Abstract

The geometrical properties of detection systems used for computerized tomography are often approximated by line integrals, despite the fact that the systems have nonnegligible beam widths as the result of a finite detector size and a finite acceptance angle. Ways to take into account these distance-dependent beam widths in algorithms for two-dimensional straight-line emission tomography are discussed. It is shown that the full three-dimensional imaging properties of the detection system, including filter functions, can be described in projection space. The relationships with the geometric matrix and the étendue (integral of solid angle over area) are discussed. Two methods to compensate for most of the beam-width effects have been developed, which can be combined with many tomography algorithms. The two methods are demonstrated to improve the quality of tomographic reconstructions of measurements by the bolometer tomography system on the Joint European Torus (JET) tokamak. The strengths and limitations of the methods are discussed.

© 1999 Optical Society of America

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    [CrossRef]
  33. J. F. Camacho, “Soft x-ray tomography on the Alcator C tokamak,” (Massachusetts Institute of Technology, Cambridge, Mass., 1985).
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    [CrossRef]
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1996

Y.-L. Hsieh, G. T. Gullberg, G. L. Zeng, R. H. Huesman, “Image reconstruction using a generalized natural pixel basis,” IEEE Trans. Nucl. Sci. 43, 2306–2319 (1996).
[CrossRef]

L. C. Ingesson, V. V. Pickalov, “An iterative projection-space reconstruction algorithm for tomography systems with irregular coverage,” J. Phys. D 29, 3009–3016 (1996).
[CrossRef]

1994

S. J. Glick, P. C. Penney, M. A. King, C. L. Byrne, “Noniterative compensation for the distance-dependent detector response and photon attenuation in SPECT imaging,” IEEE Trans. Med. Imaging 13, 363–374 (1994).
[CrossRef] [PubMed]

1992

J. R. Baker, T. F. Budinger, R. H. Huesman, “Generalized approach to inverse problems in tomography: image reconstruction for spatially variant systems using natural pixels,” Crit. Rev. Biomed. Eng. 20, 47–71 (1992).
[PubMed]

1991

G. L. Zeng, G. T. Gullberg, B. M. W. Tsui, J. A. Terry, “Three-dimensional iterative reconstruction algorithms with attenuation and geometric point response correction,” IEEE Trans. Nucl. Sci. 38, 693–702 (1991).
[CrossRef]

1990

J. L. Prince, A. S. Willsky, “A geometric projection-space reconstruction algorithm,” Linear Algebr. Appl. 130, 151–191 (1990).
[CrossRef]

B. M. W. Tsui, G. T. Gullberg, “The geometric transfer function for fan beam collimators,” Phys. Med. Biol. 35, 81–93 (1990).
[CrossRef] [PubMed]

1989

A. R. Formiconi, A. Pupi, A. Passeri, “Compensation of spatial system response in SPECT with conjugate gradient reconstruction technique,” Phys. Med. Biol. 34, 69–84 (1989).
[CrossRef] [PubMed]

C. R. Appledorn, “An analytical solution to the nonstationary reconstruction problem in single photon emission computed tomography,” Proc. 1989 Int. Conf. Inf. Proc. Med. Imaging 363, 69–79 (1989).

1988

R. S. Granetz, P. Smeulders, “X-ray tomography at JET,” Nucl. Fusion 28, 457–476 (1988).
[CrossRef]

1985

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

M. T. Madsen, C. H. Park, “Enhancement of SPECT images by Fourier filtering the projection image set,” J. Nucl. Med. 26, 395–402 (1985).
[PubMed]

M. A. King, R. B. Schwinger, B. C. Penney, “Variation of the count-dependent Metz filter with imaging system modulation transfer function,” Med. Phys. 13, 139–149 (1985).
[CrossRef]

1984

V. V. Pikalov, N. G. Preobrazhenskii, “Computer-aided tomography and physical experiment,” Sov. Phys. Usp. 26, 974–990 (1984).
[CrossRef]

1983

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

1981

M. H. Buonocore, W. R. Brody, A. Macovski, “A natural pixel decomposition for two-dimensional image reconstruction,” IEEE Trans. Biomed. Eng. BME-28, 69–78 (1981).
[CrossRef]

A. G. Lindgren, P. A. Rattey, “The inverse discrete Radon transform with applications to tomographic imaging using projection data,” Adv. Electron. Electron Phys. 56, 359–410 (1981).
[CrossRef]

1980

M. R. Teague, “Image analysis via the general theory of moments,” J. Opt. Soc. Am. 20, 920–930 (1980).
[CrossRef]

C. E. Metz, F. B. Atkins, R. N. Beck, “The geometric transfer function component for scintillation camera collimators with straight parallel holes,” Phys. Med. Biol. 25, 1059–1070 (1980).
[CrossRef] [PubMed]

1979

J. G. Verly, R. N. Bracewell, “Blurring in tomograms made with x-ray beams of finite width,” J. Comput. Assisted Tomogr. 3, 662–678 (1979).
[CrossRef]

S. Miracle, M. J. Yzuel, S. Millán, “A study of the point spread function in scintillation camera collimators based on Fourier analysis,” Phys. Med. Biol. 24, 372–384 (1979).
[CrossRef] [PubMed]

1977

D. G. McCaughey, H. C. Andrews, “Degrees of freedom for projection imaging,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-25, 63–73 (1977).
[CrossRef]

R. N. Bracewell, “Correction for collimator width (restoration) in reconstructive x-ray tomography,” J. Comput. Assisted Tomogr. 1, 6–15 (1977).
[CrossRef]

1956

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

Alper, B.

L. C. Ingesson, B. Alper, H. Chen, A. W. Edwards, G. C. Fehmers, J. C. Fuchs, R. Giannella, R. D. Gill, L. Lauro-Taroni, M. Romanelli, “Soft x-ray tomography during ELMs and impurity injection in JET,” Nucl. Fusion (to be published).

Andrews, H. C.

D. G. McCaughey, H. C. Andrews, “Degrees of freedom for projection imaging,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-25, 63–73 (1977).
[CrossRef]

Appledorn, C. R.

C. R. Appledorn, “An analytical solution to the nonstationary reconstruction problem in single photon emission computed tomography,” Proc. 1989 Int. Conf. Inf. Proc. Med. Imaging 363, 69–79 (1989).

Atkins, F. B.

C. E. Metz, F. B. Atkins, R. N. Beck, “The geometric transfer function component for scintillation camera collimators with straight parallel holes,” Phys. Med. Biol. 25, 1059–1070 (1980).
[CrossRef] [PubMed]

Baker, J. R.

J. R. Baker, T. F. Budinger, R. H. Huesman, “Generalized approach to inverse problems in tomography: image reconstruction for spatially variant systems using natural pixels,” Crit. Rev. Biomed. Eng. 20, 47–71 (1992).
[PubMed]

Bates, R. H. T.

R. H. T. Bates, M. J. McDonnel, Image Restoration and Reconstruction (Clarendon, Oxford, 1986), p. 133.

Beck, R. N.

C. E. Metz, F. B. Atkins, R. N. Beck, “The geometric transfer function component for scintillation camera collimators with straight parallel holes,” Phys. Med. Biol. 25, 1059–1070 (1980).
[CrossRef] [PubMed]

Böcker, P. J.

L. C. Ingesson, P. J. Böcker, R. Reichle, M. Romanelli, P. Smeulders, “Projection-space methods to take into account finite beam-width effects in two-dimensional tomography algorithms,” (JET Joint Undertaking, Abingdon, UK, 1998).

Bracewell, R. N.

J. G. Verly, R. N. Bracewell, “Blurring in tomograms made with x-ray beams of finite width,” J. Comput. Assisted Tomogr. 3, 662–678 (1979).
[CrossRef]

R. N. Bracewell, “Correction for collimator width (restoration) in reconstructive x-ray tomography,” J. Comput. Assisted Tomogr. 1, 6–15 (1977).
[CrossRef]

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

Brody, W. R.

M. H. Buonocore, W. R. Brody, A. Macovski, “A natural pixel decomposition for two-dimensional image reconstruction,” IEEE Trans. Biomed. Eng. BME-28, 69–78 (1981).
[CrossRef]

Budinger, T. F.

J. R. Baker, T. F. Budinger, R. H. Huesman, “Generalized approach to inverse problems in tomography: image reconstruction for spatially variant systems using natural pixels,” Crit. Rev. Biomed. Eng. 20, 47–71 (1992).
[PubMed]

Buonocore, M. H.

M. H. Buonocore, W. R. Brody, A. Macovski, “A natural pixel decomposition for two-dimensional image reconstruction,” IEEE Trans. Biomed. Eng. BME-28, 69–78 (1981).
[CrossRef]

Byrne, C. L.

S. J. Glick, P. C. Penney, M. A. King, C. L. Byrne, “Noniterative compensation for the distance-dependent detector response and photon attenuation in SPECT imaging,” IEEE Trans. Med. Imaging 13, 363–374 (1994).
[CrossRef] [PubMed]

Camacho, J. F.

J. F. Camacho, “Soft x-ray tomography on the Alcator C tokamak,” (Massachusetts Institute of Technology, Cambridge, Mass., 1985).

Chen, H.

L. C. Ingesson, B. Alper, H. Chen, A. W. Edwards, G. C. Fehmers, J. C. Fuchs, R. Giannella, R. D. Gill, L. Lauro-Taroni, M. Romanelli, “Soft x-ray tomography during ELMs and impurity injection in JET,” Nucl. Fusion (to be published).

Deans, S. R.

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

Edholm, P. R.

P. R. Edholm, R. M. Lewitt, B. Lindholm, “Novel properties of the Fourier decomposition of the sinogram,” in Physics and Engineering of Computerized Multidimensional Imaging and Processing, O. Nalcioglu, ed., Proc. SPIE671, 8–18 (1986).
[CrossRef]

R. M. Lewitt, P. R. Edholm, W. Xia, “Fourier method for correction of depth-dependent blurring,” in Medical Imaging III: Image Processing, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. SPIE1092, 232–243 (1989).
[CrossRef]

Edwards, A. W.

L. C. Ingesson, B. Alper, H. Chen, A. W. Edwards, G. C. Fehmers, J. C. Fuchs, R. Giannella, R. D. Gill, L. Lauro-Taroni, M. Romanelli, “Soft x-ray tomography during ELMs and impurity injection in JET,” Nucl. Fusion (to be published).

Fehmers, G. C.

L. C. Ingesson, B. Alper, H. Chen, A. W. Edwards, G. C. Fehmers, J. C. Fuchs, R. Giannella, R. D. Gill, L. Lauro-Taroni, M. Romanelli, “Soft x-ray tomography during ELMs and impurity injection in JET,” Nucl. Fusion (to be published).

L. C. Ingesson, R. Reichle, G. C. Fehmers, H. Guo, L. Lauro-Taroni, A. Loarte, R. Simonini, “Radiation distribution and neutral-particle loss in the JET MkI and MkIIA divertors,” in Proceedings of the 24th EPS Conference on Controlled Fusion and Plasma Physics, M. Schittenhelm, R. Bartiromo, F. Wagner, eds., Vol. 21A of Europhysics Conference Abstracts (European Physical Society, Mulhouse, France, 1997), Part 1, pp. 113–116.

Formiconi, A. R.

A. R. Formiconi, A. Pupi, A. Passeri, “Compensation of spatial system response in SPECT with conjugate gradient reconstruction technique,” Phys. Med. Biol. 34, 69–84 (1989).
[CrossRef] [PubMed]

Fuchs, J. C.

R. Reichle, J. C. Fuchs, R. M. Giannella, N. A. C. Gottardi, H. J. Jäckel, K. F. Mast, P. R. Thomas, P. van Belle, “Bolometer for ITER,” in Diagnostics for Experimental Thermonuclear Fusion Reactors, P. E. Stott, G. Gorini, E. Sindoni, eds. (Plenum, New York, 1996), pp. 560–569, and references therein.

L. C. Ingesson, B. Alper, H. Chen, A. W. Edwards, G. C. Fehmers, J. C. Fuchs, R. Giannella, R. D. Gill, L. Lauro-Taroni, M. Romanelli, “Soft x-ray tomography during ELMs and impurity injection in JET,” Nucl. Fusion (to be published).

Giannella, R.

L. C. Ingesson, B. Alper, H. Chen, A. W. Edwards, G. C. Fehmers, J. C. Fuchs, R. Giannella, R. D. Gill, L. Lauro-Taroni, M. Romanelli, “Soft x-ray tomography during ELMs and impurity injection in JET,” Nucl. Fusion (to be published).

Giannella, R. M.

R. Reichle, J. C. Fuchs, R. M. Giannella, N. A. C. Gottardi, H. J. Jäckel, K. F. Mast, P. R. Thomas, P. van Belle, “Bolometer for ITER,” in Diagnostics for Experimental Thermonuclear Fusion Reactors, P. E. Stott, G. Gorini, E. Sindoni, eds. (Plenum, New York, 1996), pp. 560–569, and references therein.

Gill, R. D.

L. C. Ingesson, B. Alper, H. Chen, A. W. Edwards, G. C. Fehmers, J. C. Fuchs, R. Giannella, R. D. Gill, L. Lauro-Taroni, M. Romanelli, “Soft x-ray tomography during ELMs and impurity injection in JET,” Nucl. Fusion (to be published).

Glick, S. J.

S. J. Glick, P. C. Penney, M. A. King, C. L. Byrne, “Noniterative compensation for the distance-dependent detector response and photon attenuation in SPECT imaging,” IEEE Trans. Med. Imaging 13, 363–374 (1994).
[CrossRef] [PubMed]

Gottardi, N. A. C.

R. Reichle, J. C. Fuchs, R. M. Giannella, N. A. C. Gottardi, H. J. Jäckel, K. F. Mast, P. R. Thomas, P. van Belle, “Bolometer for ITER,” in Diagnostics for Experimental Thermonuclear Fusion Reactors, P. E. Stott, G. Gorini, E. Sindoni, eds. (Plenum, New York, 1996), pp. 560–569, and references therein.

Granetz, R. S.

R. S. Granetz, P. Smeulders, “X-ray tomography at JET,” Nucl. Fusion 28, 457–476 (1988).
[CrossRef]

Gullberg, G. T.

Y.-L. Hsieh, G. T. Gullberg, G. L. Zeng, R. H. Huesman, “Image reconstruction using a generalized natural pixel basis,” IEEE Trans. Nucl. Sci. 43, 2306–2319 (1996).
[CrossRef]

G. L. Zeng, G. T. Gullberg, B. M. W. Tsui, J. A. Terry, “Three-dimensional iterative reconstruction algorithms with attenuation and geometric point response correction,” IEEE Trans. Nucl. Sci. 38, 693–702 (1991).
[CrossRef]

B. M. W. Tsui, G. T. Gullberg, “The geometric transfer function for fan beam collimators,” Phys. Med. Biol. 35, 81–93 (1990).
[CrossRef] [PubMed]

Guo, H.

L. C. Ingesson, R. Reichle, G. C. Fehmers, H. Guo, L. Lauro-Taroni, A. Loarte, R. Simonini, “Radiation distribution and neutral-particle loss in the JET MkI and MkIIA divertors,” in Proceedings of the 24th EPS Conference on Controlled Fusion and Plasma Physics, M. Schittenhelm, R. Bartiromo, F. Wagner, eds., Vol. 21A of Europhysics Conference Abstracts (European Physical Society, Mulhouse, France, 1997), Part 1, pp. 113–116.

Hanson, K. M.

Hashimoto, S.

K. Ogawa, S. Paek, M. Nakajima, S. Yuta, A. Kubo, S. Hashimoto, “Correction of collimator aperture using a shift-variant deconvolution filter in gamma camera emission CT,” in Medical Imaging II, R. H. Schneider, S. J. Dwyer, eds., Proc. SPIE914, 699–706 (1988).
[CrossRef]

Hsieh, Y.-L.

Y.-L. Hsieh, G. T. Gullberg, G. L. Zeng, R. H. Huesman, “Image reconstruction using a generalized natural pixel basis,” IEEE Trans. Nucl. Sci. 43, 2306–2319 (1996).
[CrossRef]

Huesman, R. H.

Y.-L. Hsieh, G. T. Gullberg, G. L. Zeng, R. H. Huesman, “Image reconstruction using a generalized natural pixel basis,” IEEE Trans. Nucl. Sci. 43, 2306–2319 (1996).
[CrossRef]

J. R. Baker, T. F. Budinger, R. H. Huesman, “Generalized approach to inverse problems in tomography: image reconstruction for spatially variant systems using natural pixels,” Crit. Rev. Biomed. Eng. 20, 47–71 (1992).
[PubMed]

Ingesson, L. C.

L. C. Ingesson, V. V. Pickalov, “An iterative projection-space reconstruction algorithm for tomography systems with irregular coverage,” J. Phys. D 29, 3009–3016 (1996).
[CrossRef]

L. C. Ingesson, R. Reichle, G. C. Fehmers, H. Guo, L. Lauro-Taroni, A. Loarte, R. Simonini, “Radiation distribution and neutral-particle loss in the JET MkI and MkIIA divertors,” in Proceedings of the 24th EPS Conference on Controlled Fusion and Plasma Physics, M. Schittenhelm, R. Bartiromo, F. Wagner, eds., Vol. 21A of Europhysics Conference Abstracts (European Physical Society, Mulhouse, France, 1997), Part 1, pp. 113–116.

L. C. Ingesson, B. Alper, H. Chen, A. W. Edwards, G. C. Fehmers, J. C. Fuchs, R. Giannella, R. D. Gill, L. Lauro-Taroni, M. Romanelli, “Soft x-ray tomography during ELMs and impurity injection in JET,” Nucl. Fusion (to be published).

L. C. Ingesson, “Visible-light tomography of tokamak plasmas,” thesis technische (Universiteit Eindhoven, Eindhoven, The Netherlands, 1995).

L. C. Ingesson, P. J. Böcker, R. Reichle, M. Romanelli, P. Smeulders, “Projection-space methods to take into account finite beam-width effects in two-dimensional tomography algorithms,” (JET Joint Undertaking, Abingdon, UK, 1998).

Jäckel, H. J.

R. Reichle, J. C. Fuchs, R. M. Giannella, N. A. C. Gottardi, H. J. Jäckel, K. F. Mast, P. R. Thomas, P. van Belle, “Bolometer for ITER,” in Diagnostics for Experimental Thermonuclear Fusion Reactors, P. E. Stott, G. Gorini, E. Sindoni, eds. (Plenum, New York, 1996), pp. 560–569, and references therein.

King, M. A.

