Abstract

Inverse problems that require the solution of integral equations are inherent in a number of indirect imaging applications, such as computerized tomography. Numerical solutions based on discretization of the mathematical model of the imaging process, or on discretization of analytic formulas for iterative inversion of the integral equations, require a discrete representation of an underlying continuous image. This paper describes discrete image representations, in n-dimensional space, that are constructed by the superposition of shifted copies of a rotationally symmetric basis function. The basis function is constructed using a generalization of the Kaiser–Bessel window function of digital signal processing. The generalization of the window function involves going from one dimension to a rotationally symmetric function in n dimensions and going from the zero-order modified Bessel function of the standard window to a function involving the modified Bessel function of order m. Three methods are given for the construction, in n-dimensional space, of basis functions having a specified (finite) number of continuous derivatives, and formulas are derived for the Fourier transform, the x-ray transform, the gradient, and the Laplacian of these basis functions. Properties of the new image representations using these basis functions are discussed, primarily in the context of two-dimensional and three-dimensional image reconstruction from line-integral data by iterative inversion of the x-ray transform. Potential applications to three-dimensional image display are also mentioned.

© 1990 Optical Society of America

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  1. D. Slepian, “Analytic solution of two apodization problems,” J. Opt. Soc. Am. 55, 1110–1115 (1965).
    [Crossref]
  2. J. W. Sherman, “Aperture-antenna analysis,” in Radar Handbook, M. I. Skolnik, ed. (McGraw-Hill, New York, 1970), Chap. 9, pp. 9.1–9.40.
  3. L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  4. D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1984).
  5. J. F. Kaiser, “Digital filters,” in System Analysis by Digital Computer, F. F. Kuo, J. F. Kaiser, eds. (Wiley, New York, 1966), Chap. 7, pp. 218–285.
  6. W. D. White, “Circular aperture distribution functions,” IEEE Trans. Antennas Propag. AP-25, 714–716 (1977).
    [Crossref]
  7. A. H. Nuttall, “A two-parameter class of Bessel weightings for spectral analysis or array processing—the ideal weighting-window pairs,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1309–1312 (1983).
    [Crossref]
  8. G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980).
  9. H. H. Barrett, W. Swindell, Radiological Imaging: Theory of Image Formation, Detection and Processing (Academic, New York, 1981).
  10. S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).
  11. F. Natterer, The Mathematics of Computerized Tomography (Wiley, Chichester, UK, 1986).
  12. R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1986).
  13. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988).
  14. G. T. Herman, A. Lent, “Iterative reconstruction algorithms,” Comp. Biol. Med. 6, 273–294 (1976).
    [Crossref]
  15. J. H. Williamson, D. E. Evans, “Computerized tomography for sparse-data plasma physics experiments,” IEEE Trans. Plasma Sci. PS-10, 82–93 (1982).
    [Crossref]
  16. K. M. Hanson, G. W. Wecksung, “Local basis-function approach to computed tomography,” Appl. Opt. 24, 4028–4039 (1985).
    [Crossref] [PubMed]
  17. K. M. Hanson, Los Alamos National Laboratory, Los Alamos, New Mexico 875454 (personal communication).
  18. R. M. Lewitt, “Reconstruction algorithms: transform methods,” Proc. IEEE 71, 390–408 (1983).
    [Crossref]
  19. M. Ravichandran, F. C. Gouldin, “Reconstruction of smooth distributions from a limited number of projections,” Appl. Opt. 27, 4084–4097 (1988).
    [Crossref] [PubMed]
  20. R. Franke, “Scattered data interpolation: tests of some methods,” Math. Computat. 38, 181–200 (1982).
  21. M. J. D. Powell, “Radial basis function approximations to polynomials,” in Numerical Analysis 1987, D. F. Griffiths, G. A. Watson, eds. (Longman, Harlow, UK, 1988), pp. 223–241.
  22. I. R. H. Jackson, “Convergence properties of radial basis functions,” Construct. Approx. 4, 243–264 (1988).
    [Crossref]
  23. M. D. Buhmann, “Multivariate cardinal interpolation with radial-basis functions,” Construct. Approx. 6, 225–255 (1990).
    [Crossref]
  24. R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).
  25. I. N. Sneddon, Fourier Transforms (McGraw-Hill, New York, 1951).
  26. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  27. D. Slepian, “Some comments on Fourier analysis, uncertainty and modeling,” SIAM Rev. 25, 379–393 (1983).
    [Crossref]
  28. D. C. Solmon, “The x-ray transform,” J. Math. Anal. Appl. 56, 61–83 (1976).
    [Crossref]
  29. K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
    [Crossref]
  30. A. K. Louis, “Nonuniqueness in inverse Radon problems: the frequency distribution of the ghosts,” Math. Z. 185, 429–440 (1984).
    [Crossref]
  31. E. J. Farrell, R. A. Zappulla, “Three-dimensional data visualization and biomedical applications,” CRC Crit. Rev. Biomed. Eng. 16, 323–363 (1989).
  32. J. K. Udupa, “Computer aspects of 3D imaging in medicine: a tutorial,” in 3D Imaging in Medicine, J. K. Udupa, G. T. Herman, eds. (CRC, Boca Raton, Fla., 1990).
  33. B. K. P. Horn, Robot Vision (MIT Press, Cambridge, Mass., 1986).
  34. K. H. Hohne, R. Bernstein, “Shading 3D-images from CT using gray-level gradients,” IEEE Trans. Med. Imaging MI-5, 45–47 (1986).
    [Crossref]
  35. M. Levoy, “Display of surfaces from volume data,” IEEE Comp. Graph. Appl. 8, (3) 29–37 (1988).
    [Crossref]
  36. G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1944).
  37. R. C. Hansen, “A one-parameter circular aperture distribution with narrow beamwidth and low sidelobes,” IEEE Trans. Antennas Propag. AP-24, 477–480 (1976).
    [Crossref]
  38. A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
    [Crossref]
  39. I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Comm. Pure Appl. Math. 41, 909–996 (1988).
    [Crossref]
  40. S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
    [Crossref]
  41. S. G. Mallat, “Multiresolution approximations and wavelet orthonormal bases of L2(R),” Trans. Am. Math. Soc. 315, 69–87 (1989).
  42. S. G. Mallat, “Multifrequency channel decompositions of images and wavelet models,” IEEE Trans. Acoust. Speech Signal Process. 37, 2091–2110 (1989).
    [Crossref]

