Abstract
Finding a match between partially available deformable shapes is a challenging problem with numerous applications. The problem is usually approached by computing local descriptors on a pair of shapes and then establishing a point-wise correspondence between the two. In this paper, we introduce an alternative correspondence-less approach to matching fragments to an entire shape undergoing a non-rigid deformation. We use diffusion geometric descriptors and optimize over the integration domains on which the integral descriptors of the two parts match. The problem is regularized using the Mumford-Shah functional. We show an efficient discretization based on the Ambrosio-Tortorelli approximation generalized to triangular meshes. Experiments demonstrating the success of the proposed method are presented.
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
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via-convergence. Comm. Pure Appl. Math. 43(8), 999–1036 (1990)
Bronstein, A.M., Bronstein, M.: Not only size matters: regularized partial matching of nonrigid shapes. In: Prof. NORDIA (2008)
Bronstein, A.M., Bronstein, M.M.: Regularized partial matching of rigid shapes. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part II. LNCS, vol. 5303, pp. 143–154. Springer, Heidelberg (2008)
Bronstein, A.M., Bronstein, M.M., Bruckstein, A.M., Kimmel, R.: Partial similarity of objects, or how to compare a centaur to a horse. IJCV 84(2), 163–183 (2009)
Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching. Proc. National Academy of Science (PNAS) 103(5), 1168–1172 (2006)
Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Numerical geometry of non-rigid shapes. Springer-Verlag New York Inc., Secaucus (2008)
Bronstein, A.M., Bronstein, M.M., Bustos, B., Castellani, U., Crisani, M., Falcidieno, B., Guibas, L.J., Kokkinos, I., Murino, V., Ovsjanikov, M., et al.: SHREC 2010: robust feature detection and description benchmark. In: Proc. 3DOR (2010)
Bronstein, A.M., Bronstein, M.M., Castellani, U., Dubrovina, A., Guibas, L.J., Horaud, R.P., Kimmel, R., Knossow, D., von Lavante, E., Mateus, D., et al.: SHREC 2010: robust correspondence benchmark. In: Eurographics Workshop on 3D Object Retrieval (2010)
Bronstein, M.M., Kokkinos, I.: Scale-invariant heat kernel signatures for non-rigid shape recognition. In: Proc. CVPR (2010)
Chan, T.F., Vese, L.A.: Active contours without edges. IEEE Trans. Image Processing 10(2), 266–277 (2001)
Clarenz, U., Rumpf, M., Telea, A.: Robust feature detection and local classification for surfaces based on moment analysis. Trans. Visualization and Computer Graphics 10(5), 516–524 (2004)
Domokos, C., Kato, Z.: Affine puzzle: Realigning deformed object fragments without correspondences. In: Daniilidis, K., Maragos, P., Paragios, N. (eds.) ECCV 2010. LNCS, vol. 6312, pp. 777–790. Springer, Heidelberg (2010)
Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey. In: Advances in Multiresolution for Geometric Modelling, vol. 1 (2005)
Gromov, M.: Structures Métriques Pour les Variétés Riemanniennes. Textes Mathématiques, vol. (1) (1981)
Jacobs, D., Weinshall, D., Gdalyahu, Y.: Class representation and image retrieval with non-metric distances. Trans. PAMI 22(6), 583–600 (2000)
Johnson, A.E., Hebert, M.: Using spin images for efficient object recognition in cluttered 3D scenes. Trans. PAMI 21(5), 433–449 (1999)
Lévy, B.: Laplace-Beltrami eigenfunctions towards an algorithm that “understands” geometry. In: Proc. Shape Modeling and Applications (2006)
Manay, S., Hong, B.W., Yezzi, A.J., Soatto, S.: Integral invariant signatures. LNCS, pp. 87–99 (2004)
Mémoli, F., Sapiro, G.: A theoretical and computational framework for isometry invariant recognition of point cloud data. Foundations of Computational Mathematics 5, 313–346 (2005)
Meyer, M., Desbrun, M., Schroder, P., Barr, A.H.: Discrete differential-geometry operators for triangulated 2-manifolds. Visualization and Mathematics III, 35–57 (2003)
Mitra, N.J., Guibas, L.J., Giesen, J., Pauly, M.: Probabilistic fingerprints for shapes. In: Proc. SGP (2006)
Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Communications on pure and applied mathematics 42(5), 577–685 (1989)
Ovsjanikov, M., Bronstein, A.M., Guibas, L.J., Bronstein, M.M.: Shape Google: a computer vision approach to invariant shape retrieval. In: Proc. NORDIA, Citeseer (2009)
Pauly, M., Keiser, R., Gross, M.: Multi-scale feature extraction on point-sampled surfaces. In: Computer Graphics Forum, vol. 22, pp. 281–289 (2003)
Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Experimental Mathematics 2(1), 15–36 (1993)
Reuter, M., Wolter, F.-E., Peinecke, N.: Laplace-spectra as fingerprints for shape matching. In: Proc. ACM Symp. Solid and Physical Modeling, pp. 101–106 (2005)
Sipiran, I., Bustos, B.: A robust 3D interest points detector based on Harris operator. In: Proc. 3DOR, pp. 7–14. Eurographics (2010)
Sivic, J., Zisserman, A.: Video Google: a text retrieval approach to object matching in videos. In: Proc. CVPR (2003)
Sun, J., Ovsjanikov, M., Guibas, L.: A Concise and Provably Informative Multi-Scale Signature Based on Heat Diffusion. In: Computer Graphics Forum, vol. 28, pp. 1383–1392 (2009)
Toldo, R., Castellani, U., Fusiello, A.: Visual vocabulary signature for 3D object retrieval and partial matching. In: Proc. 3DOR (2009)
Wardetzky, M., Mathur, S., Kälberer, F., Grinspun, E.: Discrete Laplace operators: no free lunch. In: Conf. Computer Graphics and Interactive Techniques (2008)
Zaharescu, A., Boyer, E., Varanasi, K., Horaud, R.: Surface feature detection and description with applications to mesh matching. In: Proc. CVPR (2009)
Zhang, C., Chen, T.: Efficient feature extraction for 2D/3D objects in mesh representation. In: Proc. ICIP, vol. 3 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pokrass, J., Bronstein, A.M., Bronstein, M.M. (2012). A Correspondence-Less Approach to Matching of Deformable Shapes. In: Bruckstein, A.M., ter Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2011. Lecture Notes in Computer Science, vol 6667. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24785-9_50
Download citation
DOI: https://doi.org/10.1007/978-3-642-24785-9_50
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24784-2
Online ISBN: 978-3-642-24785-9
eBook Packages: Computer ScienceComputer Science (R0)