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Applied Optics

Applied 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)

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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

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

Kennan T. Smith and F. Keinert, "Mathematical foundations of computed tomography," Appl. Opt. 24, 3950-3957 (1985)

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  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)
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  4. C. Hamaker, K. T. Smith, D. C. Solmon, S. L. Wagner, “The Divergent Beam X-Ray Transform,” Rocky Mount. J. Math. 253 (1980).
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  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.
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  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]

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