Abstract
Since the Compton camera was first introduced, various types of conical Radon transforms have been examined. Here, we derive the inversion formula for the conical Radon transform, where the cone of integration moves along a curve in three-dimensional space such as a helix. Along this three-dimensional curve, a detailed inversion formula for helical movement will be treated for Compton imaging in this paper. The inversion formula includes Hilbert transform and Radon transform. For the inversion of Compton imaging with helical movement, it is necessary to invert Hilbert transform with respect to the inner product between the vertex and the central axis of the cone of the Compton camera. However, the inner product function is not monotone. Thus, we should replace the Hilbert transform by the Riemann–Stieltjes integral over a certain monotone function related with the inner product function. We represent the Riemann–Stieltjes integral as a conventional Riemann integral over a countable union of disjoint intervals, whose end points can be computed using the Newton method. For the inversion of Radon transform, three dimensional filtered backprojection is used. For the numerical implementation, we analytically compute the Hilbert transform and Radon transform of the characteristic function of finite balls. Numerical test is given, when the density function is given by a characteristic function of a ball or three overlapping balls.
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Allmaras M, Darrow DP, Hristova Y, Kanschat G, Kuchment P. Detecting small low emission radiating sources. Inverse Probl Imaging. 2013;7(1):47–79.
Basko R, Zeng GL, Gullberg GT. Application of spherical harmonics to image reconstruction for the Compton camera. Phys Med Biol. 1998;43(4):887–94.
Cree MJ, Bones PJ. Towards direct reconstruction from a gamma camera based on Compton scattering. IEEE Trans Med Imaging. 1994;13(2):398–409.
Gouia-Zarrad R. Analytical reconstruction formula for $n$-dimensional conical Radon transform. Comput Math Appl. 2014;68(9):1016–23.
Gouia-Zarrad R, Ambartsoumian G. Exact inversion of the conical Radon transform with a fixed opening angle. Inverse Probl. 2014;30(4):045007.
Haltmeier M. Exact reconstruction formulas for a Radon transform over cones. Inverse Probl. 2014;30(3):035001.
Helgason S. The Radon transform. Progress in mathematics. Boston: Birkhäuser; 1999.
Jung C, Moon S. Inversion formulas for cone transforms arising in application of Compton cameras. Inverse Probl. 2015;31(1):015006.
Jung C, Moon S. Exact inversion of the cone transform arising in an application of a Compton camera consisting of line detectors. SIAM J Imaging Sci. 2016;9(2):520–36.
Katsevich A. Analysis of an exact inversion algorithm for spiral cone-beam CT. Phys Med Biol. 2002;47(15):2583.
Katsevich A. Theoretically exact filtered backprojection-type inversion algorithm for spiral CT. SIAM J Appl Math. 2002;62(6):2012–26.
Kuchment P, Terzioglu F. Inversion of weighted divergent beam and cone transforms. Am Inst Math Sci. 2017;11(6):1071–90.
Kuchment P, Terzioglu F. Three-dimensional image reconstruction from Compton camera data. SIAM J Imaging Sci. 2016;9(4):1708–25.
Marr RB, Chen CN, Lauterbur PC. On two approaches to 3D reconstrunction in NMR zeugmatography. In: Herman GT, Natterer F, editors. Mathematical aspects of computed tomography. Lecture notes in medical informatics, vol. 8. Berlin: Springer; 1981. p. 225–40.
Maxim V, Frandeş M, Prost R. Analytical inversion of the Compton transform using the full set of available projections. Inverse Probl. 2009;25(9):095001.
Moon S. On the determination of a function from its conical Radon transform with a fixed central axis. SIAM J Math Anal. 2016;48(3):1833–47.
Moon S. Inversion of the conical Radon transform with vertices on a surface of revolution arising in an application of a compton camera. Inverse Probl. 2017;33(6):065002.
Natterer F. The mathematics of computerized tomography. Classics in applied mathematics. Philadelphia: Society for Industrial and Applied Mathematics; 2001.
Natterer F, Wübbeling F. Mathematical methods in image reconstruction. SIAM monographs on mathematical modeling and computation. Philadelphia: Society for Industrial and Applied Mathematics; 2001.
Nguyen MK, Truong TT. The development of Radon transforms associated to compton scatter imaging concepts. Eurasian J Math Comput Appl. 2018;1(1):hal-01759845.
Nguyen MK, Truong TT, Grangeat P. Radon transforms on a class of cones with fixed axis direction. J Phys A Math Gen. 2005;38(37):8003–15.
Rigaud G, Hahn BN. 3D compton scattering imaging and contour reconstruction for a class of Radon transforms. Inverse Probl. 2018;34:075004.
Schiefeneder D, Haltmeier M. The Radon transform over cones with vertices on the sphere and orthogonal axes. SIAM J Appl Math. 2017;77(4):1335–51.
Singh M. An electronically collimated gamma camera for single photon emission computed tomography. Part I: theoretical considerations and design criteria. Med Phys. 1983;10(37):421–7.
Smith B. Reconstruction methods and completeness conditions for two Compton data models. J Opt Soc Am A. 2005;22(3):445–59.
Smith B. Computer simulations to demonstrate new inversion methods for Compton camera data. Opt Eng. 2012;51(05):053203.
Smith B. A new Compton camera imaging model to mitigate the finite spatial resolution of detectors and new camera designs for implementation. Technologies. 2015;3(4):219–37.
Terzioglu F. Some inversion formulas for the cone transform. Inverse Probl. 2015;31(11):115010.
Terzioglu F. Some analytic properties of the cone transform. Inverse Probl. 2019;35:034002.
Terzioglu F, Kuchment P, Kunyansky L. Compton camera imaging and the cone transform: a brief overview. Inverse Probl. 2018;34:054002.
Todd RW, Nightingale JM, Everett DB. A proposed gamma camera. Nature. 1974;251(6):132–4.
Truong TT, Nguyen MK, Zaidi H. The mathematical foundation of 3D Compton scatter emission imaging. Int J Biomed Imaging. 2007;2007:92780.
Zou Y, Pan X. An extended data function and its generalized backprojection for image reconstruction in helical cone-beam CT. Phys Med Biol. 2004;49(22):N383.
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The work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT (NRF-2017R1A2B4004943).
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Kwon K. declares that he has no conflict of interest in relation to the work in this article.
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Kwon, K. An inversion of the conical Radon transform arising in the Compton camera with helical movement. Biomed. Eng. Lett. 9, 233–243 (2019). https://doi.org/10.1007/s13534-019-00106-y
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DOI: https://doi.org/10.1007/s13534-019-00106-y