OSA's Digital Library

Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 64, Iss. 9 — Sep. 1, 1974
  • pp: 1215–1218

Formation of images from projections: Radon and Abel transforms

C. M. Vest  »View Author Affiliations

JOSA, Vol. 64, Issue 9, pp. 1215-1218 (1974)

View Full Text Article

Acrobat PDF (350 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



The problem of determining the structure of an object from knowledge of its projections along straight lines arises in a variety of optical contexts. The general solution to this problem is given by the inverse Radon transform of the projections. A superposition principle is developed to show the relation between this general inversion and the more familiar inverse Abel transform that is associated with radially symmetric objects.

C. M. Vest, "Formation of images from projections: Radon and Abel transforms," J. Opt. Soc. Am. 64, 1215-1218 (1974)

Sort:  Author  |  Journal  |  Reset


  1. In this paper the object field, in any coordinate system, is always denoted by ƒ, and its projection by Φ, i.e., ƒ(x,y,z) and ƒ(r,ø,z) denote the same field.
  2. M. V. Berry and D. F. Gibbs, Proc. R. Soc. (Lond.) A 314, 143 (1970).
  3. P. D. Rowley, J. Opt. Soc. Am. 59, 1496 (1969).
  4. I. J. Good, Phys. Lett. 31A, 155 (1970).
  5. H. G. Junginger and W. van Haeringer, Opt. Commun. 5, 1 (1972).
  6. C. M. Vest, J. Opt. Soc. Am. 63, 486A (1973).
  7. A. M. Cormack, Phys. Med. Biol. 18, 195 (1973).
  8. J. Radon, Berichte Saechsische Akademie der Wissenschaften (Leipzig) 69, 262 (1917).
  9. I. M. Gel'Fand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions, Vol. V (Academic, New York, 1966).
  10. F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations (Wiley—Interscience, New York, 1955).
  11. R. N. Bracewell, Australian J. Phys. 9, 198 (1956).
  12. D. P. Petersen and D. Middleton, Inf. Control 5, 279 (1962).
  13. The value of a three-dimensional field at a single point can be determined in terms of the integrals of ƒ over all planes passing through the point. This is clear from the discussions of the Radon transform in Refs. 7, 9, and 10.
  14. P. F. C. Gilbert, Proc. R. Soc. (Lond.) B 182, 89 (1972).
  15. D. W. Sweeney and C. M. Vest, Appl. Opt. 12, 2649 (1973).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited