Abstract

Five limited-data computed tomography algorithms are compared. The algorithms used are adapted versions of the algebraic reconstruction technique, the multiplicative algebraic reconstruction technique, the Gerchberg–Papoulis algorithm, a spectral extrapolation algorithm descended from that of Harris [J. Opt. Soc. Am. 54, 931–936 (1964)], and an algorithm based on the singular value decomposition technique. These algorithms were used to reconstruct phantom data with realistic levels of noise from a number of different imaging geometries. The phantoms, the imaging geometries, and the noise were chosen to simulate the conditions encountered in typical computed tomography applications in the physical sciences, and the implementations of the algorithms were optimized for these applications. The multiplicative algebraic reconstruction technique algorithm gave the best results overall; the algebraic reconstruction technique gave the best results for very smooth objects or very noisy (20-dB signal-to-noise ratio) data. My implementations of both of these algorithms incorporate a priori knowledge of the sign of the object, its extent, and its smoothness. The smoothness of the reconstruction is enforced through the use of an appropriate object model (by use of cubic B-spline basis functions and a number of object coefficients appropriate to the object being reconstructed). The average reconstruction error was 1.7% of the maximum phantom value with the multiplicative algebraic reconstruction technique of a phantom with moderate-to-steep gradients by use of data from five viewing angles with a 30-dB signal-to-noise ratio.

© 1993 Optical Society of America

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1989 (1)

1988 (4)

1987 (3)

1986 (3)

R. M. Lewitt, G. Muehllehner, “Accelerated iterative reconstruction for positron emission tomography based on the EM algorithm for maximum likelihood estimation,” IEEE Trans. Med. Imag. MI-5, 16–22 (1986).
[CrossRef]

A. B. Watson, A. Poirson, “Separable two-dimensional discrete Hartley transform,” J. Opt. Soc. Am. A 3, 2001–2004 (1986).
[CrossRef]

S. F. Gull, T. J. Newton, “Maximum entropy tomography,” Appl. Opt. 25, 156–160 (1986).
[CrossRef] [PubMed]

1985 (3)

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

S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,” IEEE Trans. Med. Imag. MI-4, 65–71 (1985).
[CrossRef]

S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,” IEEE Trans. Med. Imag. MI-4, 65–71 (1985).
[CrossRef]

1984 (2)

C. M. Vest, I. Prikryl, “Tomography by iterative convolution: empirical study and application to interferometry,” Appl. Opt. 23, 2433–2440 (1984).
[CrossRef] [PubMed]

M. Sezan, H. Stark, “Tomographic image reconstruction from incomplete view data by convex projections and direct Fourier inversion,” IEEE Trans. Med. Imag. MI-3, 91–98 (1984).
[CrossRef]

1983 (5)

1982 (6)

C. K. Rushforth, A. E. Crawford, Y. Zhou, “Least-squares reconstruction of objects with missing high-frequency components,” J. Opt. Soc. Am. 72, 204–211 (1982).
[CrossRef]

C. K. Rushforth, A. E. Crawford, Y. Zhou, “Least-squares reconstruction of objects with missing high-frequency components,” J. Opt. Soc. Am. 72, 204–211 (1982).
[CrossRef]

N. Baba, K. Murata, “Image reconstruction from limited-angle projections,” Optik 60, 327–332 (1982).

D. C. Youla, H. Webb, “Image restoration by the method of convex projections. Part 1. Theory,” IEEE Trans. Med. Imag. MI-1, 81–94 (1982).
[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections. Part 2. Applications and numerical results,” IEEE Trans. Med. Imag. MI-1, 95–101 (1982).
[CrossRef]

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag. MI-1, 113–121 (1982).
[CrossRef]

1981 (3)

1979 (2)

T. Inouye, “Image reconstruction with limited angle projection data,” IEEE Trans. Nucl. Sci. NS-26, 2666–2669 (1979).

G. Minerbo, “MENT: a maximum entropy algorithm for reconstructing a source from projection data,” Comput. Graph. Image Process. 10, 48–68 (1979).
[CrossRef]

1978 (3)

S. F. Gull, G. J. Daniel, “Image reconstruction from incomplete and noisy data,” Nature (London) 272, 686–690 (1978).
[CrossRef]

H. S. Hou, H. C. Andrews, “Cubic splines for image interpolation and digital filtering,” IEEE Trans. Acoust. Speech Signal Process. ASSP-26, 508–517 (1978).

