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The Visual Computer

, Volume 29, Issue 1, pp 53–67 | Cite as

Feature correspondences using Morse Smale complex

  • Wei FengEmail author
  • Jin Huang
  • Tao Ju
  • Hujun Bao
Original Article

Abstract

Establishing corresponding features on two non-rigidly deformed 3D surfaces is a challenging and well-studied problem in computer graphics. Unlike previous approaches that constrain the matching between feature pairs using isometry-invariant distance metrics, we constrain the matching using a discrete connectivity graph derived from the Morse–Smale complex of the Auto Diffusion Function. We observed that the graph remains stable even for surfaces differing by topology or by significant deformation. This algorithm is simple to implement and efficient to run. When tested on a range of examples, our algorithm produces comparable results with state-of-art methods on surfaces with strong isometry but with greatly improved efficiency, and often gets better correspondences on surfaces with larger shape variances.

Keywords

Point matching Correspondence Morse–Smale complex 

Supplementary material

(AVI 34.8 MB)

References

  1. 1.
    Anguelov, D., Srinivasan, P., Pang, C.H., Koller, D.: The correlated correspondence algorithm for unsupervised registration of nonrigid surfaces. In: NIPS (2004) Google Scholar
  2. 2.
    Bommes, D., Zimmer, H., Kobbelt, L.: Mixed-integer quadrangulation. In: ACM SIGGRAPH 2009 papers, SIGGRAPH ’09, pp. 77:1–77:10. ACM, New York (2009) Google Scholar
  3. 3.
    Boyer, E., Bronstein, A.M., Bronstein, M.M., Bustos, B., Darom, T., Horaud, R., Hotz, I., Keller, Y., Keustermans, J., Kovnatsky, A., Litman, R., Reininghaus, J., Sipiran, I., Smeets, D., Suetens, P., Vandermeulen, D., Zaharescu, A., Zobel, V.: Shrec 2011: robust feature detection and description benchmark. In: Proc. Workshop on 3D Object Retrieval (3DOR’11) (2011) Google Scholar
  4. 4.
    Bremer, P.T., Edelsbrunner, H., Hamann, B., Pascucci, V.: A topological hierarchy for functions on triangulated surfaces. IEEE Trans. Vis. Comput. Graph. 10, 2004 (2004) CrossRefGoogle Scholar
  5. 5.
    Bronstein, A., Bronstein, M., Bustos, B., Castellani, U., Crisani, M., Falcidieno, B., Guibas, L., Kokkinos, I., Murino, V., Sipiran, I., Ovsjanikovy, M., Patan, G., Spagnuolo, M., Sun, J.: Shrec 2010: robust feature detection and description benchmark. In: Eurographics 2010 Workshop on 3D Object Retrieval (3DOR’10), pp. 79–86. Eurographics Association, Aire-la-Ville (2010) Google Scholar
  6. 6.
    Bronstein, A.M., Bronstein, M.M., Kimmel, R., Mahmoudi, M., Sapiro, G.: A Gromov-Hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching. Int. J. Comput. Vis. 89, 266–286 (2010) CrossRefGoogle Scholar
  7. 7.
    Bronstein, M.M., Kokkinos, I.: Scale-invariant heat kernel signatures for non-rigid shape recognition. In: CVPR, pp. 1704–1711. IEEE, New York (2010) Google Scholar
  8. 8.
    Coifman, R.R., Lafon, S.: Diffusion maps. Appl. Comput. Harmon. Anal. 21(1), 5–30 (2006). Diffusion Maps and Wavelets MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dong, S., Bremer, P.T., Garland, M., Pascucci, V., Hart, J.C.: Spectral surface quadrangulation. ACM Trans. Graph. 25, 1057–1066 (2006) CrossRefGoogle Scholar
  10. 10.
    Edelsbrunner, H., Harer, J., Zomorodian, A.: Hierarchical Morse complexes for piecewise linear 2-manifolds. In: Proceedings of the Seventeenth Annual Symposium on Computational Geometry, SCG ’01, pp. 70–79. ACM, New York (2001) CrossRefGoogle Scholar
  11. 11.
    Gebal, K., Baerentzen, J.A., Aanaes, H., Larsen, R.: Shape analysis using the auto diffusion function. In: Proceedings of the Symposium on Geometry Processing, SGP ’09, pp. 1405–1413. Eurographics Association, Aire-la-Ville (2009) Google Scholar
  12. 12.
    Giorgi, D., Biasotti, S., Paraboschi, L.: Shape retrieval contest 2007: Watertight models track (2007). http://watertight.ge.imati.cnr.it/
  13. 13.
    Huang, Q.X., Adams, B., Wicke, M., Guibas, L.J.: Non-rigid registration under isometric deformations. In: Proceedings of the Symposium on Geometry Processing, SGP ’08, pp. 1449–1457. Eurographics Association, Aire-la-Ville (2008) Google Scholar
  14. 14.
