Surface Matching Using Normal Cycles

  • Pierre RoussillonEmail author
  • Joan Alexis Glaunès
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10589)


In this article we develop in the case of triangulated meshes the notion of normal cycle as a dissimilarity measure introduced in [13]. Our construction is based on the definition of kernel metrics on the space of normal cycles which take explicit expressions in a discrete setting. We derive the computational setting for discrete surfaces, using the Large Deformation Diffeomorphic Metric Mapping framework as model for deformations. We present experiments on real data and compare with the varifolds approach.


  1. 1.
    Arguillère, S., Trélat, E., Trouvé, A., Younès, L.: Shape deformation analysis from the optimal control viewpoint. Journal de Mathématiques Pures et Appliquées 104, 139–178 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Beg, M.F., Miller, M.I., Trouvé, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. J. Comput. Vision 61(2), 139–157 (2005)CrossRefGoogle Scholar
  3. 3.
    Bruveris, M., Risser, L., Vialard, F.X.: Mixture of kernels and iterated semidirect product of diffeomorphisms groups. Multiscale Modeling Simul. 10(4), 1344–1368 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Charon, N.: Analysis of geometric and fshapes with extension of currents. Application to registration and atlas estimation. Ph.D. thesis, ÉNS Cachan (2013)Google Scholar
  5. 5.
    Durrleman, S.: Statistical models of currents for measuring the variability of anatomical curves, surfaces and their evolution. Ph.D. thesis, Université Nice, Sophia Antipolis (2010)Google Scholar
  6. 6.
    Federer, H.: Curvature measures. Trans. Amer. Maths. Soc. 93, 418–491 (1959)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Glaunès, J.: Transport par difféomorphismes de points, de mesures et de courants pour la comparaison de formes et l’anatomie numérique. Ph.D. thesis, Université Paris 13 (2005)Google Scholar
  8. 8.
    Glaunès, J., Qiu, A., Miller, M., Younes, L.: Large deformation diffeomorphic metric curve mapping. Int. J. Comput. Vision 80(3), 317–336 (2008)CrossRefGoogle Scholar
  9. 9.
    Lee, S., Charon, N., Charlier, B., Popuri, K., Lebed, E., Sarunic, M., Trouvé, A., Beg, M.: Atlas-based shape analysis and classification of retinal optical coherence tomography images using the fshape framework. Med. Image Anal. 35, 570–581 (2016)CrossRefGoogle Scholar
  10. 10.
    Lee, S., Han, S.X., Young, M., Beg, M.F., Sarunic, M.V., Mackenzie, P.J.: Optic nerve head and peripapillary morphometrics in myopic glaucoma. Invest. Ophthalmol. Vis. Sci. 55(7), 4378 (2014)CrossRefGoogle Scholar
  11. 11.
    Liu, D.C., Nocedal, J.: On the limited memory BFGS method for large scale optimization. Math. Program. 45(1–3), 503–528 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Miller, M.I., Trouvé, A., Younes, L.: Geodesic shooting for computational anatomy. J. Math. Imaging Vis. 24(2), 209–228 (2006)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Roussillon, P., Glaunès, J.: Kernel metrics on normal cycles and application to curve matching. SIAM J. Imaging Sci. 9, 1991–2038 (2016)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Vaillant, M., Glaunès, J.: Surface matching via currents. In: Christensen, G.E., Sonka, M. (eds.) IPMI 2005. LNCS, vol. 3565, pp. 381–392. Springer, Heidelberg (2005). doi: 10.1007/11505730_32 CrossRefGoogle Scholar
  15. 15.
    Zähle, M.: Curvatures and currents for unions of set with positive reach. Geom. Dedicata. 23, 155–171 (1987)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.MAP5, UMR 8145 CNRS, Université Paris Descartes, Sorbonne Paris CitéParisFrance

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