Radon transform

An integral transform of a function in several variables, related to the Fourier transform. It was introduced by J. Radon (see [1]).

Let be a continuous function of the real variables that is decreasing sufficiently rapidly at infinity, , .

For any hyperplane in ,

and

the following integral is defined:

where is the Euclidean -dimensional volume in the hyperplane . The function

is called the Radon transform of the function . It is a homogeneous function of its variables of degree :

and is related to the Fourier transform , , of by

The Radon transform is immediately associated with the problem, going back to Radon, of the recovery of a function from the values of its integrals calculated over all hyperplanes of the space (that is, the problem of the inversion of the Radon transform).

References

[1]	J. Radon, "Ueber die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten" Ber. Verh. Sächs. Akad. , 69 (1917) pp. 262–277
[2]	I.M. Gel'fand, M.I. Graev, N.Ya. Vilenkin, "Generalized functions" , 5. Integral geometry and representation theory , Acad. Press (1966) (Translated from Russian)

Comments

For the far-reaching generalizations of the Radon transform to homogeneous spaces see [a3].

The Radon transform and, in particular, the corresponding inversion formula (i.e. the formula recovering from its Radon transform) is of central importance in tomography.

References

[a1]	S.R. Deans, "The Radon transform and some of its applications" , Wiley (1983)
[a2]	S. Helgason, "The Radon transform" , Birkhäuser (1980)
[a3]	S. Helgason, "Groups and geometric analysis" , Acad. Press (1984) pp. Chapt. II, Sect. 4