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Analysis and Mathematical Physics

, Volume 9, Issue 4, pp 1627–1664 | Cite as

Reconstruction of functions on the sphere from their integrals over hyperplane sections

  • Boris RubinEmail author
Article
  • 61 Downloads

Abstract

We obtain new inversion formulas for the Funk type transforms of two kinds associated to spherical sections by hyperplanes passing through a common point A which lies inside the n-dimensional unit sphere or on the sphere itself. Transforms of the first kind are defined by integration over complete subspheres and can be reduced to the classical Funk transform. Transforms of the second kind perform integration over truncated subspheres, like spherical caps or bowls, and can be reduced to the hyperplane Radon transform. The main tools are analytic families of \(\lambda \)-cosine transforms, Semyanisyi’s integrals, and modified stereorgraphic projection with the pole at A. Assumptions for functions are close to minimal.

Keywords

Radon transform Funk transform Cosine transform Semyanistyi integrals Inversion formulas 

Mathematics Subject Classification

Primary 44A12 Secondary 44A15 

Notes

Acknowledgements

The author is grateful to Prof. Mark Agranovsky for useful discussions and to the referee for valuable suggestions.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest.

References

  1. 1.
    Abouelaz, A., Daher, R.: Sur la transformation de Radon de la sphère \(S^d\). Bull. Soc. Math. Fr. 121, 353–382 (1993)Google Scholar
  2. 2.
    Agranovsky, M., Kuchment, P., Kunyansky, L.: On reconstruction formulas and algorithms for the thermoacoustic tomography. In: Wang, L.V. (ed.) Photoacoustic Imaging and Spectroscopy, pp. 89–102. CRC Press, Boca Raton, London, New York (2009)Google Scholar
  3. 3.
    Funk, P.G.: Über Flächen mit lauter geschlossenen geodätischen Linien. Thesis, Georg-August-Universitt Göttingen (1911)Google Scholar
  4. 4.
    Funk, P.G.: Über Flächen mit lauter geschlossenen geodätschen Linen. Math. Ann. 74, 278–300 (1913)Google Scholar
  5. 5.
    Gel’fand, I.M., Gindikin, S.G., Graev, M.I.: Selected Topics in Integral Geometry. Translations of Mathematical Monographs. AMS, Providence (2003)Google Scholar
  6. 6.
    Gindikin, S., Reeds, J., Shepp, L.: Spherical tomography and spherical integral geometry. In: Tomography, Impedance Imaging, and Integral Geometry (South Hadley, MA, 1993). Lectures in Applied Mathematics, vol. 30, pp. 83–92. American Mathematical Society, Providence, RI (1994)Google Scholar
  7. 7.
    Helgason, S.: Integral Geometry and Radon Transform. Springer, New York (2011)Google Scholar
  8. 8.
    Hielscher, R., Quellmalz, M.: Reconstructing a function on the sphere from its means along vertical slices. Inverse Probl. Imaging 10(3), 711–739 (2016)Google Scholar
  9. 9.
    Hristova, Y., Moon, S., Steinhauer, D.: A Radon-type transform arising in photoacoustic tomography with circular detectors: spherical geometry. Inverse Probl. Sci. Eng. 24(6), 974–989 (2016)Google Scholar
  10. 10.
    John, F.: Plane waves and spherical means applied to partial differential equations. Reprint of the 1955 original. Dover Publications, Inc., Mineola, NY (2004)Google Scholar
  11. 11.
    Kazantsev, S.G.: Funk-Minkowski transform and spherical convolution of Hilbert type in reconstructing functions on the sphere (2018). Preprint arXiv:1806.06672
  12. 12.
    Kuchment, P.: The Radon transform and medical imaging. In: CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 85. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2014)Google Scholar
  13. 13.
