Skip to main content
SpringerLink
Log in

Variational Methods in Shape Analysis

  • Reference work entry
Handbook of Mathematical Methods in Imaging

Abstract

The concept of a shape space is linked both to concepts from geometry and from physics. On one hand, a path-based viscous flow approach leads to Riemannian distances between shapes, where shapes are boundaries of objects that mainly behave like fluids. On the other hand, a state-based elasticity approach induces a (by construction) non-Riemannian dissimilarity measure between shapes, which is given by the stored elastic energy of deformations matching the corresponding objects. The two approaches are both based on variational principles. They are analyzed with regard to different applications, and a detailed comparison is given.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 679.00
Price excludes VAT (USA)
  • 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

References and Further Reading

  1. Ambrosio L, Tortorelli VM (1992) On the approximation of free discontinuity problems. B UNIONE MAT ITAL B 6(7): 105–123

    MathSciNet  MATH  Google Scholar 

  2. Ball J (1981) Global invertibility of Sobolev functions and the interpenetration of matter. Proc Roy Soc Edinburgh 88A:315–328

    Article  Google Scholar 

  3. Beg MF, Miller MI, Trouvé A, Younes L (February 2005) Computing large deformation metric mappings via geodesic flows of diffeomorphisms. Int J Comput Vis 61(2):139–157

    Article  Google Scholar 

  4. Berkels B, Linkmann G, Rumpf M (2009) An SL(2) invariant shape median (submitted)

    Google Scholar 

  5. Bornemann F, Deuflhard P (1996) The cascadic multigrid method for elliptic problems. Numer Math 75(2):135–152

    Article  MathSciNet  MATH  Google Scholar 

  6. Bronstein A, Bronstein M, Kimmel R (2008) Numerical Geometry of Non-Rigid Shapes. Monographs in computer science. Springer, New York

    MATH  Google Scholar 

  7. Burger M, Osher SJ (2005) A survey on level set methods for inverse problems and optimal design. Eur J Appl Math 16(2):263–301

    Article  MathSciNet  MATH  Google Scholar 

  8. Caselles V, Kimmel R, Sapiro G (1997) Geodesic active contours. Int J Comput Vis 22(1):61–79

    Article  MATH  Google Scholar 

  9. Chan TF, Vese LA (2001) Active contours without edges. IEEE Trans Image Process 10(2): 266–277

    Article  MATH  Google Scholar 

  10. Charpiat G, Faugeras O, Keriven R (2005) Approximations of shape metrics and application to shape warping and empirical shape statistics. Foundations Comput Math 5(1):1–58

    Article  MathSciNet  MATH  Google Scholar 

  11. Charpiat G, Faugeras O, Keriven R, Maurel P (2006) Distance-based shape statistics. In: Proceedings of the IEEE international conference on acoustics, speech and signal processing (ICASSP 2006), vol 5, pp 925–928

    Google Scholar 

  12. Chen SE, Parent RE (1989) Shape averaging and its applications to industrial design. IEEE Comput Graphics Appl 9(1):47–54

    Article  Google Scholar 

  13. Chorin AJ, Marsden JE (1990) A Mathematical introduction to fluid mechanics, vol 4 of Texts in applied mathematics. Springer, New York

    Book  Google Scholar 

  14. Christensen GE, Rabbitt RD, Miller MI (1994) 3D brain mapping using a deformable neuroanatomy. Phys Med Biol 39(3):609–618

    Article  Google Scholar 

  15. Ciarlet PG (1988) Three-dimensional elasticity. Elsevier Science B.V., New York

    MATH  Google Scholar 

  16. Cootes TF, Taylor CJ, Cooper DH, Graham J (1995) Active shape models—their training and application. Comput Vis Image Underst 61(1):38–59

    Article  Google Scholar 

  17. Cremers D, Kohlberger T, Schnörr C (2003) Shape statistics in kernel space for variational image segmentation. Pattern Recogn 36:1929–1943

