Abstract
We consider interpolation of function in two variables on the unit disk with bivariate polynomials, based on its Radon projections and function values. This is closely related to surface and image reconstruction. Due to the practical importance of these problems, recently a lot of mathematicians deal with interpolation and smoothing of bivariate functions with a data consisting of prescribed Radon projections or mixed type of data – Radon projections and function values. Here we present some new results and numerical experiments in this field.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bojanov, B., Georgieva, I.: Interpolation by bivariate polynomials based on Radon projections. Studia Math. 162, 141–160 (2004)
Bojanov, B., Petrova, G.: Numerical integration over a disc. A new Gaussian cubature formula. Numer. Math. 80, 39–59 (1998)
Bojanov, B., Petrova, G.: Uniqueness of the Gaussian cubature for a ball. J. Approx. Theory 104, 21–44 (2000)
Bojanov, B., Xu, Y.: Reconstruction of a bivariate polynomials from its Radon projections. SIAM J. Math. Anal. 37, 238–250 (2005)
Davison, M.E., Grunbaum, F.A.: Tomographic reconstruction with arbitrary directions. Comm. Pure Appl. Math. 34, 77–120 (1981)
Georgieva, I., Ismail, S.: On recovering of a bivariate polynomial from its Radon projections. In: Bojanov, B. (ed.) Constructive Theory of Functions, pp. 127–134. Marin Drinov Academic Publishing House, Sofia (2006)
Georgieva, I., Uluchev, R.: Smoothing of Radon projections type of data by bivariate polynomials. J. Comput. Appl. Math. 215(1), 167–181 (2008)
Georgieva, I., Uluchev, R.: Surface reconstruction and Lagrange basis polynomials. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2007. LNCS, vol. 4818, pp. 670–678. Springer, Heidelberg (2008)
Georgieva, I., Uluchev, R.: Interpolation of mixed type data by bivariate polynomials (to appear)
Hakopian, H.: Multivariate divided differences and multivariate interpolation of Lagrange and Hermite type. J. Approx. Theory 34, 286–305 (1982)
John, F.: Abhängigkeiten zwischen den Flächenintegralen einer stetigen Funktion. Math. Anal. 111, 541–559 (1935)
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)
Natterer, F.: The Mathematics of Computerized Tomography. Classics in Applied Mathematics, vol. 32. SIAM, Philadelphia (2001)
Nikolov, G.: Cubature formulae for the disk using Radon projections. East J. Approx. 14, 401–410 (2008)
Pickalov, V., Melnikova, T.: Plasma Tomography. Nauka, Novosibirsk (1995) (in Russian)
Radon, J.: Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten. Ber. Verch. Sächs. Akad. 69, 262–277 (1917)
Solmon, D.C.: The X-ray transform. J. Math. Anal. Appl. 56(1), 61–83 (1976)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Georgieva, I., Uluchev, R. (2010). 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. LSSC 2009. Lecture Notes in Computer Science, vol 5910. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12535-5_89
Download citation
DOI: https://doi.org/10.1007/978-3-642-12535-5_89
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-12534-8
Online ISBN: 978-3-642-12535-5
eBook Packages: Computer ScienceComputer Science (R0)