B. R. Frieden, D. C. Wells, “Restoring with maximum entropy. III. Poisson sources and backgrounds,” J. Opt. Soc. Am. 68, 93–103 (1978).
[CrossRef]

1977 (2)

B. R. Hunt, “Bayesian methods in nonlinear digital image restoration,” IEEE Trans. Comput. C-26, 219–229 (1977).
[CrossRef]

S. J. Wernecke, L. R. D’Addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C-26, 351–364 (1977).
[CrossRef]

1975 (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

1974 (3)

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

R. Gordon, G. T. Herman, “Three-dimensional reconstruction from projections: a review of algorithms,” Int. Rev. Cytol. 38, 111–151 (1974).
[CrossRef] [PubMed]

R. Gordon, “A tutorial on ART,” IEEE Trans. Nucl. Sci. NS-21, 78–93 (1974).

1973 (1)

1972 (1)

1970 (1)

R. Gordon, R. Bender, G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

1968 (1)

E. T. Jaynes, “Prior probabilities,” IEEE Trans. Syst. Sci. Cybern. SSC-4, 227–241 (1968).
[CrossRef]

1964 (1)

1956 (1)

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[CrossRef]

Abbiss, J. B.

Andrews, H. C.

H. S. Hou, H. C. Andrews, “Cubic splines for image interpolation and digital filtering,” IEEE Trans. Acoust. Speech Signal Process. ASSP-26, 508–517 (1978).

Aono, T.

Baba, N.

N. Baba, K. Murata, “Image reconstruction from limited-angle projections,” Optik 60, 327–332 (1982).

Bender, R.

R. Gordon, R. Bender, G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

Bracewell, R. N.

R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956).
[CrossRef]

Brody, W. R.

Byer, R. L.

Crawford, A. E.

D’Addario, L. R.

S. J. Wernecke, L. R. D’Addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C-26, 351–364 (1977).
[CrossRef]

Daniel, G. J.

S. F. Gull, G. J. Daniel, “Image reconstruction from incomplete and noisy data,” Nature (London) 272, 686–690 (1978).
[CrossRef]

De Mol, C.

Defrise, M.

Dhadwal, H. S.

Faris, G. W.

Frieden, B. R.

Gerchberg, R. W.

R. W. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Gordon, R.

R. Gordon, G. T. Herman, “Three-dimensional reconstruction from projections: a review of algorithms,” Int. Rev. Cytol. 38, 111–151 (1974).
[CrossRef] [PubMed]

R. Gordon, “A tutorial on ART,” IEEE Trans. Nucl. Sci. NS-21, 78–93 (1974).

R. Gordon, R. Bender, G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

Gull, S. F.

S. F. Gull, T. J. Newton, “Maximum entropy tomography,” Appl. Opt. 25, 156–160 (1986).
[CrossRef] [PubMed]

S. F. Gull, G. J. Daniel, “Image reconstruction from incomplete and noisy data,” Nature (London) 272, 686–690 (1978).
[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, G. W. Wecksung, “Bayesian approach to limited-angle reconstruction in computed tomography,” J. Opt. Soc. Am. 73, 1501–1509 (1983).
[CrossRef]

K. M. Hanson, “Computed tomographic (CT) reconstruction from limited projection angles,” in Application of Optical Instrumentation in Medicine X, G. D. Fullerton, A. G. Haus, J. A. Mulvaney, W. S. Properzio, eds., Proc. Soc. Photo-Opt. Instrum. Eng.347, 166–173 (1982).

Harris, J. L.

Herman, G. T.

