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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 16, Iss. 1 — Jan. 1, 1999
  • pp: 17–27

Projection-space methods to take into account finite beam-width effects in two-dimensional tomography algorithms

L. C. Ingesson, P. J. Böcker, R. Reichle, M. Romanelli, and P. Smeulders  »View Author Affiliations

JOSA A, Vol. 16, Issue 1, pp. 17-27 (1999)

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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

OCIS Codes
(100.3010) Image processing : Image reconstruction techniques
(100.3020) Image processing : Image reconstruction-restoration
(100.6950) Image processing : Tomographic image processing
(110.6960) Imaging systems : Tomography

Original Manuscript: March 6, 1998
Revised Manuscript: July 27, 1998
Manuscript Accepted: September 11, 1998
Published: January 1, 1999

L. C. Ingesson, P. J. Böcker, R. Reichle, M. Romanelli, and P. Smeulders, "Projection-space methods to take into account finite beam-width effects in two-dimensional tomography algorithms," J. Opt. Soc. Am. A 16, 17-27 (1999)

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  1. 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]
  2. 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]
  3. R. N. Bracewell, “Correction for collimator width (restoration) in reconstructive x-ray tomography,” J. Comput. Assisted Tomogr. 1, 6–15 (1977). [CrossRef]
  4. 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]
  5. 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]
  6. D. G. McCaughey, H. C. Andrews, “Degrees of freedom for projection imaging,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-25, 63–73 (1977). [CrossRef]
  7. 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]
  8. 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]
  9. 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]
  10. R. N. Bracewell, “Strip integration in radio astronomy,” Aust. J. Phys. 9, 198–217 (1956). [CrossRef]
  11. 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]
  12. R. H. T. Bates, M. J. McDonnel, Image Restoration and Reconstruction (Clarendon, Oxford, 1986), p. 133.
  13. 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).
  14. 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]
  15. 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]
  16. K. M. Hanson, G. W. Wecksung, “Local basis-function approach to computed tomography,” Appl. Opt. 24, 4028–4039 (1985). [CrossRef] [PubMed]
  17. 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]
  18. 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]
  19. 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]
  20. 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).
  21. 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.
  22. S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, New York, 1983).
  23. J. L. Prince, A. S. Willsky, “A geometric projection-space reconstruction algorithm,” Linear Algebr. Appl. 130, 151–191 (1990). [CrossRef]
  24. 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).
  25. R. M. Lewitt, “Reconstruction algorithms: transform methods,” Proc. IEEE 71, 390–408 (1983). [CrossRef]
  26. V. V. Pikalov, N. G. Preobrazhenskii, “Computer-aided tomography and physical experiment,” Sov. Phys. Usp. 26, 974–990 (1984). [CrossRef]
  27. 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]
  28. 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]
  29. B. M. W. Tsui, G. T. Gullberg, “The geometric transfer function for fan beam collimators,” Phys. Med. Biol. 35, 81–93 (1990). [CrossRef] [PubMed]
  30. R. S. Longhurst, Geometrical and Physical Optics, 3rd ed. (Longman, Harlow, UK, 1973), pp. 20, 450, 466.
  31. L. C. Ingesson, “Visible-light tomography of tokamak plasmas,” thesis technische (Universiteit Eindhoven, Eindhoven, The Netherlands, 1995).
  32. R. S. Granetz, P. Smeulders, “X-ray tomography at JET,” Nucl. Fusion 28, 457–476 (1988). [CrossRef]
  33. J. F. Camacho, “Soft x-ray tomography on the Alcator C tokamak,” (Massachusetts Institute of Technology, Cambridge, Mass., 1985).
  34. C. P. Tanzi, “Emission of soft x-ray and microwave radiation from tokamak plasmas,” Thesis (Universiteit Utrecht, Utrecht, The Netherlands, 1996).
  35. M. R. Teague, “Image analysis via the general theory of moments,” J. Opt. Soc. Am. 20, 920–930 (1980). [CrossRef]
  36. 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]
  37. P. Smeulders, “Second order Abel inversion with allowance for the spatial resolution,” (Max-Planck-Institut für Plasmaphysik, Garching, Germany, 1978).
  38. P. Smeulders, “A fast plasma tomography routine with second-order accuracy and compensation for spatial resolution,” (Max-Planck-Institut für Plasmaphysik, Garching, Germany, 1983).
  39. 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.

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