1990 (1)

M. D. Buhmann, “Multivariate cardinal interpolation with radial-basis functions,” Construct. Approx. 6, 225–255 (1990).
[Crossref]

1989 (4)

E. J. Farrell, R. A. Zappulla, “Three-dimensional data visualization and biomedical applications,” CRC Crit. Rev. Biomed. Eng. 16, 323–363 (1989).

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[Crossref]

S. G. Mallat, “Multiresolution approximations and wavelet orthonormal bases of L2(R),” Trans. Am. Math. Soc. 315, 69–87 (1989).

S. G. Mallat, “Multifrequency channel decompositions of images and wavelet models,” IEEE Trans. Acoust. Speech Signal Process. 37, 2091–2110 (1989).
[Crossref]

1988 (4)

M. Levoy, “Display of surfaces from volume data,” IEEE Comp. Graph. Appl. 8, (3) 29–37 (1988).
[Crossref]

I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Comm. Pure Appl. Math. 41, 909–996 (1988).
[Crossref]

I. R. H. Jackson, “Convergence properties of radial basis functions,” Construct. Approx. 4, 243–264 (1988).
[Crossref]

M. Ravichandran, F. C. Gouldin, “Reconstruction of smooth distributions from a limited number of projections,” Appl. Opt. 27, 4084–4097 (1988).
[Crossref] [PubMed]

1986 (1)

K. H. Hohne, R. Bernstein, “Shading 3D-images from CT using gray-level gradients,” IEEE Trans. Med. Imaging MI-5, 45–47 (1986).
[Crossref]

1985 (1)

1984 (1)

A. K. Louis, “Nonuniqueness in inverse Radon problems: the frequency distribution of the ghosts,” Math. Z. 185, 429–440 (1984).
[Crossref]

1983 (3)

