Three-dimensional reconstruction from projections

Jiří Jelínek; Karel Segeth; T. R. Overton

Aplikace matematiky (1985)

  • Volume: 30, Issue: 2, page 92-109
  • ISSN: 0862-7940

Abstract

top
Computerized tomograhphy is a technique for computation and visualization of density (i.e. X- or -ray absorption coefficients) distribution over a cross-sectional anatomic plane from a set of projections. Three-dimensional reconstruction may be obtained by using a system of parallel planes. For the reconstruction of the transverse section it is necessary to choose an appropriate method taking into account the geometry of the data collection, the noise in projection data, the amount of data, the computer power available, the accuracy required etc. In the paper the theory related to the convolution reconstruction methods is reviewed. The principal contribution consists in the exact mathematical treatment of Radon’s inverse transform based on the concepts of the regularization of a function and the generalized function. This approach naturally leads to the employment of the generalized Fourier transform. Reconstructions using simulated projection data are presented for both the parallel and divergent-ray collection geometries.

How to cite

top

Jelínek, Jiří, Segeth, Karel, and Overton, T. R.. "Three-dimensional reconstruction from projections." Aplikace matematiky 30.2 (1985): 92-109. <http://eudml.org/doc/15388>.

@article{Jelínek1985,
abstract = {Computerized tomograhphy is a technique for computation and visualization of density (i.e. X- or $\gamma $-ray absorption coefficients) distribution over a cross-sectional anatomic plane from a set of projections. Three-dimensional reconstruction may be obtained by using a system of parallel planes. For the reconstruction of the transverse section it is necessary to choose an appropriate method taking into account the geometry of the data collection, the noise in projection data, the amount of data, the computer power available, the accuracy required etc. In the paper the theory related to the convolution reconstruction methods is reviewed. The principal contribution consists in the exact mathematical treatment of Radon’s inverse transform based on the concepts of the regularization of a function and the generalized function. This approach naturally leads to the employment of the generalized Fourier transform. Reconstructions using simulated projection data are presented for both the parallel and divergent-ray collection geometries.},
author = {Jelínek, Jiří, Segeth, Karel, Overton, T. R.},
journal = {Aplikace matematiky},
keywords = {research survey; parallel beam; divergent beam; ill-posed problem; convolution reconstruction methods; computer tomography; Radon’s inverse transform; regularization; generalized Fourier transform; spatial filter; window functions; errors; research survey; parallel beam; divergent beam; ill-posed problem; convolution reconstruction methods; computer tomography; Radon's inverse transform; regularization; generalized Fourier transform; spatial filter; window functions; errors},
language = {eng},
number = {2},
pages = {92-109},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Three-dimensional reconstruction from projections},
url = {http://eudml.org/doc/15388},
volume = {30},
year = {1985},
}

TY - JOUR
AU - Jelínek, Jiří
AU - Segeth, Karel
AU - Overton, T. R.
TI - Three-dimensional reconstruction from projections
JO - Aplikace matematiky
PY - 1985
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 30
IS - 2
SP - 92
EP - 109
AB - Computerized tomograhphy is a technique for computation and visualization of density (i.e. X- or $\gamma $-ray absorption coefficients) distribution over a cross-sectional anatomic plane from a set of projections. Three-dimensional reconstruction may be obtained by using a system of parallel planes. For the reconstruction of the transverse section it is necessary to choose an appropriate method taking into account the geometry of the data collection, the noise in projection data, the amount of data, the computer power available, the accuracy required etc. In the paper the theory related to the convolution reconstruction methods is reviewed. The principal contribution consists in the exact mathematical treatment of Radon’s inverse transform based on the concepts of the regularization of a function and the generalized function. This approach naturally leads to the employment of the generalized Fourier transform. Reconstructions using simulated projection data are presented for both the parallel and divergent-ray collection geometries.
LA - eng
KW - research survey; parallel beam; divergent beam; ill-posed problem; convolution reconstruction methods; computer tomography; Radon’s inverse transform; regularization; generalized Fourier transform; spatial filter; window functions; errors; research survey; parallel beam; divergent beam; ill-posed problem; convolution reconstruction methods; computer tomography; Radon's inverse transform; regularization; generalized Fourier transform; spatial filter; window functions; errors
UR - http://eudml.org/doc/15388
ER -

