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Geometry of curves with application to aircraft trajectories analysis
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 3, pp. 483-504.

Cet article fait un état de l’art des métriques pouvant être utilisées sur les espaces de courbes que celles-ci soient définies à partir de points de référence ou comme immersions ou plongements. Dans ce dernier cas, l’espace final est obtenu en quotientant par l’action d’un groupe de difféormorphismes afin d’assurer l’invariance par changement de paramètrage. La détermination de la métrique adéquate pour une classe de problèmes est un sujet de recherche actif, spécialement dans les domaines de la vision par ordinateur ou de la reconnaissance de formes. Des questions similaires se posent pour l’analyse des trajectoires d’avions dans le cadre de la gestion du trafic. En dépit de son importance, peu d’études ont été menées sur ce sujet, en grande partie par absence d’un cadre théorique adapté. L’utilisation des espaces de courbes ou de formes pour représenter les vols ainsi qu’un exemple d’application à la classification des trajectoires seront présentés en seconde partie de l’article.

This article presents a survey of some metrics that can be used on geometric curve spaces which can be defined using samples points, known as landmarks, or by taking a space of immersions or embeddings and quotienting out by a group of diffeomorphisms in order to get rid of the influence of the parametrization. Finding the right metric for a class of problems is an active topic of research, with a special emphasis on applications related to computer vision or shape recognition. Similar problems arise in the field of air traffic management where the analysis of aircraft trajectories is one of the most basic issues. Despite its importance, only a few studies have been conducted on the subject, mainly due to the lack of suitable frameworks. The use of some of the shape spaces for representing aircraft flight paths, along with an example of trajectory classification will be given in the second part of the article.

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DOI : https://doi.org/10.5802/afst.1452
@article{AFST_2015_6_24_3_483_0,
     author = {St\'ephane Puechmorel},
     title = {Geometry of curves with application to aircraft trajectories analysis},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {483--504},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 24},
     number = {3},
     year = {2015},
     doi = {10.5802/afst.1452},
     mrnumber = {3403729},
     zbl = {1325.53117},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1452/}
}
Stéphane Puechmorel. Geometry of curves with application to aircraft trajectories analysis. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 3, pp. 483-504. doi : 10.5802/afst.1452. https://afst.centre-mersenne.org/articles/10.5802/afst.1452/

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