×

zbMATH — the first resource for mathematics

A relaxed approach for curve matching with elastic metrics. (English) Zbl 07194611
Summary: In this paper, we study a class of Riemannian metrics on the space of unparametrized curves and develop a method to compute geodesics with given boundary conditions. It extends previous works on this topic in several important ways. The model and resulting matching algorithm integrate within one common setting both the family of \(H^2\)-metrics with constant coefficients and scale-invariant \(H^2\)-metrics on both open and closed immersed curves. These families include as particular cases the class of first-order elastic metrics. An essential difference with prior approaches is the way that boundary constraints are dealt with. By leveraging varifold-based similarity metrics we propose a relaxed variational formulation for the matching problem that avoids the necessity of optimizing over the reparametrization group. Furthermore, we show that we can also quotient out finite-dimensional similarity groups such as translation, rotation and scaling groups. The different properties and advantages are illustrated through numerical examples in which we also provide a comparison with related diffeomorphic methods used in shape registration.

MSC:
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
Software:
femshape; HANSO
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] N. Aronszajn, Theory of reproducing kernels. Trans. Am. Math. Soc. 68 (1950) 337-404. · Zbl 0037.20701
[2] M. Bauer, M. Bruveris, N. Charon and J. Møller-Andersen, Varifold-Based Matching of Curves via Sobolev-Type Riemannian Metrics. In 6th MICCAI Workshop on Mathematical Foundations of Computational Anatomy (2017) 152-163.
[3] M. Bauer, M. Bruveris, P. Harms and P.W. Michor, Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation. Ann. Global Anal. Geom. 41 (2012) 461-472. · Zbl 1246.58005
[4] M. Bauer, M. Bruveris, P. Harms and J. Møller-Andersen, Curve Matching with Applications in Medical Imaging. In 5th MICCAI Workshop on Mathematical Foundations of Computational Anatomy (2015).
[5] M. Bauer, M. Bruveris, P. Harms and J. Møller-Andersen, A numerical framework for Sobolev metrics on the space of curves. SIAM J. Imaging Sci. 10 (2017) 47-73. · Zbl 1367.49021
[6] M. Bauer, M. Bruveris, S. Marsland and P.W. Michor, Constructing reparameterization invariant metrics on spaces of plane curves. Differ. Geom. Appl. 34 (2014) 139-165. · Zbl 1291.58002
[7] M. Bauer, M. Bruveris and P.W. Michor, Overview of the geometries of shape spaces and diffeomorphism groups. J. Math. Imaging Vis. 50 (2014) 60-97. · Zbl 1310.58005
[8] M. Bauer, M. Bruveris and P.W. Michor, Why use Sobolev metrics on the space of curves, in Riemannian Computing in Computer Vision. Springer, Cham (2016) 233-255. · Zbl 1338.65043
[9] M. Bauer, M. Eslitzbichler and M. Grasmair, Landmark-guided elastic shape analysis of human character motions. Inverse Probl. Imaging 11 (2017) 601-621. · Zbl 1368.65028
[10] M. Bauer and P. Harms, Metrics on spaces of immersions where horizontality equals normality. Differ. Geom. Appl. 39 (2015) 166-183. · Zbl 1321.58007
[11] M.F. Beg, M.I. Miller, A. Trouvé and L. Younes, Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int. J. Comput. Vis. 61 (2005) 139-157.
[12] J. Benn, S. Marsland, R. McLachlan, K. Modin and O. Verdier, Currents and finite elements as tools for shapespace (2017). Preprint . · Zbl 07122009
[13] M. Bruveris, Completeness properties of Sobolev metrics on the space of curves. J. Geom. Mech. 7 (2015) 125-150. · Zbl 1328.58007
[14] M. Bruveris, Regularity of maps between Sobolev spaces. Ann. Global Anal. Geom. 52 (2017) 11-24. · Zbl 1373.58005
[15] M. Bruveris, P.W. Michor and D. Mumford, Geodesic completeness for Sobolev metrics on the space of immersed plane curves. Forum Math. Sigma 2 (2014) e19. · Zbl 1315.58009
[16] M. Bruveris and J. Møller-Andersen, Completeness of length-weighted Sobolev metrics on the space ofcurves (2017). Preprint .
[17] C. Carmeli, E. De Vito, A. Toigo and V. Umanita, Vector valued reproducing kernel Hilbert spaces and universality. Anal. Appl. 8 (2010) 19-61. · Zbl 1195.46025
[18] V. Cervera, F. Mascaró and P.W. Michor, The action of the diffeomorphism group on the space of immersions. Differ. Geom. Appl. 1 (1991) 391-401. · Zbl 0783.58012
[19] B. Charlier, N. Charon and A. Trouvé, Fshapes tool kit, 2014. Available at (2019).
[20] N. Charon, Analysis of geometric and functional shapes with extensions of currents: application to registration and atlas estimation. PhD thesis, ENS Cachan (2013).
[21] N. Charon and A. Trouvé, The varifold representation of non-oriented shapes for diffeomorphic registration. SIAM J. Imaging Sci. 6 (2013) 2547-2580. · Zbl 1279.68313
[22] S. Durrleman, P. Fillard, X. Pennec, A. Trouvé and N. Ayache, Registration, atlas estimation and variability analysis of white matter fiber bundles modeled as currents. NeuroImage 55 (2010) 1073-1090.
