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Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Vol. 27, Iss. 19 — Oct. 1, 1988
  • pp: 4084–4097

Reconstruction of smooth distributions from a limited number of projections

M. Ravichandran and F. C. Gouldin  »View Author Affiliations


Applied Optics, Vol. 27, Issue 19, pp. 4084-4097 (1988)
http://dx.doi.org/10.1364/AO.27.004084


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Abstract

Two reconstruction methods that recognize smoothness to be a priori information and use numerically space-limited basis functions have been developed. The first method is a modification of the well-known convolution method and uses such basis functions for the projections. The second method is a continuous algebraic reconstruction technique that employs consistent basis functions for the projections as well as the distribution and makes use of other a priori information like the constraints on the domain as well as the range of the distribution in a rigorous way. The efficacy of these methods has been demonstrated using a limited number of projections synthetically generated from a distribution phantom.

© 1988 Optical Society of America

History
Original Manuscript: November 16, 1987
Published: October 1, 1988

Citation
M. Ravichandran and F. C. Gouldin, "Reconstruction of smooth distributions from a limited number of projections," Appl. Opt. 27, 4084-4097 (1988)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-27-19-4084


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References

  1. J. Radon, “Uber die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Ber. Sachsische Akad. Wiss. Leipzig Math. Phys. Kl. 69, 262 (1917).
  2. S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).
  3. R. M. Lewitt, “Reconstruction Algorithms: Transform Methods,” Proc. IEEE 71, 390 (1983). [CrossRef]
  4. Y. Censor, “Finite Series-Expansion Reconstruction Methods,” Proc. IEEE 71, 409 (1983). [CrossRef]
  5. P. J. Emmerman, R. Goulard, R. J. Santoro, H. G. Semerjian, “Multiangular Absorption Diagnosis of a Turbulent Argon-Methane Jet,” J. Energy 4, 70 (1980). [CrossRef]
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  7. C. M. Vest, Ivan Prikryl, “Tomography by Iterative Convolution: Empirical Study and Application to Interferometry,” Appl. Opt. 23, 2433 (1984). [CrossRef] [PubMed]
  8. D. W. Sweeney, C. M. Vest, “Reconstruction of Three-Dimensional Refractive Index Fields from Multidirectional Interferometric Data,” Appl. Opt. 12, 2649 (1973). [CrossRef] [PubMed]
  9. L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1975).
  10. A. Macovski, “Physical Problems of Computerized Tomography,” Proc. IEEE 71, 373 (1983). [CrossRef]
  11. G. T. Herman, Image Reconstructions from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980).
  12. The Kaiser window, which is an approximation to prolate spheroidal wave functions which are closed form solutions to the mathematical problem of finding a time-limited (or space-limited in the present case) function whose Fourier transform best approximates a band-limited function, has been identified as an ideal candidate for a window (Rabiner and Gold9); and its application to reconstruction problems has previously been suggested by Lewitt.3
  13. It should be noted that the modified inversion formula is not equivalent to the standard CBP inversion formula [Eq. (7)] with W(R) in Eq. (8) replaced by the Kaiser window.
  14. G. N. Ramachandran, A. V. Lakshminarayanan, “Three Dimensional Reconstruction from Radiographs and Electron Micrographs: Application of Convolutions Instead of Fourier Transforms,” Proc. Natl. Acad. Sci. USA 68, 2236 (1970). [CrossRef]
  15. Profile error is defined as ||frecons − f||/||f − f˜Ω ||, where f˜Ω = ∫Ωf(x,y)dxdy/∫Ωdxdy.
  16. Maximum deviation is defined as maxΩ|frecons − f|.
  17. This method essentially involves implementing Eq. (4) by explicitly computing the forward and inverse Fourier transforms. Its usual implementation is error prone due to interpolation in the Fourier space between polar and Cartesian grids.
  18. R. C. Hansen, “A One-Parameter Aperture Distribution with Narrow Beamwidth and Low Sidelobes,” IEEE Trans. Antennas Propag. AP-24, 477 (1976). [CrossRef]
  19. This is equivalent to what is referred to as hexagonal sampling in signal processing.20
  20. D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1984).
  21. G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U.P., Baltimore, 1985).
  22. In the context of ARTs, Censor (1983) characterizes typical projection matrices to be ~105 × 105 with <1% nonzero elements. In our test problems, C is ~102 × 102 with ~30% nonzero elements.
  23. C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, NJ, 1974).
  24. For example, some of the 2-D basis function coefficients representing certain regions of the domain of the distribution may be known to be more important than the others.
  25. C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind (Pitman, Boston, 1984).
  26. Contributions can be stored as GJK(∈RJ×K), which is used in f¯ = GJKf, where f˜ (∈RJ) is the discrete representation of f(x,y) at J locations.
  27. For example, S. No. 4 of Table I with a D matrix size of 102 × 97 and ninety-seven non-negativity constraints required ≈30s for this step on an IBM 4341 machine.
  28. The quantity 1/σi has been factored out from the coefficients as well as their standard deviations.
  29. The projection data are free of external errors, but the use of numerically space-limited basis functions introduces internal errors whose standard deviation is estimated to be ≈1.5 × 10−4.

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