Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T22:05:52.220Z Has data issue: false hasContentIssue false
  • English
  • Français

Hierarchical Framework for Shape Correspondence

Published online by Cambridge University Press:  28 May 2015

Dan Raviv ,
Anastasia Dubrovina  and
Ron Kimmel
Dan Raviv*
Affiliation:
Technion, Israel Institute of Technology, Technion City, Haifa 32000, Israel
Anastasia Dubrovina*
Affiliation:
Technion, Israel Institute of Technology, Technion City, Haifa 32000, Israel
Ron Kimmel*
Affiliation:
Technion, Israel Institute of Technology, Technion City, Haifa 32000, Israel
*
Corresponding author.Email address:darav@cs.technion.ac.il
Corresponding author.Email address:nastyad@tx.technion.ac.il
Corresponding author.Email address:ron@cs.technion.ac.il
Get access

Abstract

Detecting similarity between non-rigid shapes is one of the fundamental problems in computer vision. In order to measure the similarity the shapes must first be aligned. As opposite to rigid alignment that can be parameterized using a small number of unknowns representing rotations, reflections and translations, non-rigid alignment is not easily parameterized. Majority of the methods addressing this problem boil down to a minimization of a certain distortion measure. The complexity of a matching process is exponential by nature, but it can be heuristically reduced to a quadratic or even linear for shapes which are smooth two-manifolds. Here we model the shapes using both local and global structures, employ these to construct a quadratic dissimilarity measure, and provide a hierarchical framework for minimizing it to obtain sparse set of corresponding points. These correspondences may serve as an initialization for dense linear correspondence search.

