Abstract
In this paper, we consider the conical Radon transform on all one-sided circular cones in \(\mathbf{R}^3\) with horizontal central axis whose vertices are on a vertical line. We derive an explicit inversion formula for such transform. The inversion makes use of the vertical slice transform on a sphere and V-line transform on a plane.
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We do not generate any data for or from this research.
Notes
In this paper, a cone means a surface, not a solid object.
References
Allmaras, M., Darrow, D.P., Hristova, Y., Kanschat, G., Kuchment, P.: Detecting small low emission radiating sources (2010). arXiv:1012.3373
Ambartsoumian, G.: Inversion of the v-line radon transform in a disc and its applications in imaging. Comput. Math. Appl. 64(3), 260–265 (2012)
Basko, R., Zeng, G.L., Gullberg, G.T.: Analytical reconstruction formula for one-dimensional Compton camera. IEEE Trans. Nucl. Sci. 44(3), 1342–1346 (1997)
Basko, R., Zeng, G.L., Gullberg, G.T.: Application of spherical harmonics to image reconstruction for the Compton camera. Phys. Med. Biol. 43(4), 887 (1998)
Cree, M.J., Bones, P.J.: Towards direct reconstruction from a gamma camera based on Compton scattering. IEEE Trans. Med. Imaging 13(2), 398–407 (1994)
Everett, D.B., Fleming, J.S., Todd, R.W., Nightingale, J.M.: Gamma-radiation imaging system based on the Compton effect. In: Proceedings of the Institution of Electrical Engineers, vol. 124, pp. 995–1000. IET (1977)
Florescu, L., Markel, V.A., Schotland, J.C.: Inversion formulas for the broken-ray radon transform. Inverse Probl. 27(2), 025002 (2011)
Florescu, L., Schotland, J.C., Markel, V.A.: Single-scattering optical tomography. Phys. Rev. E 79(3), 036607 (2009)
Gindikin, S., Reeds, S., Shepp, L.: Spherical tomography and spherical integral geometry. Tomography, impedance imaging, and integral geometry (South Hadley, MA, 1993). Lectures Appl. Math. 30, 83–92 (1994)
Gouia-Zarrad, R.: Analytical reconstruction formula for n-dimensional conical radon transform. Comput. Math. Appl. 68(9), 1016–1023 (2014)
Gouia-Zarrad, R., Ambartsoumian, G.: Exact inversion of the conical radon transform with a fixed opening angle. Inverse Probl. 30(4), 045007 (2014)
Haltmeier, M.: Exact reconstruction formulas for a radon transform over cones. Inverse Probl. 30(3), 035001 (2014)
Helgason, S., Helgason, S.: The Radon Transform, vol. 2. Springer, Berlin (1999)
Hielscher, R., Quellmalz, M.: Reconstructing a function on the sphere from its means along vertical slices. Inverse Probl. Imaging 10(3), 711–739 (2016)
Hristova, Y.: Inversion of a v-line transform arising in emission tomography. J. Coupled Syst. Multiscale Dyn. 3(3), 272–277 (2015)
Jung, C.-Y., Moon, S.: Inversion formulas for cone transforms arising in application of Compton cameras. Inverse Probl. 31(1), 015006 (2015)
Kuchment, P., Terzioglu, F.: Inversion of weighted divergent beam and cone transforms. Inverse Probl. Imaging 11(6), 1071 (2017)
Kuchment, P., Terzioglu, F.: Three-dimensional image reconstruction from Compton camera data. SIAM J. Imaging Sci. 9(4), 1708–1725 (2016)
Moon, S., Haltmeier, M.: Analytic inversion of a conical radon transform arising in application of Compton cameras on the cylinder. SIAM J. Imaging Sci. 10(2), 535–557 (2017)
Morvidone, M., Nguyen, M.K., Truong, T.T., Zaidi, H.: On the v-line radon transform and its imaging applications. J. Biomed. Imaging 2010, 11 (2010)
Nguyen, M.K., Truong, T.T., Grangeat, P.: Radon transforms on a class of cones with fixed axis direction. J. Phys. A Math. Gen. 38(37), 8003 (2005)
Rubin, B.: The vertical slice transform in spherical tomography (2018). arXiv:1807.07689
Rubin, B.: The vertical slice transform on the unit sphere. Fract. Calc. Appl. Anal. 22(4), 899–917 (2019)
Schiefeneder, D., Haltmeier, M.: The radon transform over cones with vertices on the sphere and orthogonal axes. SIAM J. Appl. Math. 77(4), 1335–1351 (2017)
Singh, M.: An electronically collimated gamma camera for single photon emission computed tomography. part i: theoretical considerations and design criteria. Med. Phys. 10(4), 421–427 (1983)
Terzioglu, F.: Some inversion formulas for the cone transform. Inverse Probl. 31(11), 115010 (2015)
Terzioglu, F., Kuchment, P., Kunyansky, L.: Compton camera imaging and the cone transform: a brief overview. Inverse Probl. 34(5), 054002 (2018)
Truong, T.T., Nguyen, M.K.: On new v-line radon transforms in \({\bf R}^2\) and their inversion. J. Phys. A Math. Theor. 44(7), 075206 (2011)
Acknowledgements
Linh Nguyen’s research is partially supported by the NSF grants DMS 1212125 and DMS 1616904. The authors are thankful to the anonymous referee for helpful comments and suggestions.
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Nguyen, D.N., Nguyen, L.V. An inversion formula for the horizontal conical Radon transform. Anal.Math.Phys. 11, 42 (2021). https://doi.org/10.1007/s13324-020-00468-y
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DOI: https://doi.org/10.1007/s13324-020-00468-y