S. J. Glick, P. C. Penney, M. A. King, C. L. Byrne, “Noniterative compensation for the distance-dependent detector response and photon attenuation in SPECT imaging,” IEEE Trans. Med. Imaging 13, 363–374 (1994).
[CrossRef] [PubMed]

M. A. King, R. B. Schwinger, B. C. Penney, “Variation of the count-dependent Metz filter with imaging system modulation transfer function,” Med. Phys. 13, 139–149 (1985).
[CrossRef]

Kubo, A.

K. Ogawa, S. Paek, M. Nakajima, S. Yuta, A. Kubo, S. Hashimoto, “Correction of collimator aperture using a shift-variant deconvolution filter in gamma camera emission CT,” in Medical Imaging II, R. H. Schneider, S. J. Dwyer, eds., Proc. SPIE914, 699–706 (1988).
[CrossRef]

Lauro-Taroni, L.

L. C. Ingesson, R. Reichle, G. C. Fehmers, H. Guo, L. Lauro-Taroni, A. Loarte, R. Simonini, “Radiation distribution and neutral-particle loss in the JET MkI and MkIIA divertors,” in Proceedings of the 24th EPS Conference on Controlled Fusion and Plasma Physics, M. Schittenhelm, R. Bartiromo, F. Wagner, eds., Vol. 21A of Europhysics Conference Abstracts (European Physical Society, Mulhouse, France, 1997), Part 1, pp. 113–116.

L. C. Ingesson, B. Alper, H. Chen, A. W. Edwards, G. C. Fehmers, J. C. Fuchs, R. Giannella, R. D. Gill, L. Lauro-Taroni, M. Romanelli, “Soft x-ray tomography during ELMs and impurity injection in JET,” Nucl. Fusion (to be published).

Lewitt, R. M.

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

P. R. Edholm, R. M. Lewitt, B. Lindholm, “Novel properties of the Fourier decomposition of the sinogram,” in Physics and Engineering of Computerized Multidimensional Imaging and Processing, O. Nalcioglu, ed., Proc. SPIE671, 8–18 (1986).
[CrossRef]

R. M. Lewitt, P. R. Edholm, W. Xia, “Fourier method for correction of depth-dependent blurring,” in Medical Imaging III: Image Processing, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. SPIE1092, 232–243 (1989).
[CrossRef]

Lindgren, A. G.

A. G. Lindgren, P. A. Rattey, “The inverse discrete Radon transform with applications to tomographic imaging using projection data,” Adv. Electron. Electron Phys. 56, 359–410 (1981).
[CrossRef]

Lindholm, B.

P. R. Edholm, R. M. Lewitt, B. Lindholm, “Novel properties of the Fourier decomposition of the sinogram,” in Physics and Engineering of Computerized Multidimensional Imaging and Processing, O. Nalcioglu, ed., Proc. SPIE671, 8–18 (1986).
[CrossRef]

Loarte, A.

L. C. Ingesson, R. Reichle, G. C. Fehmers, H. Guo, L. Lauro-Taroni, A. Loarte, R. Simonini, “Radiation distribution and neutral-particle loss in the JET MkI and MkIIA divertors,” in Proceedings of the 24th EPS Conference on Controlled Fusion and Plasma Physics, M. Schittenhelm, R. Bartiromo, F. Wagner, eds., Vol. 21A of Europhysics Conference Abstracts (European Physical Society, Mulhouse, France, 1997), Part 1, pp. 113–116.

Longhurst, R. S.

R. S. Longhurst, Geometrical and Physical Optics, 3rd ed. (Longman, Harlow, UK, 1973), pp. 20, 450, 466.

Macovski, A.

M. H. Buonocore, W. R. Brody, A. Macovski, “A natural pixel decomposition for two-dimensional image reconstruction,” IEEE Trans. Biomed. Eng. BME-28, 69–78 (1981).
[CrossRef]

Madsen, M. T.

M. T. Madsen, C. H. Park, “Enhancement of SPECT images by Fourier filtering the projection image set,” J. Nucl. Med. 26, 395–402 (1985).
[PubMed]

Mast, K. F.

R. Reichle, J. C. Fuchs, R. M. Giannella, N. A. C. Gottardi, H. J. Jäckel, K. F. Mast, P. R. Thomas, P. van Belle, “Bolometer for ITER,” in Diagnostics for Experimental Thermonuclear Fusion Reactors, P. E. Stott, G. Gorini, E. Sindoni, eds. (Plenum, New York, 1996), pp. 560–569, and references therein.

McCaughey, D. G.

D. G. McCaughey, H. C. Andrews, “Degrees of freedom for projection imaging,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-25, 63–73 (1977).
[CrossRef]

McDonnel, M. J.

R. H. T. Bates, M. J. McDonnel, Image Restoration and Reconstruction (Clarendon, Oxford, 1986), p. 133.

Metz, C. E.

C. E. Metz, F. B. Atkins, R. N. Beck, “The geometric transfer function component for scintillation camera collimators with straight parallel holes,” Phys. Med. Biol. 25, 1059–1070 (1980).
[CrossRef] [PubMed]

Millán, S.

S. Miracle, M. J. Yzuel, S. Millán, “A study of the point spread function in scintillation camera collimators based on Fourier analysis,” Phys. Med. Biol. 24, 372–384 (1979).
[CrossRef] [PubMed]

Miracle, S.

S. Miracle, M. J. Yzuel, S. Millán, “A study of the point spread function in scintillation camera collimators based on Fourier analysis,” Phys. Med. Biol. 24, 372–384 (1979).
[CrossRef] [PubMed]

Nakajima, M.

K. Ogawa, S. Paek, M. Nakajima, S. Yuta, A. Kubo, S. Hashimoto, “Correction of collimator aperture using a shift-variant deconvolution filter in gamma camera emission CT,” in Medical Imaging II, R. H. Schneider, S. J. Dwyer, eds., Proc. SPIE914, 699–706 (1988).
[CrossRef]

Ogawa, K.

K. Ogawa, S. Paek, M. Nakajima, S. Yuta, A. Kubo, S. Hashimoto, “Correction of collimator aperture using a shift-variant deconvolution filter in gamma camera emission CT,” in Medical Imaging II, R. H. Schneider, S. J. Dwyer, eds., Proc. SPIE914, 699–706 (1988).
[CrossRef]

Paek, S.