References

top
  1. R. N. Bracewell A. C. Riddle, 10.1086/149346, Astrophys. J. 150 (1967), 427-434. (1967) DOI10.1086/149346
  2. T. F. Budinger G. T. Gullberg, 10.1109/TNS.1974.6499234, IEEE Trans. Nucl. Sci. NS-21 (1974), 2-20. (1974) DOI10.1109/TNS.1974.6499234
  3. D. De Rosier A. Klug, 10.1038/217130a0, Nature 217 (1968), 130-138. (1968) DOI10.1038/217130a0
  4. I. M. Geľfand G. E. Šilov, Generalized functions and operations on them, (Russian.) Gos. izd. fiz.-mat. lit., Moskva 1959. (1959) MR0097715
  5. G. T. Gullberg, 10.1016/0146-664X(79)90033-9, Comput. Graphics and Image Process. 10 (1979), 30-47. (1979) DOI10.1016/0146-664X(79)90033-9
  6. S. Helgason, The Radon transform, Birkhäuser, Boston 1980. (Russian translation: Mir, Moskva 1983.) (1980) Zbl0453.43011MR0741182
  7. G. T. Herman A. V. Lakshminarayanan A. Naparstek, 10.1016/0010-4825(76)90065-2, Comput. Biol. Med. 6 (1976), 259 - 271. (1976) DOI10.1016/0010-4825(76)90065-2
  8. В. K. Р. Horn, Fan-beam reconstruction methods, A. I. Memo No. 448, Massachusetts Institute of Technology, 1977. (1977) 
  9. G. N. Hounsfield, 10.1259/0007-1285-46-552-1016, British J. Radiol. 46 (1973), 1016-1022. (1973) DOI10.1259/0007-1285-46-552-1016
  10. R. H. Huesman G. T. Gullberg W. L. Greenberg T. Budinger, Donner algorithms for reconstruction tomography, Tech. Rpt. Pub. 214, Lawrence Berkeley Laboratory, 1977. (1977) 
  11. J. E. Husband I. K. Fry, Computed tomography of the body, Mac Millan Publishers 1981. (1981) 
  12. V. Jarník, Integrální počet II, Nakladatelství ČSAV, Praha 1955. (1955) 
  13. J. Jelínek T. R. Overton, Comparative study on reconstruction from projections: Correction functions, Proceedings Colloquium on Mathematical Morphology, Stereology and Image Analysis, ČSAV, Prague 1982, 113-116. (1982) 
  14. P. M. Joseph R. D. Spital, 10.1088/0031-9155/26/3/010, Phys. Med. Biol. 3 (1981), 473-487. (1981) DOI10.1088/0031-9155/26/3/010
  15. P. M. Joseph R. D. Spital C. D. Stockham, 10.1016/0363-8235(80)90023-X, Comput. Tomogr. 4 (1980), 189-206. (1980) DOI10.1016/0363-8235(80)90023-X
  16. R. M. Lewitt, Ultra-fast convolution approximations for image reconstruction from parallel and fan beam projection data, Technical Report No. MIPG25, Medical Imaging Processing Group, Dept. of Computer Science, State University of New York at Buffalo, 1979. (1979) 
  17. R. M. Lewitt R. H. T. Bates, Image reconstruction from projections II: Modified back-projection methods, Optik 50 (1978), 85-109. (1978) 
  18. R. M. Mersereau, 10.1016/0010-4825(76)90064-0, Comput. Biol. Med. 6 (1976), 247-258. (1976) DOI10.1016/0010-4825(76)90064-0
  19. J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte langs gewisser Manningfaltigkeiten, Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math.-Nat. Kl., 69 (1917), 262-277. (1917) MR0692055
  20. G. N. Ramachandran A. V. Lakshminarayanan, 10.1073/pnas.68.9.2236, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 2236-2240. (1971) MR0287750DOI10.1073/pnas.68.9.2236
  21. I. S. Reed Y. S. Kwoh T. K. Truong E. L. Hall, 10.1109/TNS.1977.4328792, IEEE Trans. Nucl. Sci. NS-24 (1977), 843-849. (1977) DOI10.1109/TNS.1977.4328792
  22. P. Rüegsegger T. N. Hangartner H. U. Keller, et al., 10.1097/00004728-197804000-00012, J. Comput. Assist. Tomogr. 2 (1978), 184-188. (1978) DOI10.1097/00004728-197804000-00012
  23. L. Schwartz, Théorie des distributions 1 & 2, Hermann, Paris 1950, 1951. (1950) MR0209834
  24. L. A. Shepp R. F. Logan, 10.1109/TNS.1974.6499235, IEEE Trans. Nucl. Sci. NS-21 (1974), 21-43. (1974) DOI10.1109/TNS.1974.6499235
  25. E. Tanaka, 10.1088/0031-9155/24/1/013, Phys. Med. Biol. 24 (1979), 157-161. (1979) DOI10.1088/0031-9155/24/1/013
  26. K. Yosida, Functional analysis, Springer-Verlag, Berlin 1965. (1965) Zbl0126.11504

NotesEmbed ?

top

You must be logged in to post comments.