    Article  MATH  Google Scholar 

  18. Dacorogna B (1989) Direct methods in the calculus of variations. Springer, New York

    MATH  Google Scholar 

  19. Dambreville S, Rathi Y, Tannenbaum A (2006) A shape-based approach to robust image segmentation. In: Campilho A, Kamel M (eds) IEEE computer society conference on computer vision and pattern recognition, vol 4141 of LNCS, pp 173–183

    Google Scholar 

  20. Delfour MC, Zolésio J (2001) Geometries and shapes: analysis, differential calculus and optimization. Advance in design and control 4. SIAM, Philadelphia

    MATH  Google Scholar 

  21. do Carmo MP (1992) Riemannian geometry. Birkhäuser, Boston

    MATH  Google Scholar 

  22. Droske M, Rumpf M (2007) Multi scale joint segmentation and registration of image morphology. IEEE Trans Pattern Recogn Mach Intell 29(12):2181–2194

    Article  Google Scholar 

  23. Dupuis D, Grenander U, Miller M (1998) Variational problems on flows of diffeomorphisms for image matching. Quart Appl Math 56: 587–600

    MathSciNet  MATH  Google Scholar 

  24. Eckstein I, Pons JP, Tong Y, Kuo CC, Desbrun M (2007) Generalized surface flows for mesh processing. In: Eurographics symposium on geometry processing

    Google Scholar 

  25. Elad (Elbaz) A, Kimmel R (2003) On bending invariant signatures for surfaces. IEEE Trans Pattern Anal Mach Intell 25(10):1285–1295

    Article  Google Scholar 

  26. Fletcher P, Lu C, Pizer S, Joshi S (2004) Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Trans Med Imaging 23(8):995–1005

    Article  Google Scholar 

  27. Fletcher PT, Lu C, Joshi S (2003) Statistics of shape via principal geodesic analysis on Lie groups. In: IEEE computer society conference on computer vision and pattern recognition (CVPR), vol 1. Los Alamitos, CA, pp 95–101

    Google Scholar 

  28. Fletcher T, Venkatasubramanian S, Joshi S (2008) Robust statistics on Riemannian manifolds via the geometric median. In: IEEE conference on computer vision and pattern recognition (CVPR)

    Google Scholar 

  29. Fletcher P, Whitaker R (2006) Riemannian metrics on the space of solid shapes. In: Medical image computing and computer assisted intervention – MICCAI 2006

    Google Scholar 

  30. Fréchet M (1948) Les éléments aléatoires de nature quelconque dans un espace distancié. Ann Inst H Poincaré 10:215–310

    Google Scholar 

  31. Fuchs M, Jüttler B, Scherzer O, Yang H (2009) Shape metrics based on elastic deformations. J Math Imaging Vis 35(1):86–102

    Article  Google Scholar 

  32. Fuchs M, Scherzer O (May 2007) Segmentation of biologic image data with a-priori knowledge. FSP Report, Forschungsschwerpunkt S92 52, Universität Innsbruck, Austria

    Google Scholar 

  33. Fuchs M, Scherzer O (2008) Regularized reconstruction of shapes with statistical a priori knowledge. Int J Comput Vis 79(2):119–135

    Article  Google Scholar 

  34. Glaunès J, Qiu A, Miller MI, Younes L (2008) Large deformation diffeomorphic metric curve mapping. Int J Comput Vis 80(3):317–336

    Article  Google Scholar 

  35. Hafner B, Zachariah S, Sanders J (2000) Characterisation of three-dimensional anatomic shapes using principal components: application to the proximal tibia. Med Biol Eng Comput 38:9–16

    Article  Google Scholar 

  36. Hong BW, Soatto S, Vese L (2008) Enforcing local context into shape statistics. Adv Comput Math (online first)

    Google Scholar 

  37. Joshi S, Davis B, Jomier M, Gerig G (2004) Unbiased diffeomorphic atlas construction for computational anatomy. NeuroImage 23 (Suppl 1):151–160

