Journal of the Korean Society for Industrial and Applied Mathematics

Korean Society for Industrial and Applied Mathematics (한국산업응용수학회)
권호 표지
DOI QR코드

DOI QR Code

OPTIMAL INVERSION OF THE NOISY RADON TRANSFORM ON CLASSES DEFINED BY A DEGREE OF THE LAPLACE OPERATOR

  • Received : 2017.01.05
  • Accepted : 2017.03.04
  • Published : 2017.03.25
Download PDF
⟨ Previous Next ⟩

Abstract

A general optimal recovery problem is to approximate a value of a linear operator on a subset (class) in linear space from a value of another linear operator (called information), measured with an error in given metric. We use this formulation to investigate the classical computerized tomography problem of inversion of the noisy Radon transform.

Keywords

References

  1. S. A. Smolyak, Candidate dissertation on optimal restoration of functions and functionals of them, Moscow Sate University, Moscow, 1965.
  2. C. A. Michelli and T. J. Rivlin, Optimal estimation in approximation theory, A Survey of Optimal Recovery, 1977, 1-54.
  3. C. A. Michelli and T. J. Rivlin, Lectures on optimal recovery, Numerical Analysis 1129, Springer, Berlin/Hidelberg, 1984.
  4. J. F. Traub and H. Wozniakowski, A general theory of optimal algorithms, Academic Press, New York, 1980.
  5. K. Yu.Osipenko, Optimal interpolation of analytic functions, Mathematical Notes of the Academy of Sciences of the USSR, 12(4) (1972), 712-719. https://doi.org/10.1007/BF01093679
  6. G. G. Magaril-Il'yaev and K. Yu. Osipenko, Hardy-Littlewood-Polya inequality and recovery of derivatives from inaccurate data, Doklady Mathematics, 83(3) (2011), 337-339. https://doi.org/10.1134/S1064562411030203
  7. G. G. Magaril-Il'yaev and K. Yu. Osipenko, Optimal recovery of functions and their derivatives from Fourier coefficients prescribed with an error, Sbornik: Mathematics, 193 (2002), 387-407. https://doi.org/10.1070/SM2002v193n03ABEH000637
  8. A. Sard, Best approximate integration formulas; best approximation formulas, American Journal of Mathematics, 71(1) (1949), 80-91. https://doi.org/10.2307/2372095
  9. S. M. Nikol'skii, Quadrature formulas, Nauka, Moscow, 1988.
  10. F. Natterer, The mathematics of computerized tomography, John Wiley & Sons, Stuttgart, 1986.
  11. B. F. Logan and L. A. Shepp, Optimal reconstruction of a function from its projections, Duke Mathematical Journal, 42(4) (1975), 645-659. https://doi.org/10.1215/S0012-7094-75-04256-8
  12. A. J. Degraw, Optimal recovery of holomorphic functions from inaccurate information about radial integration, American Journal of Computational Mathematics, 2 (2012), 258-268. https://doi.org/10.4236/ajcm.2012.24035
  13. T. E. Bagramyan, Optimal recovery of harmonic functions from inaccurate information on the values of the radial integration operator, Vladikavkazskii Matematicheskii Zhurnal, 14 (2012), 22-36.
  14. T. E. Bagramyan, Optimal recovery of harmonic functions in the ball from inaccurate information on the Radon transform, Mathematical Notes, 98(1) (2015), 195-203. https://doi.org/10.1134/S0001434615070196
  15. T. E. Bagramyan, The optimal recovery of a function from an inaccurate information on its k-plane transform, Inverse Problems, 32(6) (2016), 13-27.