D. Slepian, “Some comments on Fourier analysis, uncertainty and modeling,” SIAM Rev. 25, 379–393 (1983).
[Crossref]

R. M. Lewitt, “Reconstruction algorithms: transform methods,” Proc. IEEE 71, 390–408 (1983).
[Crossref]

A. H. Nuttall, “A two-parameter class of Bessel weightings for spectral analysis or array processing—the ideal weighting-window pairs,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1309–1312 (1983).
[Crossref]

1982 (2)

R. Franke, “Scattered data interpolation: tests of some methods,” Math. Computat. 38, 181–200 (1982).

J. H. Williamson, D. E. Evans, “Computerized tomography for sparse-data plasma physics experiments,” IEEE Trans. Plasma Sci. PS-10, 82–93 (1982).
[Crossref]

1977 (3)

K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[Crossref]

W. D. White, “Circular aperture distribution functions,” IEEE Trans. Antennas Propag. AP-25, 714–716 (1977).
[Crossref]

A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[Crossref]

1976 (3)

R. C. Hansen, “A one-parameter circular aperture distribution with narrow beamwidth and low sidelobes,” IEEE Trans. Antennas Propag. AP-24, 477–480 (1976).
[Crossref]

G. T. Herman, A. Lent, “Iterative reconstruction algorithms,” Comp. Biol. Med. 6, 273–294 (1976).
[Crossref]

D. C. Solmon, “The x-ray transform,” J. Math. Anal. Appl. 56, 61–83 (1976).
[Crossref]

1965 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Barrett, H. H.

H. H. Barrett, W. Swindell, Radiological Imaging: Theory of Image Formation, Detection and Processing (Academic, New York, 1981).

Bates, R. H. T.

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1986).

Bernstein, R.

K. H. Hohne, R. Bernstein, “Shading 3D-images from CT using gray-level gradients,” IEEE Trans. Med. Imaging MI-5, 45–47 (1986).
[Crossref]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).

Buhmann, M. D.

M. D. Buhmann, “Multivariate cardinal interpolation with radial-basis functions,” Construct. Approx. 6, 225–255 (1990).
[Crossref]

Daubechies, I.

I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Comm. Pure Appl. Math. 41, 909–996 (1988).
[Crossref]

Deans, S. R.

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

Dudgeon, D. E.

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Evans, D. E.

J. H. Williamson, D. E. Evans, “Computerized tomography for sparse-data plasma physics experiments,” IEEE Trans. Plasma Sci. PS-10, 82–93 (1982).
[Crossref]

Farrell, E. J.

E. J. Farrell, R. A. Zappulla, “Three-dimensional data visualization and biomedical applications,” CRC Crit. Rev. Biomed. Eng. 16, 323–363 (1989).

Franke, R.

R. Franke, “Scattered data interpolation: tests of some methods,” Math. Computat. 38, 181–200 (1982).

Gold, B.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Gouldin, F. C.

Hansen, R. C.

R. C. Hansen, “A one-parameter circular aperture distribution with narrow beamwidth and low sidelobes,” IEEE Trans. Antennas Propag. AP-24, 477–480 (1976).
[Crossref]

Hanson, K. M.

K. M. Hanson, G. W. Wecksung, “Local basis-function approach to computed tomography,” Appl. Opt. 24, 4028–4039 (1985).
[Crossref] [PubMed]

K. M. Hanson, Los Alamos National Laboratory, Los Alamos, New Mexico 875454 (personal communication).

Herman, G. T.

G. T. Herman, A. Lent, “Iterative reconstruction algorithms,” Comp. Biol. Med. 6, 273–294 (1976).
[Crossref]

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980).

Hohne, K. H.

K. H. Hohne, R. Bernstein, “Shading 3D-images from CT using gray-level gradients,” IEEE Trans. Med. Imaging MI-5, 45–47 (1986).
[Crossref]

Horn, B. K. P.

B. K. P. Horn, Robot Vision (MIT Press, Cambridge, Mass., 1986).

Jackson, I. R. H.

I. R. H. Jackson, “Convergence properties of radial basis functions,” Construct. Approx. 4, 243–264 (1988).
[Crossref]

Jerri, A. J.

A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[Crossref]

Kaiser, J. F.

J. F. Kaiser, “Digital filters,” in System Analysis by Digital Computer, F. F. Kuo, J. F. Kaiser, eds. (Wiley, New York, 1966), Chap. 7, pp. 218–285.

