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Representation of surfaces with normal cycles and application to surface registration. (English) Zbl 1468.68270

Summary: In this paper, we present a framework for computing dissimilarities between surfaces which is based on the mathematical model of normal cycle from geometric measure theory. This model allows to take into account all the curvature information of the surface without explicitly computing it. By defining kernel metrics on normal cycles, we define explicit distances between surfaces that are sensitive to curvature. This mathematical framework also has the advantage of encompassing both continuous and discrete surfaces (triangulated surfaces). We then use this distance as a data attachment term for shape matching, using the large deformation diffeomorphic metric mapping for modelling deformations. We also present an efficient numerical implementation of this problem in PyTorch, using the KeOps library, which allows both the use of auto-differentiation tools and a parallelization of GPU calculations without memory overflow. We show that this method can be scalable on data up to a million points, and we present several examples on surfaces, comparing the results with those obtained with the similar varifold framework.

MSC:

68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
28A75 Length, area, volume, other geometric measure theory
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
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[1] Allard, W.: On the first variation of a varifold. Ann. Math. 95(3), 417-491 (1972) · Zbl 0252.49028 · doi:10.2307/1970868
[2] Almgren, F.: Plateau’s Problem: An Invitation to Varifold Geometry. Mathematical Library, Amsterdam (1966) · Zbl 0165.13201
[3] Arguillère, S., Trélat, E., Trouvé, A., Younès, L.: Shape deformation analysis from the optimal control viewpoint. J. Math. Pures Appl. 104(1), 139-178 (2015) · Zbl 1319.49064 · doi:10.1016/j.matpur.2015.02.004
[4] Aronszajn, N.: Theory of reproducing kernels. Trans. Am. Math. Soc. 68, 337-404 (1950) · Zbl 0037.20701 · doi:10.1090/S0002-9947-1950-0051437-7
[5] Buet, B., Leonardi, G.P., Masnou, S.: A varifold approach to surface approximation. Arch. Ration. Mech. Anal. 226(2), 639-694 (2017) · Zbl 1378.49052 · doi:10.1007/s00205-017-1141-0
[6] Carmeli, C., De Vito, E., Toigo, A., Umanit, V.: Vector valued reproducing kernel Hilbert spaces and universality. arXiv:0807.1659 [math] (2008) · Zbl 1195.46025
[7] Charlier, B., Charon, N., Trouvé, A.: The fshape framework for the variability analysis of functional shapes. In: Foundations of Computational Mathematics, pp. 1-71 (2015). https://doi.org/10.1007/s10208-015-9288-2 · Zbl 1376.49054
[8] Charlier, B., Feydy, J., Glaunès, J.: Kernel operations on the GPU, with autodiff, without memory overflows. http://www.kernel-operations.io/. Accessed 21 Dec 2018 · Zbl 07370591
[9] Charlier, B., Feydy, J., Glaunès, J.: KeOps: Calcul rapide sur GPU dans les espaces à noyaux. In: Proceedings of Journées de Statistique de la SFdS. Paris, France (2018)
[10] Charon, N.: Analysis of geometric and functionnal shapes with extension of currents. Application to registration and atlas estimation. Ph.D. thesis, École Normale Supérieure de Cachan (2013)
[11] Charon, N., Trouvé, A.: The varifold representation of nonoriented shapes for diffeomorphic registration. SIAM J. Imaging Sci. 6(4), 2547-2580 (2013) · Zbl 1279.68313 · doi:10.1137/130918885
[12] Chazal, F., Cohen-Steiner, D., Lieutier, A., Thibert, B.: Stability of curvature measures. CoRR arXiv:0812.1390 (2008) · Zbl 1388.53007
[13] Chazal, F., Cohen-Steiner, D., Lieutier, A., Thibert, B.: Stability of curvature measures. Comput. Graph. Forum (2009). https://doi.org/10.1111/j.1467-8659.2009.01525.x · Zbl 1388.53007 · doi:10.1111/j.1467-8659.2009.01525.x