[23] D.G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92 (1970) 102-163. · Zbl 0211.57401
[24] M. Eslitzbichler, Modelling character motions on infinite-dimensional manifolds. Vis. Comput. 31 (2015) 1179-1190.
[25] J. Glaunès, A. Qiu, M. Miller and L. Younes, Large deformation diffeomorphic metric curve mapping. Int. J. Comput. Vis. 80 (2008) 317-336.
[26] J. Glaunès, A. Trouvé and L. Younes, Diffeomorphic matching of distributions: a new approach for unlabelled point-sets and sub-manifolds matching. IEEE Comput. Soc. Conf. Comput. Vis. Pattern Recognit. 2 (2004) 712-718.
[27] J. Glaunès and M. Vaillant, Surface matching via currents. Proc. Inform. Process. Med. Imaging (IPMI). In Vol. 3565 of Lect. Notes Comput. Sci. 3565 (2006) 381-392.
[28] U. Grenander, General Pattern Theory: A Mathematical Study of Regular Structures. Oxford Mathematical Monographs. Clarendon Press, Oxford (1993).
[29] I. Kaltenmark, B. Charlier and N. Charon, A general framework for curve and surface comparison and registration with oriented varifolds. IEEE Conf. Comput. Vis. Pattern Recognit. (2017) 4580-4589.
[30] E. Klassen, A. Srivastava, W. Mio and S.H. Joshi, Analysis of planar shapes using geodesic paths on shape spaces. IEEE Trans. Pattern Anal. Mach. Intell. 26 (2004) 372-383.
[31] A. Kriegl and P.W. Michor, The Convenient Setting of Global Analysis. Vol. 53 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (1997). · Zbl 0889.58001
[32] S. Kurtek and T. Needham, Simplifying transforms for general elastic metrics on the space of planecurves (2018). Preprint .
[33] H. Laga, S. Kurtek, A. Srivastava and S.J. Miklavcic, Landmark-free statistical analysis of the shape of plant leaves. J. Theor. Biol. 363 (2014) 41-52. · Zbl 1309.92015
[34] A.C. Mennucci, A. Yezzi and G. Sundaramoorthi, Properties of Sobolev-type metrics in the space of curves. Interface. Free Bound. 10 (2008) 423-445. · Zbl 1168.58005
[35] P.W. Michor and D. Mumford, Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. 8 (2006) 1-48. · Zbl 1101.58005
[36] P.W. Michor and D. Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. Appl. Comput. Harmon. Anal. 23 (2007) 74-113. · Zbl 1116.58007
[37] W. Mio, J.C. Bowers and X. Liu, Shape of elastic strings in euclidean space. Int. J. Comput. Vis. 82 (2009) 96-112.
[38] W. Mio and A. Srivastava, Elastic-String Models for Representation and Analysis of Planar Shapes. Vol. 2 of Proceedings ofthe 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, CVPR (2004) II-10-II-15.
[39] W. Mio, A. Srivastava and S. Joshi, On shape of plane elastic curves. Int. J. Comput. Vis. 73 (2007) 307-324.
[40] G. Nardi, G. Peyré and F.-X. Vialard, Geodesics on shape spaces with bounded variation and Sobolev metrics. SIAM J. Imaging Sci. 9 (2016) 238-274. · Zbl 1339.49037
[41] J. Nocedal and S. Wright, Numerical Optimization. Springer, New York (2006). · Zbl 1104.65059
[42] M. Overton, HANSO: hybrid algorithm for non-smooth optimization 2.2, 2016. Available at (2019).
[43] J. Shah, \(H^0\)-type Riemannian metrics on the space of planar curves. Quart. Appl. Math. 66 (2008) 123-137. · Zbl 1144.58005
[44] B. Sriperumbudur, K. Fukumizu and G. Lanckriet, On the Relation Between Universality, Characteristic Kernels and RKHS Embedding of Measures. Vol. 9 of Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics (2010) 773-780.
[45] A. Srivastava and E. Klassen, Functional and Shape Data Analysis. Springer Series in Statistics. Springer-Verlag, New York (2016). · Zbl 1376.62003
[46] A. Srivastava, E. Klassen, S.H. Joshi and I.H. Jermyn, Shape analysis of elastic curves in Euclidean spaces. IEEE Trans. Pattern Anal. Mach. Intell. 33 (2011) 1415-1428.
[47] J. Su, S. Kurtek, E. Klassen and A. Srivastava, Statistical analysis of trajectories on Riemannian manifolds: bird migration, hurricane tracking and video surveillance. Ann. Appl. Stat. 8 (2014) 530-552. · Zbl 1454.62554
[48] J. Su, A. Srivastava, F.D.M. de Souza and S. Sarkar, Rate-invariant analysis of trajectories on Riemannian manifolds with application in visual speech recognition. IEEE Conf. Comput. Vis. Pattern Recognit. 6 (2014) 620-627.
[49] Z. Su, E. Klassen and M. Bauer, The square root velocity framework for curves in a homogeneous space. In Proceedingsof 2017 IEEE Conference on Computer Vision and Pattern Recognition Workshops. 07 (2017) 680-689.
[50] L. Younes, Hybrid Riemannian metrics for diffeomorphic shape registration. Ann. Math. Sci. Appl. 3 (2018) 189-210. · Zbl 1387.49065
[51] L. Younes, Computable elastic distances between shapes. SIAM J. Appl. Math. 58 (1998) 565-586. · Zbl 0907.68158
[52] L. Younes, P.W. Michor, J. Shah and D. Mumford, A metric on shape space with explicit geodesics. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 19 (2008) 25-57. · Zbl 1142.58013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.