K. Ogawa, S. Paek, M. Nakajima, S. Yuta, A. Kubo, S. Hashimoto, “Correction of collimator aperture using a shift-variant deconvolution filter in gamma camera emission CT,” in Medical Imaging II, R. H. Schneider, S. J. Dwyer, eds., Proc. SPIE914, 699–706 (1988).
[CrossRef]

Park, C. H.

M. T. Madsen, C. H. Park, “Enhancement of SPECT images by Fourier filtering the projection image set,” J. Nucl. Med. 26, 395–402 (1985).
[PubMed]

Passeri, A.

A. R. Formiconi, A. Pupi, A. Passeri, “Compensation of spatial system response in SPECT with conjugate gradient reconstruction technique,” Phys. Med. Biol. 34, 69–84 (1989).
[CrossRef] [PubMed]

Penney, B. C.

M. A. King, R. B. Schwinger, B. C. Penney, “Variation of the count-dependent Metz filter with imaging system modulation transfer function,” Med. Phys. 13, 139–149 (1985).
[CrossRef]

Penney, P. C.

S. J. Glick, P. C. Penney, M. A. King, C. L. Byrne, “Noniterative compensation for the distance-dependent detector response and photon attenuation in SPECT imaging,” IEEE Trans. Med. Imaging 13, 363–374 (1994).
[CrossRef] [PubMed]

Pickalov, V. V.

L. C. Ingesson, V. V. Pickalov, “An iterative projection-space reconstruction algorithm for tomography systems with irregular coverage,” J. Phys. D 29, 3009–3016 (1996).
[CrossRef]

Pikalov, V. V.

V. V. Pikalov, N. G. Preobrazhenskii, “Computer-aided tomography and physical experiment,” Sov. Phys. Usp. 26, 974–990 (1984).
[CrossRef]

Preobrazhenskii, N. G.

V. V. Pikalov, N. G. Preobrazhenskii, “Computer-aided tomography and physical experiment,” Sov. Phys. Usp. 26, 974–990 (1984).
[CrossRef]

Prince, J. L.

J. L. Prince, A. S. Willsky, “A geometric projection-space reconstruction algorithm,” Linear Algebr. Appl. 130, 151–191 (1990).
[CrossRef]

Pupi, A.

A. R. Formiconi, A. Pupi, A. Passeri, “Compensation of spatial system response in SPECT with conjugate gradient reconstruction technique,” Phys. Med. Biol. 34, 69–84 (1989).
[CrossRef] [PubMed]

Rattey, P. A.

A. G. Lindgren, P. A. Rattey, “The inverse discrete Radon transform with applications to tomographic imaging using projection data,” Adv. Electron. Electron Phys. 56, 359–410 (1981).
[CrossRef]

Reichle, R.

L. C. Ingesson, R. Reichle, G. C. Fehmers, H. Guo, L. Lauro-Taroni, A. Loarte, R. Simonini, “Radiation distribution and neutral-particle loss in the JET MkI and MkIIA divertors,” in Proceedings of the 24th EPS Conference on Controlled Fusion and Plasma Physics, M. Schittenhelm, R. Bartiromo, F. Wagner, eds., Vol. 21A of Europhysics Conference Abstracts (European Physical Society, Mulhouse, France, 1997), Part 1, pp. 113–116.

R. Reichle, J. C. Fuchs, R. M. Giannella, N. A. C. Gottardi, H. J. Jäckel, K. F. Mast, P. R. Thomas, P. van Belle, “Bolometer for ITER,” in Diagnostics for Experimental Thermonuclear Fusion Reactors, P. E. Stott, G. Gorini, E. Sindoni, eds. (Plenum, New York, 1996), pp. 560–569, and references therein.

L. C. Ingesson, P. J. Böcker, R. Reichle, M. Romanelli, P. Smeulders, “Projection-space methods to take into account finite beam-width effects in two-dimensional tomography algorithms,” (JET Joint Undertaking, Abingdon, UK, 1998).

Romanelli, M.

L. C. Ingesson, P. J. Böcker, R. Reichle, M. Romanelli, P. Smeulders, “Projection-space methods to take into account finite beam-width effects in two-dimensional tomography algorithms,” (JET Joint Undertaking, Abingdon, UK, 1998).

L. C. Ingesson, B. Alper, H. Chen, A. W. Edwards, G. C. Fehmers, J. C. Fuchs, R. Giannella, R. D. Gill, L. Lauro-Taroni, M. Romanelli, “Soft x-ray tomography during ELMs and impurity injection in JET,” Nucl. Fusion (to be published).

Schwinger, R. B.

M. A. King, R. B. Schwinger, B. C. Penney, “Variation of the count-dependent Metz filter with imaging system modulation transfer function,” Med. Phys. 13, 139–149 (1985).
[CrossRef]

Simonini, R.

L. C. Ingesson, R. Reichle, G. C. Fehmers, H. Guo, L. Lauro-Taroni, A. Loarte, R. Simonini, “Radiation distribution and neutral-particle loss in the JET MkI and MkIIA divertors,” in Proceedings of the 24th EPS Conference on Controlled Fusion and Plasma Physics, M. Schittenhelm, R. Bartiromo, F. Wagner, eds., Vol. 21A of Europhysics Conference Abstracts (European Physical Society, Mulhouse, France, 1997), Part 1, pp. 113–116.

Smeulders, P.

R. S. Granetz, P. Smeulders, “X-ray tomography at JET,” Nucl. Fusion 28, 457–476 (1988).
[CrossRef]

L. C. Ingesson, P. J. Böcker, R. Reichle, M. Romanelli, P. Smeulders, “Projection-space methods to take into account finite beam-width effects in two-dimensional tomography algorithms,” (JET Joint Undertaking, Abingdon, UK, 1998).

P. Smeulders, “A fast plasma tomography routine with second-order accuracy and compensation for spatial resolution,” (Max-Planck-Institut für Plasmaphysik, Garching, Germany, 1983).

P. Smeulders, “Second order Abel inversion with allowance for the spatial resolution,” (Max-Planck-Institut für Plasmaphysik, Garching, Germany, 1978).

Tanzi, C. P.

C. P. Tanzi, “Emission of soft x-ray and microwave radiation from tokamak plasmas,” Thesis (Universiteit Utrecht, Utrecht, The Netherlands, 1996).

Teague, M. R.

