Advertisement

Journal of Mathematical Imaging and Vision

, Volume 35, Issue 1, pp 86–102 | Cite as

Shape Metrics Based on Elastic Deformations

  • Matthias FuchsEmail author
  • Bert Jüttler
  • Otmar Scherzer
  • Huaiping Yang
Article

Abstract

Deformations of shapes and distances between shapes are an active research topic in computer vision. We propose an energy of infinitesimal deformations of continuous 1- and 2-dimensional shapes that is based on the elastic energy of deformed objects. This energy defines a shape metric which is inherently invariant with respect to Euclidean transformations and yields very natural deformations which preserve details. We compute shortest paths between planar shapes based on elastic deformations and apply our approach to the modeling of 2-dimensional shapes.

Keywords

Shape space Shape metric Shape modeling Elastic deformation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975) zbMATHGoogle Scholar
  2. 2.
    Allaire, G., de Gournay, F., Jouve, F., Toader, A.-M.: Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194(1), 363–393 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Boothby, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geometry. Pure and Applied Mathematics, vol. 63. Academic Press, New York (1975) zbMATHGoogle Scholar
  4. 4.
    Charpiat, G., Faugeras, O., Keriven, R.: Approximations of shape metrics and application to shape warping and empirical shape statistics. Found. Comput. Math. 5(1), 1–58 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Charpiat, G., Maurel, P., Pons, J.-P., Keriven, R., Faugeras, O.: Generalized gradients: Priors on minimization flows. Int. J. Comput. Vis. 73(3), 325–344 (2007) CrossRefGoogle Scholar
  6. 6.
    Ciarlet, P.G.: Mathematical Elasticity, Volume I: Three-Dimensional Elasticity. North-Holland, Amsterdam (1988) zbMATHGoogle Scholar
  7. 7.
    Ciarlet, P.G., Ciarlet, P. Jr.: Another approach to linearized elasticity and Korn’s inequality. C. R. Acad. Sci. I(339), 307–312 (2004) MathSciNetGoogle Scholar
  8. 8.
    Keeling, S.L.: Generalized rigid and generalized affine image registration and interpolation by geometric multigrid. J. Math. Imaging Vis. 29(2–3), 163–183 (2007) CrossRefMathSciNetGoogle Scholar
  9. 9.
    Keeling, S.L., Ring, W.: Medical image registration and interpolation by optical flow with maximal rigidity. J. Math. Imaging Vis. 23(1), 47–65 (2005) CrossRefMathSciNetGoogle Scholar
  10. 10.
    Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16(2), 81–121 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kilian, M., Mitra, N.J., Pottmann, H.: Geometric modeling in shape space. ACM Trans. Graph. 26(3), 1–8 (2007) CrossRefGoogle Scholar
  12. 12.
    Klassen, E., Srivastava, A., Mio, M., Joshi, S.H.: Analysis of planar shapes using geodesic paths on shape spaces. IEEE Trans. Pattern Anal. Mach. Intell. 26(3), 372–383 (2004) CrossRefGoogle Scholar
  13. 13.
    Lang, S.: Fundamentals of Differential Geometry. Springer, Berlin (1999) zbMATHGoogle Scholar
  14. 14.
    Liu, D.C., Nocedal, J.: On the limited memory BFGS method for large scale optimization. Math. Program. B 45(3), 503–528 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Michor, P.W., Mumford, D.B.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10, 217–245 (2005) zbMATHMathSciNetGoogle Scholar
  16. 16.
    Michor, P.W., Mumford, D.B.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. 8(1), 1–48 (2006) zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Miller, M.I., Younes, L.: Group actions, homeomorphisms, and matching: A general framework. Int. J. Comput. Vis. 41(1–2), 61–84 (2001) zbMATHCrossRefGoogle Scholar
  18. 18.
    Mio, W., Srivastava, A., Joshi, S.: On shape of plane elastic curves. Int. J. Comput. Vis. 73(3), 307–324 (2007) CrossRefGoogle Scholar
  19. 19.
    Sederberg, T.W., Zheng, J., Bakenov, A., Nasri, A.: T-splines and T-NURCCs. ACM Trans. Graph. 22(3), 477–484 (2003) CrossRefGoogle Scholar
  20. 20.
    Sundaramoorthi, G., Yezzi, A., Mennucci, A.: Sobolev active contours. Int. J. Comput. Vis. 73(3), 345–366 (2007) CrossRefGoogle Scholar
  21. 21.
    van der Vorst, H.A.: Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Comput. 13(2), 631–644 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Yang, H., Fuchs, M., Jüttler, B., Scherzer, O.: Evolution of T-spline level sets with distance field constraints for geometry reconstruction and image segmentation. In: IEEE International Conference on Shape Modeling and Applications 2006 (SMI’06), pp. 247–252. IEEE Computer Society, Los Alamitos (2006) Google Scholar
  23. 23.
    Yezzi, A., Mennucci, A.: Metrics in the Space of Curves (2004) Google Scholar
  24. 24.
    Yosida, K.: Functional Analysis. Springer, Berlin (1965) zbMATHGoogle Scholar
  25. 25.
    Younes, L.: Computable elastic distances between shapes. SIAM J. Appl. Math. 58(2), 565–586 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Zolésio, J.-P.: Control of moving domains, shape stabilization and variational tube formulations. Int. Ser. Numer. Math. 155, 329–382 (2007) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Matthias Fuchs
    • 1
    Email author
  • Bert Jüttler
    • 3
  • Otmar Scherzer
    • 1
    • 2
  • Huaiping Yang
    • 3
  1. 1.Dept. of MathematicsUniv. of InnsbruckInnsbruckAustria
  2. 2.RICAMAustrian Acad. of ScienceLinzAustria
  3. 3.Inst. of Applied GeometryUniv. of LinzLinzAustria

Personalised recommendations