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988).

Lent, A.

G. T. Herman, A. Lent, “Iterative reconstruction algorithms,” Comp. Biol. Med. 6, 273–294 (1976).
[Crossref]

Levoy, M.

M. Levoy, “Display of surfaces from volume data,” IEEE Comp. Graph. Appl. 8, (3) 29–37 (1988).
[Crossref]

Lewitt, R. M.

R. M. Lewitt, “Reconstruction algorithms: transform methods,” Proc. IEEE 71, 390–408 (1983).
[Crossref]

Louis, A. K.

A. K. Louis, “Nonuniqueness in inverse Radon problems: the frequency distribution of the ghosts,” Math. Z. 185, 429–440 (1984).
[Crossref]

Mallat, S. G.

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[Crossref]

S. G. Mallat, “Multiresolution approximations and wavelet orthonormal bases of L2(R),” Trans. Am. Math. Soc. 315, 69–87 (1989).

S. G. Mallat, “Multifrequency channel decompositions of images and wavelet models,” IEEE Trans. Acoust. Speech Signal Process. 37, 2091–2110 (1989).
[Crossref]

McDonnell, M. J.

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1986).

Mersereau, R. M.

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Wiley, Chichester, UK, 1986).

Nuttall, A. H.

A. H. Nuttall, “A two-parameter class of Bessel weightings for spectral analysis or array processing—the ideal weighting-window pairs,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1309–1312 (1983).
[Crossref]

Powell, M. J. D.

M. J. D. Powell, “Radial basis function approximations to polynomials,” in Numerical Analysis 1987, D. F. Griffiths, G. A. Watson, eds. (Longman, Harlow, UK, 1988), pp. 223–241.

Rabiner, L. R.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Ravichandran, M.

Sherman, J. W.

J. W. Sherman, “Aperture-antenna analysis,” in Radar Handbook, M. I. Skolnik, ed. (McGraw-Hill, New York, 1970), Chap. 9, pp. 9.1–9.40.

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988).

Slepian, D.

D. Slepian, “Some comments on Fourier analysis, uncertainty and modeling,” SIAM Rev. 25, 379–393 (1983).
[Crossref]

D. Slepian, “Analytic solution of two apodization problems,” J. Opt. Soc. Am. 55, 1110–1115 (1965).
[Crossref]

Smith, K. T.

K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[Crossref]

Sneddon, I. N.

I. N. Sneddon, Fourier Transforms (McGraw-Hill, New York, 1951).

Solmon, D. C.

K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[Crossref]

D. C. Solmon, “The x-ray transform,” J. Math. Anal. Appl. 56, 61–83 (1976).
[Crossref]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

Swindell, W.

H. H. Barrett, W. Swindell, Radiological Imaging: Theory of Image Formation, Detection and Processing (Academic, New York, 1981).

Udupa, J. K.

J. K. Udupa, “Computer aspects of 3D imaging in medicine: a tutorial,” in 3D Imaging in Medicine, J. K. Udupa, G. T. Herman, eds. (CRC, Boca Raton, Fla., 1990).

Wagner, S. L.

K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[Crossref]

Watson, G. N.

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1944).

Wecksung, G. W.

White, W. D.

W. D. White, “Circular aperture distribution functions,” IEEE Trans. Antennas Propag. AP-25, 714–716 (1977).
[Crossref]

Williamson, J. H.

J. H. Williamson, D. E. Evans, “Computerized tomography for sparse-data plasma physics experiments,” IEEE Trans. Plasma Sci. PS-10, 82–93 (1982).
[Crossref]

Zappulla, R. A.

E. J. Farrell, R. A. Zappulla, “Three-dimensional data visualization and biomedical applications,” CRC Crit. Rev. Biomed. Eng. 16, 323–363 (1989).