M. R. Teague, “Image analysis via the general theory of moments,” J. Opt. Soc. Am. 20, 920–930 (1980).
[CrossRef]

Terry, J. A.

G. L. Zeng, G. T. Gullberg, B. M. W. Tsui, J. A. Terry, “Three-dimensional iterative reconstruction algorithms with attenuation and geometric point response correction,” IEEE Trans. Nucl. Sci. 38, 693–702 (1991).
[CrossRef]

Thomas, P. R.

R. Reichle, J. C. Fuchs, R. M. Giannella, N. A. C. Gottardi, H. J. Jäckel, K. F. Mast, P. R. Thomas, P. van Belle, “Bolometer for ITER,” in Diagnostics for Experimental Thermonuclear Fusion Reactors, P. E. Stott, G. Gorini, E. Sindoni, eds. (Plenum, New York, 1996), pp. 560–569, and references therein.

Tsui, B. M. W.

G. L. Zeng, G. T. Gullberg, B. M. W. Tsui, J. A. Terry, “Three-dimensional iterative reconstruction algorithms with attenuation and geometric point response correction,” IEEE Trans. Nucl. Sci. 38, 693–702 (1991).
[CrossRef]

B. M. W. Tsui, G. T. Gullberg, “The geometric transfer function for fan beam collimators,” Phys. Med. Biol. 35, 81–93 (1990).
[CrossRef] [PubMed]

van Belle, P.

R. Reichle, J. C. Fuchs, R. M. Giannella, N. A. C. Gottardi, H. J. Jäckel, K. F. Mast, P. R. Thomas, P. van Belle, “Bolometer for ITER,” in Diagnostics for Experimental Thermonuclear Fusion Reactors, P. E. Stott, G. Gorini, E. Sindoni, eds. (Plenum, New York, 1996), pp. 560–569, and references therein.

Verly, J. G.

J. G. Verly, R. N. Bracewell, “Blurring in tomograms made with x-ray beams of finite width,” J. Comput. Assisted Tomogr. 3, 662–678 (1979).
[CrossRef]

Wecksung, G. W.

Willsky, A. S.

J. L. Prince, A. S. Willsky, “A geometric projection-space reconstruction algorithm,” Linear Algebr. Appl. 130, 151–191 (1990).
[CrossRef]

Xia, W.

R. M. Lewitt, P. R. Edholm, W. Xia, “Fourier method for correction of depth-dependent blurring,” in Medical Imaging III: Image Processing, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. SPIE1092, 232–243 (1989).
[CrossRef]

Yuta, S.

K. Ogawa, S. Paek, M. Nakajima, S. Yuta, A. Kubo, S. Hashimoto, “Correction of collimator aperture using a shift-variant deconvolution filter in gamma camera emission CT,” in Medical Imaging II, R. H. Schneider, S. J. Dwyer, eds., Proc. SPIE914, 699–706 (1988).
[CrossRef]

Yzuel, M. J.

S. Miracle, M. J. Yzuel, S. Millán, “A study of the point spread function in scintillation camera collimators based on Fourier analysis,” Phys. Med. Biol. 24, 372–384 (1979).
[CrossRef] [PubMed]

Zeng, G. L.

Y.-L. Hsieh, G. T. Gullberg, G. L. Zeng, R. H. Huesman, “Image reconstruction using a generalized natural pixel basis,” IEEE Trans. Nucl. Sci. 43, 2306–2319 (1996).
[CrossRef]

G. L. Zeng, G. T. Gullberg, B. M. W. Tsui, J. A. Terry, “Three-dimensional iterative reconstruction algorithms with attenuation and geometric point response correction,” IEEE Trans. Nucl. Sci. 38, 693–702 (1991).
[CrossRef]

Adv. Electron. Electron Phys.

A. G. Lindgren, P. A. Rattey, “The inverse discrete Radon transform with applications to tomographic imaging using projection data,” Adv. Electron. Electron Phys. 56, 359–410 (1981).
[CrossRef]

Appl. Opt.

Aust. J. Phys.

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

Crit. Rev. Biomed. Eng.

J. R. Baker, T. F. Budinger, R. H. Huesman, “Generalized approach to inverse problems in tomography: image reconstruction for spatially variant systems using natural pixels,” Crit. Rev. Biomed. Eng. 20, 47–71 (1992).
[PubMed]

IEEE Trans. Acoust., Speech, Signal Process.

D. G. McCaughey, H. C. Andrews, “Degrees of freedom for projection imaging,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-25, 63–73 (1977).
[CrossRef]

IEEE Trans. Biomed. Eng.

M. H. Buonocore, W. R. Brody, A. Macovski, “A natural pixel decomposition for two-dimensional image reconstruction,” IEEE Trans. Biomed. Eng. BME-28, 69–78 (1981).
[CrossRef]

IEEE Trans. Med. Imaging

S. J. Glick, P. C. Penney, M. A. King, C. L. Byrne, “Noniterative compensation for the distance-dependent detector response and photon attenuation in SPECT imaging,” IEEE Trans. Med. Imaging 13, 363–374 (1994).
[CrossRef] [PubMed]

IEEE Trans. Nucl. Sci.

G. L. Zeng, G. T. Gullberg, B. M. W. Tsui, J. A. Terry, “Three-dimensional iterative reconstruction algorithms with attenuation and geometric point response correction,” IEEE Trans. Nucl. Sci. 38, 693–702 (1991).
[CrossRef]

Y.-L. Hsieh, G. T. Gullberg, G. L. Zeng, R. H. Huesman, “Image reconstruction using a generalized natural pixel basis,” IEEE Trans. Nucl. Sci. 43, 2306–2319 (1996).
[CrossRef]

J. Comput. Assisted Tomogr.

J. G. Verly, R. N. Bracewell, “Blurring in tomograms made with x-ray beams of finite width,” J. Comput. Assisted Tomogr. 3, 662–678 (1979).
[CrossRef]

R. N. Bracewell, “Correction for collimator width (restoration) in reconstructive x-ray tomography,” J. Comput. Assisted Tomogr. 1, 6–15 (1977).
[CrossRef]

J. Nucl. Med.

M. T. Madsen, C. H. Park, “Enhancement of SPECT images by Fourier filtering the projection image set,” J. Nucl. Med. 26, 395–402 (1985).
[PubMed]

J. Opt. Soc. Am.

M. R. Teague, “Image analysis via the general theory of moments,” J. Opt. Soc. Am. 20, 920–930 (1980).
[CrossRef]

J. Phys. D

L. C. Ingesson, V. V. Pickalov, “An iterative projection-space reconstruction algorithm for tomography systems with irregular coverage,” J. Phys. D 29, 3009–3016 (1996).
[CrossRef]

Linear Algebr. Appl.

