Numerische Mathematik

, Volume 127, Issue 3, pp 423–445 | Cite as

Interpolation of harmonic functions based on Radon projections

Article

Abstract

We consider an algebraic method for reconstruction of a harmonic function in the unit disk via a finite number of values of its Radon projections. The approach is to seek a harmonic polynomial which matches given values of Radon projections along some chords of the unit circle. We prove an analogue of the famous Marr’s formula for computing the Radon projection of the basis orthogonal polynomials in our setting of harmonic polynomials. Using this result, we show unique solvability for a family of schemes where all chords are chosen at equal distance to the origin. For the special case of chords forming a regular convex polygon, we prove error estimates on the unit circle and in the unit disk. We present an efficient reconstruction algorithm which is robust with respect to noise in the input data and provide numerical examples.

Mathematics Subject Classification (2000)

41A05 41A63 42A10 44A12 65D05 

Notes

Acknowledgments

The authors acknowledge the support by the Bulgarian National Science Fund through Grant DMU 03/17. The research of the first author was also supported by Bulgarian National Science Fund Grant DDVU 0230/11. The second author was supported by the project AComIn “Advanced Computing for Innovation”, Grant 316087, funded by the FP7 Capacity Programme “Research Potential of Convergence Regions”, and by the Austrian Science Fund (FWF): W1214-N15, project DK04.

References

  1. 1.
    Bojanov, B., Georgieva, I.: Interpolation by bivariate polynomials based on Radon projections. Studia Math. 162, 141–160 (2004)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bojanov, B., Jayne, C.: Surface approximation by piecewise harmonic functions. In: Nikolov, G., Uluchev, R. (eds.) Constructive Theory of Functions, Sozopol 2010: In memory of Borislav Bojanov, pp. 46–52. Prof. Marin Drinov Academic Publishing House, Sofia (2012)Google Scholar
  3. 3.
    Bojanov, B., Petrova, G.: Numerical integration over a disc. A new Gaussian cubature formula. Numer. Math. 80, 39–59 (1998)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bojanov, B., Petrova, G.: Uniqueness of the Gaussian cubature for a ball. J. Approx. Theory 104, 21–44 (2000)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bojanov, B., Xu, Y.: Reconstruction of a bivariate polynomial from its Radon projections. SIAM J. Math. Anal. 37, 238–250 (2005)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Cavaretta, A.S., Goodman, T.N.T., Micchelli, C.A., Sharma, A.: Multivariate interpolation and the Radon transform, part III: Lagrange representation. In: Canadian Mathematical Society Conference Proceedings, pp. 37–50. American Mathematical Society, Providence (1982)Google Scholar
  7. 7.
    Cavaretta, A.S., Micchelli, C.A., Sharma, A.: Multivariate interpolation and the Radon transform. Part I. Math. Z. 174, 263–279 (1980)MATHMathSciNetGoogle Scholar
  8. 8.
    Cavaretta, A.S., Micchelli, C.A., Sharma, A.: Multivariate interpolation and the Radon transform, part II. In: Quantitive Approximation, pp. 49–62. Academic Press, New York (1980)Google Scholar
  9. 9.
    Davison, M., Grunbaum, F.: Tomographic reconstruction with arbitrary directions. Comm. Pure Appl. Math. 34, 77–120 (1981)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Gasca, M., Sauer, T.: Polynomial interpolation in several variables. Adv. Comput. Math. 12, 377–410 (2000). doi: 10.1023/A:1018981505752 CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Georgieva, I., Hofreither, C., Koutschan, C., Pillwein, V., Thanatipanonda, T.: Harmonic interpolation based on Radon projections along the sides of regular polygons. Cent. Eur. J. Math. 11(4), 609–620 (2013). doi: 10.2478/s11533-012-0160-1. Also available as Technical Report 2011–12 in the series of the DK Computational Mathematics Linz. https://www.dk-compmath.jku.at/publications/dk-reports/2011-10-20/view Google Scholar
  12. 12.
    Georgieva, I., Hofreither, C., Uluchev, R.: Interpolation of mixed type data by bivariate polynomials. In: Nikolov, G., Uluchev, R. (eds.) Constructive Theory of Functions, Sozopol 2010: In memory of Borislav Bojanov, pp. 93–107. Prof. Marin Drinov Academic Publishing House, Sofia (2012). Also available as Technical Report 2010–14 in the series of the DK Computational Mathematics Linz. https://www.dk-compmath.jku.at/publications/dk-reports/2010-12-10/view
  13. 13.
    Georgieva, I., Ismail, S.: On recovering of a bivariate polynomial from its Radon projections. In: Bojanov, B. (ed.) Constructive Theory of Functions, Varna 2005, pp. 127–134. Marin Drinov Academic Publishing House, Sofia (2006)Google Scholar
  14. 14.
    Georgieva, I., Uluchev, R.: Smoothing of Radon projections type of data by bivariate polynomials. J. Comput. Appl. Math. 215, 167–181 (2008). doi: 10.1016/j.cam.2007.04.002. http://portal.acm.org/citation.cfm?id=1349899.1350213
  15. 15.
    Georgieva, I., Uluchev, R.: Surface reconstruction and Lagrange basis polynomials. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) Large-Scale Scientific Computing 2007, pp. 670–678. Springer, Berlin (2008). http://dx.doi.org/10.1007/978-3-540-78827-0_77
  16. 16.
    Georgieva, I., Uluchev, R.: On interpolation in the unit disk based on both Radon projections and function values. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) Large-Scale Scientific Computing 2009, pp. 747–755. Springer, Berlin (2010)Google Scholar
  17. 17.
    Hakopian, H.: Multivariate divided differences and multivariate interpolation of Lagrange and Hermite type. J. Approx. Theory 34, 286–305 (1982)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Hamaker, C., Solmon, D.: The angles between the null spaces of \(x\)-rays. J. Math. Anal. Appl. 62, 1–23 (1978)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Jain, A.K.: Fundamentals of Digital Image Processing. In: Kailath, T. (ed.) Prentice Hall Information and System Sciences Series, 5th edn. Prentice Hall, Englewood Cliffs (1989)Google Scholar
  20. 20.
    John, F.: Abhängigkeiten zwischen den Flächenintegralen einer stetigen Funktion. Math. Anal. 111, 541–559 (1935)CrossRefGoogle Scholar
  21. 21.
    Logan, B., Shepp, L.: Optimal reconstruction of a function from its projections. Duke Math. J. 42, 645–659 (1975)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Marr, R.: On the reconstruction of a function on a circular domain from a sampling of its line integrals. J. Math. Anal. Appl. 45, 357–374 (1974)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Natterer, F.: The mathematics of computerized tomography. In: Classics in Applied Mathematics, vol. 32. SIAM, Philadelphia (2001)Google Scholar
  24. 24.
    Nikolov, G.: Cubature formulae for the disk using Radon projections. East J. Approx. 14, 401–410 (2008)MATHMathSciNetGoogle Scholar
  25. 25.
    Radon, J.: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Verch. Sächs. Akad. 69, 262–277 (1917)Google Scholar
  26. 26.
    Walsh, J.L.: The approximation of harmonic functions by harmonic polynomials and by harmonic rational functions. Bull. Am. Math. Soc 35(4), 499–544 (1929)Google Scholar
  27. 27.
    Zygmund, A.: Trigonometric Series. Cambridge Mathematical Library, 3rd edn. Cambridge University Press, Cambridge (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria
  2. 2.Institute of Information and Communication TechnologiesBulgarian Academy of SciencesSofiaBulgaria

Personalised recommendations