Appl. Opt. (2)

Bull. Am. Math. Soc. (1)

K. T. Smith, D. C. Solmon, S. L. Wagner, “Practical and mathematical aspects of the problem of reconstructing objects from radiographs,” Bull. Am. Math. Soc. 83, 1227–1270 (1977).
[Crossref]

Comm. Pure Appl. Math. (1)

I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Comm. Pure Appl. Math. 41, 909–996 (1988).
[Crossref]

Comp. Biol. Med. (1)

G. T. Herman, A. Lent, “Iterative reconstruction algorithms,” Comp. Biol. Med. 6, 273–294 (1976).
[Crossref]

Construct. Approx. (2)

I. R. H. Jackson, “Convergence properties of radial basis functions,” Construct. Approx. 4, 243–264 (1988).
[Crossref]

M. D. Buhmann, “Multivariate cardinal interpolation with radial-basis functions,” Construct. Approx. 6, 225–255 (1990).
[Crossref]

CRC Crit. Rev. Biomed. Eng. (1)

E. J. Farrell, R. A. Zappulla, “Three-dimensional data visualization and biomedical applications,” CRC Crit. Rev. Biomed. Eng. 16, 323–363 (1989).

IEEE Comp. Graph. Appl. (1)

M. Levoy, “Display of surfaces from volume data,” IEEE Comp. Graph. Appl. 8, (3) 29–37 (1988).
[Crossref]

IEEE Trans. Acoust. Speech Signal Process. (2)

A. H. Nuttall, “A two-parameter class of Bessel weightings for spectral analysis or array processing—the ideal weighting-window pairs,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1309–1312 (1983).
[Crossref]

S. G. Mallat, “Multifrequency channel decompositions of images and wavelet models,” IEEE Trans. Acoust. Speech Signal Process. 37, 2091–2110 (1989).
[Crossref]

IEEE Trans. Antennas Propag. (2)

W. D. White, “Circular aperture distribution functions,” IEEE Trans. Antennas Propag. AP-25, 714–716 (1977).
[Crossref]

R. C. Hansen, “A one-parameter circular aperture distribution with narrow beamwidth and low sidelobes,” IEEE Trans. Antennas Propag. AP-24, 477–480 (1976).
[Crossref]

IEEE Trans. Med. Imaging (1)

K. H. Hohne, R. Bernstein, “Shading 3D-images from CT using gray-level gradients,” IEEE Trans. Med. Imaging MI-5, 45–47 (1986).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Mach. Intell. 11, 674–693 (1989).
[Crossref]

IEEE Trans. Plasma Sci. (1)

J. H. Williamson, D. E. Evans, “Computerized tomography for sparse-data plasma physics experiments,” IEEE Trans. Plasma Sci. PS-10, 82–93 (1982).
[Crossref]

J. Math. Anal. Appl. (1)

D. C. Solmon, “The x-ray transform,” J. Math. Anal. Appl. 56, 61–83 (1976).
[Crossref]

J. Opt. Soc. Am. (1)

Math. Computat. (1)

R. Franke, “Scattered data interpolation: tests of some methods,” Math. Computat. 38, 181–200 (1982).

Math. Z. (1)

A. K. Louis, “Nonuniqueness in inverse Radon problems: the frequency distribution of the ghosts,” Math. Z. 185, 429–440 (1984).
[Crossref]

Proc. IEEE (2)

A. J. Jerri, “The Shannon sampling theorem—its various extensions and applications: a tutorial review,” Proc. IEEE 65, 1565–1596 (1977).
[Crossref]

R. M. Lewitt, “Reconstruction algorithms: transform methods,” Proc. IEEE 71, 390–408 (1983).
[Crossref]

SIAM Rev. (1)

D. Slepian, “Some comments on Fourier analysis, uncertainty and modeling,” SIAM Rev. 25, 379–393 (1983).
[Crossref]

Trans. Am. Math. Soc. (1)

S. G. Mallat, “Multiresolution approximations and wavelet orthonormal bases of L2(R),” Trans. Am. Math. Soc. 315, 69–87 (1989).

Other (18)

J. W. Sherman, “Aperture-antenna analysis,” in Radar Handbook, M. I. Skolnik, ed. (McGraw-Hill, New York, 1970), Chap. 9, pp. 9.1–9.40.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1984).

J. F. Kaiser, “Digital filters,” in System Analysis by Digital Computer, F. F. Kuo, J. F. Kaiser, eds. (Wiley, New York, 1966), Chap. 7, pp. 218–285.

K. M. Hanson, Los Alamos National Laboratory, Los Alamos, New Mexico 875454 (personal communication).

G. T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic, New York, 1980).

H. H. Barrett, W. Swindell, Radiological Imaging: Theory of Image Formation, Detection and Processing (Academic, New York, 1981).