J. L. Prince, A. S. Willsky, “A geometric projection-space reconstruction algorithm,” Linear Algebr. Appl. 130, 151–191 (1990).
[CrossRef]

Med. Phys.

M. A. King, R. B. Schwinger, B. C. Penney, “Variation of the count-dependent Metz filter with imaging system modulation transfer function,” Med. Phys. 13, 139–149 (1985).
[CrossRef]

Nucl. Fusion

R. S. Granetz, P. Smeulders, “X-ray tomography at JET,” Nucl. Fusion 28, 457–476 (1988).
[CrossRef]

Phys. Med. Biol.

S. Miracle, M. J. Yzuel, S. Millán, “A study of the point spread function in scintillation camera collimators based on Fourier analysis,” Phys. Med. Biol. 24, 372–384 (1979).
[CrossRef] [PubMed]

C. E. Metz, F. B. Atkins, R. N. Beck, “The geometric transfer function component for scintillation camera collimators with straight parallel holes,” Phys. Med. Biol. 25, 1059–1070 (1980).
[CrossRef] [PubMed]

B. M. W. Tsui, G. T. Gullberg, “The geometric transfer function for fan beam collimators,” Phys. Med. Biol. 35, 81–93 (1990).
[CrossRef] [PubMed]

A. R. Formiconi, A. Pupi, A. Passeri, “Compensation of spatial system response in SPECT with conjugate gradient reconstruction technique,” Phys. Med. Biol. 34, 69–84 (1989).
[CrossRef] [PubMed]

Proc. 1989 Int. Conf. Inf. Proc. Med. Imaging

C. R. Appledorn, “An analytical solution to the nonstationary reconstruction problem in single photon emission computed tomography,” Proc. 1989 Int. Conf. Inf. Proc. Med. Imaging 363, 69–79 (1989).

Proc. IEEE

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

Sov. Phys. Usp.

V. V. Pikalov, N. G. Preobrazhenskii, “Computer-aided tomography and physical experiment,” Sov. Phys. Usp. 26, 974–990 (1984).
[CrossRef]

Other

J. F. Camacho, “Soft x-ray tomography on the Alcator C tokamak,” (Massachusetts Institute of Technology, Cambridge, Mass., 1985).

C. P. Tanzi, “Emission of soft x-ray and microwave radiation from tokamak plasmas,” Thesis (Universiteit Utrecht, Utrecht, The Netherlands, 1996).

P. Smeulders, “Second order Abel inversion with allowance for the spatial resolution,” (Max-Planck-Institut für Plasmaphysik, Garching, Germany, 1978).

P. Smeulders, “A fast plasma tomography routine with second-order accuracy and compensation for spatial resolution,” (Max-Planck-Institut für Plasmaphysik, Garching, Germany, 1983).

L. C. Ingesson, R. Reichle, G. C. Fehmers, H. Guo, L. Lauro-Taroni, A. Loarte, R. Simonini, “Radiation distribution and neutral-particle loss in the JET MkI and MkIIA divertors,” in Proceedings of the 24th EPS Conference on Controlled Fusion and Plasma Physics, M. Schittenhelm, R. Bartiromo, F. Wagner, eds., Vol. 21A of Europhysics Conference Abstracts (European Physical Society, Mulhouse, France, 1997), Part 1, pp. 113–116.

R. S. Longhurst, Geometrical and Physical Optics, 3rd ed. (Longman, Harlow, UK, 1973), pp. 20, 450, 466.

L. C. Ingesson, “Visible-light tomography of tokamak plasmas,” thesis technische (Universiteit Eindhoven, Eindhoven, The Netherlands, 1995).

L. C. Ingesson, B. Alper, H. Chen, A. W. Edwards, G. C. Fehmers, J. C. Fuchs, R. Giannella, R. D. Gill, L. Lauro-Taroni, M. Romanelli, “Soft x-ray tomography during ELMs and impurity injection in JET,” Nucl. Fusion (to be published).

R. H. T. Bates, M. J. McDonnel, Image Restoration and Reconstruction (Clarendon, Oxford, 1986), p. 133.

L. C. Ingesson, P. J. Böcker, R. Reichle, M. Romanelli, P. Smeulders, “Projection-space methods to take into account finite beam-width effects in two-dimensional tomography algorithms,” (JET Joint Undertaking, Abingdon, UK, 1998).

R. Reichle, J. C. Fuchs, R. M. Giannella, N. A. C. Gottardi, H. J. Jäckel, K. F. Mast, P. R. Thomas, P. van Belle, “Bolometer for ITER,” in Diagnostics for Experimental Thermonuclear Fusion Reactors, P. E. Stott, G. Gorini, E. Sindoni, eds. (Plenum, New York, 1996), pp. 560–569, and references therein.

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

K. Ogawa, S. Paek, M. Nakajima, S. Yuta, A. Kubo, S. Hashimoto, “Correction of collimator aperture using a shift-variant deconvolution filter in gamma camera emission CT,” in Medical Imaging II, R. H. Schneider, S. J. Dwyer, eds., Proc. SPIE914, 699–706 (1988).
[CrossRef]

P. R. Edholm, R. M. Lewitt, B. Lindholm, “Novel properties of the Fourier decomposition of the sinogram,” in Physics and Engineering of Computerized Multidimensional Imaging and Processing, O. Nalcioglu, ed., Proc. SPIE671, 8–18 (1986).
[CrossRef]

R. M. Lewitt, P. R. Edholm, W. Xia, “Fourier method for correction of depth-dependent blurring,” in Medical Imaging III: Image Processing, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. SPIE1092, 232–243 (1989).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Contour plot of the sinogram of a typical emission profile in the JET tokamak [calculated from the phantom in Fig. 2(a)]. The origin was chosen on the magnetic axis of the tokamak cross section. The shaded area indicates where lines of sight pass outside the boundary of the tokamak. The points (solid and open squares) mark the (p, ξ) coordinates of the average lines of sight of the JET bolometer system. Regions Di of the detectors are shown around these points (visible only as thin lines). (b)–(d) Blowups of three regions Di covered by bolometers, where the contours indicate ki(p, ξ): (b) a rectangular pinhole system with parallel detector and aperture (contours at intervals of 0.2% of the maximum); (c) a rectangular pinhole system with nonparallel detector and aperture (contours at 0.2% intervals); (d) a system with a rectangular detector, a round collimator, and several other bounding apertures (contours at 5% intervals). In (b)–(d) the p axis has been stretched by subtracting the pap(ξ) values that correspond to the center of the front aperture as a function of ξ. In (b) and (c) the maximum contour is in the lower-left corner of the shape, but for detectors straight behind the aperture the maximum is near the middle of the shape.

