Journal of Fourier Analysis and Applications

, Volume 19, Issue 1, pp 140–166 | Cite as

Generalized Splines for Radon Transform on Compact Lie Groups with Applications to Crystallography

  • Swanhild Bernstein
  • Svend Ebert
  • Isaac Z. Pesenson
Article

Abstract

The Radon transform \(\mathcal{R}f\) of functions f on SO(3) has recently been applied extensively in texture analysis, i.e. the analysis of preferred crystallographic orientation. In practice one has to determine the orientation probability density function fL2(SO(3)) from \(\mathcal{R}f\in L_{2}(S^{2}\times S^{2})\) which is known only on a discrete set of points. Since one has only partial information about \(\mathcal{R}f\) the inversion of the Radon transform becomes an ill-posed inverse problem. Motivated by this problem we define a new notion of the Radon transform \(\mathcal{R}f\) of functions f on general compact Lie groups and introduce two approximate inversion algorithms which utilize our previously developed generalized variational splines on manifolds. Our new algorithms fit very well to the application of Radon transform on SO(3) to texture analysis.

Keywords

Radon transform Lie groups Generalized variational splines Sampling theorem 

Mathematics Subject Classification

44A12 43A85 58E30 41A99 

Notes

Acknowledgements

We thank Professor S. Helgason who brought our attention to reference [22].