    Article  Google Scholar 

  38. Karcher H (1977) Riemannian center of mass and mollifier smoothing. Commun Pure Appl Math 30(5):509–541

    Article  MathSciNet  MATH  Google Scholar 

  39. Kendall DG (1984) Shape manifolds, procrustean metrics, and complex projective spaces. Bull Lond Math Soc 16:81–121

    Article  MathSciNet  MATH  Google Scholar 

  40. Kilian M, Mitra NJ, Pottmann H (2007) Geometric modeling in shape space. In: ACM Trans Graph 26(64):1–8

    Google Scholar 

  41. Klassen E, Srivastava A, Mio W, Joshi SH (2004) Analysis of planar shapes using geodesic paths on shape spaces. IEEE Trans Pattern Anal Mach Intell 26(3):372–383

    Article  Google Scholar 

  42. Klingenberg WPA (1995) Riemannian geometry. Walter de Gruyter, Berlin

    Book  MATH  Google Scholar 

  43. Leventon ME, Grimson WEL, Faugeras O (2002) Statistical shape influence in geodesic active contours. In: 5th IEEE EMBS international summer school on biomedical imaging

    Google Scholar 

  44. Ling H, Jacobs DW (2007) Shape classification using the inner-distance. IEEE Trans Pattern Anal Mach Intell 29(2):286–299

    Article  Google Scholar 

  45. Liu X, Shi Y, Dinov I, Mio W (2010) A computational model of multidimensional shape. Int J Comput Vis (online first)

    Google Scholar 

  46. Manay S, Cremers D, Hong BW, Yezzi AJ, Soatto S (2006) Integral invariants for shape matching. IEEE Trans Pattern Anal Mach Intell 28(10): 1602–1618

    Article  Google Scholar 

  47. Marsden JE, Hughes TJR (1983) Mathematical foundations of elasticity. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  48. McNeill G, Vijayakumar S (2005) 2d shape classification and retrieval. In: Proceedings of the 19th international joint conference on artificial intelligence, pp 1483–1488

    Google Scholar 

  49. Mémoli F (2008) Gromov-Hausdorff distances in euclidean spaces. In: Workshop on non-rigid shape analysis and deformable image alignment (CVPR workshop, NORDIA’08)

    Google Scholar 

  50. Mémoli F, Sapiro G (2005) A theoretical and computational framework for isometry invariant recognition of point cloud data. Foundations Comput Math 5:313–347

    Article  MATH  Google Scholar 

  51. Michor PW, Mumford D (2006) Riemannian geometries on spaces of plane curves. J Eur Math Soc 8:1–48

    Article  MathSciNet  MATH  Google Scholar 

  52. Michor PW, Mumford D, Shah J, Younes L (2008) A metric on shape space with explicit geodesics. Rend Lincei Mat Appl 9:25–57

    MathSciNet  Google Scholar 

  53. Miller M, Trouvé A, Younes L (2002) On the metrics and Euler-Lagrange equations of computational anatomy. Annu Rev Biomed Engg 4:375–405

    Article  Google Scholar 

  54. Miller MI, Younes L (2001) Group actions, homeomorphisms, and matching: a general framework. Int J Comput Vis 41(1–2):61–84

    Article  MATH  Google Scholar 

  55. Modica L, Mortola S (1977) Un esempio di-\({\Gamma }^{-}\)-convergenza. Boll Un Mat Ital B (5) 14(1): 285–299

    Google Scholar 

  56. Mumford D, Shah J (1989) Optimal approximation by piecewise smooth functions and associated variational problems. Commun Pure Appl Math 2:577–685

