, 55:2 | Cite as

Image reconstruction from scattered Radon data by weighted positive definite kernel functions



We propose a novel kernel-based method for image reconstruction from scattered Radon data. To this end, we employ generalized Hermite–Birkhoff interpolation by positive definite kernel functions. For radial kernels, however, a straightforward application of the generalized Hermite–Birkhoff interpolation method fails to work, as we prove in this paper. To obtain a well-posed reconstruction scheme for scattered Radon data, we introduce a new class of weighted positive definite kernels, which are symmetric but not radially symmetric. By our construction, the resulting weighted kernels are combinations of radial positive definite kernels and positive weight functions. This yields very flexible image reconstruction methods, which work for arbitrary distributions of Radon lines. We develop suitable representations for the weighted basis functions and the symmetric positive definite kernel matrices that are resulting from the proposed reconstruction scheme. For the relevant special case, where Gaussian radial kernels are combined with Gaussian weights, explicit formulae for the weighted Gaussian basis functions and the kernel matrices are given. Supporting numerical examples are finally presented.


Image reconstruction Kernel-based approximation Generalized Hermite–Birkhoff interpolation Radon transform Positive definite kernels 

Mathematics Subject Classification

65D05 65D15 



The authors thank one anonymous referee for suggesting an improvement concerning Proposition 1. The first author was supported by the project “Multivariate approximation with application to image reconstruction” of the University of Padova, years 2013–2014. The third author thanks the German Research Foundation (DFG) for financial support within the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart. The research of the first and the third author has been accomplished within RITA (Rete ITaliana di Approssimazione).


  1. 1.
    Beatson, R.K., Castell, W.: Scattered data interpolation of Radon data. Calcolo 48, 5–19 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bjørck, Å.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia (1996)CrossRefMATHGoogle Scholar
  3. 3.
    Bochner, S.: Vorlesungen über Fouriersche Integrale. Akademische Verlagsgesellschaft, Leipzig (1932)MATHGoogle Scholar
  4. 4.
    De Marchi, S., Iske, A., Sironi, A.: Kernel-based image reconstruction from scattered Radon data. Dolomites Res. Notes Approx. 9, 19–31 (2016)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Feeman, T.G.: The Mathematics of Medical Imaging. A Beginner’s Guide. Springer Undergraduate Texts in Mathematics and Technology, 2nd edn. Springer, New York (2015)CrossRefMATHGoogle Scholar
  6. 6.
    Gordon, R., Bender, R., Herman, G.: Algebraic reconstruction techniques (ART) for three dimensional electron microscopy and X-ray photography. J. Theor. Biol. 29(3), 471–481 (1970)CrossRefGoogle Scholar
  7. 7.
    Guillemard, M., Iske, A.: Interactions between kernels, frames, and persistent homology. In: Pesenson, I., Gia, Q.T.Le, Mayeli, A., Mhaskar, H., Zhou, D.-X. (eds.) Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science. Volume 2: Novel Methods in Harmonic Analysis, pp. 861–888. Birkhäuser, Basel (2017)CrossRefGoogle Scholar
  8. 8.
    Helgason, S.: The Radon Transform. Progress in Mathematics, vol. 5, 2nd edn. Birkhäuser, Basel (1999)CrossRefMATHGoogle Scholar
  9. 9.
    Hertle, A.: On the problem of well-posedness for the Radon transform. In: Herman, G.T., Natterer, F. (eds.) Mathematical Aspects of Computerized Tomography. Lecture Notes in Medical Informatics, vol. 8, pp. 36–44. Springer, Berlin (1981)CrossRefGoogle Scholar
  10. 10.
    Iske, A.: Charakterisierung bedingt positiv definiter Funktionen für multivariate Interpolations methoden mit radialen Basisfunktionen. Dissertation, University of Göttingen (1994)Google Scholar
  11. 11.
    Iske, A.: Reconstruction of functions from generalized Hermite–Birkhoff data. In: Chui, C.K., Schumaker, L.L. (eds.) Approximation Theory VIII, Vol 1: Approximation and Interpolation, pp. 257–264. World Scientific, Singapore (1995)Google Scholar
  12. 12.
    Iske, A.: Scattered data approximation by positive definite kernel functions. Rend. Sem. Mat. Univ. Pol. Torino 69(3), 217–246 (2011)MathSciNetMATHGoogle Scholar
  13. 13.
    Natterer, F.: The Mathematics of Computerized Tomography. Classics in Applied Mathematics, vol. 32. SIAM, Philadelphia (2001)CrossRefMATHGoogle Scholar
  14. 14.
    Radon, J.: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Berichte Sächsische Akademie der Wissenschaften 69, 262–277 (1917)MATHGoogle Scholar
  15. 15.
    Schaback, R., Wendland, H.: Characterization and construction of radial basis functions. In: Dyn, N., Leviatan, D., Levin, D., Pinkus, A. (eds.) Multivariate Approximation and Applications, pp. 1–24. Cambridge University Press, Cambridge (2001)Google Scholar
  16. 16.
    Wang, Z., Bovik, A.C., Sheikh, H.R., Simoncelli, E.P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)CrossRefGoogle Scholar
  17. 17.
    Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)MATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PaduaPaduaItaly
  2. 2.Department of MathematicsUniversity of HamburgHamburgGermany
  3. 3.Institute of Applied Analysis and Numerical SimulationUniversity of StuttgartStuttgartGermany

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