Mathematics in Biomedical ImagingView this Special Issue
Research Article | Open Access
T. T. Truong, M. K. Nguyen, H. Zaidi, "The Mathematical Foundations of 3D Compton Scatter Emission Imaging", International Journal of Biomedical Imaging, vol. 2007, Article ID 092780, 11 pages, 2007. https://doi.org/10.1155/2007/92780
The Mathematical Foundations of 3D Compton Scatter Emission Imaging
The mathematical principles of tomographic imaging using detected (unscattered) X- or gamma-rays are based on the two-dimensional Radon transform and many of its variants. In this paper, we show that two new generalizations, called conical Radon transforms, are related to three-dimensional imaging processes based on detected Compton scattered radiation. The first class of conical Radon transform has been introduced recently to support imaging principles of collimated detector systems. The second class is new and is closely related to the Compton camera imaging principles and invertible under special conditions. As they are poised to play a major role in future designs of biomedical imaging systems, we present an account of their most important properties which may be relevant for active researchers in the field.
- F. Natterer, The Mathematics of Computerized Tomography, Classics in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 2001.
- A. M. Cormack, “My connection with the Radon transform,” in 75 Years of Radon Transform, S. Gindikin and P. Michor, Eds., vol. 4 of Conference Proceedings and Lecture Notes in Mathematical Physics, pp. 32–35, International Press, Boston, Mass, USA, 1994.
- A. M. Cormack, “Representation of a function by its line integrals, with some radiological applications,” Journal of Applied Physics, vol. 34, no. 9, pp. 2722–2727, 1963.
- A. M. Cormack, “Representation of a Function by Its Line Integrals, with Some Radiological Applications. II,” Journal of Applied Physics, vol. 35, no. 10, pp. 2908–2913, 1964.
- J. Radon, “Über die Bestimmung von Funktionnen durch ihre Integralwerte längs gewisser Mannigfaltikeiten,” Berichte über die Verhandlungen der Säschsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Naturwissenschaftliche Klasse, vol. 69, pp. 262–277, 1917.
- P. Grangeat, “Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform,” in Mathematical Methods in Tomography, G. T. Herman, A. K. Louis, and F. Natterer, Eds., vol. 1497 of Lecture Notes in Mathematics, pp. 66–97, Springer, New York, NY, USA, 1991.
- I. M. Gelfand, M. I. Graev, and N. Y. Vilenkin, Generalized Functions V, Academic Press, New York, NY, USA, 1965.
- F. Natterer, “Inversion of the attenuated Radon transform,” Inverse Problems, vol. 17, no. 1, pp. 113–119, 2001.
- R. G. Novikov, “Une formule d'inversion pour la transformation d'un rayonnement X atténué,” Comptes Rendus l'Académie des Sciences, vol. 332, pp. 1059–1063, 2001.
- R. H. Morgan, “An analysis of the physical factors controlling the diagnostic,” American Journal of Roentgenology, vol. 55, pp. 67–89, 1946.
- M. K. Nguyen, C. Faye, L. Eglin, and T. T. Truong, “Apparent image formation by Compton-scattered photons in gamma-ray imaging,” IEEE Signal Processing Letters, vol. 8, no. 9, pp. 248–251, 2001.
- M. K. Nguyen and T. T. Truong, “On an integral transform and its inverse in nuclear imaging,” Inverse Problems, vol. 18, no. 1, pp. 265–277, 2002.
- R. W. Todd, J. M. Nightingale, and D. B. Everett, “A proposed camera,” Nature, vol. 251, no. 5471, pp. 132–134, 1974.
- D. B. Everett, J. S. Fleming, R. W. Todd, and J. M. Nightingale, “Gamma-radiation imaging system based on the Compton effect,” Proceedings of the Institution of Electrical Engineers, vol. 124, no. 11, pp. 995–1000, 1977.
- M. Singh, “An electronically collimated gamma camera for single photon emission computed tomography—part 1: theoretical considerations and design criteria,” Medical Physics, vol. 10, no. 4, pp. 421–427, 1983.
- H. Zaidi, “Relevance of accurate Monte Carlo modeling in nuclear medical imaging,” Medical Physics, vol. 26, no. 4, pp. 574–608, 1999.
- A. M. Cormack, “A paraboloidal Radon transform,” in 75 Years of Radon Transform, S. Gindikin and P. Michor, Eds., vol. 4 of Conference Proceedings and Lecture Notes in Mathematical Physics, pp. 105–109, International Press, Boston, Mass, USA, 1994.
- A. M. Cormack and E. T. Quinto, “A Radon transform on spheres through the origin in and applications to the Darboux equation,” Transactions of the American Mathematical Society, vol. 260, no. 2, pp. 575–581, 1980.
- A. M. Cormack, “Radon's problem for some surfaces in ,” Proceedings of the American Mathematical Society, vol. 99, no. 2, pp. 305–312, 1987.
- K. Denecker, J. Van Overloop, and F. Sommen, “The general quadratic Radon transform,” Inverse Problems, vol. 14, no. 3, pp. 615–633, 1998.
