Geometry of curves with application to aircraft trajectories analysis

Puechmorel, Stéphane

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.

[1] Bauer (M.), M. Bruveris (M.), S. Marsland (S.), and Michor (P. W.).— Constructing reparametrization invariant metrics on spaces of plane curves. ArXiv e-prints, July 2012. | MR

[2] Carne (T.K.).— The geometry of shape space. Proc. of the London Math. Society, p. 407-432 (1989). | MR | Zbl

[3] Enriquez (M.).— Identifying temporally persistent ows in the terminal airspace via spectral clustering. In Tenth USA/Europe Air Trafic Management Research and Development Seminar (ATM2013), (2013).

[4] EUROCONTROL/NMD/STATFOR. Eurocontrol seven-year forecast. https: //www.eurocontrol.int/sites/default/files/content/documents/official- documents/forecasts/seven-year-flights-service-units-forecast-2014-2020- sep2014.pdf (2014).

[5] Harman (W.H.).— Air trafic density and distribution measurements. Lincoln Laboratory, MIT, Report ATC-80, May 1979.

[6] Hurter (C.), Ersoy (O.), and Telea (A.).— Smooth bundling of large streaming and sequence graphs. In Visualization Symposium (Pacific Vis), 2013 IEEE Pacific, p. 41-48, Feb 2013.

[7] Kendall (D. G.).— Shape manifolds, procrustean metrics, and complex projective spaces. Bulletin of the London Mathematical Society, 16(2), p. 81-121 (1984). | MR | Zbl

[8] Krakowski (K.), Hüper (K.), and Manton (J.).— On the computation of the karcher mean on spheres and special orthogonal groups. Proc. Workshop Robot. Math. (RoboMat07) (2007).

[9] Le (H.).— On geodesics in euclidean shape spaces. Journal of the London Mathematical Society, s2-44(2), p. 360-372 (1991). | MR | Zbl

[10] Mennucci (A. C. G.), Soatto (S.), Sundaramoorthi (G.), and Yezzi (A.).— A new geometric metric in the space of curves, and applications to tracking deforming objects by prediction and filtering. SIAM Journal on Imaging Sciences, 4(1) (2011). | MR | Zbl

[11] Michor (P. W.) and David Mumford (D.).— Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Documenta Math, p. 217-245 (2005). | EuDML | MR | Zbl

[12] Michor (P. W.) and Mumford (D.).— Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. (JEMS), 8, p. 1-48 (2006). | EuDML | MR | Zbl

[13] Michor (P. W.) and Mumford (D.).— An overview of the riemannian metrics on spaces of curves using the hamiltonian approach. Applied and Computational Harmonic Analysis, 23(1), p. 74-113 (2007). | MR | Zbl

[14] O’Neill (B.).— The fundamental equations of a submersion. The Michigan Mathematical Journal, 13(4), p. 459-469, 12 (1966). | MR | Zbl

[15] Sibson (R.).— Studies in the robustness of multidimensional scaling: Perturbational analysis of classical scaling. Journal of the Royal Statistical Society. Series B (Methodological), 41(2), p. 217-229 (1979). | MR | Zbl

[16] Srivastava (A.), Eric Klassen (E.), Shantanu H. Joshi (S. H.), and Jermyn (I. H.).— Shape analysis of elastic curves in euclidean spaces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(7) p. 1415-1428 (2011).