In a recent paper by Iglesias et al. [Found. Comput. Math. 18 (2018) 891–927] a variational model for deformations matching a pair of shapes given as level set functions was proposed. Its main feature is the presence of anisotropic energies active only in a narrow band around the hypersurfaces that resemble the behavior of elastic shells. In this work we consider some extensions and further analysis of that model. First, we present a symmetric energy functional such that given two particular shapes, it assigns the same energy to any given deformation as to its inverse when the roles of the shapes are interchanged, and introduce the adequate parameter scaling to recover a surface problem when the width of the narrow band vanishes. Then, we obtain existence of minimizing deformations for the symmetric energy in classes of bi-Sobolev homeomorphisms for small enough widths, and prove a -convergence result for the corresponding non-symmetric energies as the width tends to zero. Finally, numerical results on realistic shape matching applications demonstrating the effect of the symmetric energy are presented.
Keywords: Variational shape matching, implicit surfaces, thin shells, invertibility, scaling limits
@article{M2AN_2021__55_3_1133_0,
author = {Iglesias, Jos\'e A.},
title = {Symmetry and scaling limits for matching of implicit surfaces based on thin shell energies},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1133--1161},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {3},
doi = {10.1051/m2an/2021018},
mrnumber = {4269465},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021018/}
}
TY - JOUR AU - Iglesias, José A. TI - Symmetry and scaling limits for matching of implicit surfaces based on thin shell energies JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 1133 EP - 1161 VL - 55 IS - 3 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021018/ DO - 10.1051/m2an/2021018 LA - en ID - M2AN_2021__55_3_1133_0 ER -
%0 Journal Article %A Iglesias, José A. %T Symmetry and scaling limits for matching of implicit surfaces based on thin shell energies %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 1133-1161 %V 55 %N 3 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021018/ %R 10.1051/m2an/2021018 %G en %F M2AN_2021__55_3_1133_0
Iglesias, José A. Symmetry and scaling limits for matching of implicit surfaces based on thin shell energies. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 3, pp. 1133-1161. doi: 10.1051/m2an/2021018
[1] , and , Thin inclusions in linear elasticity: a variational approach. J. Reine Angew. Math. 386 (1988) 99–115. | MR | Zbl
[2] and , Sobolev Spaces, 2nd edition. In: Vol. 140 of Pure and Applied Mathematics (Amsterdam). Elsevier/Academic Press (2003). | MR | Zbl
[3] and , Relaxation and 3D–2D passage theorems in hyperelasticity. J. Convex Anal. 19 (2012) 759–794. | MR | Zbl
[4] , Constitutive inequalities and existence theorems in nonlinear elastostatics. Nonlinear Analysis and Mechanics: Heriot–Watt Symposium, Vol. I. In: Vol. 17 of Research Notes in Math. Pitman (1977) 187–241. | MR | Zbl
[5] , Global invertibility of Sobolev functions and the interpenetration of matter. Proc. Roy. Soc. Edinburgh Sect. A 88 (1981) 315–328. | MR | Zbl | DOI
[6] , Progress and puzzles in nonlinear elasticity, edited by and . In: Poly-, Quasi- and Rank-One Convexity in Applied Mechanics. Springer (2010) 1–15.
[7] , and , Overview of the geometries of shape spaces and diffeomorphism groups. J. Math. Imaging Vision 50 (2014) 60–97. | MR | Zbl | DOI
[8] , , and , A relaxed approach for curve matching with elastic metrics. ESAIM COCV 25 (2019) 72. | MR | Numdam | DOI
[9] , , and , A numerical framework for elastic surface matching, comparison, and interpolation. Preprint arXiv:2006.11652 [cs.CV] (2020).
[10] , and , Multi-materials with strong interface: variational modelings. Asymptot. Anal. 61 (2009) 1–19. | MR | Zbl
[11] , -Convergence for Beginners. In: Vol. 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press (2002). | MR | Zbl
[12] , and , A hyperelastic regularization energy for image registration. SIAM J. Sci. Comput. 35 (2013) B132–B148. | MR | Zbl | DOI
[13] and , Symmetrization of the non-rigid registration problem using inversion-invariant energies: application to multiple sclerosis. In: Medical Image Computing and Computer-Assisted Intervention–MICCAI 2000. Springer (2000) 472–481.