Keywords

65M5065M6068T45Shape correspondenceLaplace-Beltramidiffusion geometry local signatures
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 1 , February 2013 , pp. 245 - 261
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bemporad, A.. Hybrid Toolbox - User’s Guide, 2004.Google Scholar
[2] Bérard, P., Besson, G., and Gallot, S.. Embedding riemannian manifolds by their heat kernel. Geometric and Functional Analysis, 4(4):373–398, 1994.Google Scholar
[3] Boyd, S. and Vandenberghe, L.. Convex Optimization. Cambridge University Press, 2006.Google Scholar
[4] Bronstein, A. M., Bronstein, M. M., Devir, Y. S., Kimmel, R., and Weber, O.. Parallel algorithms for approximation of distance maps on parametric surfaces. In Proc. ACM Transactions on Graphics (SIGGRAPH), volume 27, 2008.Google Scholar
[5] Bronstein, A. M., Bronstein, M. M., and Kimmel, R.. Expression-invariant face recognition via spherical embedding. In Proc. Int’l Conf. Image Processing (ICIP), volume 3, pages 756–759, 2005.Google Scholar
[6] Bronstein, A. M., Bronstein, M. M., and Kimmel, R.. Efficient computation of isometry-invariant distances between surfaces. SIAM J. Scientific Computing, 28(5):1812–1836, 2006.Google Scholar
[7] Bronstein, M. M. and Bronstein, A. M.. Shape recognition with spectral distances. Trans. on Pattern Analysis and Machine Intelligence (PAMI), 2010.CrossRefGoogle Scholar
[8] Burago, D., Burago, Y., Ivanov, S., and American Mathematical Society. A course in metric geometry. American Mathematical Society Providence, 2001.CrossRefGoogle Scholar
[9] Coifman, R. R. and Lafon, S.. Diffusion maps. Applied and Computational Harmonic Analysis, 21:5–30, July 2006.Google Scholar
[10] Coifman, R. R. and Lafon, S.. Diffusion maps. Applied and Computational Harmonic Analysis, 21(1):5–30, 2006. Definition of diffusion distance.Google Scholar
[11] Coifman, R. R., Lafon, S., Lee, A. B., Maggioni, M., Nadler, B., Warner, F., and Zucker, S. W.. Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps. PNAS, 102(21):7426–7431, 2005.Google Scholar
[12] Dubrovina, A. and Kimmel, R.. Matching shapes by eigendecomposition of the laplace_belrami operator. In Proc. Symposium on 3D Data Processing Visualization and Transmission (3DPVT), 2010.Google Scholar
[13] Durrleman, S., Pennec, X., Trouv, A.è, and Ayache, N.. Measuring brain variability via sulcal lines registration: a diffeomorphic approach. In Proc. Medical Image Computing and Computer Assisted Intervention (MICCAI), pages 675–682, 2007.Google Scholar
[14] Dziuk, G.. Finite elements for the Beltrami operator on arbitrary surfaces. In Hildebrandt, S. and Leis, R., editors, Partial differential equations and calculus of variations, pages 142–155. 1988.Google Scholar
[15] Elad, A. and Kimmel, R.. On bending invariant signatures for surfaces. Trans. on Pattern Analysis and Machine Intelligence (PAMI), 25(10):1285–1295, 2003.Google Scholar
[16] Glaunès, J., Vaillant, M., and Miller, M. I.. Landmark matching via large deformation diffeo-morphisms on the sphere. Journal of Mathematical Imaging and Vision, 20:179–200, 2004.Google Scholar
[17] Hochbaum, D.S. and Shmoys, D.B.. A best possible heuristic for the k-center problem. Mathematics of Operations Research, pages 180–184, 1985.Google Scholar
[18] Hu, J. and Hua, J.. Salient spectral geometric features for shape matching and retrieval. Vis. Comput., 25(5-7):667–675, 2009.Google Scholar
[19] Jain, V and Zhang, H.. A spectral approach to shape-based retrieval of articulated 3D models. Computer-Aided Design, 39:398–407, 2007.Google Scholar
[20] Kimmel, R. and Sethian, J. A.. Computing geodesic paths on manifolds. Proc. National Academy of Sciences (PNAS), 95(15):8431–8435, 1998.Google Scholar
[21] Lai, R., Shi, Y., Scheibel, K., Fears, S., Woods, R., Toga, A. W., and Chan, T. F.. Metric-induced optimal embedding for intrinsic 3d shape analysis. In Proc. International Conference of Computer Vision (CVPR), 2010.Google Scholar
[22] Leow, A., Yu, C. L., Lee, S. J., Huang, S. C., Protas, H., Nicolson, R., Hayashi, K. M., Toga, A. W., and Thompson, P. M.. Brain structural mapping using a novel hybrid implicit/explicit framework based on the level-set method. NeuroImage, 24(3):910–927, 2005.Google Scholar
[23] Lipman, Y and Funkhouser, T.. Mobius voting for surface correspondence. In Proc. ACM Transactions on Graphics (SIGGRAPH), volume 28, 2009.Google Scholar
[24] Mateus, D., Horaud, R. P., Knossow, D., Cuzzolin, F., and Boyer, E.. Articulated shape matching using laplacian eigenfunctions and unsupervised point registration. In Proc. of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2008.Google Scholar
[25] Mémoli, F. and Sapiro, G.. A theoretical and computational framework for isometry invariant recognition of point cloud data. Foundations of Computational Mathematics, 5:313–346, 2005.Google Scholar
[26] Miller, M. I., Faisal Beg, M., Ceritoglu, C., and Stark, C.. Increasing the power of functional maps of the medial temporal lobe by using large deformation diffeomorphic metric mapping. Proc. National Academy of Science (PNAS), 102(27):9685–9690, 2005.Google Scholar
[27] Ovsjanikov, M., Bronstein, A. M., Bronstein, M. M., and Guibas, L. J.. Shape Google: a computer vision approach to invariant shape retrieval. In Proc. Non-Rigid Shape Analysis and Deformable Image Alignment (NORDIA), 2009.Google Scholar
[28] Ovsjanikov, M., Mérigot, Q., Mémoli, F., and Guibas, L. J.. One point isometric matching with the heat kernel. Proc. Symposium on Geometry Processing (SGP), 29(5): 1555–1564, Jul 2010.Google Scholar
[29] Ovsjanikov, M., Sun, J., and Guibas, L. J.. Global intrinsic symmetries of shapes. In Computer Graphics Forum, volume 27, pages 1341–1348, 2008.CrossRefGoogle Scholar
[30] Raviv, D., Bronstein, A. M., Bronstein, M. M., and Kimmel, R.. Full and partial symmetries of non-rigid shapes. International Journal of Computer Vision (IJCV), 2009.Google Scholar
[31] Raviv, D., Bronstein, A. M., Bronstein, M. M., and Kimmel, R.. Volumetric heat kernel signatures. In Proc. 3D Object recognition (3DOR), part of ACM Multimedia., 2010.Google Scholar
[32] Raviv, D., Bronstein, A. M., Bronstein, M. M., Kimmel, R., and Sapiro, G.. Diffusion symmetries of non-rigid shapes. In Proc. International Symposium on 3D Data Processing, Visualization and Transmission (3DPVT), 2010.Google Scholar
[33] Reuter, M., Biasotti, S., Giorgi, D., Patan, G.è, and Spagnuolo, M.. Discrete Laplace-Beltrami operators for shape analysis and segmentation. Computers & Graphics, 33(3):381–390, 2009.Google Scholar
[34] Rubner, Y, Tomasi, C, and Guibas, L.J.. The earth mover’s distance as a metric for image retrieval. IJCV, 40(2):99–121, 2000.Google Scholar
[35] Rustamov, R. M.. Laplace-Beltrami eigenfunctions for deformation invariant shape representation. In Proc. Symposium on Geometry Processing (SGP), pages 225–233, 2007.Google Scholar
[36] Shi, Y, Lai, R., Gill, R., Pelletier, D., Mohr, D., Sicotte, N., and Toga, A. W.. Conformal metric optimization on surface (cmos) for deformation and mapping in laplace-beltrami embedding space. In Proc. MedicalImage Computing and Computer Assisted Intervention (MICCAI), 2011.Google Scholar
[37] Spira, A. and Kimmel, R.. An efficient solution to the eikonal equation on parametric manifolds. Interfaces and Free Boundaries, 6(4):315–327, 2004.Google Scholar
[38] Spira, A. and Kimmel, R.. An efficient solution to the eikonal equation on parametric manifolds. Interfaces and Free Boundaries, 6(3):315–327, 2004.Google Scholar
[39] Sun, J., Ovsjanikov, M., and Guibas, L. J.. A concise and provably informative multi-scale signature based on heat diffusion. In Proc. Symposium on Geometry Processing (SGP), 2009.Google Scholar
[40] Tosun, D., Rettmann, M. E., and Prince, J. L.. Mapping techniques for aligning sulci across multiple brains. Medical Image Analysis, 8:295–309, 2004.Google Scholar
[41] Tsitsiklis, J. N.. Efficient algorithms for globally optimal trajectories. IEEE Trans. Automatic Control, 40(9):1528–1538, 1995.CrossRefGoogle Scholar
[42] Wang, C., Bronstein, M. M., and Paragios, N.. Discrete minimum distortion correspondence problems for non-rigid shape matching. Research Report 7333, INRIA, 2010.Google Scholar
[43] Zaharescu, A., Boyer, E., Varanasi, K., and Horaud, R. Surface feature detection and description with applications to mesh matching. In Proc. Computer Vision and Pattern Recognition (CVPR), 2009.Google Scholar
[44] Zigelman, G., Kimmel, R., and Kiryati, N.. Texture mapping using surface flattening via multidimensional scaling. IEEE Trans. Visualization and Computer Graphics (TVCG), 9(2):198–207, 2002.Google Scholar