Point-set registration in bounded domains via the Fokker–Planck equation

Angelo Iollo; Tommaso Taddei

doi:10.5802/crmath.753

Research article - Numerical analysis

Point-set registration in bounded domains via the Fokker–Planck equation

Angelo Iollo ^1,²; Tommaso Taddei ^1,²

¹ IMB, UMR 5251, Univ. Bordeaux, 33400 Talence, France
² Centre Inria de l’Université de Bordeaux, Team MEMPHIS, 33400 Talence, France

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 809-824.

Abstract (Original language)
Résumé

We present a point set registration method in bounded domains based on the solution to the Fokker–Planck equation. Our approach leverages (i) density estimation based on Gaussian mixture models; (ii) a stabilized finite element discretization of the Fokker–Planck equation; (iii) a specialized method for the integration of the particles. We review relevant properties of the Fokker–Planck equation that provide the foundations for the numerical method. We discuss two strategies for the integration of the particles and we propose a regularization technique to control the distance of the particles from the boundary of the domain. We perform extensive numerical experiments for two two-dimensional model problems to illustrate the many features of the method.

Nous présentons une méthode d’alignement de nuages de points dans des domaines bornés basée sur la solution de l’équation de Fokker–Planck. Notre approche repose sur : (i) l’estimation de densité à l’aide de modèles de mélange gaussien ; (ii) une discrétisation par éléments finis stabilisée de l’équation de Fokker–Planck ; (iii) une méthode spécialisée pour l’intégration des particules. Nous examinons les propriétés pertinentes de l’équation de Fokker–Planck qui sous-tendent la méthode numérique. Nous discutons deux stratégies pour l’intégration des particules et proposons une technique de régularisation pour contrôler la distance des particules à la frontière du domaine. Nous réalisons de nombreuses expériences numériques pour deux problèmes modèles bidimensionnels afin d’illustrer l’efficacité de la méthode.

Received: 2024-12-12
Revised: 2025-03-13
Accepted: 2025-04-21
Published online: 2025-07-04

DOI: 10.5802/crmath.753

Keywords: Point set registration, Fokker–Planck, model order reduction
Mots-clés : Alignement de nuages de points, équation de Fokker–Planck, réduction de modèle

Author's affiliations:

Angelo Iollo ^{1, 2}; Tommaso Taddei ^{1, 2}

¹ IMB, UMR 5251, Univ. Bordeaux, 33400 Talence, France
² Centre Inria de l’Université de Bordeaux, Team MEMPHIS, 33400 Talence, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Angelo Iollo and Tommaso Taddei},
     title = {Point-set registration in bounded domains via the {Fokker{\textendash}Planck} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {809--824},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.753},
     language = {en},
}

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Angelo Iollo; Tommaso Taddei. Point-set registration in bounded domains via the Fokker–Planck equation. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 809-824. doi : 10.5802/crmath.753. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.753/

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