An Integral Transform whose inverse is used to reconstruct images from medical CT scans. A technique for using Radon transforms to reconstruct a map of a planet's polar regions using a spacecraft in a polar orbit has also been devised (Roulston and Muhleman 1997).

The Radon transform can be defined by

(1) |

(2) |

(3) |

Using the identity

(4) |

(5) |

(6) |

(7) |

Nievergelt (1986) uses the inverse formula

(8) |

(9) |

(10) | |||

(11) |

The Radon transform satisfies superposition

(12) |

(13) |

(14) |

(15) |

(16) |

The line integral along is

(17) |

(18) |

(19) |

(20) |

(21) |

**References**

Anger, B. and Portenier, C. *Radon Integrals.* Boston, MA: Birkhäuser, 1992.

Armitage, D. H. and Goldstein, M. ``Nonuniqueness for the Radon Transform.'' *Proc. Amer. Math. Soc.* **117**, 175-178, 1993.

Deans, S. R. *The Radon Transform and Some of Its Applications.* New York: Wiley, 1983.

Durrani, T. S. and Bisset, D. ``The Radon Transform and its Properties.'' *Geophys.* **49**, 1180-1187, 1984.

Esser, P. D. (Ed.). *Emission Computed Tomography: Current Trends.* New York: Society of Nuclear Medicine, 1983.

Gindikin, S. (Ed.). *Applied Problems of Radon Transform.* Providence, RI: Amer. Math. Soc., 1994.

Gradshteyn, I. S. and Ryzhik, I. M. *Tables of Integrals, Series, and Products, 5th ed.* San Diego, CA:
Academic Press, 1979.

Helgason, S. *The Radon Transform.* Boston, MA: Birkhäuser, 1980.

Kunyansky, L. A. ``Generalized and Attenuated Radon Transforms: Restorative Approach to the Numerical Inversion.''
*Inverse Problems* **8**, 809-819, 1992.

Nievergelt, Y. ``Elementary Inversion of Radon's Transform.'' *SIAM Rev.* **28**, 79-84, 1986.

Rann, A. G. and Katsevich, A. I. *The Radon Transform and Local Tomography.* Boca Raton, FL: CRC Press, 1996.

Robinson, E. A. ``Spectral Approach to Geophysical Inversion Problems by Lorentz, Fourier, and Radon
Transforms.'' *Proc. Inst. Electr. Electron. Eng.* **70**, 1039-1053, 1982.

Roulston, M. S. and Muhleman, D. O. ``Synthesizing Radar Maps of Polar Regions with a Doppler-Only Method.'' *Appl. Opt.*
**36**, 3912-3919, 1997.

Shepp, L. A. and Kruskal, J. B. ``Computerized Tomography: The New Medical X-Ray Technology.'' *Amer. Math. Monthly* **85**,
420-439, 1978.

Strichartz, R. S. ``Radon Inversion--Variation on a Theme.'' *Amer. Math. Monthly* **89**, 377-384 and 420-423, 1982.

Zalcman, L. ``Uniqueness and Nonuniqueness for the Radon Transform.'' *Bull. London Math. Soc.* **14**, 241-245, 1982.

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1999-05-25