[14] Cignoni, P., Callieri, M., Corsini, M., Dellepiane, M., Ganovelli, F., Ranzuglia, G.: MeshLab: an open-source mesh processing tool. In: Scarano, V., Chiara, R.D., Erra, U. (eds.) Eurographics Italian Chapter Conference. The Eurographics Association (2008). https://doi.org/10.2312/LocalChapterEvents/ItalChap/ItalianChapConf2008/129-136
[15] Cohen-Steiner, D., Morvan, J.M.: Restricted Delaunay triangulations and normal cycle. In: SoCG’03 (2003) · Zbl 1422.65051
[16] Cohen-Steiner, D., Morvan, J.M.: Second fundamental measure of geometric sets and local approximation of curvatures. J. Differ. Geom. 74(3), 363-394 (2006). https://doi.org/10.4310/jdg/1175266231 · Zbl 1107.49029 · doi:10.4310/jdg/1175266231
[17] Csernansky, J., Wang, L., Swank, J., Miller, J., Gado, M., McKeel, D., Miller, M., Morris, J.: Preclinical detection of Alzheimer’s disease: hippocampal shape and volume predict dementia onset in the elderly. NeuroImage 25(3), 783-792 (2005) · doi:10.1016/j.neuroimage.2004.12.036
[18] Csernansky, J.G., Wang, L., Joshi, S.C., Ratnanather, J.T., Miller, M.I.: Computational anatomy and neuropsychiatric disease: probabilistic assessment of variation and statistical inference of group difference, hemispheric asymmetry, and time-dependent change. NeuroImage 23(Supplement 1), S56-S68 (2004). Mathematics in Brain Imaging · doi:10.1016/j.neuroimage.2004.07.025
[19] Durrleman, S., Allassonnière, S., Joshi, S.: Sparse adaptive parameterization of variability in image ensembles. Int. J. Comput. Vis. 101(1), 161-183 (2013) · Zbl 1259.68196 · doi:10.1007/s11263-012-0556-1
[20] Durrleman, S., Fillard, P., Pennec, X., Trouvé, A., Ayache, N.: Registration, atlas estimation and variability analysis of white matter fiber bundles modeled as currents. NeuroImage 55(3), 1073-1090 (2011) · doi:10.1016/j.neuroimage.2010.11.056
[21] Durrleman, S., Prastawa, M., Charon, N., Korenberg, J.R., Joshi, S., Gerig, G., Trouvé, A.: Morphometry of anatomical shape complexes with dense deformations and sparse parameters. NeuroImage 101, 35-49 (2014) · doi:10.1016/j.neuroimage.2014.06.043
[22] Federer, H.: Curvature measures. Trans. Am. Math. Soc. 93, 418-491 (1959) · Zbl 0089.38402 · doi:10.1090/S0002-9947-1959-0110078-1
[23] Federer, H.: Geometric Measure Theory. Springer, Berlin (1969) · Zbl 0176.00801
[24] Federer, H., Fleming, W.: Normal and integral currents. Ann. Math. 72, 458-520 (1960) · Zbl 0187.31301 · doi:10.2307/1970227
[25] Feydy, J., Charlier, B., Vialard, F.X., Peyré, G.: Optimal transport for diffeomorphic registration. In: MICCAI 2017, Proceedings of MICCAI 2017, Quebec, Canada (2017)
[26] 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)
[27] Glaunès, J., Qiu, A., Miller, M., Younes, L.: Large deformation diffeomorphic metric curve mapping. Int. J. Comput. Vis. 80(3), 317-336 (2008). https://doi.org/10.1007/s11263-008-0141-9 · Zbl 1477.68471 · doi:10.1007/s11263-008-0141-9
[28] Grenander, U., Miller, M.I.: Computational anatomy: an emerging discipline. Q. Appl. Math. LVI(4), 617-694 (1998) · Zbl 0952.92016 · doi:10.1090/qam/1668732
[29] Helm, P.A., Younes, L., Beg, M.F., Ennis, D.B., Leclercq, C., Faris, O.P., McVeigh, E., Kass, D., Miller, M.I., Winslow, R.L.: Evidence of structural remodeling in the dyssynchronous failing heart. Circ. Res. 98(1), 125-132 (2006). https://doi.org/10.1161/01.RES.0000199396.30688.eb · doi:10.1161/01.RES.0000199396.30688.eb
[30] Kaltenmark, I., Charlier, B., Charon, N.: A general framework for curve and surface comparison and registration with oriented varifolds. In: The IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (2017)
[31] Lee, S., Charon, N., Charlier, B., Popuri, K., Lebed, E., Sarunic, M.V., Trouvé, A., Beg, M.F.: Atlas-based shape analysis and classification of retinal optical coherence tomography images using the functional shape (fshape) framework. Med. Image Anal. 35, 570-581 (2017) · doi:10.1016/j.media.2016.08.012
[32] Lee, S., Heisler, M.L., Popuri, K., Charon, N., Charlier, B., Trouvé, A., Mackenzie, P.J., Sarunic, M.V., Beg, M.F.: Age and glaucoma-related characteristics in retinal nerve fiber layer and choroid: localized morphometrics and visualization using functional shapes registration. Front. Neurosci. 11, 381 (2017) · doi:10.3389/fnins.2017.00381