[14] , and , Approximations of shape metrics and application to shape warping and empirical shape statistics. Found. Comput. Math. 5 (2004) 1–58. | MR | Zbl | DOI
[15] , Riemannian Geometry, 2nd edition. In: Vol. 98 of Cambridge Studies in Advanced Mathematics. Cambridge University Press (2006). | MR | Zbl
[16] and , On the continuity of the -th root of a continuous nonnegative definite matrix-valued function. J. Math. Anal. Appl. 209 (1997) 60–66. | MR | Zbl | DOI
[17] , Mathematical Elasticity, Volume III: Theory of Shells. North-Holland (2000). | MR | Zbl
[18] , Direct Methods in the Calculus of Variations, 2nd edition. In: Vol. 78 ofApplied Mathematical Sciences. Springer (2008). | MR | Zbl
[19] and , Smooth approximation of bi-Lipschitz orientation-preserving homeomorphisms. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014) 567–589. | MR | Zbl | Numdam | DOI
[20] , , , , , and , A variational model dedicated to joint segmentation, registration, and atlas generation for shape analysis. SIAM J. Imaging Sci. 13 (2020) 351–380. | MR | DOI
[21] and , Shape analysis via oriented distance functions. J. Funct. Anal. 123 (1994) 129–201. | MR | Zbl | DOI
[22] and , A boundary differential equation for thin shells. J. Differ. Equ. 119 (1995) 426–449. | MR | Zbl | DOI
[23] and , Shapes and Geometries, 2nd edition. In: Vol. 22 of Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM) (2011). | MR | Zbl
[24] and , Functional Spaces for the Theory of Elliptic Partial Differential Equations. Universitext. Springer/EDP Sciences (2012). | MR | Zbl | DOI
[25] and , A variational approach to nonrigid morphological image registration. SIAM J. Appl. Math. 64 (2003/04) 668–687. | MR | Zbl | DOI
[26] , , , and , Elastic correspondence between triangle meshes. Comput. Graph. Forum 38 (2019) 121–134. | DOI
[27] , Regularity of the distance function. Proc. Am. Math. Soc. 92 (1984) 153–155. | MR | Zbl
[28] , Variation of tensor powers and spectra. Linear Multilinear Algebra 12 (1982/83) 81–98. | MR | Zbl | DOI
[29] , and , A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence. Arch. Ration. Mech. Anal. 180 (2006) 183–236. | MR | Zbl | DOI
[30] , , and , Shape metrics based on elastic deformations. J. Math. Imaging Vision 35 (2009) 86–102. | MR | DOI
[31] and , Elliptic Partial Differential Equations of Second Order. In: Classics in Mathematics. Springer-Verlag (2001). Reprint of the 1998 edition. | MR | Zbl
[32] and , Lectures on Mappings of Finite Distortion. In: Vol. 2096 of Lecture Notes in Mathematics. Springer(2014). | MR | Zbl
[33] , , and , A thin shell approach to the registration of implicit surfaces. In: Proceedings of the Vision, Modeling, and Visualization Workshop 2013. Eurographics Association (2013) 89–96.
[34] , and , Shape-aware matching of implicit surfaces based on thin shell energies. Found. Comput. Math. 18 (2018) 891–927. | MR | DOI
[35] and , Hyperelastic deformations of smallest total energy. Arch. Ration. Mech. Anal. 194 (2009) 927–986. | MR | Zbl | DOI
[36] , and , Diffeomorphic approximation of Sobolev homeomorphisms. Arch. Ration. Mech. Anal. 201 (2011) 1047–1067. | MR | Zbl | DOI
[37] , and , Conditions for the invertibility of the isoparametric mapping for hexahedral finite elements. Finite Elem. Anal. Des. 40 (2003) 159–172. | MR | DOI
[38] , and , A symmetric deformation-based similarity measure for shape analysis. In: IEEE 12th International Symposium on Biomedical Imaging (ISBI). IEEE (2015) 314–318. | DOI
[39] , Nonlinear Elliptic Partial Differential Equations. Universitext. Springer (2018). | MR
[40] and , The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity. J. Math. Pures Appl. 74 (1995) 549–578. | MR | Zbl
[41] and , The membrane shell model in nonlinear elasticity: a variational asymptotic derivation. J. Nonlinear Sci. 6 (1996) 59–84. | MR | Zbl | DOI
[42] , , and , An image processing approach to surface matching. In: Symposium on Geometry Processing, edited by and (2005) 207–216.
[43] , Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. Manuscripta Math. 51 (1985) 1–28. | MR | Zbl | DOI
[44] , Morse Theory. In: Annals of Mathematics Studies, No. 51. Princeton University Press (1963). | MR | Zbl
[45] , An extended interpolation inequality. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 20 (1966) 733–737. | MR | Zbl | Numdam
[46] Quocmesh library, AG rumpf, Institute for Numerical Simulation, Universität Bonn. http://numod.ins.uni-bonn.de/software/quocmesh/index.html.
[47] and , A nonlinear elastic shape averaging approach. SIAM J. Imaging Sci. 2 (2009) 800–833. | MR | Zbl | DOI
[48] and , Variational methods in shape analysis. In: Handbook of Mathematical Methods in Imaging, edited by . Springer (2011) 1363–1401. | Zbl | DOI
[49] and , Discrete geodesic calculus in shape space and applications in the space of viscous fluidic objects. SIAM J. Imaging Sci. 6 (2013) 2581–2602. | MR | Zbl | DOI
[50] and , Variational time discretization of geodesic calculus. IMA J. Numer. Anal. 35 (2015) 1011–1046. | MR | Zbl | DOI
[51] , Level Set Methods and Fast Marching Methods, 2nd edition. Cambridge University Press (1999). | MR | Zbl
[52] , , and , Geometrically consistent elastic matching of 3D shapes: a linear programming solution. In: International Conference on Computer Vision (2011) 2134–2141.
[53] , Weakly Differentiable Functions. In: Vol. 120 of Graduate Texts in Mathematics. Springer-Verlag (1989). | MR | Zbl | DOI
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