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


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.


Radon transform Lie groups Generalized variational splines Sampling theorem 

Mathematics Subject Classification

44A12 43A85 58E30 41A99 



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


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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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