Duke Mathematical Journal

Inversion formulas for the k-dimensional Radon transform in real hyperbolic spaces

Carlos A. Berenstein and Tarabusi Enrico Casadio

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Article information

Source
Duke Math. J., Volume 62, Number 3 (1991), 613-631.

Dates
First available in Project Euclid: 20 February 2004

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1077296509

Digital Object Identifier
doi:10.1215/S0012-7094-91-06227-7

Mathematical Reviews number (MathSciNet)
MR1104811

Zentralblatt MATH identifier
0742.44002

Subjects
Primary: 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]
Secondary: 58G99

Citation

Berenstein, Carlos A.; Casadio, Tarabusi Enrico. Inversion formulas for the $k$ -dimensional Radon transform in real hyperbolic spaces. Duke Math. J. 62 (1991), no. 3, 613--631. doi:10.1215/S0012-7094-91-06227-7. https://projecteuclid.org/euclid.dmj/1077296509


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References

  • [BB1] D. C. Barber and B. H. Brown, Applied potential tomography, J. Phys. E. 17 (1984), 723–733.
  • [BB2] D. C. Barber and B. H. Brown, Recent developments in applied potential tomography—APT, Information Processing in Medical Imaging ed. S. Bacharach, Nijhoff, Dordrecht, 1986, pp. 106–121.
  • [BCCP] C. A. Berenstein, Casadio E. Tarabusi, J. M. Cohen, and M. A. Picardello, Integral geometry on trees, Amer. J. Math., To appear.
  • [BZ] C. A. Berenstein and L. Zalcman, Pompeiu's problem on symmetric spaces, Comment. Math. Helv. 55 (1980), no. 4, 593–621.
  • [B] G. Beylkin, The inversion problem and applications of the generalized Radon transform, Comm. Pure Appl. Math. 37 (1984), no. 5, 579–599.
  • [BS] W. Bray and D. C. Solmon, The horocycle transform and harmonic analysis on the Poincaré disk, preprint.
  • [Ci] Casadio E. Tarabusi, Inversion of the X-ray transform: continuous vs. discrete, Contemp. Math. To appear.
  • [CK]1 A. M. Cormack, The Radon transform on a family of curves in the plane, Proc. Amer. Math. Soc. 83 (1981), no. 2, 325–330.
  • [CK]2 A. M. Cormack, The Radon transform on a family of curves in the plane. II, Proc. Amer. Math. Soc. 86 (1982), no. 2, 293–298.
  • [CQ] A. M. Cormack and E. T. Quinto, A Radon transform on spheres through the origin in $\bf R\spn$ and applications to the Darboux equation, Trans. Amer. Math. Soc. 260 (1980), no. 2, 575–581.
  • [Gv] A. B. Goncharov, Dopustimye semeĭstva $k$-mernykh podmnogoobraziĭ, Dokl. Akad. Nauk SSSR 300 (1988), no. 3, 535–539, Admissible families of $k$-dimensional surfaces, Soviet Math. Dokl. 37 (1988), 683–687.
  • [Gz] F. Gonzalez, Radon transforms on Grassmann manifolds, Ph.D. thesis, Mass. Inst. Technology, 1984.
  • [GR] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1980, corrected and enlarged edition, Academic Press, Orlando.
  • [GU] A. Greenleaf and G. Uhlmann, Nonlocal inversion formulas for the X-ray transform, Duke Math. J. 58 (1989), no. 1, 205–240.
  • [Gg] E. L. Grinberg, Spherical harmonics and integral geometry on projective spaces, Trans. Amer. Math. Soc. 279 (1983), no. 1, 187–203.
  • [H1] S. Helgason, Differential operators on homogenous spaces, Acta Math. 102 (1959), 239–299.
  • [H2] S. Helgason, Groups and geometric analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, Pure and Applied Mathematics, vol. 113, Academic Press Inc., Orlando, FL, 1984.
  • [H3] S. Helgason, The totally-geodesic Radon transform on constant curvature spaces, Contemp. Math. To appear.
  • [KS] A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging, IEEE Press Selected Reprint Ser., New York, 1988.
  • [K] F. Keinert, Inversion of $k$-plane transforms and applications in computer tomography, SIAM Rev. 31 (1989), no. 2, 273–298.
  • [N] F. Natterer, The mathematics of computerized tomography, B. G. Teubner, Stuttgart, 1986.
  • [Q]1 E. T. Quinto, The invertibility of rotation invariant Radon transforms, J. Math. Anal. Appl. 91 (1983), no. 2, 510–522.
  • [Q]2 E. T. Quinto, Erratum: “The invertibility of rotation invariant Radon transforms”, J. Math. Anal. Appl. 94 (1983), no. 2, 602–603.
  • [SV] F. Santosa and M. Vogelius, A backprojection algorithm for electrical impedance imaging, SIAM J. Appl. Math. 50 (1990), no. 1, 216–243.
  • [Sĭ] V. I. Semjanistyĭ, Nekotorye zadachi integral'noĭgeometrii v psevdoevklidovykh i neevklidovykh prostranstvakh, Trudy Sem. Vektor. Tenzor. Anal. 13 (1966), 244–302, Certain problems of integral geometry in pseudo-Euclidean and non-Euclidean spaces, abstract #7125, MR 35 (1968), p. 1324.
  • [Sn] D. C. Solmon, The $X$-ray transform, J. Math. Anal. Appl. 56 (1976), no. 1, 61–83.
  • [Sz1] R. S. Strichartz, $L\sp p$ estimates for Radon transforms in Euclidean and non-Euclidean spaces, Duke Math. J. 48 (1981), no. 4, 699–727.
  • [Sz2] R. S. Strichartz, Radon inversion—variations on a theme, Amer. Math. Monthly 89 (1982), no. 6, 377–384, 420–423.
  • [WY] D. I. Wallace and Ryuji Yamaguchi, The Radon transform on $\rm SL(2,\bf R)/\rm SO(2,\bf R)$, Trans. Amer. Math. Soc. 297 (1986), no. 1, 305–318.