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).

F. Natterer, The Mathematics of Computerized Tomography (Wiley, Chichester, UK, 1986).

R. H. T. Bates, M. J. McDonnell, Image Restoration and Reconstruction (Clarendon, Oxford, 1986).

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, New York, 1988).

M. J. D. Powell, “Radial basis function approximations to polynomials,” in Numerical Analysis 1987, D. F. Griffiths, G. A. Watson, eds. (Longman, Harlow, UK, 1988), pp. 223–241.

R. N. Bracewell, The Fourier Transform and Its Applications, 2nd ed. (McGraw-Hill, New York, 1978).

I. N. Sneddon, Fourier Transforms (McGraw-Hill, New York, 1951).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, 1944).

J. K. Udupa, “Computer aspects of 3D imaging in medicine: a tutorial,” in 3D Imaging in Medicine, J. K. Udupa, G. T. Herman, eds. (CRC, Boca Raton, Fla., 1990).

B. K. P. Horn, Robot Vision (MIT Press, Cambridge, Mass., 1986).

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Figures (3)

Fig. 1
Fig. 1

Illustration of the geometry of the x-ray transform for n = 3. ω is the plane through the origin perpendicular to the unit vector ω. A line is specified by its direction ω and by the point ξ in ω at which it intersects the plane, ξ is the point in ω at which the line through a given point x and having the direction ω intersects the plane.

Fig. 2
Fig. 2

Example of the Kaiser–Bessel window function with parameters α = 6, a = 1 in Eq. (4.1).

Fig. 3
Fig. 3

Radial profiles (log scale) of the n-dimensional Fourier transform of the Kaiser–Bessel window function in Fig. 2 for (a) n = 1, (b) n = 2, (c) n = 3, (d) n = 4.

Equations (85)

Equations on this page are rendered with MathJax. Learn more.