    Jain, V., Zhang, H.: A spectral approach to shape-based retrieval of articulated 3d models. Comput. Aided Des. 39, 398–407 (2007) CrossRefGoogle Scholar
  15. 15.
    Kim, V., Lipman, Y., Chen, X., Funkhouser, T.: Mobius transformations for global intrinsic symmetry analysis. Comput. Graph. Forum (Symposium on Geometry Processing) 29(5) (2010) Google Scholar
  16. 16.
    Kim, V.G., Lipman, Y., Funkhouser, T.: Blended intrinsic maps. Trans. Graph. (Proc. of SIGGRAPH 2011) (2011) Google Scholar
  17. 17.
    Leordeanu, M., Hebert, M.: A spectral technique for correspondence problems using pairwise constraints. In: ICCV ’05: Proceedings of the Tenth IEEE International Conference on Computer Vision, pp. 1482–1489. IEEE Computer Society, Washington (2005) Google Scholar
  18. 18.
    Lipman, Y., Funkhouser, T.: Möbius voting for surface correspondence. ACM Trans. Graph. 28(3), 1–12 (2009) CrossRefGoogle Scholar
  19. 19.
    Milnor, J.: Morse Theory. Princeton Univ. Press, Princeton (1963) zbMATHGoogle Scholar
  20. 20.
    Ovsjanikov, M., Mérigot, Q., Mémoli, F., Guibas, L.: One point isometric matching with the heat kernel. Comput. Graph. Forum 29(5), 1555–1564 (2010) CrossRefGoogle Scholar
  21. 21.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Inc., Upper Saddle River (1982) zbMATHGoogle Scholar
  22. 22.
    Praun, E., Sweldens, W., Schröder, P.: Consistent mesh parameterizations. In: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’01, pp. 179–184. ACM, New York (2001). http://doi.acm.org/10.1145/383259.383277 CrossRefGoogle Scholar
  23. 23.
    Reuter, M.: Hierarchical shape segmentation and registration via topological features of Laplace-Beltrami eigenfunctions. Int. J. Comput. Vis. 89, 287–308 (2010). doi: 10.1007/s11263-009-0278-1 CrossRefGoogle Scholar
  24. 24.
    Ruggeri, M.R., Patanè, G., Spagnuolo, M., Saupe, D.: Spectral-driven isometry-invariant matching of 3d shapes. Int. J. Comput. Vis. 89, 248–265 (2010) CrossRefGoogle Scholar
  25. 25.
    Rustamov, R.M.: Laplace-Beltrami eigenfunctions for deformation invariant shape representation. In: Proceedings of the fifth Eurographics Symposium on Geometry Processing, pp. 225–233. Eurographics Association, Aire-la-Ville (2007) Google Scholar
  26. 26.
    Sharma, A., Horaud, R.P.: Shape matching based on diffusion embedding and on mutual isometric consistency. In: Workshop on Nonrigid Shape Analysis and Deformable Image Alignment, NORDIA 2010, June, 2010, pp. 29–36. IEEE, San Francisco, Etats-Unis (2010) Google Scholar
  27. 27.
    Sun, J., Ovsjanikov, M., Guibas, L.: A concise and provably informative multi-scale signature based on heat diffusion. In: Proceedings of the Symposium on Geometry Processing, SGP ’09, pp. 1383–1392. Eurographics Association, Aire-la-Ville (2009) Google Scholar
  28. 28.
    Sun, J., Chen, X., Funkhouser, T.: Fuzzy geodesics and consistent sparse correspondences for deformable shapes. Comput. Graph. Forum (Symposium on Geometry Processing) 29(5) (2010) Google Scholar
  29. 29.
    Tevs, A., Berner, A., Wand, M., Ihrke, I., Seidel, H.P.: Intrinsic shape matching by planned landmark sampling. Comput. Graph. Forum 30, 543–552 (2011) CrossRefGoogle Scholar
  30. 30.
    van Kaick, O., Zhang, H., Hamarneh, G., Cohen-Or, D.: A survey on shape correspondence. In: Proc. of Eurographics State-of-the-Art Report, pp. 1–22 (2010) Google Scholar
  31. 31.
    Weinkauf, T., Gingold, Y.I., Sorkine, O.: Topology-based smoothing of 2d scalar fields with c1-continuity. Comput. Graph. Forum 29(3), 1221–1230 (2010) CrossRefGoogle Scholar
  32. 32.
    Zeng, Y., Gu, X., Samaras, D., Wang, C., Wang, Y., Paragios, N., Galen, E., de France, I.S.I.: Dense non-rigid surface registration using high-order graph matching. In: CVPR (2010) Google Scholar
  33. 33.
    Zhang, H., Sheffer, A., Cohen-Or, D., Zhou, Q., van Kaick, O., Tagliasacchi, A.: Deformation-driven shape correspondence. In: Proceedings of the Symposium on Geometry Processing, SGP ’08, pp. 1431–1439. Eurographics Association, Aire-la-Ville (2008) Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.CAD&CG labZhejiang UniversityHangzhouChina
  2. 2.Department of Computer Science and EngineeringWashington University in St. LouisSt. LouisUSA

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