SpringerLink
Log in

Manifold Intrinsic Similarity

  • Reference work entry
Handbook of Mathematical Methods in Imaging

Abstract

Non-rigid shapes are ubiquitous in Nature and are encountered at all levels of life, from macro to nano. The need to model such shapes and understand their behavior arises in many applications in imaging sciences, pattern recognition, computer vision, and computer graphics. Of particular importance is understanding which properties of the shape are attributed to deformations and which are invariant, i.e., remain unchanged. This chapter presents an approach to non-rigid shapes from the point of view of metric geometry. Modeling shapes as metric spaces, one can pose the problem of shape similarity as the similarity of metric spaces and harness tools from theoretical metric geometry for the computation of such a similarity.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
EUR 29.95
Price includes VAT (Germany)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
EUR 533.93
Price includes VAT (Germany)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References and Further Reading

  1. Adams CC, Franzosa R (2008) Introduction to topology: pure and applied, Prentice-Hall, Harlow

    Google Scholar 

  2. Aizawa A (2003) An information-theoretic perspective of tf–idf measures. Inform Process Manage 39(1):45–65

    Article  MathSciNet  MATH  Google Scholar 

  3. Alt H, Mehlhorn K, Wagener H, Welzl E (1988) Congruence, similarity, and symmetries of geometric objects. Discrete Comput Geom 3: 237–256

    Article  MathSciNet  MATH  Google Scholar 

  4. Andreetto M, Brusco N, Cortelazzo GM (2004) Automatic 3D modeling of textured cultural heritage objects. Trans Image Process 13(3):335–369

    Google Scholar 

  5. Assfalg J, Bertini M, Pala P, Del Bimbo A (2007) Content-based retrieval of 3d objects using spin image signatures. Trans Multimedia 9(3): 589–599

    Article  Google Scholar 

  6. Atallah MJ (1985) On symmetry detection. IEEE Trans Comput c-34(7):663–666

    Google Scholar 

  7. Aurenhammer F (1991) Voronoi diagramsa survey of a fundamental geometric data structure. ACM Comput Surv 23(3):345–405

    Article  Google Scholar 

  8. Bay H, Tuytelaars T, Van Gool L (2006) SURF: speeded up robust features. Proceedings of ECCV6, pp 404–417

    Google Scholar 

  9. Belkin M, Niyogi P (2003) Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput 13:1373–1396, Introduction of Laplacian embeddings

    Article  Google Scholar 

  10. Bellman RE (2003) Dynamic programming. Dover, New York

    MATH  Google Scholar 

  11. Belongie S, Malik J, Puzicha J (2002) Shape matching and object recognition using shape contexts. Trans PAMI 24:509–522

    Article  Google Scholar 

  12. Ben-Chen M, Weber O, Gotsman C (2008) Characterizing shape using conformal factors. Proceedings of 3DOR

    Google Scholar 

  13. Bérard P, Besson G, Gallot S (1994) Embedding Riemannian manifolds by their heat kernel. Geom Funct Anal 4(4):373–398

    Article  MathSciNet  MATH  Google Scholar 

  14. Besl PJ, McKay ND (1992) A method for registration of 3D shapes, IEEE Trans Pattern Anal Mach Intell (PAMI) 14(2):239–256, Introduction of ICP

    Article  Google Scholar 

  15. Bjorck AA (1996) Numerical methods for least squares problems. Society for Industrial Mathematics, Philadelphia

    Book  Google Scholar 

  16. Bernstein M, de Silva V, Langford JC, Tenenbaum JB (2000) Graph approximations to geodesics on embedded manifolds, Technical report

    Google Scholar 

  17. Borg I, Groenen P (1997) Modern multidimensional scaling - theory and applications. Comprehensive overview of MDS problems and their numerical solution. Springer, New York

    MATH  Google Scholar 

  18. Bronstein AM, Bronstein MM (2008) Not only size matters: regularized partial matching of nonrigid shapes. IEEE computer society conference on computer vision and pattern recognition workshops, 2008 CVPR Workshops 2008

    Google Scholar 

  19. Bronstein AM, Bronstein MM (2008) Regularized partial matching of rigid shapes. Proceedings of European conference on computer vision (ECCV), pp 143–154

    Google Scholar 

  20. Bronstein AM, Bronstein MM, Kimmel R (2003) Expression-invariant 3D face recognition. Proceedings of audio and video-based biometric person authentication. Lecture notes in computer science, vol 2688, 3D face recognition using metric model. Springer, Berlin, pp 62–69