x = ( i = 1 n x i 2 ) 1 / 2 .
f ˆ n ( X ) = x 1 x n f n ( x ) exp [ i 2 π ( X · x ) ] d x ,
b ˆ n ( R ) = 2 π R n / 2 1 0 b n ( r ) J n / 2 1 ( 2 π R r ) r n / 2 d r .
x 1 x n | f n ( x ) | 2 d x = X 1 X n | f ˆ n ( X ) | 2 d X .
2 π n / 2 Γ ( n / 2 ) 0 | b n ( r ) | 2 r n 1 d r = 2 π n / 2 Γ ( n / 2 ) 0 | b ˆ n ( R ) | 2 R n 1 d R ,
λ n ( a ) = 0 a | b n ( r ) | 2 r n 1 d r 0 | b n ( r ) | 2 r n 1 d r ,
Λ n ( A ) = 0 A | b ˆ n ( R ) | 2 R n 1 d R 0 | b ˆ n ( R ) | 2 R n 1 d R .
S n 1 = { x E n | x = 1 } .
ω = { x E n | x · ω = 0 } .
[ P n f n ] ( ξ , ω ) = f n ( ξ + t ω ) d t .
g n ( x ) = f n ( x x ) ,
[ P n g n ] ( ξ , ω ) = [ P n f n ] ( ξ ξ , ω ) ,
ξ = x ( x · ω ) ω .
p ( ξ ) = [ P n b n ] ( ξ , · ) .
p ( s ) = 2 0 ( a 2 s 2 ) 1 / 2 b n [ ( s 2 + t 2 ) 1 / 2 ] d t .
[ P n f n ] ˆ ( η , ω ) = f ˆ n ( η ) , η ω ,
[ P n f n ] ˆ ( η , ω ) = ξ ω [ P n f n ] ( ξ , ω ) exp [ i 2 π ( ξ · η ) ] d ξ , η ω
H n 1 P n b n = H n b n .
P n b n = H n 1 H n b n .
P n , i f n = [ P n f n ] ( ξ i , ω i ) , ω i S n 1 , ξ i ω i .
f ¯ n ( x ) = j = 1 J c j Φ n , j ( x ) ,
[ L f ¯ n ] ( · ) = j = 1 J c j [ L Φ n , j ] ( · ) .
P n , i f ¯ n = j = 1 J c j P n , i Φ n , j , i = 1 I .
Φ n , j ( x ) = b n ( x x j ) , x j E n ,
f ¯ n ( x ) = j = 1 J c j b n ( x x j ) .
[ P n f ¯ n ] ( ξ , ω ) = j = 1 J c j [ P n b n ] [ ξ ξ ( x j , ω ) ] , ξ , ξ ω ,
ξ ( x j , ω ) = x j ( x j · ω ) ω .
f ¯ n ( x ) = j = 1 J c j b n ( x x j ) .
b n ( r ) = b n r ( r ) .
2 f ¯ n ( x ) = j = 1 J c j 2 b n ( x x j ) ,
2 b n ( r ) = 2 b n r 2 ( r ) + ( n 1 ) r b n r ( r ) .
γ ( x ) = j = 1 J c j δ n ( x x j )
[ γ * b n ] ( x ) = E n γ ( x ) b n ( x x ) d x .
[ f ¯ n ] ˆ = γ ˆ b ˆ n .
w n ( r ) = { I 0 { α [ 1 ( r / a ) 2 ] 1 / 2 } I 0 ( α ) 0 r a 0 otherwise ,
I ν ( z ) = e i ν π / 2 J ν ( i z ) ,
w ˆ n ( R ) = 2 π I 0 ( α ) a n / 2 + 1 R n / 2 1 0 π / 2 J 0 ( i α sin θ ) × J n / 2 1 ( 2 π a R cos θ ) cos n / 2 θ sin θ d θ .
w ˆ n ( R ) = { ( 2 π ) n / 2 a n I 0 ( α ) I n / 2 [ α 2 ( 2 π a R ) 2 ] 1 / 2 { [ α 2 ( 2 π a R ) 2 ] 1 / 2 } n / 2 , 2 π a R α ( 2 π ) n / 2 a n I 0 ( α ) J n / 2 [ ( 2 π a R ) 2 α 2 ] 1 / 2 { [ ( 2 π a R ) 2 α 2 ] 1 / 2 } n / 2 , 2 π a R α .
σ = [ ( 2 π a R ) 2 α 2 ] 1 / 2 ,
σ = [ α 2 ( 2 π a R ) 2 ] 1 / 2 = ± i σ .
J 1 / 2 ( z ) = 2 π z sin z ,
J 3 / 2 ( z ) = 2 π z ( sin z z cos z ) .
w ˆ 1 ( R ) = { 2 a I 0 ( α ) sinh ( σ ) σ , 2 π a R α 2 a I 0 ( α ) sinh ( σ ) σ , 2 π a R α .
w ˆ 2 ( R ) = { 2 π a 2 I 0 ( α ) I 1 ( σ ) σ , 2 π a R α 2 π a 2 I 0 ( α ) J 1 ( σ ) σ , 2 π a R α .
w ˆ 3 ( R ) = { 4 π a 3 I 0 ( α ) ( σ ) 2 [ cosh ( σ ) sinh ( σ ) σ ] , 2 π a R α 4 π a 3 I 0 ( α ) σ 2 [ sin ( σ ) σ cos σ ] , 2 π a R α .
