OSA's Digital Library

Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Vol. 24, Iss. 23 — Dec. 1, 1985
  • pp: 3950–3957

Mathematical foundations of computed tomography

Kennan T. Smith and F. Keinert  »View Author Affiliations


Applied Optics, Vol. 24, Issue 23, pp. 3950-3957 (1985)
http://dx.doi.org/10.1364/AO.24.003950


View Full Text Article

Enhanced HTML    Acrobat PDF (1351 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Along with a review of some of the mathematical foundations of computed tomography, the article contains new results on derivation of reconstruction formulas in a general setting encompassing all standard formulas; discussion and examples of the role of the point spread function with recipes for producing suitable ones; formulas for, and examples of, the reconstruction of certain functions of the attenuation coefficient, e.g., sharpened versions of it, some of them with the property that reconstruction at a point requires only the attenuation along rays meeting a small neighborhood of the point.

© 1985 Optical Society of America

History
Original Manuscript: August 12, 1984
Published: December 1, 1985

Citation
Kennan T. Smith and F. Keinert, "Mathematical foundations of computed tomography," Appl. Opt. 24, 3950-3957 (1985)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-24-23-3950


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. K. T. Smith, “Inversion of the X-Ray Transform,” SIAM-AMS Proc. 14, 41 (1984).
  2. K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and Mathematical Aspects of Reconstructing Objects from Radiographs,” BAMS1227 (1977)
  3. J. V. Leahy, K. T. Smith, D. C. Solmon, “Uniqueness, Nonuniqueness, and Inversion in the X-ray and Radon Problems,” at International Symposium on Ill-Posed Problems, U. Delaware, Newark, 1979 (to appear).
  4. C. Hamaker, K. T. Smith, D. C. Solmon, S. L. Wagner, “The Divergent Beam X-Ray Transform,” Rocky Mount. J. Math. 253 (1980).
  5. B. E. Petersen, K. T. Smith, D. C. Solmon, “Sums of Plane Waves and the Range of the Radon Transform,” Math. Ann. 163 (1979).
  6. J. Boman, “On the Closure of Sums of Plane Waves and the Range of the X-Ray Transform, I, II,”. Mathematics Department, U. Stockholm (1981), (1982). To appear in Ann. Inst. Fourier. Grenoble.
  7. D. V. Finch, D. C. Solmon, “Sums of Homogeneous Functions and the Range of the Divergent Beam X-Ray Transform,” Numer. Functional Anal. Optim. 5, 363 (1983). [CrossRef]
  8. B. F. Logan, L. A. Shepp, “Optimal Reconstruction of a Function from its Projections,” Duke Math. J. XX, 645 (1975). [CrossRef]
  9. C. Hamaker, D. C. Solmon, “The Angles Between the Null Spaces of X-Rays,” J. Math. Anal. Appl. XX, 1 (1978). [CrossRef]
  10. M. E. Davison, F. A. Grunbaum, “Tomographic Reconstruction with Arbitrary Directions,” Commun. Pure. Appl. Math. XX, 77 (1981). [CrossRef]
  11. R. B. Marr, “An Overview of Image Reconstruction,” at International Symposium of Ill-Posed Problems, U. Delaware, Newark, 1979. (To appear.)
  12. K. J. Falconer, “Consistency Conditions for a Finite Set of Projections of a Function,” Math. Proc. Cambridge Philos. Soc. XX, 61 (1979). [CrossRef]
  13. M. Riesz, “Intégrales de Riemann-Liouville et Potentiels,” Acta Szeged. XX, 1 (1938).
  14. J. Radon, “Über die Bestimmung von Functionen durch ihre Integralwerte langs gewissen Mannigfaltigkeiten,” Ber. Verh. Sach. Akad. Wiss. Leipzig. Math.-Nat. kl. XX, 262, (1977).
  15. N. Ramachandran, A. V. Lakshminarayanan, “Three-Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions Instead of Fourier Transforms,” Proc. Nat. Acad. Sci. U.S.A. 2236 (1971).
  16. A. V. Lakshminarayanan, “Reconstruction from Divergent X-Ray Data,” SUNY Tech. Report 92, Computer Sciences Department, Buffalo, N.Y. (1975).
  17. H. J. Scudder, “Introduction to Computer Aided Tomography,” Proc. IEEE, 66, 628 (1978). [CrossRef]
  18. K. T. Smith, “Reconstruction Formulas in Computed Tomography. Computed Tomography,” in Proceedings Symposium on Applied Mathematics No. 27, L. A. Shepp, Ed. (American Mathematical Society, Providence, R.I., 1983). [CrossRef]
  19. L. Schwartz, “Théorie des Distributions, I, II,” Ac. Sci. Ind. 1091, 1122 (1950–51).
  20. A. P. Calderón, A. Zygmund, “On the Existence of Certain Singular Integrals,” Acta Math. 88, 85 (1952). [CrossRef]
  21. O. Frostman, “Potentiels d’équilibre et capacité des ensembles,” Thesis, Lund (1935).
  22. N. Aronszajn, K. T. Smith, “Theory of Bessel Potentials, Part I,” Ann. Inst. Fourier Grenoble XI, 385 (1961). [CrossRef]

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.

Figures

Fig. 1 Fig. 2 Fig. 3
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited