OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 22, Iss. 3 — Mar. 1, 2005
  • pp: 445–459

Reconstruction methods and completeness conditions for two Compton data models

Bruce Smith  »View Author Affiliations

JOSA A, Vol. 22, Issue 3, pp. 445-459 (2005)

View Full Text Article

Enhanced HTML    Acrobat PDF (372 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



Two different models are proposed for the data produced in a Compton imaging device. A sequence of equations, which relate the model to the distribution of radioactivity that is being imaged, is developed for each of the two models. No series expansions are used in these developments. On the basis of these sequences of equations, a completeness condition is developed for each of the two models. These completeness conditions may prove useful in the future in determining appropriate shapes, configurations, and motions of the device’s detectors. A computer simulation is performed to verify one of these sequences of equations. A computer simulation is also performed to demonstrate that this sequence of equations can produce more accurate images than a backprojection reconstruction method. In addition, a procedure is proposed that could mitigate the effects of the Klein–Nishina distribution, the Doppler broadening, and the variability in the data due to the random generation of photons.

© 2005 Optical Society of America

OCIS Codes
(100.3010) Image processing : Image reconstruction techniques
(100.6890) Image processing : Three-dimensional image processing
(100.6950) Image processing : Tomographic image processing

Original Manuscript: April 26, 2004
Revised Manuscript: September 17, 2004
Manuscript Accepted: September 22, 2004
Published: March 1, 2005