w ˆ 4 ( R ) = { 4 π 2 a 4 I 0 ( α ) I 2 ( σ ) ( σ ) 2 , 2 π a R α 4 π 2 a 4 I 0 ( α ) J 2 ( σ ) σ 2 , 2 π a R α .
SL R n = 20 log 10 [ w ˆ n ( 0 ) | w ˆ n ( R * ) | ] .
SL R n = 20 log 10 { w ˆ n ( 0 ) w ˆ n [ α / ( 2 π a ) ] } + 20 log 10 { w ˆ n [ α / ( 2 π a ) ] | w ˆ n ( R * ) | } .
SL R 1 = 13.26 + 20 log 10 [ sinh α α ] ,
SL R 2 = 17.57 + 20 log 10 [ 2 I 1 ( α ) α ] ,
SL R 3 = 21.29 + 20 log 10 [ 3 α 2 ( cosh α sinh α α ) ] ,
SL R 4 = 24.64 + 20 log 10 [ 8 I 2 ( α ) α 2 ] .
f ¯ n ( x ) = j = 1 J c j b n ( x x j ) , x , x j E n ,
b n ( r ) = w n ( r ) ,
p ( s ) = 2 a α I 0 ( α ) sinh ( α 1 ( s / a ) 2 ) .
b n ( r ) = w ˆ n ( r )
b ˆ n ( R ) = w n ( R ) ,
P n b n = H n 1 H n b n .
P n b n = P n w ˆ n = w ˆ n 1 = b n 1 .
x = Δ ( j 1 , , j n ) T .
g n ( x ) = Δ n j 1 = j n = g n ( x ) h n ( x x ) ,
h ˆ n ( R ) = { 1 , | R | A 0 , otherwise .
h n ( r ) = A n / 2 J n / 2 ( 2 π A r ) r n / 2 .
b n ( r ) = h n ( r ) w n ( r ) ,
b ˆ n ( R ) = [ h ˆ n * w ˆ n ] ( R ) .
f ¯ n ( x ) = Δ n j = 1 J f n ( x j ) b n ( x x j ) ,
b n = h n w n + h n w n
b n = h n w n + 2 h n w n + h n w n .
w n m ( r ) = { [ 1 ( r / a ) 2 ] m I m [ α 1 ( r / a ) 2 ] I m ( α ) , 0 r a 0 , otherwise .
w ˆ n m ( R ) = 2 π i m I m ( α ) a n / 2 + 1 R n / 2 1 0 π / 2 J m ( i α sin θ ) × J n / 2 1 ( 2 π a R cos θ ) cos n / 2 θ sin m + 1 θ d θ .
w ˆ n m ( R ) = { ( 2 π ) n / 2 a n α m I m ( α ) I n / 2 + m [ α 2 ( 2 π a R ) 2 ] [ α 2 ( 2 π a R ) 2 ] n / 2 + m , 2 π a R α ( 2 π ) n / 2 a n α m I m ( α ) J n / 2 + m [ ( 2 π a R ) 2 α 2 ] [ ( 2 π a R ) 2 α 2 ] n / 2 + m , 2 π a R α .
w ˆ n m ( R ) ~ 1 R ( n + 1 ) / 2 + m .
p m ( s ) = 2 0 ( a 2 s 2 ) 1 / 2 w n m [ ( s 2 + t 2 ) ] d t .
p m ( s ) = 2 a ( i ) m I m ( α ) [ 1 ( s / a ) 2 ] m + 1 × 0 π / 2 J m [ i α 1 ( s / a ) 2 sin ϕ ] sin m + 1 ϕ d ϕ .
p m ( s ) = a I m ( α ) ( 2 π α ) 1 / 2 [ 1 ( s / a ) 2 ] m + 1 / 2 × I m + 1 / 2 [ α 1 ( s / a ) 2 ] .
d d z { z ± ν J ν ( z ) } = ± z ± ν J v 1 ( z ) ,
d d z { z ± ν I ν ( z ) } = z ± ν I v 1 ( z ) .
J m ( z ) = ( 1 ) m J m ( z ) , I m ( z ) = I m ( z ) .
r w n m ( r ) = [ 1 / a α m 2 I m ( α ) ] ( r / a ) z m 1 I m 1 ( z ) , 0 r a .
2 w n m ( r ) = [ 1 / a 2 α m 2 I m ( α ) ] × [ ( α r a ) 2 z m 2 I m 2 ( z ) n z m 1 I m 1 ( z ) ] , 0 r a .
K n m = ( 2 π ) n / 2 a n α m I m ( α )
B ν ( R ) = { I ν [ α 2 ( 2 π a R ) 2 ] [ α 2 ( 2 π a R ) 2 ] ν , 2 π a R α J ν [ ( 2 π a R ) 2 α 2 ] [ ( 2 π a R ) 2 α 2 ] ν , 2 π a R α .
w ˆ n m ( R ) = K n m B n / 2 + m ( R ) ,
r w ˆ n m ( R ) = K n m ( 2 π a ) [ ( 2 π a R ) B n / 2 + m + 1 ( R ) ] ,
2 w ˆ n m ( R ) = K n m ( 2 π a ) 2 [ ( 2 π a R ) 2 B n / 2 + m + 2 ( R ) n B n / 2 + m + 1 ( R ) ] .

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