Pattern Analysis and Applications

, Volume 20, Issue 4, pp 1261–1273 | Cite as

Biharmonic density estimate: a scale-space descriptor for 3-D deformable surfaces

  • Anirban Mukhopadhyay
  • Suchendra M. BhandarkarEmail author
Short Paper


The wide variability in deformable three-dimensional (3-D) shapes calls for the formulation of a multiscale surface signature for effective characterization and analysis of the underlying 3-D intrinsic geometry. To this end, a novel intrinsic geometric scale-space descriptor for 3-D deformable surfaces, termed as the biharmonic density estimate (BDE), is proposed. The BDE, derived from the biharmonic distance measure, is shown to provide an intrinsic geometric scale-space signature for multiscale surface feature-based representation of deformable 3-D shapes that is both effective and useful for practical applications. The proposed BDE signature provides a theoretical framework for the concept of intrinsic geometric scale space, resulting in a highly descriptive characterization of both the local surface structure and the global metric of the underlying 3-D shape. The compactness and robustness of the BDE are experimentally demonstrated on two standard benchmark datasets. The applications of the BDE in the detection of key components on a deformable 3-D surface and determination of sparse point correspondences between two deformable 3-D shapes are also demonstrated.


Shape analysis Biharmonic distance Diffusion geometry 


  1. 1.
    Aubry M, Schlickewei U, Cremers D (2011) The wave kernel signature: a quantum mechanical approach to shape analysis. In: Proceedings of the IEEE conference computer vision pattern recognition (CVPR)Google Scholar
  2. 2.
    Bansal M, Daniilidis K (2013) Joint spectral correspondence for disparate image matching. In: Proceedings of the IEEE conference computer vision pattern recognition (CVPR), pp 2802–2809Google Scholar
  3. 3.
    Belkin MM, Sun J, Wang Y (2008) Laplace operator on meshed surface. In: Proceedings of the symposium on computational geometry (SCG), pp 278–287Google Scholar
  4. 4.
    Belongie S, Malik J, Puzicha J (2000) Shape context: a new descriptor for shape matching and object recognition. In: Proceedings of the neural information processing systems (NIPS)Google Scholar
  5. 5.
    Boscaini D, Castellani U (2014) A sparse coding approach for local-to-global 3D shape description. Vis Comput 30(11):1233–1245CrossRefGoogle Scholar
  6. 6.
    Bronstein AM, Bronstein MM, Kimmel R (2007) Calculus of non-rigid surfaces for geometry and texture manipulation. IEEE Trans Vis Comput Graph 13(5):902–913CrossRefGoogle Scholar
  7. 7.
    Bronstein AM, Bronstein MM, Kimmel R (2008) Numerical geometry of non-rigid shapes. Springer, BerlinzbMATHGoogle Scholar
  8. 8.
    Bronstein AM, Bronstein MM, Castellani U, Falcidieno B, Fusiello A, Godil A, Guibas LJ, Kokkinos I, Lian Z, Ovsjanikov M, Patane G, Spagnuolo M, Toldo R (2010) SHREC 2010: robust large-scale shape retrieval benchmark. In: Proceedings of the eurographics workshop 3D object retrieval (3DOR)Google Scholar
  9. 9.
    Bronstein MM, Bronstein AM (2011) Shape recognition with spectral distances. IEEE Trans Pattern Anal Mach Intell 33(5):1065–1071CrossRefGoogle Scholar
  10. 10.
    Bronstein AM, Bronstein MM, Guibas LJ, Ovsjanikov M (2011) Shape Google: geometric words and expressions for invariant shape retrieval. ACM Trans Graph 30(1):1–20CrossRefGoogle Scholar
  11. 11.
    Fang Y, Sun M, Ramani K (2012) Temperature distribution descriptor for robust 3D shape retrieval. In: Proceedings of the IEEE conference computer vision and pattern recognition (CVPR), pp 9–16Google Scholar
  12. 12.
    Johnson A, Hebert M (2002) Using spin images for efficient object recognition in cluttered 3D scenes. IEEE Trans Pattern Anal Mach Intell 21(5):433–449CrossRefGoogle Scholar
  13. 13.
    Karni Z, Gotsman C (2000) Spectral compression of mesh geometry. In: Proceedings of the ACM SIGGRAPHGoogle Scholar
  14. 14.
    Levy B (2006) Laplace–Beltrami eigenfunctions: towards an algorithm that understands geometry. In: Proceedings of the IEEE international conference on shape modeling and applications, p 13Google Scholar
  15. 15.
    Li X, Guskov I (2005) Multi-scale features for approximate alignment of point-based surfaces. In: Proceedings of the eurographics symposium on geometry processing (SGP), p 217Google Scholar
  16. 16.
