Let be a continuous function of the real variables that is decreasing sufficiently rapidly at infinity, , .
For any hyperplane in ,
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).
|||J. Radon, "Ueber die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten" Ber. Verh. Sächs. Akad. , 69 (1917) pp. 262–277|
|||I.M. Gel'fand, M.I. Graev, N.Ya. Vilenkin, "Generalized functions" , 5. Integral geometry and representation theory , Acad. Press (1966) (Translated from Russian)|
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.
|[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|
Radon transform. R.A. Minlos (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Radon_transform&oldid=14894