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

Remco Duits, Erik J. Bekkers, Alexey Mashtakov

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Uittreksel

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.

TaalEngels
Artikelnummer38
Aantal pagina's38
TijdschriftEntropy
Volume21
Nummer van het tijdschrift1
DOI's
StatusGepubliceerd - 8 jan 2019

Vingerafdruk

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

Bibliografische nota

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.

Trefwoorden

    Citeer dit

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    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",
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    doi = "10.3390/e21010038",
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    volume = "21",
    journal = "Entropy",
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    Fourier transform on the homogeneous space of 3D positions and orientations for exact solutions to linear PDEs. / Duits, Remco; Bekkers, Erik J.; Mashtakov, Alexey.

    In: Entropy, Vol. 21, Nr. 1, 38, 08.01.2019.

    Onderzoeksoutput: Bijdrage aan tijdschriftTijdschriftartikelAcademicpeer 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

    VL - 21

    JO - Entropy

    T2 - Entropy

    JF - Entropy

    SN - 1099-4300

    IS - 1

    M1 - 38

    ER -