    Google Scholar 

  21. Bronstein AM, Bronstein MM, Kimmel R (2005) On isometric embedding of facial surfaces into S3 (2005) Proceedings of international conference scale space and pde methods in computer vision. Lecture notes in computer science, vol 3459, MDS with spherical geometry. Springer, New York, pp 622–631

    Google Scholar 

  22. Bronstein AM, Bronstein MM, Kimmel R (2005) Three-dimensional face recognition. Int J Comput Vis (IJCV) 64(1):5–30, 3D face recognition using metric model

    Article  Google Scholar 

  23. Bronstein AM, Bronstein MM, Kimmel R (2006) Efficient computation of isometry-invariant distances between surfaces. SIAM J Sci Comput 28(5):1812–1836, computation of the Gromov-Hausdorff distance using GMDS

    Article  MathSciNet  MATH  Google Scholar 

  24. Bronstein AM, Bronstein MM, Kimmel R (2006) Generalized multidimensional scaling: a framework for isometry-invariant partial surface matching. Proc Natl Acad Sci (PNAS) 103(5):1168–1172, Introduction of generalized MDS

    Article  MathSciNet  MATH  Google Scholar 

  25. Bronstein AM, Bronstein MM, Kimmel R (2006) Robust expression-invariant face recognition from partially missing data. Proceedings of European Conference on Computer Vision (ECCV), 3D face recognition with partially missing data, pp 396–408

    Google Scholar 

  26. Bronstein AM, Bronstein MM, Kimmel R (2008) Numerical geometry of non-rigid shapes. Springer, New York, first systematic treatment of non-rigid shapes

    MATH  Google Scholar 

  27. Bronstein AM, Bronstein MM, Kimmel R, Mahmoudi M, Sapiro G (2010) A Gromov-Hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching. Int J Comput Vis (IJCV) 89:266–286

    Article  Google Scholar 

  28. Bronstein AM, Bronstein MM, Ovsjanikov M, Guibas LJ (2009) Shape google: a computer vision approach to invariant shape retrieval. Proceedings of NORDIA

    Google Scholar 

  29. Bronstein AM, Bronstein MM, Bruckstein AM, Kimmel R (2009) Partial similarity of objects, or how to compare a centaur to a horse. Int J Comput Vis 84(2):163–183

    Article  Google Scholar 

  30. Bronstein AM, Bronstein MM, Bustos B, Castellani U, Crisani M, Falcidieno B, Guibas LJ, Isipiran I, Kokkinos I, Murino V, Ovsjanikov M, Patané G, Spagnuolo M, Sun J (2010) Robust feature detection and description benchmark. Proceedings of 3DOR

    Google Scholar 

  31. Bronstein AM, Bronstein MM, Kimmel R, Mahmoudi M, Sapiro G (2010) A Gromov-Hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching. IJCV 89(2–3):266–286

    Article  Google Scholar 

  32. Bronstein MM, Bronstein AM (2010) Shape recognition with spectral Distances. Trans PAMI (in press)

    Google Scholar 

  33. Bronstein MM, Bronstein AM, Kimmel R, Yavneh I (2006) Multigrid multidimensional scaling. Num Linear Algebra Appl 13(2–3): 149–171, Multigrid solver for MDS problems

    Article  MathSciNet  MATH  Google Scholar 

  34. Bronstein MM, Kokkinos I (2010) Scale-invariant heat kernel signatures for non-rigid shape recognition. Proceedings of CVPR

    Google Scholar 

  35. Burago D, Burago Y, Ivanov S (2001) A course in metric geometry. Graduate studies in mathematics, vol 33, Systematic introduction to metric geometry. AMS, Providence

    Google Scholar 

  36. Chan TF, Vese LA (2001) A level set algorithm for minimizing the Mumford-Shah functional in image processing. IEEE workshop on variational and level set methods, pp 161–168

    Chapter  Google Scholar 

  37. Chen Y, Medioni G (1991) Object modeling by registration of multiple range images. Proceedings of conference on robotics and automation, Introduction of ICP

    Google Scholar 

  38. Chum O, Philbin J, Sivic J, Isard M, Zisserman A (2007) Total recall: automatic query expansion with a generative feature model for object retrieval. Proceedings of ICCV