E. Levitan, G. T. Herman, “A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography,” IEEE Trans. Med. Imag. MI-6, 185–192 (1987).
[CrossRef]

R. Gordon, G. T. Herman, “Three-dimensional reconstruction from projections: a review of algorithms,” Int. Rev. Cytol. 38, 111–151 (1974).
[CrossRef] [PubMed]

R. Gordon, R. Bender, G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[CrossRef] [PubMed]

G. T. Herman, Image Reconstruction from Projections (Academic, New York, 1980).

Hino, M.

Hirama, M.

Hou, H. S.

H. S. Hou, H. C. Andrews, “Cubic splines for image interpolation and digital filtering,” IEEE Trans. Acoust. Speech Signal Process. ASSP-26, 508–517 (1978).

Hunt, B. R.

B. R. Hunt, “Bayesian methods in nonlinear digital image restoration,” IEEE Trans. Comput. C-26, 219–229 (1977).
[CrossRef]

Ikeda, O.

Inouye, T.

T. Inouye, “Image reconstruction with limited angle projection data,” IEEE Trans. Nucl. Sci. NS-26, 2666–2669 (1979).

Jaynes, E. T.

E. T. Jaynes, “Prior probabilities,” IEEE Trans. Syst. Sci. Cybern. SSC-4, 227–241 (1968).
[CrossRef]

Joklik, R. G.

R. E. Snyder, R. G. Joklik, H. G. Semerjian, “Laser tomographic measurements in an unsteady jet-diffusion flame,” presented at the Annual Meeting of the American Society of Mechanical Engineers, San Francisco, Calif., 10–15 December 1989.

Kawata, S.

O. Nakamura, S. Kawata, S. Minami, “Optical microscope tomography. II. Nonnegative constraint by a gradient-projection method,” J. Opt. Soc. Am. A 5, 554–561 (1988).
[CrossRef]

S. Kawata, O. Nakamura, S. Minami, “Optical microscope tomography. I. Support constraint,” J. Opt. Soc. Am. A 4, 292–297 (1987).
[CrossRef]

S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,” IEEE Trans. Med. Imag. MI-4, 65–71 (1985).
[CrossRef]

S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,” IEEE Trans. Med. Imag. MI-4, 65–71 (1985).
[CrossRef]

Lent, A.

A. Lent, “A convergent algorithm for maximum entropy image restoration, with a medical x-ray application,” in 1976 SPSE Conference Proceedings, R. Shaw, ed., (Society of Photographic Scientists and Engineers, Washington, D.C., 1977), pp. 249–257.

Levitan, E.

E. Levitan, G. T. Herman, “A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography,” IEEE Trans. Med. Imag. MI-6, 185–192 (1987).
[CrossRef]

Lewitt, R. M.

R. M. Lewitt, G. Muehllehner, “Accelerated iterative reconstruction for positron emission tomography based on the EM algorithm for maximum likelihood estimation,” IEEE Trans. Med. Imag. MI-5, 16–22 (1986).
[CrossRef]

Linzer, M.

Macovski, A.

Mammone, R. J.

C. I. Podilchuk, R. J. Mammone, “Step size for the general iterative image recovery algorithm,” Opt. Eng. 27, 806–811 (1983).

Medoff, B. P.

Meinel, E. S.

Minami, S.

Minerbo, G.

G. Minerbo, “MENT: a maximum entropy algorithm for reconstructing a source from projection data,” Comput. Graph. Image Process. 10, 48–68 (1979).
[CrossRef]

Muehllehner, G.

R. M. Lewitt, G. Muehllehner, “Accelerated iterative reconstruction for positron emission tomography based on the EM algorithm for maximum likelihood estimation,” IEEE Trans. Med. Imag. MI-5, 16–22 (1986).
[CrossRef]

Murata, K.

N. Baba, K. Murata, “Image reconstruction from limited-angle projections,” Optik 60, 327–332 (1982).

Nakajima, M.

Nakamura, O.

Nalcioglu, O.