Fig. 2
Fig. 2

Gray-scale plots of the emission in the divertor region of a poloidal cross section of the tokamak (scale: white, 0 W m-3; black, 3 MW). The separatrix, i.e., the last closed flux surface in the tokamak, is shown in each of (a)–(d) as a close-dashed curve for reference. In (a)–(c) lines of sight are shown as dashed straight lines; the box in the lower-left corner indicates the grid size used in the reconstructions. (a) Phantom and the 118 lines of sight. (b) Tomogram of a reconstruction with a reduced number of lines of sight, assuming unmodified line integrals (method B). (c) Tomogram of a reconstruction with a reduced number of lines of sight after three iterations of beam-width corrections (method D3). (d) Lines of sight with method D of reconstructions with a limited number of lines of sight during two iterations (solid lines, original; dashed lines, first iteration; dotted lines, second iteration).

Fig. 3
Fig. 3

(a) Sinogram values along the aperture curve of the vertical fan for reconstructions with a reduced number of lines of sight. Sinograms of phantom (solid curve), reconstructions with the proper geometric matrix (method A; short-dashed curve); line integrals (method B; dotted curve); the first and third iterations of the iterative method (D1 and D3; dashed–three-dotted and long-dashed curves, respectively); and the simplified method (C; dashed–dotted curve). Symbols: filled squares, pseudomeasurements divided by the unweighted étendue; open squares, corrected values from the simplified method (B); and pluses and crosses, respectively, corrected values in the first and third iterations of the iterative method (D1 and D3). Note the shift in ξ for the points of D1 and D3. (b) For reference, the angular étendue of the relevant channels along the aperture curve.

Tables (1)

Tables Icon

Table 1 Reconstruction Errors σg for Phantom Simulations Using Various Beam-Width Correction Methodsa

Equations (36)

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

f(p, ξ)=[Rg(x, y)]=g(x, y)δ[p+(x-x0)×sin ξ-(y-y0)cos ξ]dxdy,
fˆi=Ki(x, y)g(x, y)dxdy
g(x, y)jgjbj(x, y).
fˆi=jKijgj.
Kij=Ki(x, y)bj(x, y)dxdy.
fˆi=Diki(p, ξ)f(p, ξ)dpdξ=Dig(x, y)ki(p, ξ)×δ(p+x sin ξ-y cos ξ)dxdydpdξ=g(x, y)Diki(p, ξ)×δ(p+x sin ξ-y cos ξ)dpdξdxdy.
Ki(x, y)=Diki(p, ξ)δ(p+x sin ξ-y cos ξ)dpdξ=0π-ki(p, ξ)i(p, ξ)δ(p+x sin ξ-y cos ξ)dpdξ,
i(p, ξ)=0(p, ξ)Di1(p, ξ)Di.
Ki(x, y)=k0πi(-x sin ξ+y cos ξ, ξ)dξ=kΔξ,
ki(p, ξ)=-12π2 Ki(x, y)(p+x sin ξ-y cos ξ)2dxdy,
fˆi=fˆ(p¯i, ξ¯i)=k(p-p¯i, ξ-ξ¯i)×f(p, ξ)dpdξ=K˜(p¯i, ξ¯i|x, y)g(x, y)dxdy,
L(s)=x(s)y(s)=x1+sxdy1+syd,
LKi(x(s), y(s))dL(s)dsds
=-0π-ki(p, ξ)
×δ[p+x(s)sin ξ-y(s)cos ξ]xd2+yd2d pdξds
=0π-xd2+yd2ki(p, ξ)
×-δ(p+x1 sin ξ-y1 cos ξ
+s[xd sin ξ-yd cos ξ])dsd pdξ
=0π- xd2+yd2|xd sin ξ-yd cos ξ|ki(p, ξ)d pdξ=Ei(xd, yd),
Ki(x, y, z)=0π-π/2π/2ηi[p(x, y, ξ), ξ, zL(x, y, z, ξ, θ), θ]cos θdθdξ,
Ki(x, y)=-Ki(x, y, z)dz=0π--π/2π/2ηi[p(x, y, ξ), ξ, zL(x, y, z, ξ, θ), θ]cos θdθdzdξ,
ki(p, ξ)=ηi(p, ξ, z, θ)cos θ dθ dz.
fˆi=Ki(x, y)g(x, y)dxdy=g(x)Ki(x, y)dydx=Ei(ξ=0)g(x)dx=Ei(0)fi,
Ei= dΩdAΩeff Aeff,
Ei(ξ¯i)= ki(p, ξ)cos(ξ-ξ¯i)dpdξ.
p¯i=0π-pf(p, ξ)ki(p, ξ)dpdξ0π-f(p, ξ)ki(p, ξ)dpdξ,
ξ¯i=0π-ξ f(p, ξ)ki(p, ξ)dpdξ0π-f(p, ξ)ki(p, ξ)dpdξ.
Ewifˆif(p¯i, ξ¯i).
Ewig(x, y)Ki(x, y)dxdyg(x, y)δ(p¯i+x sin ξ¯i-y cos ξ¯i)dxdy,
Ewif(p, ξ)ki(p, ξ)dpdξf(p¯i, ξ¯i),
f(p, ξ)ki(p, ξ)dpdξf(pi(ξ), ξ)ki(p, ξ)dpdξ.
ei(ξ)=ki(p, ξ)dp.
fˆi=ki(p, ξ)f(p, ξ)dpdξ ei(t)f(t)dt=ai t2ei(t)dt+bi tei(t)dt+ci ei(t)dt.
fˆi=qi,j-1 fi-1+qi,i fi+qi,i+1 fi+1,
fˆ=Qf,
σg=g-g0g0.

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