    Google Scholar 

  39. Clarenz U, Rumpf M, Telea A (2004) Robust feature detection and local classification for surfaces based on moment analysis. Trans Visual Comput Graphics 10(5):516–524

    Article  Google Scholar 

  40. Coifman RR, Lafon S (2006) Diffusion maps. Appl Comput Harmon Anal 21(1):5–30, Definition of diffusion distance

    Article  MathSciNet  MATH  Google Scholar 

  41. Coifman RR, Lafon S, Lee AB, Maggioni M, Nadler B, Warner F, Zucker SW (2005) Geometric diffusions as a tool for harmonic analysis and structure definition of data: diffusion maps. Proc Natl Acad Sci (PNAS) 102(21):7426–7431, Introduction of diffusion maps and diffusion distances

    Article  Google Scholar 

  42. Cox TF, Cox MAA (1994) Multidimensional scaling. Chapman & Hall, London

    MATH  Google Scholar 

  43. Crandal MG, Lions P-L (1983) Viscosity solutions of Hamilton–Jacobi Equations. Trans AMS 277:1–43

    Article  Google Scholar 

  44. Dalai N, Triggs B (2005) Histograms of oriented gradients for human Detection. Proceedings of CVPR

    Google Scholar 

  45. De Leeuw J (1977) Recent developments in statistics, ch Applications of convex analysis to multidimensional scaling. North-Holland, Amsterdam, pp 133–145

    Google Scholar 

  46. Du Q, Faber V, Gunzburger M (2006) Centroidal Voronoi tessellations: applications and algorithms. SIAM Rev 41(4):637–676

    Article  MathSciNet  Google Scholar 

  47. Dubrovina A, Kimmel R (2010) Matching shapes by eigendecomposition of the Laplace-Beltrami operator. Proceedings of 3DPVT

    Google Scholar 

  48. Elad A, Kimmel R (2001) Bending invariant representations for surfaces. Proceedings on computer vision and pattern recognition (CVPR), Introduction of canonical forms, pp 168–174

    Google Scholar 

  49. Elad A, Kimmel R (2003) On bending invariant signatures for surfaces. IEEE Trans Pattern Anal Mach Intell (PAMI) 25(10):1285–1295, Introduction of canonical forms

    Article  Google Scholar 

  50. Gebal K, Bærentzen JA, Aanæs H, Larsen R (2009) Shape analysis using the auto diffusion function. Computer Graphics Forum 28(5):1405–1413

    Article  Google Scholar 

  51. Gelfand N, Mitra NJ, Guibas LJ, Pottmann H (2005) Robust global registration. Proceedings of SGP

    Google Scholar 

  52. Gersho A, Gray RM (1992) Vector quantization and signal compression. Kluwer, Boston

    Book  MATH  Google Scholar 

  53. Glomb P (May 2009) Detection of interest points on 3D data: extending the harris Operator. Computer recognition systems 3. Advances in soft computing, vol 57. Springer, Berlin/Heidelberg, pp 103–111

    Google Scholar 

  54. Gold S, Rangarajan A (1996) A graduated assignment algorithm for graph matching. Trans PAMI 18:377–388

    Article  Google Scholar 

  55. Gordon C, Webb DL, Wolpert S (1992) One cannot hear the shape of the drum. Bull AMS 27(1):134–138, Example of isospectral but non-isometric shapes

    Article  MathSciNet  MATH  Google Scholar 

  56. Gromov M (1981) Structures Métriques Pour les Variétés Riemanniennes. Textes Mathématiques, vol 1, Introduction of the Gromov-Hausdorff distance

    MATH  Google Scholar 

  57. Gu X, Gortler S, Hoppe H (2002) Geometry images. Proceedings of SIGGRAPH, pp 355–361

    Google Scholar 

  58. Harris C, Stephens M (1988) A combined corner and edge detection. Proceedings of fourth Alvey vision conference, pp 147–151

    Google Scholar 

  59. Hausdorff F (1914) Grundzüge der Mengenlehre, Definition of the Hausdorff distance. Verlag Veit & Co, Leipzig,

    Google Scholar 

  60. Hochbaum DS, Shmoys DB (1985) A best possible heuristic for the k-center problem. Math Oper Res 10:180–184

    Article  MathSciNet  MATH  Google Scholar 

  61. Indyk P, Thaper N (2003) Fast image retrieval via embeddings. 3rd International workshop on statistical and computational theories of vision

    Google Scholar 

  62. Johnson AE, Hebert M (1999) Using spin images for efficient object recognition in cluttered 3D scenes. Trans PAMI 21(5):433–449

