##### Actions

The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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)

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.