Advertisement

Intrinsic Regularity Detection in 3D Geometry

  • Niloy J. Mitra
  • Alex Bronstein
  • Michael Bronstein
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6313)

Abstract

Automatic detection of symmetries, regularity, and repetitive structures in 3D geometry is a fundamental problem in shape analysis and pattern recognition with applications in computer vision and graphics. Especially challenging is to detect intrinsic regularity, where the repetitions are on an intrinsic grid, without any apparent Euclidean pattern to describe the shape, but rising out of (near) isometric deformation of the underlying surface. In this paper, we employ multidimensional scaling to reduce the problem of intrinsic structure detection to a simpler problem of 2D grid detection. Potential 2D grids are then identified using an autocorrelation analysis, refined using local fitting, validated, and finally projected back to the spatial domain. We test the detection algorithm on a variety of scanned plaster models in presence of imperfections like missing data, noise and outliers. We also present a range of applications including scan completion, shape editing, super-resolution, and structural correspondence.

Keywords

Isometric Embedding Descriptor Image Intrinsic Structure Landmark Point Geometry Processing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Leyton, M.: Shape as memory. Springer, Heidelberg (2006)Google Scholar
  2. 2.
    White, R., Forsyth, D.A.: Combining cues: Shape from shading and texture. In: IEEE CVPR, vol. II, pp. 1809–1816 (2006)Google Scholar
  3. 3.
    Elad, A., Kimmel, R.: Bending invariant representations for surfaces. In: CVPR, pp. 168–174 (2001)Google Scholar
  4. 4.
    Ling, H., Jacobs, D.W.: Deformation invariant image matching. In: IEEE ICCV, Washington, DC, USA, pp. 1466–1473. IEEE Computer Society, Los Alamitos (2005)Google Scholar
  5. 5.
    Shechtman, E., Irani, M.: Matching local self-similarities across images and videos. In: CVPR, pp. 511–518 (2007)Google Scholar
  6. 6.
    Zaharescu, A., Boyer, E., Varanasi, K., Horaud, R.: Surface feature detection and description with applications to mesh matching. In: IEEE CVPR, pp. 373–380 (2009)Google Scholar
  7. 7.
    Mitra, N.J., Flory, S., Ovsjanikov, M., Gelfand, N., Guibas, L., Pottmann, H.: Dynamic geometry registration. In: Symposium on Geometry Processing, pp. 173–182 (2007)Google Scholar
  8. 8.
    Pauly, M., Mitra, N.J., Wallner, J., Pottmann, H., Guibas, L.: Discovering structural regularity in 3D geometry. ACM ToG (Proc. SIGGRAPH) 27, #43, 1–11 (2008)Google Scholar
  9. 9.
    Thrun, S., Wegbreit, B.: Shape from symmetry. In: IEEE ICCV, pp. 1824–1831 (2005)Google Scholar
  10. 10.
    Mitra, N.J., Guibas, L., Pauly, M.: Partial and approximate symmetry detection for 3d geometry. ACM ToG (Proc. SIGGRAPH) 25, 560–568 (2006)Google Scholar
  11. 11.
    Zabrodsky, H., Peleg, S., Avnir, D.: Symmetry as a continuous feature. IEEE Trans. Pattern Anal. Mach. Intell. 17, 1154–1166 (1995)CrossRefGoogle Scholar
  12. 12.
    Kazhdan, M., Funkhouser, T., Rusinkiewicz, S.: Symmetry descriptors and 3D shape matching. In: Proc. of Symp. of Geometry Processing (2004)Google Scholar
  13. 13.
    Liu, Y., Collins, R.T., Tsin, Y.: A computational model for periodic pattern perception based on frieze and wallpaper groups. IEEE Trans. Pattern Anal. Mach. Intell. 26, 354–371 (2004)CrossRefGoogle Scholar
  14. 14.
    Martinet, A., Soler, C., Holzschuch, N., Sillion, F.: Accurate detection of symmetries in 3d shapes. ACM ToG 25, 439–464 (2006)Google Scholar
  15. 15.
    Podolak, J., Shilane, P., Golovinskiy, A., Rusinkiewicz, S., Funkhouser, T.: A planar-reflective symmetry transform for 3D shapes. ACM ToG (Proc. SIGGRAPH) 25 (2006)Google Scholar
  16. 16.
    Loy, G., Eklundh, J.O.: Detecting symmetry and symmetric constellations of features. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3952, pp. 508–521. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Bokeloh, M., Berner, A., Wand, M., Seidel, H.P., Schilling, A.: Symmetry detection using line features. Computer Graphics Forum (Proc. EUROGRAPHICS) 28, 697–706 (2009)CrossRefGoogle Scholar