    Article  Google Scholar 

  63. Kac M (1966) Can one hear the shape of a drum? Am Math Mon 73:1–23, Kac’s conjecture about isospectral but non-isometric shapes

    Article  MATH  Google Scholar 

  64. Kimmel R, Sethian JA (1998) Computing geodesic paths on manifolds. Proc Natl Acad Sci (PNAS) 95(15):8431–8435

    Article  MathSciNet  MATH  Google Scholar 

  65. Kolomenkin M, Shimshoni I, Tal A (2009) On edge detection on surfaces. Proceedings of CVPR

    Google Scholar 

  66. Komodakis N, Paragios N, Tziritas G (2007) MRF optimization via dual decomposition: message-passing revisited. Proceedings of ICCV

    Google Scholar 

  67. Leibon G, Letscher D (2000) Delaunay triangulations and Voronoi diagrams for Riemannian manifolds. Proceedings of symposium on computational geometry, pp 341–349

    Google Scholar 

  68. Lévy B (2006) Laplace-Beltrami eigenfunctions towards an algorithm that “understands” geometry. International conference on shape modeling and applications, The use of Laplace-Belrami operator for shape analysis and synthesis

    Google Scholar 

  69. Lloyd SP (1957) Least squares quantization in PCM. Bell telephone laboratories paper

    Google Scholar 

  70. Losasso F, Hoppe H, Schaefer S, Warren J (2003) Smooth geometry Images. Proceedings of symposium on geometry processing (SGP), pp 138–145

    Google Scholar 

  71. Lowe D (2004) Distinctive image features from scale-invariant keypoint. Int J Comput Vis 60:91–110

    Article  Google Scholar 

  72. Matas J, Chum O, Urban M, Pajdla T (2004) Robust wide-baseline stereo from maximally stable extremal regions. Image Vis Comput 22(10):761–767

    Article  Google Scholar 

  73. Mateus D, Horaud RP, Knossow D, Cuzzolin F, Boyer E (2008) Articulated shape matching using Laplacian eigenfunctions and unsupervised point registration. Proceedings of CVPR

    Google Scholar 

  74. Max J (1960) Quantizing for minimum distortion. IEEE Trans Inform Theory 6(1):7–12

    Article  MathSciNet  Google Scholar 

  75. Mémoli F (2007) On the use of Gromov-Hausdorff distances for shape Comparison. Proceedings of point based graphics, Prague, Definition of the Gromov-Wasserstein distance

    Google Scholar 

  76. Mémoli F (2008) Gromov-Hausdorff distances in Euclidean spaces. Proceedings of non-rigid shapes and deformable image alignment (NORDIA), Relation of Gromov-Hausdorff distances in Euclidean spaces to Hausdorff and ICP distances

    Google Scholar 

  77. Mémoli F, Sapiro G (2001) Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces. J Comput Phys 173(1):764–795

    Google Scholar 

  78. Memoli F, Sapiro G (2005) Distance functions and geodesics on submanifolds of rd and point clouds. SIAM J Appl Math 65(4):1227

    Article  MathSciNet  MATH  Google Scholar 

  79. Mémoli F, Sapiro G (2005) A theoretical and computational framework for isometry invariant recognition of point cloud data. Found Comput Math 5:313–346, First use of the Gromov-Hausdorff distance in shape recognition

    Article  MathSciNet  MATH  Google Scholar 

  80. Meyer M, Desbrun M, Schroder P, Barr AH (2003) Discrete differential-geometry operators for triangulated 2-manifolds. Visual Math III:35–57, Cotangent weights discretization of the Laplace-Beltrami operator

    MathSciNet  Google Scholar 

  81. Mitra NJ, Bronstein AM, Bronstein MM (2010) Intrinsic regularity detection in 3D geometry, Proc. ECCV

    Google Scholar 

  82. Mitra NJ, Gelfand N, Pottmann H, Guibas L (2004) Registration of point cloud data from a geometric optimization perspective. Proceedings of Eurographics symposium on geometry processing, pp 23–32, Analysis of ICP algorithms from optimization standpoint

    Google Scholar 

  83. Mitra NJ, Guibas LJ, Giesen J, Pauly M (2006) Probabilistic fingerprints for shapes. Proceedings of SGP

    Google Scholar 

  84. Mitra NJ, Guibas LJ, Pauly M (2006) Partial and approximate symmetry detection for 3D geometry. ACM Trans Graphics 25(3): 560–568

