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On the Bijectivity of Thin-Plate Splines

  • Anders P. Erikson
  • Kalle Åström
Conference paper
Part of the Springer Proceedings in Mathematics book series (PROM, volume 6)

Abstract

The thin-plate spline (TPS) has been widely used in a number of areas such as image warping, shape analysis and scattered data interpolation. Introduced by Bookstein (IEEE Trans. Pattern Anal. Mach. Intell. 11(6):567–585 1989), it is a natural interpolating function in two dimensions, parameterized by a finite number of landmarks. However, even though the thin-plate spline has a very intuitive interpretation as well as an elegant mathematical formulation, it has no inherent restriction to prevent folding, i.e. a non-bijective interpolating function. In this chapter we discuss some of the properties of the set of parameterizations that form bijective thin-plate splines, such as convexity and boundness. Methods for finding sufficient as well as necessary conditions for bijectivity are also presented. The methods are used in two settings (a) to register two images using thin-plate spline deformations, while ensuring bijectivity and (b) group-wise registration of a set of images, while enforcing bijectivity constraints.

Keywords

Image Registration Minimum Description Length Quadratic Constraint Double Cone Active Appearance Model 
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.

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References

  1. 1.
    Bookstein, F.L.: Principal warps: Thin-plate splines and the decomposition of deformations. IEEE Trans. Pattern Anal. Mach. Intell. 11(6), 567–585 (1989)CrossRefMATHGoogle Scholar
  2. 2.
    Cootes, T.F., Edwards, G.J., Taylor, C.J.: Active appearance models. In: Proceedings of the 5th European Conference on Computer Vision. Freiburg, Germany (1998)Google Scholar
  3. 3.
    Cootes, T.: Statistical models of shape and appearance. Technical Report, Imaging Science and Biomedical Engineering (2004)Google Scholar
  4. 4.
    Davies, R.H., Twining, C.J., Cootes, T.F., Waterton, J.C., Taylor, C.J.: A minimum description length approach to statistical shape modeling. IEEE Trans. Med. Imaging 21(5), 525–537 (2002)CrossRefGoogle Scholar
  5. 5.
    Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. Wiley, New York (1998)MATHGoogle Scholar
  6. 6.
    Duchon, J.: Splines minimizing rotation-invariant semi-norms in sobolev spaces. In: Schempp, W., Zeller, K. (eds.) Constructive theory of functions of several variables, pp. 85–100. Springer, Berlin (1977)CrossRefGoogle Scholar
  7. 7.
    Green, P.J., Silverman, B.W.: Nonparametric regression and generalized linear models. Number 58 in Monographs on Statistics and Applied Probability. Chapman & Hall, London (1994)Google Scholar
  8. 8.
    Hettich, R., Kortanek, K.O.: Semi-infinite programming: Theory, methods, and applications. SIAM Rev. 35(3), 380–429 (1993)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Ericsson, A., Karlsson, J., Åström, K.: Parameterisation invariant statistical shape models. In: Proceedings of the International Conference on Pattern Recognition. Cambridge, UK (2004)Google Scholar
  10. 10.
    Boyd, S., Vandenberghe, L., Wu, S.W.: Determinant maximization with linear matrix inequality constraints. SIAM J. Matrix Anal. Appl. 19(2), 499–533 (1998)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Meinguet, J.: Multivariate interpolation at arbitrary points made simple. J. Appl. Math. Phys. 30, 292–304 (1979)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Feron, E., Boyd, S., El Ghaoui, L., Balakrishnan, V.: Linear matrix inequalities in system and control theory. Society for Industrial and Applied Mathematics, Philadelphia (1994)Google Scholar
  13. 13.
    Thodberg, H.H.: Minimum description length shape and appearance models. In: Image processing medical imaging, IPMI 2003 (2003)Google Scholar
  14. 14.
    Vandenberghe, L., Boyd, S.: Convex Optimization. Cambridge University Press, Cambridge (2004)MATHGoogle Scholar
  15. 15.
    Wahba, G.: Spline models for observational data. Society for Industrial and Applied Mathematics, Philadelphia (1990)CrossRefMATHGoogle Scholar
  16. 16.
    Zitova, B., Flusser, J.: Image registration methods: a survey. Image Vis. Comput. 21(11), 977–1000 (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Centre for Mathematical SciencesLund UniversityLundSweden

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