[33] Liu, D.C., Nocedal, J.: On the limited memory BFGS method for large scale optimization. Math. Program. 45(1-3), 503-528 (1989). https://doi.org/10.1007/BF01589116 · Zbl 0696.90048 · doi:10.1007/BF01589116
[34] Mansi, T., Voigt, I., Leonardi, B., Pennec, X., Durrleman, S., Sermesant, M., Delingette, H., Taylor, A.M., Boudjemline, Y., Pongiglione, G., Ayache, N.: A statistical model for quantification and prediction of cardiac remodelling: application to tetralogy of fallot. IEEE Trans. Med. Imaging 30(9), 1605-1616 (2011). https://doi.org/10.1109/TMI.2011.2135375 · doi:10.1109/TMI.2011.2135375
[35] Miller, M.I., Trouvé, A., Younes, L.: Geodesic shooting for computational anatomy. J. Math. Imaging Vis. 24(2), 209-228 (2006) · Zbl 1478.92084 · doi:10.1007/s10851-005-3624-0
[36] Morvan, J.M.: Generalized Curvatures. Springer, Berlin (2008) · Zbl 1149.53001 · doi:10.1007/978-3-540-73792-6
[37] Paszke, A., Gross, S., Chintala, S., Chanan, G., Yang, E., DeVito, Z., Lin, Z., Desmaison, A., Antiga, L., Lerer, A.: Automatic differentiation in pytorch. In: NIPS-W (2017)
[38] Pennec, X.: Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vis. 25(1), 127-154 (2006) · Zbl 1478.94072 · doi:10.1007/s10851-006-6228-4
[39] Qiu, A., Younes, L., Miller, M.I., Csernansky, J.G.: Parallel transport in diffeomorphisms distinguishes the time-dependent pattern of hippocampal surface deformation due to healthy aging and the dementia of the Alzheimer’s type. NeuroImage 40(1), 68-76 (2008) · doi:10.1016/j.neuroimage.2007.11.041
[40] Rataj, J., Zähle, M.: Curvatures and currents for unions of sets with positive reach, II. Ann. Global Anal. Geom. 20(1), 1-21 (2001) · Zbl 0997.53062 · doi:10.1023/A:1010624214933
[41] Rekik, I., Li, G., Lin, W., Shen, D.: Multidirectional and topography-based dynamic-scale varifold representations with application to matching developing cortical surfaces. NeuroImage 135, 152-162 (2016) · doi:10.1016/j.neuroimage.2016.04.037
[42] Roussillon, P.: Modèle de cycles normaux pour l’analyse des dformations. Ph.D. thesis, Université Paris Descartes (2017)
[43] Roussillon, P., Glaunès, J.: Kernel metrics on normal cycles and application to curve matching. SIAM J. Imaging Sci. 9, 1991-2038 (2016) · Zbl 1364.53071 · doi:10.1137/16M1070529
[44] Roussillon, P., Glaunès, J.: Surface matching using normal cycles. In: GSI’17: Geometric Science Information, 2017, Paris (2017) · Zbl 1425.53017
[45] Tang, X., Holland, D., Dale, A.M., Younes, L., Miller, M.I.: for the Alzheimer’s disease neuroimaging initiative: shape abnormalities of subcortical and ventricular structures in mild cognitive impairment and Alzheimer’s disease: detecting, quantifying, and predicting. Hum. Brain Mapp. 35(8), 3701-3725 (2014) · doi:10.1002/hbm.22431
[46] Thäle, C.: 50 years sets with positive reach, a survey. Surv. Math. Appl. 3, 125-165 (2008) · Zbl 1173.49039
[47] Vaillant, M.; Glaunès, J.; Christensen, GE (ed.); Sonka, M. (ed.), Surface matching via currents, 381-392 (2005), Berlin
[48] Wang, L., Beg, F., Ratnanather, T., Ceritoglu, C., Younes, L., Morris, J.C., Csernansky, J.G., Miller, M.I.: Large deformation diffeomorphism and momentum based hippocampal shape discrimination in dementia of the Alzheimer type. IEEE Trans. Med. Imaging 26(4), 462-470 (2007). https://doi.org/10.1109/TMI.2006.887380 · doi:10.1109/TMI.2006.887380
[49] Younès, L.: Shapes and Diffeomorphisms. Springer, Berlin (2010) · Zbl 1205.68355 · doi:10.1007/978-3-642-12055-8
[50] Zähle, M.: Integral and current representation of Federer’s curvature measure. Arch. Maths. 23, 557-567 (1986) · Zbl 0598.53058 · doi:10.1007/BF01195026
[51] Zähle, M.: Curvatures and currents for unions of set with positive reach. Geom. Dedicata 23, 155-171 (1987) · Zbl 0627.53053 · doi:10.1007/BF00181273
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