    Li B, Godil A, Johan H (2014) Hybrid shape descriptor and meta similarity generation for non-rigid and partial 3D model retrieval. Multimed Tools Appl 72(2):1531–1560CrossRefGoogle Scholar
  17. 17.
    Ling H, Okada K (2006) Diffusion distance for histogram comparison. In: Proceedings of the IEEE conference computer vision pattern recognition (CVPR), vol 1Google Scholar
  18. 18.
    Lipman Y, Rustamov RM, Funkhouser TA (2010) Biharmonic distance. ACM Trans Graph 29(3):27:1–27:11CrossRefGoogle Scholar
  19. 19.
    Litman R, Bronstein AM (2014) Learning spectral descriptors for deformable shape correspondence. IEEE Trans Pattern Anal Mach Intell 36(1):171–180CrossRefGoogle Scholar
  20. 20.
    Manay S, Cremers D, Hong BW, Yezzi AJ, Soatto S (2006) Integral invariants for shape matching. IEEE Trans Pattern Anal Mach Intell 28(10):1602–1618CrossRefzbMATHGoogle Scholar
  21. 21.
    Moenning C, Dodgson NA (2003) Fast marching farthest point sampling. In: Proceedings of the EurographicsGoogle Scholar
  22. 22.
    Mukhopadhyay A, Bhandarkar SM (2014) Biharmonic density estimate—a scale space signature for deformable surfaces. In: Proceedings of the IEEE international conference on image processing (ICIP)Google Scholar
  23. 23.
    Mukhopadhyay A, Arun Kumar CS, Bhandarkar SM (2016) Joint geometric graph embedding for partial shape matching in images. In: Proceedings of the IEEE Winter conference on applications of computer vision (WACV)Google Scholar
  24. 24.
    New AT, Mukhopadhyay A, Arabnia HR, Bhandarkar SM (2012) Non-rigid shape correspondence and description using geodesic field estimate distribution. In: Proceedings of the ACM SIGGRAPH, posterGoogle Scholar
  25. 25.
    Osada R, Funkhouser T, Chazelle B, Dobkin D (2002) Shape distributions. ACM Trans Graph 21(4):807–832CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Ovsjanikov M, Merigot Q, Memoli F, Guibas L (2010) One point isometric matching with the heat kernel. In: Proceedings of the Eurographics symposium on geometry processing (SGP)Google Scholar
  27. 27.
    Peyre G (2009) Toolbox Graph - A toolbox to process graph and triangulated meshes. In: Matlab Central.
  28. 28.
    Pinkall U, Polthier K (1993) Computing discrete minimal surfaces and their conjugates. Exp Maths 2(1):15–36CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Reuter M, Wolter F-E, Peinecke N (2006) Laplace–Beltrami spectra as “Shape-DNA” of surfaces and solids. Comput-Aided Des 38(4):342–366CrossRefGoogle Scholar
  30. 30.
    Rustamov R (2007) Laplace–Beltrami eigenfunctions for deformation invariant shape representation. In: Proceedings of the eurographics symposium on geometry processing (SGP), pp 225–233Google Scholar
  31. 31.
    Sipiran I, Bustos B (2012) Key-component detection on 3D meshes using local features. In: Proceedings of the 5th Eurographics conference 3D Object Retrieval (3DOR)Google Scholar
  32. 32.
    Sun J, Ovsjanikov M, Guibas L (2009) A concise and provably informative multi-scale signature based on heat diffusion. Comput Graph Forum 28(5):1383–1392CrossRefGoogle Scholar
  33. 33.
    Taubin G (1995) A signal processing approach to fair surface design. In: Proceedings of the ACM SIGGRAPHGoogle Scholar
  34. 34.
    Wardetzkey M (2005) Convergence of the cotangent formula: an overview. In: Discrete differential geometry. Birkhäuser Basel, pp 89–112Google Scholar
  35. 35.
    Xu G (2004) Discrete Laplace–Beltrami operators and their convergence. Comput Aided Geom Des 21(8):767–784CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Yen L, Fouss F, Decaestecker C, Francq P, Saerens M (2007) Graph nodes clustering based on the commute-time kernel. In: Proceedings of the 11th Pacific-Asia conference knowledge discovery and data mining (PAKDD)Google Scholar
  37. 37.
    Zaharescu A, Boyer E, Horaud R (2012) Keypoints and local descriptors of scalar functions on 2D manifolds. Int J Comput Vis 100(1):78–98CrossRefzbMATHGoogle Scholar
  38. 38.
    Zou G, Hua J, Lai Z, Gu X, Dong M (2009) Intrinsic geometric scale space by shape diffusion. IEEE Trans Vis Comput Graph 15(6):1193–1200CrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2017

Authors and Affiliations

  • Anirban Mukhopadhyay
    • 1
  • Suchendra M. Bhandarkar
    • 1
    Email author
  1. 1.Department of Computer ScienceThe University of GeorgiaAthensUSA

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