Skip to main content
SpringerLink
Log in

Inversion formula for the conical Radon transform arising in a single first semicircle second Compton camera with rotation

  • Special Feature: Original Paper
  • The Seventh China-Japan-Korea Joint Conference on Numerical Mathematics (CJK2018), 20-24 August 2018, Kanazawa, Japan
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

Since a Compton camera and its data type were introduced, several types of conical Radon transforms have been studied. To increase the accuracy, new Compton camera designs have been proposed (Smith, Technologies 3(4):219–237, 2015). Among these, we consider the Single First Semicircle Second (SFSS) camera design consisting of a first (scattering) detector element and a second (absorption) detector shaped as a semicircle. Here we introduce a new type of a conical Radon transform which this SFSS Compton camera design brings about. To obtain sufficient data, we rotate the SFSS camera around the object of interest. Also, we provide an inversion formula for this conical Radon transform and generalize this result to an n-dimensional conical Radon transform. To demonstrate our suggested algorithm, numerical simulations for the 3-dimensional case are presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. The radius \(\rho \) of the semicircle of the absorption detector could be any positive number.

  2. The domain of our conical Radon transform is different from the conical Radon transforms studied in [13, 29].

  3. This theorem is also presented in [11].

References

  1. Allmaras, M., Darrow, D.P., Hristova, Y., Kanschat, G., Kuchment, P.: Detecting small low emission radiating sources. Inverse Probl. Imaging 7(1), 47–79 (2013)

    Article  MathSciNet  Google Scholar 

  2. Basko, R., Zeng, G.L., Gullberg, G.T.: Application of spherical harmonics to image reconstruction for the Compton camera. Phys. Med. Biol. 43(4), 887–894 (1998)

    Article  Google Scholar 

  3. Butzer, P.L., Nessel, R.J.: Fourier Analysis and Approximation: One-dimensional Theory. Fourier Analysis and Approximation. Academic Press, New York (1971)

    Book  Google Scholar 

  4. Cree, M.J., Bones, P.J.: Towards direct reconstruction from a gamma camera based on Compton scattering. IEEE Trans. Med. Imaging 13(2), 398–409 (1994)

    Article  Google Scholar 

  5. Everett, D.B., Fleming, J.S., Todd, R.W., Nightingale, J.M.: Gamma-radiation imaging system based on the Compton effect. Electr. Eng. Proc. Inst. 124(11), 995 (1977)

    Article  Google Scholar 

  6. Giang, D.V.: Finite Hilbert transforms. J. Approx. 200(12), 221–226 (2015)

    Article  MathSciNet  Google Scholar 

  7. 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 

  8. Haltmeier, M., Moon, S., Schiefeneder, D.: Inversion of the attenuated v-line transform with vertices on the circle. IEEE Trans. Comput. Imaging 3(4), 853–863 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. 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. 9(2), 520–536 (2016)

    Article  MathSciNet  Google Scholar 

  11. Kwon, K.: An inversion of the conical Radon transform arising in the compton camera with helical movement. Biomed. Eng. Lett. 9, 233–243 (2019)

    Article  Google Scholar 

  12. 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 

  13. Kuchment, P., Terzioglu, F.: Inversion of weighted divergent beam and cone transforms. Inverse Probl. Imaging 11(6), 1071–1090 (2017)

    Article  MathSciNet  Google Scholar 

  14. Marr, R.B., Chen, C.-N., Lauterbur, P.C.: On two approaches to 3D reconstrunction in NMR zeugmatography. Math. Asp. Comput. Tomogr. Lect. Notes Med. Inf. 8, 225–240 (1981)

    MATH  Google Scholar 

  15. Maxim, V., Frandeş, M., Prost, R.: Analytical inversion of the Compton transform using the full set of available projections. Inverse Probl. 25(9), 095001 (2009)

    Article  MathSciNet  Google Scholar 

  16. 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. 33(6), 065002 (2017)

    Article  MathSciNet  Google Scholar 

  17. 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 

  18. Natterer, F.: The Mathematics of Computerized Tomography. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (2001)

    Book  Google Scholar 

  19. Natterer, F., Wübbeling, F.: Mathematical Methods in Image Reconstruction. SIAM Monographs on Mathematical Modeling and Computation. SIAM, Society of Industrial and Applied Mathematics, Philadelphia (2001)

  20. Nguyen, M.K., Truong, T.T., Bui, H.D., Delarbre, J.L.: A novel inverse problem in \(\gamma \)-rays emission imaging. Inverse Probl. Sci. Eng. 12(2), 225–246 (2004)

    Article  MathSciNet  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), 8003–8015 (2005)

    Article  MathSciNet  Google Scholar 

  22. Okada, S., Elliott, D.: The finite Hilbert transform in \(L^2\). Math. Nachr. 153(1), 43–56 (1991)

    Article  MathSciNet  Google Scholar 

  23. Rosenblum, M., Rovnyak, J.: Two theorems on finite Hilbert transforms. J. Math. Anal. Appl. 48, 708–720 (1974)

    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(37), 421–427 (1983)

    Article  Google Scholar 

  26. Smith, B.: Reconstruction methods and completeness conditions for two Compton data models. J. Opt. Soc. Am. A 22(3), 445–459 (2005)

    Article  Google Scholar 

  27. Smith, B.: A new Compton camera imaging model to mitigate the finite spatial resolution of detectors and new camera designs for implementation. Technologies 3(4), 219–237 (2015)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  29. 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 

  30. Tricomi, F.G.: Integral Equations. Pure and Applied Mathematics, vol. 5. Dover Publications, Mineola (1957)

    Google Scholar 

  31. Truong, T.T., Nguyen, M.K., Zaidi, H.: The mathematical foundation of 3D Compton scatter emission imaging. Int. J. Biomed. Imaging 2007, 92780 (2007). https://doi.org/10.1155/2007/92780

    Article  Google Scholar 

  32. Xun, X., Mallick, B., Carroll, R.J., Kuchment, P.: A bayesian approach to the detection of small low emission sources. Inverse Probl. 27(11), 115009 (2011)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The work of S. Moon was supported by the National Research Foundation of Korea grant funded by the Korean government (MSIP) (NRF-2018R1D1A3B07041149). The work of K. Kwon was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (NRF-2017R1A2B4004943).

Author information

Authors and Affiliations

  1. Department of Mathematics, Kyungpook National University, Daegu, 41566, South Korea

    Sunghwan Moon

  2. Department of Mathematics, Dongguk University, Seoul, 04620, South Korea

    Kiwoon Kwon

Authors
  1. Sunghwan Moon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kiwoon Kwon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kiwoon Kwon.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Moon, S., Kwon, K. Inversion formula for the conical Radon transform arising in a single first semicircle second Compton camera with rotation. Japan J. Indust. Appl. Math. 36, 989–1004 (2019). https://doi.org/10.1007/s13160-019-00379-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-019-00379-x

Keywords

Mathematics Subject Classification

Search

Navigation