    Minkowski, H.: Über die Körper konstanter Breite. Mat. Sbornik 25, 505–508 (1904). [in Russian] (German translation in Gesammelte Abhandlungen 2, Bd. Teubner, Leipzig, 277–279 (1911))Google Scholar
  14. 14.
    Oberlin, D.M., Stein, E.M.: Mapping properties of the Radon transform. Indiana Univ. Math. J. 31, 641–650 (1982)Google Scholar
  15. 15.
    Ólafsson, G., Pasquale, A., Rubin, B.: Analytic and group-theoretic aspects of the cosine transform. Contemp. Math. 598, 167–188 (2013)Google Scholar
  16. 16.
    Ournycheva, E., Rubin, B.: Semyanistyi’s integrals and Radon transforms on matrix spaces. J. Fourier Anal. Appl. 14, 60–88 (2008)Google Scholar
  17. 17.
    Palamodov, V.P.: Reconstructive Integral Geometry. Monographs in Mathematics, vol. 98. Birkhäuser Verlag, Basel (2004)Google Scholar
  18. 18.
    Palamodov, V.P.: Reconstruction from Integral Data. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2016)Google Scholar
  19. 19.
    Quellmalz, M.: A generalization of the Funk-Radon transform. Inverse Probl. 33(3), 035016 (2017)Google Scholar
  20. 20.
    Quellmalz, M.: The Funk-Radon transform for hyperplane sections through a common point. Preprint arXiv:1810.08105 (2018)
  21. 21.
    Rubin, B.: Fractional calculus and wavelet transforms in integral geometry. Fract. Calc. Appl. Anal. 1, 193–219 (1998)Google Scholar
  22. 22.
    Rubin, B.: Inversion formulas for the spherical Radon transform and the generalized cosine transform. Adv. Appl. Math. 29, 471–497 (2002)Google Scholar
  23. 23.
    Rubin, B.: Radon, cosine, and sine transforms on real hyperbolic space. Adv. Math. 170, 206–223 (2002)Google Scholar
  24. 24.
    Rubin, B.: Intersection bodies and generalized cosine transforms. Adv. Math. 218, 696–727 (2008)Google Scholar
  25. 25.
    Rubin, B.: Semyanistyi fractional integrals and Radon transforms. Contemp. Math. 598, 221–237 (2013)Google Scholar
  26. 26.
    Rubin, B.: Funk, cosine, and sine transforms on Stiefel and Grassmann manifolds. J. Geom. Anal. 23(3), 1441–1497 (2013)Google Scholar
  27. 27.
    Rubin, B.: Introduction to Radon Transforms: With Elements of Fractional Calculus and Harmonic Analysis (Encyclopedia of Mathematics and its Applications). Cambridge University Press, Cambridge (2015)Google Scholar
  28. 28.
    Rubin, B.: Radon transforms and Gegenbauer–Chebyshev integrals, II; examples. Anal. Math. Phys. 7(4), 349–375 (2017)Google Scholar
  29. 29.
    Rubin, B.: The Vertical Slice Transform in Spherical Tomography. Preprint arXiv:1807.07689 (2018)
  30. 30.
    Salman, Y.: An inversion formula for the spherical transform in \(S^2\) for a special family of circles of integration. Anal. Math. Phys. 6(1), 43–58 (2016)Google Scholar
  31. 31.
    Salman, Y.: Recovering functions defined on the unit sphere by integration on a special family of sub-spheres. Anal. Math. Phys. 7(2), 165–185 (2017)Google Scholar
  32. 32.
    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach Sc. Publ., New York (1993)Google Scholar
  33. 33.
    Semyanistyi, V.I.: On some integral transformations in Euclidean space. Dokl. Akad. Nauk SSSR 134, 536–539 (1960). (In Russian) Google Scholar
  34. 34.
    Stepanov, V.N.: The method of spherical harmonics for integral transforms on a sphere. Math. Struct. Model. 2(42), 36–48 (2017)Google Scholar
  35. 35.
    Zangerl, G., Scherzer, O.: Exact reconstruction in photoacoustic tomography with circular integrating detectors II: spherical geometry. Math. Methods Appl. Sci. 33(15), 1771–1782 (2010)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA

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