- P. C. Johns, R. J. Leclair, and M. P. Wismayer, “Medical X-ray imaging with scattered photons,” in Opto-Canada: SPIE Regional Meeting on Optoelectronics, Photonics, and Imaging SPIE TD01, pp. 355–357, Ottawa, Canada, May 2002.
- H. Zaidi and B. H. Hasegawa, “Determination of the attenuation map in emission tomography,” Journal of Nuclear Medicine, vol. 44, no. 2, pp. 291–315, 2003.
- H. Zaidi and K. F. Koral, “Scatter modelling and compensation in emission tomography,” European Journal of Nuclear Medicine and Molecular Imaging, vol. 31, no. 5, pp. 761–782, 2004.
- C. E. Floyd, R. J. Jaszczak, C. C. Harris, and R. E. Coleman, “Energy and spatial distribution of multiple order Compton scatter in SPECT: a Monte Carlo investigation,” Physics in Medicine and Biology, vol. 29, no. 10, pp. 1217–1230, 1984.
- A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions II, R. E. Krieger, Malabar, Fla, USA, 1981.
- M. J. Yaffe and P. C. Johns, “Scattered radiation in diagnostic radiology: magnitudes, effects, and methods of reduction,” Journal of Applied Photographic Engineering, vol. 9, no. 6, pp. 184–195, 1983.
- A. H. Compton, “A quantum theory of the scattering of X-rays by light elements,” Physical Review, vol. 21, no. 5, pp. 483–502, 1923.
- H. H. Barrett, “The Radon transform and its applications,” in Progress in Optics, E. Wolf, Ed., vol. 21, pp. 219–286, North Holland, Amsterdam, The Netherlands, 1984.
- O. Klein and T. Nishina, “Über die Streuung von Strahlung durch freie Elektronen nach der neuen relativistischen Quantendynamik von Dirac,” Zeitschrift für Physik, vol. 52, no. 11-12, pp. 853–868, 1929.
- M. K. Nguyen, T. T. Truong, H. D. Bui, and J. L. Delarbre, “A novel inverse problem in -rays emission imaging,” Inverse Problems in Science and Engineering, vol. 12, no. 2, pp. 225–246, 2004.
- M. K. Nguyen, T. T. Truong, and P. Grangeat, “Radon transforms on a class of cones with fixed axis direction,” Journal of Physics A: Mathematical and General, vol. 38, no. 37, pp. 8003–8015, 2005.
- M. Singh and D. Doria, “An electronically collimated gamma camera for single photon emission computed tomography—part 2: image reconstruction and preliminary experimental measurements,” Medical Physics, vol. 10, no. 4, pp. 428–435, 1983.
- M. J. Cree and P. J. Bones, “Towards direct reconstruction from a gamma camera based on Compton scattering,” IEEE Transactions on Medical Imaging, vol. 13, no. 2, pp. 398–407, 1994.
- R. Basko, G. L. Zeng, and G. T. Gullberg, “Application of spherical harmonics to image reconstruction for the Compton camera,” Physics in Medicine and Biology, vol. 43, no. 4, pp. 887–894, 1998.
- L. C. Parra, “Reconstruction of cone-beam projections from Compton scattered data,” IEEE Transactions on Nuclear Science, vol. 47, no. 4, part 2, pp. 1543–1550, 2000.
- J. Pauli, E.-M. Pauli, and G. Anton, “ITEM—QM solutions for EM problems in image reconstruction exemplary for the Compton camera,” Nuclear Instruments and Methods in Physics Research A: Accelerators, Spectrometers, Detectors and Associated Equipment, vol. 488, no. 1-2, pp. 323–331, 2002.
- M Hirasawa, T. Tomitani, and S. Shibata, “New analytical method for three dimensional image reconstruction in multitracer gamma-ray emission imaging; Compton camera for multitracer,” RIKEN Review, no. 35, pp. 118–119, 2001.
- T. Tomitani and M. Hirasawa, “Image reconstruction from limited angle Compton camera data,” Physics in Medicine and Biology, vol. 47, no. 12, pp. 2129–2145, 2002.
- M. Hirasawa and T. Tomitani, “An analytical image reconstruction algorithm to compensate for scattering angle broadening in Compton cameras,” Physics in Medicine and Biology, vol. 48, no. 8, pp. 1009–1026, 2003.
- J. D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York, NY, USA, 3rd edition, 1999.
- S. R. Deans, “A unified Radon inversion formula,” Journal of Mathematical Physics, vol. 19, no. 11, pp. 2346–2349, 1978.
- S. Helgason, The Radon Transform, Birkhäuser, Boston, Mass, USA, 1999.
- G. Beyklin, “The inversion problem and applications of the generalized Radon transform,” Communications on Pure and Applied Mathematics, vol. 37, no. 5, pp. 579–599, 1984.
- G. Beylkin, “Inversion and applications of the generalized Radon transform,” in 75 Years of Radon Transform, S. Gindikin and P. Michor, Eds., vol. 4 of Conference Proceedings and Lecture Notes in Mathematical Physics, pp. 71–79, International Press, Boston, Mass, USA, 1994.
Copyright © 2007 T. T. Truong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.