    Article  Google Scholar 

  85. Nash J (1956) The imbedding problem for Riemannian manifolds. Ann Math 63:20–63, Nash embedding theorem

    Article  MathSciNet  MATH  Google Scholar 

  86. Osada R, Funkhouser T, Chazelle B, Dobkin D (2002) Shape distributions. ACM Trans Graphics (TOG) 21(4):807–832, Introduction of the shape distributions method for rigid shapes

    Article  Google Scholar 

  87. Ovsjanikov M, Sun J, Guibas L (2008) Global intrinsic symmetries of Shapes. Computer graphics forum, vol 27. Spectral method for non-rigid symmetry detection, pp 1341–1348

    Google Scholar 

  88. Ovsjanikov M, Sun J, Guibas LJ (2008) Global intrinsic symmetries of shapes. Proceedings of SGP, pp 1341–1348

    Google Scholar 

  89. Pauly M, Keiser R, Gross M (2003) Multi-scale feature extraction on point-sampled surfaces. Computer graphics forum, vol 22, pp 281–289

    Google Scholar 

  90. Pauly M, Mitra NJ, Wallner J, Pottmann H, Guibas LJ (2008) Discovering structural regularity in 3D geometry, ACM trans. Graphics 27(3)

    Google Scholar 

  91. Peyre G, Cohen L (2004) Surface segmentation using geodesic centroidal Tesselation. Proceedings of international symposium on 3D data processing visualiztion transmission, pp  995–1002

    Google Scholar 

  92. Pinkall U, Polthier K (1993) Computing discrete minimal surfaces and their conjugates. Exp Math 2(1):15–36, Cotangent weights discretization of the Laplace-Beltrami operator

    Article  MathSciNet  MATH  Google Scholar 

  93. Raviv D, Bronstein AM, Bronstein MM, Kimmel R (2007) Symmetries of non-rigid shapes, Proceedings of workshop on non-rigid registration and tracking through learning (NRTL)

    Google Scholar 

  94. Raviv D, Bronstein AM, Bronstein MM, Kimmel R (2010) Full and partial symmetries of non-rigid shapes. Intl J Comput Vis (IJCV) 89(1): 18–39

    Article  Google Scholar 

  95. Reuter M, Biasotti S, Giorgi D, Patanè G, Spagnuolo M (2009) Discrete Laplace-Beltrami operators for shape analysis and segmentation. Comput Graphics 33:381–390, FEM approximation of the Laplace-Beltrami operator

    Article  Google Scholar 

  96. Reuter M, Wolter F-E, Peinecke N (2006) Laplace-beltrami spectra as “shape-DNA” of surfaces and solids. Comput Aided Design 38(4):342–366, Shape recognition using Laplace-Beltrami spectrum

    Article  Google Scholar 

  97. Rosman G, Bronstein AM, Bronstein MM, Sidi A, Kimmel R (2008) Fast multidimensional scaling using vector extrapolation. Technical report CIS-2008-01, Department of Computer Science, Technion, Israel, Introduction of vector extrapolation methods for MDS problems

    Google Scholar 

  98. Rubner Y, Guibas LJ, Tomasi C (1997) The earth movers distance, multi-dimensional scaling, and color-based image retrieval. Proceedings of the ARPA image understanding workshop, pp 661–668

    Google Scholar 

  99. Rustamov RM (2007) Laplace-Beltrami eigenfunctions for deformation invariant shape representation. Proceedings of SGP, Introduction of GPS embedding, pp 225–233

    Google Scholar 

  100. Sander P, Wood Z, Gortler S, Snyder J, Hoppe H (2003) Multichart geometry images. Proceedings of Symposium on geometry processing (SGP), pp 146–155

    Google Scholar 

  101. Sethian JA (1996) A fast marching level set method for monotonically advancing fronts. Proc Natl Acad Sci (PNAS) 93(4):1591–1595

    Article  MathSciNet  MATH  Google Scholar 

  102. Shilane P, Funkhauser T (2006) Selecting distinctive 3D shape descriptors for similarity retrieval. Proceedings of Shape Modelling and Applications

    Google Scholar 

  103. Shirdhonkar S, Jacobs DW (2008) Approximate earth movers distance in linear time. IEEE Conference on Computer Vision and Pattern Recognition, 2008. CVPR 2008

    Google Scholar 

  104. Sivic J, Zisserman A (2003) Video google: a text retrieval approach to object matching in videos. Proceedings of CVPR

    Google Scholar 

  105. Spira A, Kimmel R (2004) An efficient solution to the eikonal equation on parametric manifolds. Interfaces Free Boundaries 6(4): 315–327

    Article  MathSciNet  MATH  Google Scholar 

  106. Starck J, Hilton A (2007) Correspondence labelling for widetimeframe free-form surface matching. Proceedings of ICCV

    Google Scholar 

  107. Sun J, Ovsjanikov M, Guibas LJ (2009) A concise and provably informative multi-scale signature based on heat diffusion. Proceedings of SGP

    Google Scholar 

  108. Thorstensen N, Keriven R (2009) Non-rigid shape matching using geometry and photometry. Proceedings of CVPR

    Google Scholar 

  109. Thrun S, Wegbreit B (2005) Shape from symmetry. Procedings of ICCV

    Google Scholar 

  110. Toldo R, Castellani U, Fusiello A (2009) Visual vocabulary signature for 3D object retrieval and partial matching. Proceedings of 3DOR

    Google Scholar 

  111. Torresani L, Kolmogorov V, Rother C (2008) Feature correspondence via graph matching: Models and global optimization. Proceedings of ECCV, pp 596–609

    Google Scholar 

  112. Tsai YR, Cheng LT, Osher S, Zhao HK (2003) Fast swee** algorithms for a class of Hamilton-Jacobi equations. SIAM J Num Anal (SINUM) 41(2):673–694

    Article  MathSciNet  MATH  Google Scholar 

  113. Tsitsiklis JN (1995) Efficient algorithms for globally optimal trajectories. IEEE Trans Automatic Control 40(9):1528–1538

    Article  MathSciNet  MATH  Google Scholar 

  114. Tutte WT (1963) How to draw a graph. Proc Lond Math Soc 13(3):743–768, Tutte Laplacian operator

    Article  MathSciNet  MATH  Google Scholar 

  115. Walter J, Ritter H (2002) On interactive visualization of high-dimensional data using the hyperbolic plane. Proceedings of international conference on knowledge discovery and data mining (KDD), MDS with hyperbolic geometry, pp 123–131

    Google Scholar 

  116. Wang C, Bronstein MM, Paragios N (2010) Discrete minimum distortion correspondence problems for non-rigid shape matching, Research report 7333, INRIA

    Google Scholar 

  117. Wardetzky M, Mathur S, Kälberer F, Grinspun E (2008) Discrete Laplace operators: no free lunch. Conference on computer graphics and interactive techniques, Analysis of different discretizations of the Laplace-Beltrami operator

    Google Scholar 

  118. Weber O, Devir YS, Bronstein AM, Bronstein MM, Kimmel R (2008) Parallel algorithms for approximation of distancelm maps on parametric surfaces. ACM Trans Graph 27(4):1–16

    Article  Google Scholar 

  119. Wolter JD, Woo TC, Volz RA (1985) Optimal algorithms for symmetry detection in two and three dimensions. Visual Comput 1:37–48

    Article  MATH  Google Scholar 

  120. Zaharescu A, Boyer E, Varanasi K, Horaud R (2009) Surface feature detection and description with applications to mesh matching. Proceedings of CVPR

    Google Scholar 

  121. Zhang H (2004) Discrete combinatorial Laplacian operators for digital geometry processing. SIAM Conference on Geometric Design. Combinatorial Laplace-Beltrami operator, pp 575–592

    Google Scholar 

  122. Zhao HK (2005) Fast swee** method for Eikonal equations. Math Comput 74:603–627

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Technion-Israel Institute of Technology, Haifa, Israel

    Alexander M. Bronstein

  2. Technion-Israel Institute of Technology, Haifa, Israel

    Michael M. Bronstein

Authors
  1. Alexander M. Bronstein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael M. Bronstein
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Computational Science Center, University of Vienna, Nordbergstrasse 15, Vienna, Austria

    Otmar Scherzer

  2. RICAM, Austrian Academy of Sciences, Linz, Austria

    Otmar Scherzer

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this entry

Cite this entry

Bronstein, A.M., Bronstein, M.M. (2011). Manifold Intrinsic Similarity. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging. Springer, New York, NY. https://doi.org/10.1007/978-0-387-92920-0_32

Download citation

Publish with us

Policies and ethics

Search

Navigation