The integral equations of Compton Scatter Tomography

M. K. Nguyen, T. T. Truong

Equipes Traitement de l’Information et Systèmes (ETIS) UMR CNRS 8051/ ENSEA / University of Cergy-Pontoise F-95014 Cergy-Pontoise, France.
Laboratoire de Physique Théorique et Modélisation (LPTM) UMR CNRS 8089 / University of Cergy-Pontoise F-95302 Cergy-Pontoise.

DOI: 10.4236/ojapps.2012.24B013                

Abstract

Two new Compton Scatter Tomography modalities, which are aimed at imaging hidden structures in bulk matter for industrial non-destructive control (or testing) and for medical diagnostics are shown to be based on the solutions of a special class of Chebyshev integral transforms. Besides their remarkable analytic properties, they can be inverted by existing methods which lend themselves nicely to numerical treatment and provide convergent, stable and fast computation algorithms. The existence of explicit inversion formulas implies that viable new imaging techniques can be developed, which may take over the current ones in a near future.

Keywords

Integral transforms; Radon transforms; Compton scatter imaging; Non-destructive testing; Medical imaging.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Ta. Li, “A new class of integral transforms,” Proc. Amer. Math. Soc., vol. 11, pp. 290–298, 1960.
[2]	R.G. Buschman, “Integrals of hypergeometric functions,” Math. Zeitschr. vol 89, pp. 74-76, 1965.
[3]	J. Radon, “über die Bestimmung von Funktionnen durch ihre Integralwerte l?ngst gewisser Mannigfaltigkeiten,” Ber.Verh.Sachs.Akad.Wiss. Leipzig-Math.-Natur.Kl. vol. 69, pp. 262-277, 1917.
[4]	M. K. Nguyen and T. T. Truong, “Inversion of a new circular arc Radon transform for Compton scattering tomography,” Inverse Problems vol. 26, pp. 065005, 2010.
[5]	T. T. Truong and M. K. Nguyen, “Radon transforms on generalized Cormack’s curves and a new Compton scatter tomography modality,” Inverse Problems, vol. 27, pp. 125001, 2011.
[6]	A. M. Cormack, “The Radon transform on a family of curves in the plane ,” Proc. Amer. Math. Soc., vol. 83, pp. 325-330, October 1981.
[7]	A. M. Cormack, “Radon’s problem – Old and new”, SIAM Proceedings, vol. 14, pp. 33-39, 1984.
[8]	S. J; Norton, “Compton scattering tomography,” J. Appl. Phys., vol. 70, pp. 2007-2015, 1994.
[9]	C. H. Chapman and P. W. Carey, “The circular harmonic Radon transform,” Inverse Problems, vol. 2, pp. 23-79, 1986.
[10]	M. K. Nguyen, C. Faye, G. Rigaud and T. T. Truong, “A novel technological imaging process using ionizing radiation properties,” Proceedings of the 9th IEEE - RVIF International Conference on Computing and Communications Technologies (RIVF 12), Ho-chi-Minh City, Vietnam, February 2012.