Abstract
Isometric 3D shape partial matching has attracted a great amount of interest, with a plethora of applications ranging from shape recognition to texture mapping. In this paper, we propose a novel isometric 3D shape partial matching algorithm using the geodesic disk Laplace spectrum (GD-DNA). It transforms the partial matching problem into the geodesic disk matching problem. Firstly, the largest enclosed geodesic disk extracted from the partial shape is matched with geodesic disks from the full shape by the Laplace spectrum of the geodesic disk. Secondly, Generalized Multi-Dimensional Scaling algorithm (GMDS) and Euclidean embedding are conducted to establish final point correspondences between the partial and the full shape using the matched geodesic disk pair. The proposed GD-DNA is discriminative for matching geodesic disks, and it can well solve the anchor point selection problem in challenging partial shape matching tasks. Experimental results on the Shape Retrieval Contest 2016 (SHREC’16) benchmark validate the proposed method, and comparisons with isometric partial matching algorithms in the literature show that our method has a higher precision.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bronstein A M, Bronstein M, Kimmel R. Generalized multidimensional scaling: A framework for isometry-invariant partial surface matching. Processings of National Academy of Sciences of the United States of America, 2006, 103(5): 1168-1172.
Sahillioğlu Y, Yemez Y. Scale normalization for isometric shape matching. Computer Graphics Forum, 2012, 31(7): 2233-2240.
Sahillioğlu Y, Yemez Y. Partial 3-D correspondence from shape extremities. Computer Graphics Forum, 2014, 33(6): 63-76.
Rodolà E, Cosmo L, Bronstein M M, Torsello A, Cremers D. Partial functional correspondence. Computer Graphics Forum, 2017, 36(1): 222-236.
Litany O, Rodolà E, Bronstein A M, Bronstein M M, Cremers D. Non-rigid puzzles. Computer Graphics Forum, 2016, 35(5): 135-143.
Litany O, Rodolà E, Bronstein A M, Bronstein M M. Fully spectral partial shape matching. Computer Graphics Forum, 2017, 36(2): 247-258.
Belongie S J, Malik J, Puzicha J. Shape matching and object recognition using shape contexts. IEEE Transactions on Pattern Analysis & Machine Intelligence, 2002, 24(4): 509-522.
Sun J, Ovsjanikov M, Guibas L. A concise and provably informative multi-scale signature based on heat diffusion. Computer Graphics Forum, 2010, 28(5): 1383-1392.
Aubry M, Schlickewei U, Cremers D. The wave kernel signature: A quantum mechanical approach to shape analysis. In Proc. IEEE International Conference on Computer Vision Workshops, Nov. 2011, pp.1626-1633.
van Kaick O, Zhang H, Hamarneh G, Cohen-Or D. A survey on shape correspondence. Computer Graphics Forum, 2011, 30(6): 1681-1707.
Masci J, Boscaini D, Bronstein M M, Vandergheynst P. Geodesic convolutional neural networks on Riemannian manifolds. In Proc. the 2015 IEEE International Conference on Computer Vision Workshop, Dec. 2015, pp.832-840.
Masci J, Rodolà, Emanuele, Boscaini D, Bronstein M M, Li H. Geometric deep learning. In Proc. SIGGRAPH Asia 2016 Courses, Dec. 2016, Article No. 1.
Borg I, Groenen P J F. Modern Multidimensional Scaling. Springer-Verlag New York, 2005.
Lipman Y, Funkhouser T. Möbius voting for surface correspondence. ACM Transactions on Graphics, 2009, 28(3): Article No. 72.
Rodolà E, Bronstein A M, Albarelli A, Bergamasco F. A game-theoretic approach to deformable shape matching. In Proc. IEEE Conference on Computer Vision and Pattern Recognition, Jun. 2012, pp.182-189.
Rodolà E, Torsello A, Harada T, Kuniyoshi Y, Cremers D. Elastic net constraints for shape matching. In Proc. IEEE International Conference on Computer Vision, Dec. 2013, pp.1169-1176.
Ovsjanikov M, Ben-Chen M, Solomon J, Butscher A, Guibas L. Functional maps: A flexible representation of maps between shapes. ACM Transactions on Graphics, 2012, 31(4): Article No. 30.
Kovnatsky A, Bronstein M M, Bresson X, Vandergheynst P. Functional correspondence by matrix completion. In Proc. IEEE Conference on Computer Vision and Pattern Recognition, Jun. 2015, pp.905-914.
Zaharescu A, Boyer E, Varanasi K, Horaud R. Surface feature detection and description with applications to mesh matching. In Proc. IEEE Conference on Computer Vision and Pattern Recognition, Jun. 2009, pp.373-380.
Pokrass J, Bronstein A M, Bronstein M M. Partial shape matching without point-wise correspondence. Numerical Mathematics: Theory Methods & Applications, 2013, 6(1): 223-244.
Brunton A, Wand M, Wuhrer S, Seidel H P, Weinkauf T. A low-dimensional representation for robust partial isometric correspondences computation. Graphical Models, 2014, 76(2): 70-85.
van Kaick O, Zhang H, Hamarneh G. Bilateral maps for partial matching. Computer Graphics Forum, 2013, 32(6): 189-200.
Ovsjanikov M, Mérigot Q, Mémoli F, Guibas L. One point isometric matching with the heat kernel. Computer Graphics Forum, 2010, 29(5): 1555-1564.
Reuter M, Wolter F E, Peinecke N. Laplace-Beltrami spectra as ‘Shape-DNA’ of surfaces and solids. Computer-Aided Design, 2006, 38(4): 342-366.
Reuter M, Wolter F E, Shenton M, Niethammer M. Laplace-Beltrami eigenvalues and topological features of eigenfunctions for statistical shape analysis. Computer-Aided Design, 2009, 41(10): 739-755.
Reuter M, Spagnuolo M. Discrete Laplace-Beltrami operators for shape analysis and segmentation. In Proc. IEEE International Conference on Shape Modeling and Applications, June 2009, pp.381-390.
Mémoli F, Sapiro G. A theoretical and computational framework for isometry invariant recognition of point cloud data. Foundations of Computational Mathematics, 2005, 5(3): 313-347.
Cosmo L, Rodolà E, Bronstein M, Torsello A, Cremers D, Sahillioğlu Y. SHREC’16: Partial matching of deformable shapes. In Proc. Eurographics Workshop on 3D Object Retrieval, May 2016, pp.61-67.
Bronstein A M, Bronstein M M, Kimmel R. Numerical Geometry of Non-Rigid Shapes. Springer-Verlag, New York, 2009.
Kim V G, Lipman Y, Funkhouser T. Blended intrinsic maps. ACM Transactions on Graphics, 2011, 30(4): Article No. 79.
Mitchell J S B, Mount D M, Papadimitriou C H. The discrete geodesic problem. SIAM Journal on Computing, 1987, 16(4): 647-668.
Author information
Authors and Affiliations
Corresponding authors
Electronic supplementary material
Below is the link to the electronic supplementary material.
ESM 1
(PDF 650 kb)
Rights and permissions
About this article
Cite this article
Du, GG., Yin, CL., Zhou, MQ. et al. Isometric 3D Shape Partial Matching Using GD-DNA. J. Comput. Sci. Technol. 33, 1178–1191 (2018). https://doi.org/10.1007/s11390-018-1880-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11390-018-1880-x