "1",
day = "8",
doi = "10.3390/e21010038",
language = "English",
volume = "21",
journal = "Entropy",
issn = "1099-4300",
publisher = "Multidisciplinary Digital Publishing Institute (MDPI)",
number = "1",

}

Fourier transform on the homogeneous space of 3D positions and orientations for exact solutions to linear PDEs. / Duits, Remco (Corresponding author); Bekkers, Erik J.; Mashtakov, Alexey.

In: Entropy, Vol. 21, No. 1, 38, 08.01.2019.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Fourier transform on the homogeneous space of 3D positions and orientations for exact solutions to linear PDEs

AU - Duits, Remco

AU - Bekkers, Erik J.

AU - Mashtakov, Alexey

N1 - This article belongs to the Special Issue Joseph Fourier 250th Birthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst century.

PY - 2019/1/8

Y1 - 2019/1/8

N2 - 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.

AB - 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.

KW - Fourier transform

KW - Homogeneous spaces

KW - Lie Groups

KW - Lévy processes

KW - Partial differential equations

KW - Rigid body motions

KW - Stochastic differential equations

UR - http://www.scopus.com/inward/record.url?scp=85060402262&partnerID=8YFLogxK

U2 - 10.3390/e21010038

DO - 10.3390/e21010038

M3 - Article

AN - SCOPUS:85060402262

VL - 21

JO - Entropy

JF - Entropy

SN - 1099-4300

IS - 1

M1 - 38

ER -