    Article  MathSciNet  Google Scholar 

  57. Nečas J, Čsilhavý M (1991) Multipolar viscous fluids. Quart Appl Math 49(2):247–265

    MathSciNet  MATH  Google Scholar 

  58. Ogden RW (1984) Non-linear elastic deformations. Wiley, New York

    Google Scholar 

  59. Osher S, Fedkiw R (2003) Level set methods and dynamic implicit surfaces, vol 153 of Applied mathematical sciences. Springer, New York

    Google Scholar 

  60. Osher S, Sethian JA (1988) Fronts propagating with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J Comput Phys 79(1):12–49

    Article  MathSciNet  MATH  Google Scholar 

  61. Pennec X (2006) Left-invariant Riemannian elasticity: a distance on shape diffeomorphisms? In: Mathematical foundations of computational anatomy - MFCA 2006, pp 1–14

    Google Scholar 

  62. Pennec X, Stefanescu R, Arsigny V, Fillard P, Ayache N (2005) Riemannian elasticity: a statistical regularization framework for non-linear registration. In: Medical image computing and computer-assisted intervention – MICCAI 2005. LNCS, Palm Springs, pp 943–950

    Google Scholar 

  63. Perperidis D, Mohiaddin R, Rueckert D (2005) Construction of a 4d statistical atlas of the cardiac anatomy and its use in classification. In: Duncan J, Gerig G (eds) Medical image computing and computer assisted intervention, vol 3750 of LNCS, pp 402–410

    Chapter  Google Scholar 

  64. Rathi Y, Dambreville S, Tannenbaum A (2006) Statistical shape analysis using kernel PCA. In: Proceedings of SPIE, vol 6064, pp 425–432

    Google Scholar 

  65. Rathi Y, Dambreville S, Tannenbaum A (2006) Comparative analysis of kernel methods for statistical shape learning. In: Beichel R, Sonka M (eds) Computer vision approaches to medical image analysis, vol 4241 of LNCS, pp 96–107

    Chapter  Google Scholar 

  66. Rueckert D, Frangi AF, Schnabel JA (2003) Automatic construction of 3-d statistical deformation models of the brain using nonrigid registration. IEEE Trans Med Imaging 22(8):1014–1025

    Article  Google Scholar 

  67. Rumpf M, Wirth B (2009) A nonlinear elastic shape averaging approach. SIAM J Imaging Sci 2(3):800–833

    Article  MathSciNet  MATH  Google Scholar 

  68. Rumpf M, Wirth B (2009) An elasticity approach to principal modes of shape variation. In: Proceedings of the second international conference on scale space methods and variational methods in computer vision (SSVM 2009), vol 5567 of LNCS, pp 709–720

    Google Scholar 

  69. Rumpf M, Wirth B (2009) An elasticity-based covariance analysis of shapes. Int J Comput Vis (accepted)

    Google Scholar 

  70. Schmidt FR, Clausen M, Cremers D (2006) Shape matching by variational computation of geodesics on a manifold. In: Pattern recognition, vol 4174 of LNCS. Springer, Berlin, pp 142–151

    Chapter  Google Scholar 

  71. Sethian JA (1999) Level set methods and fast marching methods. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  72. Shilane P, Min P, Kazhdan M, Funkhouser T (2004) The Princeton shape benchmark. In: Proceedings of the shape modeling international, 2004, Genova, pp 167–178

    Google Scholar 

  73. Söhn M, Birkner M, Yan D, Alber M (2005) Modelling individual geometric variation based on dominant eigenmodes of organ deformation: implementation and evaluation. Phys Med Biol 50:5893–5908

    Article  Google Scholar 

  74. Spivak M (1970) A comprehensive introduction to differential geometry, vol 1. Publish or Perish, Boston

    Google Scholar 

  75. Srivastava A, Jain A, Joshi S, Kaziska D (2006) Statistical shape models using elastic-string representations. In Narayanan P (ed) Asian conference on computer vision, vol 3851 of LNCS. Springer, Heidelberg, pp 612–621