  18. 18.
    Park, M., Lee, S., Chen, P.C., Kashyap, S., Butt, A.A., Liu, Y.: Performance evaluation of state-of-the-art discrete symmetry detection algorithms. In: IEEE CVPR, pp. 1–8 (2008)Google Scholar
  19. 19.
    Gong, Y., Wang, Q., Yang, C., Gao, Y., Li, C.: Symmetry detection for multi-object using local polar coordinate. In: Jiang, X., Petkov, N. (eds.) Computer Analysis of Images and Patterns. LNCS, vol. 5702, pp. 277–284. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  20. 20.
    Simari, P., Kalogerakis, E., Singh, K.: Folding meshes: hierarchical mesh segmentation based on planar symmetry. In: Proc. of Symp. of Geometry Processing, pp. 111–119 (2006)Google Scholar
  21. 21.
    Mitra, N.J., Guibas, L.J., Pauly, M.: Symmetrization. ACM ToG (Proc. SIGGRAPH) 26, #63, 1–8 (2007)Google Scholar
  22. 22.
    Liu, Y., Belkina, T., Hays, J.H., Lublinerman, R.: Image de-fencing. In: IEEE CVPR (2008)Google Scholar
  23. 23.
    Cheng, M.M., Zhang, F.L., Mitra, N.J., Huang, X., Hu, S.M.: Repfinder: Finding approximately repeated scene elements for image editing. ACM ToG (Proc. SIGGRAPH) 29 (2010) (to appear)Google Scholar
  24. 24.
    Park, M., Collins, R., Liu, Y.: Deformed lattice detection via mean-shift belief propagation. In: Forsyth, D., Torr, P., Zisserman, A. (eds.) ECCV 2008, Part II. LNCS, vol. 5303, pp. 474–485. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  25. 25.
    Raviv, D., Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Full and partial symmetries of non-rigid shapes. IJCV (preprint)Google Scholar
  26. 26.
    Mémoli, F., Sapiro, G.: A theoretical and computational framework for isometry invariant recognition of point cloud data. Foundations of Comp. Mathematics 5, 313–346 (2005)zbMATHCrossRefGoogle Scholar
  27. 27.
    Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching. PNAS 103, 1168–1172 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Ovsjanikov, M., Sun, J., Guibas, L.J.: Global intrinsic symmetries of shapes. Comput. Graph. Forum (Proc. SGP) 27, 1341–1348 (2008)Google Scholar
  29. 29.
    Xu, K., Zhang, H., Tagliasacchi, A., Liu, L., Li, G., Meng, M., Xiong, Y.: Partial intrinsic reflectional symmetry of 3D shapes. ACM ToG (Proc. SIGGRAPH Asia)  28 (2009)Google Scholar
  30. 30.
    Zigelman, G., Kimmel, R., Kiryati, N.: Texture mapping using surface flattening via multi-dimensional scaling. IEEE Trans. on Vis. and Comp. Graphics 9, 198–207 (2002)CrossRefGoogle Scholar
  31. 31.
    Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Calculus of non-rigid surfaces for geometry and texture manipulation. IEEE Trans. on Vis. and Comp. Graphics (2008)Google Scholar
  32. 32.
    Borg, I., Groenen, P.: Modern multidimensional scaling - theory and apps. Springer, Heidelberg (1997)Google Scholar
  33. 33.
    Kimmel, R., Sethian, J.A.: Computing geodesic paths on manifolds. PNAS, 8431–8435 (1998)Google Scholar
  34. 34.
    De Silva, V., Tenenbaum, J.: Global versus local methods in nonlinear dimensionality reduction. In: NIPS, pp. 721–728 (2003)Google Scholar
  35. 35.
    Hochbaum, D., Shmoys, D.: A best possible heuristic for the k-center problem. Mathematics of Operations Research 10, 180–184 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Bronstein, M.M., Bronstein, A.M., Kimmel, R., Yavneh, I.: Multigrid multidimensional scaling. Numerical Linear Algebra with Applications 13, 149–171 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Borg, I., Groenen, P.: Modern multidimensional scaling: Theory and applications (2005)Google Scholar
  38. 38.
    Rosman, G., Bronstein, A., Bronstein, M., Kimmel, R.: Topologically constrained isometric embedding. Human Motion: Understanding, Modelling, Capture, and Animation (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Niloy J. Mitra
    • 1
  • Alex Bronstein
    • 2
  • Michael Bronstein
    • 3
  1. 1.Indian Institute of TechnologyDelhi
  2. 2.Tel Aviv University 
  3. 3.Technion 

Personalised recommendations