Bruce Smith, "Reconstruction methods and completeness conditions for two Compton data models," J. Opt. Soc. Am. A 22, 445-459 (2005)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. J. W. Leblanc, “A Compton camera for low energy gamma ray imaging in nuclear medicine applications,” Ph.D. thesis (University of Michigan, Ann Arbor, Mich., 1999).
  2. N. H. Clinthorne, C. Ng, C. Hua, J. E. Gormley, “Theoretical performance comparison of a Compton-scatter aperture and parallel-hole collimator,” in Nuclear Science Symposium, 1996. Conference Record (IEEE Press, Piscataway, N.J., 1996), Vol. 2, pp. 788–792.
  3. J. E. Gormley, W. L. Rogers, N. H. Clinthorne, D. K. Wehe, “Experimental Comparison of Mechanical and Electronic Collimation,” in Nuclear Science Symposium, 1996. Conference Record (IEEE Press, Piscataway, N.J., 1996), Vol. 2, pp. 798–802.
  4. G. E. Fakhri, S. C. Moore, P. Maksud, A. Aurengo, M. F. Kijewski, “Absolute activity quantitation in simultaneous  123I/99mTc Brain SPECT,” J. Nucl. Med. 42, 300–308 (2001). [PubMed]
  5. G. E. Fakhri, P. Maksud, M. F. Kijewski, R. E. Zimmerman, S. C. Moore, “Quantitative simultaneous  99mTc/123I SPECT: design study and validation with Monte Carlo simulations and physical acquisitions,” IEEE Trans. Nucl. Sci. 49, 2315–2321 (2002). [CrossRef]
  6. Y. F. Yang, Y. Gono, S. Motomura, S. Enomoto, Y. Yano, “A Compton camera for multi-tracer imaging,” IEEE Trans. Nucl. Sci. 48, 656–661 (2001). [CrossRef]
  7. R. W. Todd, J. M. Nightingale, D. B. Everett, “A proposed Gamma camera,” Nature 251, 132–134 (1974). [CrossRef]
  8. J. W. LeBlanc, N. H. Clinthorne, C.-H. Hua, E. Nygard, “Experimental results from the C-SPRINT prototype Compton camera,” IEEE Trans. Nucl. Sci. 46, 201–204 (1999). [CrossRef]
  9. R. C. Rohe, J. D. Valentine, “A novel Compton scatter camera design for in vivo medical imaging of radiopharmaceuticals. Part II,” IEEE Trans. Nucl. Sci. 43, 3256–3263 (1996). [CrossRef]
  10. A. H. Compton, “A quantum theory of the scattering of x-rays by light elements,” Phys. Rev. 21, 483–502 (1923). [CrossRef]
  11. C. E. Ordonez, A. Bolozdynya, W. Chang, “Doppler broadening of energy spectra in Compton cameras,” in Nuclear Science Symposium, 1997. Conference Record (IEEE Press, Piscataway, N.J., 1997), Vol. 2, pp. 1361–1365.
  12. S. E. King, G. W. Phillips, P. S. Haskins, J. E. McKission, “A solid-state Compton camera for three-dimensional imaging,” Nucl. Instrum. Methods Phys. Res. A 353, 320–323 (1994). [CrossRef]
  13. L. Junqiang, J. D. Valentine, J. N. Aarsvold, M. Khamzin, “A rebinning technique for 3D reconstruction of Compton camera data,” in Nuclear Science Symposium, 2001. Conference Record (IEEE Press, Piscataway, N.J., 2001), Vol. 4, pp. 1877–1881.
  14. P. Antich, R. Parkey, N. Slavin, E. Tsyganov, A. Zinchenko, “Compact Compton camera design: parameters and imaging algorithms,” in Nuclear Science Symposium, 2000. Conference Record (IEEE Press, Piscataway, N.J., 2000), Vol. 3, pp. 15–20.
  15. L. C. Parra, “Reconstruction of cone-beam projections from Compton scattered data,” IEEE Trans. Nucl. Sci. 47, 1543–1550 (2000). [CrossRef]
  16. R. Basko, G. L. Zeng, G. T. Gullberg, “Application of spherical harmonics to image reconstruction for the Compton camera,” Phys. Med. Biol. 43, 887–894 (1998). [CrossRef] [PubMed]
  17. T. Tomitani, M. Hirasawa, “Image reconstruction from limited angle Compton camera data,” Phys. Med. Biol. 47, 1009–1026 (2002). [CrossRef]
  18. M. Hirasawa, T. Tomitani, “An analytical image reconstruction algorithm to compensate for scattering angle broadening in Compton cameras,” Phys. Med. Biol. 48, 1009–1026 (2003). [CrossRef] [PubMed]
  19. A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications,” J. Appl. Phys. 34, 2722–2727 (1963). [CrossRef]
  20. A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications: II,” J. Appl. Phys. 35, 2908–2917 (1964). [CrossRef]
  21. E. Zeitler, “The reconstruction of objects from their projections,” Optik 39, 396–445 (1974).
  22. S. R. Deans, The Radon Transform and Some of Its Applications (Wiley-Interscience, New York, 1983).
  23. F. Natterer, The Mathematics of Computerized Tomography (Wiley, New York, 1986).
  24. T. Hebert, R. Leahy, M. Singh, “Three-dimensional maximum-likelihood reconstruction for a electronically collimated single-photon-emission imaging system,” J. Opt. Soc. Am. A 7, 1305–1313 (1990). [CrossRef] [PubMed]
  25. A. C. Sauve, A. O. Hero, W. L. Rogers, S. J. Wilderman, N. H. Clinthorne, “3D image reconstruction for Compton SPECT camera model,” IEEE Trans. Nucl. Sci. 46, 2075–2084 (1999). [CrossRef]
  26. R. A. Kroeger, W. N. Johnson, R. L. Kinzer, J. D. Kurfess, S. E. Inderhees, B. Phlips, N. Gehrels, “Spatial and Spectral Resolution of a Germanium Strip Detector,” in Imaging in High Energy Astronomy, L. Bassani, G. di Cocco, eds. (Kluwer, Dordrecht, The Netherlands, 1995), pp. 325–329.
  27. C. H. Hua, N. H. Clinthorne, S. J. Wilderman, J. W. LeBlanc, W. L. Rogers, “Quantitative evaluation of information loss for Compton cameras,” IEEE Trans. Nucl. Sci. 45, (1999).
  28. B. D. Smith, “Image reconstruction from cone-beam projections: necessary and sufficient conditions and reconstruction methods,” IEEE Trans. Med. Imaging MI-4, 14–28 (1985). [CrossRef]
  29. B. D. Smith, “Cone beam tomography: recent advances and a tutorial,” Opt. Eng. 29, 524–534 (1990). [CrossRef]
  30. M. J. Cree, P. J. Bones, “Towards direct reconstruction from a gamma camera based on Compton scattering,” IEEE Trans. Med. Imaging 13, 398–407 (1994). [CrossRef] [PubMed]
  31. L. A. Shepp, Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging MI-1, 113–122 (1982). [CrossRef]
  32. K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assist. Tomogr. 8, 306–316 (1984). [PubMed]
  33. B. L. Evans, J. B. Martin, M. C. Roggemann, “Deconvolution of shift-variant broadening for Compton scatter imaging,” Nucl. Instrum. Methods Phys. Res. A 422, 661–666 (1999). [CrossRef]
  34. W. Feller, An Introduction to Probability Theory and Its Applications, 3rd ed. (Wiley, New York, 1968), Vol. 1.
  35. J. A. Fessler, “Penalized weighted least-square image re-construction for positron emission tomography,” IEEE Trans. Med. Imaging 13, 290–300 (1994). [CrossRef]
  36. I. M. Gel’fand, G. E. Shilov, Generalized Functions (Academic, New York, 1964), Vol. 1.
  37. R. N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965).
  38. B. K. P. Horn, “Density reconstruction using arbitrary ray-sampling schemes,” Proc. IEEE 66, 551–562 (1978). [CrossRef]
  39. B. D. Smith, “Computer-aided tomography imaging from cone-beam data,” Ph.D. thesis (University of Rhode Island, Kingston, R.I., 1987).
  40. M. V. Ranganath, A. P. Dhawan, N. Mullani, “A multigrid expectation maximization reconstruction algorithm for positron tomography,” IEEE Trans. Med. Imaging 7, 273–278 (1988). [CrossRef]
  41. T. S. Pan, A. E. Yagle, “Numerical study of multigrid implementations of some iterative image reconstruction algorithms,” IEEE Trans. Med. Imaging 10, 572–588 (1991). [CrossRef] [PubMed]
  42. A. N. Tikhonov, V. Y. Arsenin, Solution of Ill-Prosed Problems (Winston, Washington, D.C., 1977).
  43. I. M. Gel’fand, M. I. Graev, N. Y. Valenkin, Generalized Functions: Integral Geometry and Representation Theory (Academic, New York, 1966), Vol. 2. Translated by Eugene Saletan.
  44. G. W. Phillips, “Applications of Compton imaging in nuclear waste characterization and treaty verification,” in Nuclear Science Symposium, 1996. Conference Record (IEEE Press, Piscataway, N.J., 1996), Vol. 1, pp. 362–364.
  45. S. Chelikani, J. Gore, G. Zuba, “Optimizing Compton camera geometries,” Phys. Med. Biol. 49, 1387–1408 (2004). [CrossRef] [PubMed]
  46. J. R. D. Earnhart, “A Compton camera for spectroscopic imaging from 100 keV to 1 MeV,” Ph.D. thesis (North Carolina State University, Raleigh, N.C., 1999).
  47. K. T. Smith, D. C. Solomon, S. L. Wagner, “Mathematical aspects of divergent beam radiography,” Bull. Am. Math. Soc. 83, 1227–1270 (1977). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited