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 $ f ( x _ {1} \dots x _ {n} ) $ be a continuous function of the real variables $ x _ {i} \in \mathbf R ^ {1} $ that is decreasing sufficiently rapidly at infinity, $ i = 1 \dots n $, $ n = 1 , 2 ,\dots $.

For any hyperplane in $ \mathbf R ^ {n} $,

$$ \Gamma = \{ {( x _ {1} \dots x _ {n} ) } : {\xi _ {1} x _ {1} + \dots + \xi _ {n} x _ {n} = C } \} , $$

$$ \xi _ {i} \in \mathbf R ^ {1} ,\ i = 1 \dots n , $$

and

$$ \sum_{i=1}^ { n } \xi _ {i} ^ {2} > 0 ,\ C \in \mathbf R ^ {1} , $$

the following integral is defined:

$$ F ( \xi _ {1} \dots \xi _ {n} ; C ) = \ \frac{1}{\left ( \sum_{i=1}^ { n } \xi _ {j} \right ) ^ {1/2} } \int\limits _ \Gamma f ( x _ {1} \dots x _ {n} ) d V _ \Gamma , $$

where $ V _ \Gamma $ is the Euclidean $ ( n - 1 ) $- dimensional volume in the hyperplane $ \Gamma $. The function

$$ F ( \xi _ {1} \dots \xi _ {n} ; C ) ,\ \ ( \xi _ {1} \dots x _ {n} , C ) \in \mathbf R ^ {n+} 1 , $$

is called the Radon transform of the function $ f $. It is a homogeneous function of its variables of degree $ - 1 $:

$$ F ( \alpha \xi _ {1} \dots \alpha \xi _ {n} ; \alpha C ) = \ \frac{1}{| \alpha | } F ( \xi _ {1} \dots \xi _ {n} ; C ) , $$

and is related to the Fourier transform $ \widetilde{f} ( \xi _ {1} \dots \xi _ {n} ) $, $ \xi _ {i} \in \mathbf R ^ {1} $, of $ f $ by

$$ F ( \xi _ {1} \dots \xi _ {n} ; C ) = \frac{1}{2 \pi } \int\limits _ {- \infty } ^ \infty \widetilde{f} ( \alpha \xi _ {1} \dots \alpha \xi _ {n} ) e ^ {- i \alpha C } d \alpha . $$

The Radon transform is immediately associated with the problem, going back to Radon, of the recovery of a function $ f $ from the values of its integrals calculated over all hyperplanes of the space $ \mathbf R ^ {n} $( 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 $ f $ 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