    Google Scholar 

  76. Sundaramoorthi G, Yezzi A, Mennucci A (2007) Sobolev active contours. Int J Comput Vis 73(3):345–366

    Article  Google Scholar 

  77. Thorstensen N, Segonne F, Keriven R (2009) Pre-image as karcher mean using diffusion maps: application to shape and image denoising. In: Proceedings of the second international conference on scale space methods and variational methods in computer vision (SSVM 2009), vol 5567 of LNCS, pp 721–732

    Google Scholar 

  78. Truesdell C, Noll W (2004) The non-linear field theories of mechanics. Springer, Berlin

    Google Scholar 

  79. Tsai A, Yezzi A, Wells W, Tempany C, Tucker D, Fan A, Grimson WE, Willsky A (2003) A shape-based approach to the segmentation of medical imagery using level sets. IEEE Trans Med Imaging 22(2):137–154

    Article  Google Scholar 

  80. Vaillant M, Glaunès J (2005) Surface matching via currents. In: IPMI 2005: Information processing in medical imaging, vol 3565 of LNCS. Springer, Glenwood Springs, pp 381–392

    Google Scholar 

  81. Wirth B (2009) Variational methods in shape space. Dissertation, University Bonn, Bonn

    Google Scholar 

  82. Wirth B, Bar L, Rumpf M, Sapiro G (2009) Geodesics in shape space via variational time discretization. In: Proceedings of the 7th international conference on energy minimization methods in computer vision and pattern recognition (EMMCVPR’09), vol 5681 of LNCS, pp 288–302

    Google Scholar 

  83. Wirth B, Bar L, Rumpf M, Sapiro G (2010) A continuum mechanical approach to geodesics in shape space (submitted to IJCV)

    Google Scholar 

  84. Yezzi AJ, Mennucci A (2005) Conformal metrics and true “gradient flows” for curves. In: ICCV 2005: Proceedings of the 10th IEEE international conference on computer vision, pp 913–919

    Google Scholar 

  85. Younes L (April 1998) Computable elastic distances between shapes. SIAM J Appl Math 58(2):565–586

    Article  MathSciNet  MATH  Google Scholar 

  86. Younes L, Qiu A, Winslow RL, Miller MI (2008) Transport of relational structures in groups of diffeomorphisms. J Math Imaging Vis 32(1):41–56

    Article  MathSciNet  Google Scholar 

  87. Yushkevich P, Fletcher PT, Joshi S, Thalla A, Pizer SM (2003) Continuous medial representations for geometric object modeling in 2d and 3d. Image Vis Comput 21(1):17–27

    Article  Google Scholar 

  88. Zolésio JP (2004) Shape topology by tube geodesic. In: IFIP conference on system modeling and optimization No 21, pp 185–204

    Google Scholar 

Download references

Acknowledgments

The model proposed in Sect. 31.4.2 has been developed in cooperation with Leah Bar and Guillermo Sapiro from the University of Minnesota. Benedikt Wirth has been funded by the Bonn International Graduate School in Mathematics. Furthermore, the work was supported by the Deutsche Forschungsgemeinschaft, SPP 1253 “Optimization with Partial Differential Equations.” Part of Figs. 31-3 – 31-4 , and 31-19 – 31-23 have been taken from [83], the results from Figs. 31-6 , 31-8 , and 31-10 – 31-18 stem from [67, 69].

Author information

Authors and Affiliations

  1. Bonn University, Bonn, Germany

    Martin Rumpf

  2. Bonn University, Bonn, Germany

    Benedikt Wirth

Authors
  1. Martin Rumpf
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Benedikt Wirth
    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

Rumpf, M., Wirth, B. (2011). Variational Methods in Shape Analysis. In: Scherzer, O. (eds) Handbook of Mathematical Methods in Imaging. Springer, New York, NY. https://doi.org/10.1007/978-0-387-92920-0_31

Download citation

Publish with us

Policies and ethics

Search

Navigation