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An inversion formula for the horizontal conical Radon transform

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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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Data Availability Statement

We do not generate any data for or from this research.

Notes

  1. In this paper, a cone means a surface, not a solid object.

References

  1. Allmaras, M., Darrow, D.P., Hristova, Y., Kanschat, G., Kuchment, P.: Detecting small low emission radiating sources (2010). arXiv:1012.3373

  2. 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)

    Article  MathSciNet  Google Scholar 

  3. 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)

    Article  Google Scholar 

  4. 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)

    Article  Google Scholar 

  5. 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)

    Article  Google Scholar 

  6. 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)

  7. Florescu, L., Markel, V.A., Schotland, J.C.: Inversion formulas for the broken-ray radon transform. Inverse Probl. 27(2), 025002 (2011)

    Article  MathSciNet  Google Scholar 

  8. Florescu, L., Schotland, J.C., Markel, V.A.: Single-scattering optical tomography. Phys. Rev. E 79(3), 036607 (2009)

    Article  Google Scholar 

  9. 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)

    MATH  Google Scholar 

  10. Gouia-Zarrad, R.: Analytical reconstruction formula for n-dimensional conical radon transform. Comput. Math. Appl. 68(9), 1016–1023 (2014)

    Article  MathSciNet  Google Scholar 

  11. Gouia-Zarrad, R., Ambartsoumian, G.: Exact inversion of the conical radon transform with a fixed opening angle. Inverse Probl. 30(4), 045007 (2014)

    Article  MathSciNet  Google Scholar 

  12. Haltmeier, M.: Exact reconstruction formulas for a radon transform over cones. Inverse Probl. 30(3), 035001 (2014)

    Article  MathSciNet  Google Scholar 

  13. Helgason, S., Helgason, S.: The Radon Transform, vol. 2. Springer, Berlin (1999)

    Book  Google Scholar 

  14. Hielscher, R., Quellmalz, M.: Reconstructing a function on the sphere from its means along vertical slices. Inverse Probl. Imaging 10(3), 711–739 (2016)

    Article  MathSciNet  Google Scholar 

  15. Hristova, Y.: Inversion of a v-line transform arising in emission tomography. J. Coupled Syst. Multiscale Dyn. 3(3), 272–277 (2015)

    Article  Google Scholar 

  16. Jung, C.-Y., Moon, S.: Inversion formulas for cone transforms arising in application of Compton cameras. Inverse Probl. 31(1), 015006 (2015)

    Article  MathSciNet  Google Scholar 

  17. Kuchment, P., Terzioglu, F.: Inversion of weighted divergent beam and cone transforms. Inverse Probl. Imaging 11(6), 1071 (2017)

    Article  MathSciNet  Google Scholar 

  18. Kuchment, P., Terzioglu, F.: Three-dimensional image reconstruction from Compton camera data. SIAM J. Imaging Sci. 9(4), 1708–1725 (2016)

    Article  MathSciNet  Google Scholar 

  19. 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)

    Article  MathSciNet  Google Scholar 

  20. 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)

    Google Scholar 

  21. 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)

    Article  MathSciNet  Google Scholar 

  22. Rubin, B.: The vertical slice transform in spherical tomography (2018). arXiv:1807.07689

  23. Rubin, B.: The vertical slice transform on the unit sphere. Fract. Calc. Appl. Anal. 22(4), 899–917 (2019)

    Article  MathSciNet  Google Scholar 

  24. 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)

    Article  MathSciNet  Google Scholar 

  25. 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)

    Article  Google Scholar 

  26. Terzioglu, F.: Some inversion formulas for the cone transform. Inverse Probl. 31(11), 115010 (2015)

    Article  MathSciNet  Google Scholar 

  27. Terzioglu, F., Kuchment, P., Kunyansky, L.: Compton camera imaging and the cone transform: a brief overview. Inverse Probl. 34(5), 054002 (2018)

    Article  MathSciNet  Google Scholar 

  28. 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)

    Article  Google Scholar 

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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

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