S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,” IEEE Trans. Med. Imag. MI-4, 65–71 (1985).
[CrossRef]

S. Kawata, O. Nalcioglu, “Constrained iterative reconstruction by the conjugate gradient method,” IEEE Trans. Med. Imag. MI-4, 65–71 (1985).
[CrossRef]

Nassi, M.

Newton, T. J.

Norton, S. J.

Oskoui-Fard, P.

P. Oskoui-Fard, H. Sark, “Tomographic image reconstruction using the theory of convex projections,” IEEE Trans. Med. Imag. 7, 45–58 (1988).
[CrossRef]

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735–742 (1975).
[CrossRef]

Perez-Mendex, V.

K. C. Tam, V. Perez-Mendex, “Limited angle three-dimensional reconstructions using Fourier transform iterations and Radon transform iterations,” Opt. Eng. 20, 586–589 (1981).

Perez-Mendez, V.

Podilchuk, C. I.

C. I. Podilchuk, R. J. Mammone, “Step size for the general iterative image recovery algorithm,” Opt. Eng. 27, 806–811 (1983).

Poirson, A.

Prikryl, I.

Rushforth, C. K.

Sark, H.

P. Oskoui-Fard, H. Sark, “Tomographic image reconstruction using the theory of convex projections,” IEEE Trans. Med. Imag. 7, 45–58 (1988).
[CrossRef]

Sato, T.

Semerjian, H. G.

R. E. Snyder, R. G. Joklik, H. G. Semerjian, “Laser tomographic measurements in an unsteady jet-diffusion flame,” presented at the Annual Meeting of the American Society of Mechanical Engineers, San Francisco, Calif., 10–15 December 1989.

Sezan, M.

M. Sezan, H. Stark, “Tomographic image reconstruction from incomplete view data by convex projections and direct Fourier inversion,” IEEE Trans. Med. Imag. MI-3, 91–98 (1984).
[CrossRef]

Sezan, M. I.

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections. Part 2. Applications and numerical results,” IEEE Trans. Med. Imag. MI-1, 95–101 (1982).
[CrossRef]

Shepp, L. A.

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag. MI-1, 113–121 (1982).
[CrossRef]

Snyder, R. E.

R. E. Snyder, R. G. Joklik, H. G. Semerjian, “Laser tomographic measurements in an unsteady jet-diffusion flame,” presented at the Annual Meeting of the American Society of Mechanical Engineers, San Francisco, Calif., 10–15 December 1989.

Stark, H.

M. Sezan, H. Stark, “Tomographic image reconstruction from incomplete view data by convex projections and direct Fourier inversion,” IEEE Trans. Med. Imag. MI-3, 91–98 (1984).
[CrossRef]

M. I. Sezan, H. Stark, “Image restoration by the method of convex projections. Part 2. Applications and numerical results,” IEEE Trans. Med. Imag. MI-1, 95–101 (1982).
[CrossRef]

Sweeney, D. W.

Tam, K. C.

K. C. Tam, V. Perez-Mendez, “Tomographical imaging with limited angle input,” J. Opt. Soc. Am. 71, 582–592 (1981).
[CrossRef]

K. C. Tam, V. Perez-Mendex, “Limited angle three-dimensional reconstructions using Fourier transform iterations and Radon transform iterations,” Opt. Eng. 20, 586–589 (1981).

Vardi, Y.

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag. MI-1, 113–121 (1982).
[CrossRef]

Verhoeven, D. D.

D. D. Verhoeven, “MART-type CT algorithms for the reconstruction of multidirectional interferometric data,” in Laser Interferometry TV: Computer-Aided Interferometry, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1553, 376–387 (1992).

D. D. Verhoeven, “Application of computed tomography and holographic interferometry to the study of transparent media,” Rep. 38501 (Institut Frangais du Pétrole, Rueil-Malmaison, France, 1990).

D. D. Verhoeven, “An experimental study of the performance of an optical tomography system,” in Laser Interferometry: Quantitave Analysis of Interferograms, R. J. Pryputniewicz, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1162, 369–377 (1990).

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