References

  1. 1.
    Alfeld, P., Neamtu, M., Schumaker, L.L.: Fitting scattered data on sphere-like surfaces using spherical splines. J. Comput. Appl. Math. 73, 35–43 (1996) MathSciNetCrossRefGoogle Scholar
  2. 2.
    Asgeirsson, L.: Über eine Mittelwerteigenschaft von Lösungen homogener linearer partieller Differentialgleichungen zweiter Ordnung mit konstanten Koeffizienten. Ann. Math. 113, 321–346 (1937) MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bernstein, S., Ebert, S.: Wavelets on S 3 and SO(3)—their construction, relation to each other and Radon transform of wavelets on SO(3). Math. Methods Appl. Sci. 33(16), 1895–1909 (2010) MathSciNetMATHGoogle Scholar
  4. 4.
    Bernstein, S., Schaeben, H.: A one-dimensional Radon transform on SO(3) and its application to texture goniometry. Math. Methods Appl. Sci. 28, 1269–1289 (2005) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bernstein, S., Hielscher, R., Schaeben, H.: The generalized totally geodesic Radon transform and its application to texture analysis. Math. Methods Appl. Sci. 32, 379–394 (2009) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    van den Boogaart, K.G., Hielscher, R., Prestin, J., Schaeben, H.: Kernel-based methods for inversion of the Radon transform on SO(3) and their applications to texture analysis. J. Comput. Appl. Math. 199, 122–140 (2007) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bunge, H.J.: Texture Analysis in Material Science. Mathematical Methods. Butterworths, London (1982) Google Scholar
  8. 8.
    Cerejeiras, P., Ferreira, M., Kèhler, U., Teschke, G.: Inversion of the noisy Radon transform on SO(3) by Gabor frames and sparse recovery principles. Appl. Comput. Harmon. Anal. 31(3), 325–345 (2011) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Davydov, O., Schumaker, L.L.: Interpolation and scattered data fitting on manifolds using projected Powell-Sabin splines. IMA J. Numer. Anal. 28(4), 785–805 (2008) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Ebert, S.: Wavelets and Lie groups and homogeneous spaces. Ph.D. thesis, TU Bergakademie Freiberg, Department of Mathematics and Computer Sciences (2011) Google Scholar
  11. 11.
    Ebert, S., Wirth, J.: Diffusive wavelets on groups and homogeneous spaces. Proc. R. Soc. Edinb. 141A, 497–520 (2011) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fasshauer, G.E.: Hermite interpolation with radial basis functions on spheres. Adv. Comput. Math. 10(1), 81–96 (1999) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Fasshauer, G.E., Schumaker, L.L.: Scattered data fitting on the sphere. In: Mathematical methods for curves and surfaces I, Lillehammer, 1997. Innov. Appl. Math., pp. 117–166. Vanderbilt University Press, Nashville (1998) Google Scholar
  14. 14.
    Fasshauer, G.E., Schumaker, L.L.: Scattered data fitting on the sphere. In: Dhlen, M., Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods for Curves and Surfaces II, pp. 117–166. Vanderbilt University Press, Nashville (1998) Google Scholar
  15. 15.
    John, F.: The ultrahyperbolic differential with four independent variables. Duke Math. J. 4, 300–322 (1938) MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hangelbroek, T., Narcowich, F.J., Ward, J.D.: Polyharmonic and related kernels on manifolds: interpolation and approximation. arXiv:1012.4852
  17. 17.
    Hangelbroek, T., Narcowich, F.J., Sun, X., Ward, J.D.: Kernel approximation on manifolds II: the L norm of the L 2 projector. SIAM J. Math. Anal. 43(2), 662–684 (2011) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Helgason, S.: The Radon Transform, 2nd edn. Progress in Mathematics, vol. 5. Birkhäuser, Boston (1999) MATHGoogle Scholar
  19. 19.
    Helgason, S.: Integral Geometry and Radon Transforms. Springer, New York (2010) Google Scholar
  20. 20.
    Hielscher, R.: Die Radontransformation auf der Drehgruppe—Inversion und Anwendung in der Texturanalyse. Ph.D. thesis, University of Mining and Technology Freiberg (2007) Google Scholar
  21. 21.
    Hielscher, R., Potts, D., Prestin, J., Schaeben, H., Schmalz, M.: The Radon transform on SO(3): a Fourier slice theorem and numerical inversion. Inverse Probl. 24(2), 025011 (2008), 21 pp. MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kakehi, T., Tsukamoto, C.: Characterization of images of Radon transform. Adv. Stud. Pure Math. 1993(22), 101–116 (1993) MathSciNetGoogle Scholar
  23. 23.
    Lai, M.-J., Schumaker, L.L.: Spline Functions on Triangulations. Encyclopedia of Mathematics and its Applications, vol. 110. Cambridge University Press, Cambridge (2007) MATHCrossRefGoogle Scholar
  24. 24.
    Matthies, S.: On the reproducibility of the orientation distribution function of texture samples from pole figures (ghost phenomena). Phys. Status Solidi B 92, K135–K138 (1979) CrossRefGoogle Scholar
  25. 25.
    Meister, L., Schaeben, H.: A concise quaternionic geometry of rotations. Math. Methods Appl. Sci. 28, 101–126 (2004) MathSciNetCrossRefGoogle Scholar
  26. 26.
    Minakshisundaram, S., Pleijel, Å.: Some properties of the eigenfunctions of the Laplace operator on Riemannian manifolds. Can. J. Math. 1, 242–256 (1949) MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Muller, J., Esling, C., Bunge, H.J.: An inversion formula expressing the texture function in terms of angular distribution functions. J. Phys. 42, 161–165 (1982) MathSciNetGoogle Scholar
  28. 28.
    Nikolayev, D.I., Schaeben, H.: Characteristics of the ultra-hyperbolic differential equation governing pole density functions. Inverse Probl. 15, 1603–1619 (1999) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Palamodov, V.P.: Reconstruction from a sampling of circle integrals in SO(3). Inverse Probl. 26, 095-008 (2010), 10 pp. MathSciNetCrossRefGoogle Scholar
  30. 30.
    Pesenson, I.: A sampling theorem on homogeneous manifolds. Trans. Am. Math. Soc. 9, 4257–4269 (2000) MathSciNetCrossRefGoogle Scholar
  31. 31.
    Pesenson, I., Grinberg, E.: Invertion of the spherical Radon transform by a Poisson type formula. Contemp. Math. 278, 137–147 (2001) MathSciNetCrossRefGoogle Scholar
  32. 32.
    Pesenson, I.: Poincare-type inequalities and reconstruction of Paley-Wiener functions on manifolds. J. Geom. Anal. 4(1), 101–121 (2004) MathSciNetCrossRefGoogle Scholar
  33. 33.
    Pesenson, I.: An approach to spectral problems on Riemannian manifolds. Pac. J. Math. 215(1), 183–199 (2004) MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Pesenson, I.: Variational splines on Riemannian manifolds with applications to integral geometry. Adv. Appl. Math. 33(3), 548–572 (2004) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Vilenkin, N.J.: Special Functions and the Theory of Group Representations. Translations of Mathematical Monographs, vol. 22. Am. Math. Soc., Providence (1978) Google Scholar
  36. 36.
    Vilenkin, N.J., Klimyk, A.U.: Representations of Lie Groups and Special Functions, vol. 2. Kluwer Academic, Norwell (1993) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Swanhild Bernstein
    • 1
  • Svend Ebert
    • 1
  • Isaac Z. Pesenson
    • 2
  1. 1.TU Bergakademie FreibergInstitute of Applied AnalysisFreibergGermany
  2. 2.Department of MathematicsTemple UniversityPhiladelphiaUSA

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