Abstract
A major task in non-rigid shape analysis is to retrieve correspondences between two almost isometric 3D objects. An important tool for this task are geometric feature descriptors. Ideally, a feature descriptor should be invariant under isometric transformations and robust to small elastic deformations. A successful class of feature descriptors employs the spectral decomposition of the Laplace-Beltrami operator. Important examples are the heat kernel signature using the heat equation and the more recent wave kernel signature applying the Schrödinger equation from quantum mechanics.
In this work we propose a novel feature descriptor which is based on the classic wave equation that describes e.g. sound wave propagation. We explore this new model by discretizing the underlying partial differential equation. Thereby we consider two different time integration methods. By a detailed evaluation at hand of a standard shape data set we demonstrate that our approach may yield significant improvements over state of the art methods for finding correct shape correspondences.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Aubry, M., Schlickewei, U., Cremers, D.: The wave kernel signature: a quantum mechanical approach to shape analysis. In: IEEE Computer Vision Workshops (ICCV Workshops), pp. 1626–1633 (2011)
Belkin, M., Sun, J., Wang, Y.: Discrete Laplace operator on meshed surfaces. In: Proceedings of SOCG, pp. 278–287 (2008)
Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Numerical Geometry of Non-rigid Shapes. Springer, New York (2009). doi:10.1007/978-0-387-73301-2
Bronstein, M.M., Kokkinos, I.: Scale-invariant heat kernel signatures for non-rigid shape recognition. In: Proceedings of the International Conference on Computer Vision and Pattern Recognition, pp. 1704–1711 (2010)
Dachsel, R., Breuß, M., Hoeltgen, L.: Shape matching by time integration of partial differential equations. In: Lauze, F., Dong, Y., Dahl, A.B. (eds.) SSVM 2017. LNCS, vol. 10302, pp. 669–680. Springer, Cham (2017). doi:10.1007/978-3-319-58771-4_53
Davis, T.A.: Algorithm 930: factorize: an object-oriented linear system solver for Matlab. ACM Trans. Math. Softw. 39(4), 1–18 (2013)
Johnson, A., Herbert, M.: Using spin images for efficient object recognition in cluttered 3D scenes. IEEE Trans. Pattern Anal. Mach. Intell. 21(5), 433–449 (1999)
Kac, M.: Can one hear the shape of a drum? Am. Math. Mon. 73(4), 1–23 (1966)
Kim, V.G., Lipman, Y., Funkhouser, T.A.: Blended intrinsic maps. ACM Trans. Graph. 30, 79 (2011)
Lipman, Y., Funkhouser, T.A.: Möbius voting for surface correspondence. ACM Trans. Graph. 28(3), 1–12 (2009)
Lévy, B.: Laplace-Beltrami eigenfunctions towards an algorithm that ‘understands’ geometry. In: International Conference on Shape Modeling and Applications (2006)
Manay, S., Hong, B.W., Yezzi, A.J., Soatto, S.: Integral invariant signatures. In: Proceedings of ECCV, pp. 87–99 (2004)
Meyer, M., Desbrun, M., Schröder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. In: Hege, H.C., Polthier, K. (eds.) Visualization and Mathematics III, pp. 35–57. Springer, Heidelberg (2002). doi:10.1007/978-3-662-05105-4_2
Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2(1), 15–36 (1993)
Reuter, M., Wolter, F.E., Peinecke, N.: Laplace-Beltrami spectra as shape-DNA of surfaces and solids. Comput. Aided Des. 38(4), 342–366 (2006)
Rustamov, R.: Laplace-Beltrami eigenfunctions for deformation invariant shape representation. In: Symposium on Geometry Processing, pp. 225–233 (2007)
Sun, J., Ovsjanikov, M., Guibas, L.: A concise and provably informative multi-scale signature based on heat diffusion. Comput. Graph. Forum 28(5), 1383–1392 (2009)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Dachsel, R., Breuß, M., Hoeltgen, L. (2017). The Classic Wave Equation Can Do Shape Correspondence. In: Felsberg, M., Heyden, A., Krüger, N. (eds) Computer Analysis of Images and Patterns. CAIP 2017. Lecture Notes in Computer Science(), vol 10424. Springer, Cham. https://doi.org/10.1007/978-3-319-64689-3_22
Download citation
DOI: https://doi.org/10.1007/978-3-319-64689-3_22
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-64688-6
Online ISBN: 978-3-319-64689-3
eBook Packages: Computer ScienceComputer Science (R0)