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An integral transform of a function in several variables, related to the [[Fourier transform|Fourier transform]]. It was introduced by J. Radon (see [[#References|[1]]]).
 
An integral transform of a function in several variables, related to the [[Fourier transform|Fourier transform]]. It was introduced by J. Radon (see [[#References|[1]]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r0771901.png" /> be a continuous function of the real variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r0771902.png" /> that is decreasing sufficiently rapidly at infinity, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r0771903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r0771904.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r0771905.png" />,
+
For any hyperplane in $  \mathbf R  ^ {n} $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r0771906.png" /></td> </tr></table>
+
$$
 +
\Gamma  = \{ {( x _ {1} \dots x _ {n} ) } : {\xi _ {1} x _ {1} + \dots + \xi _ {n} x _ {n} = C } \}
 +
,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r0771907.png" /></td> </tr></table>
+
$$
 +
\xi _ {i}  \in  \mathbf R  ^ {1} ,\  i  = 1 \dots n ,
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r0771908.png" /></td> </tr></table>
+
$$
 +
\sum_{i=1}^ { n }  \xi _ {i}  ^ {2}  > 0 ,\ C  \in  \mathbf R  ^ {1} ,
 +
$$
  
 
the following integral is defined:
 
the following integral is defined:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r0771909.png" /></td> </tr></table>
+
$$
 +
F ( \xi _ {1} \dots \xi _ {n} ; C )  = \
 +
 
 +
\frac{1}{\left ( \sum_{i=1}^ { n }  \xi _ {j} \right )  ^ {1/2} }
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r07719010.png" /> is the Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r07719011.png" />-dimensional volume in the hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r07719012.png" />. The function
+
\int\limits _  \Gamma  f ( x _ {1} \dots x _ {n} )  d V _  \Gamma  ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r07719013.png" /></td> </tr></table>
+
where  $  V _  \Gamma  $
 +
is the Euclidean  $  ( n - 1 ) $-
 +
dimensional volume in the hyperplane  $  \Gamma $.  
 +
The function
  
is called the Radon transform of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r07719014.png" />. It is a homogeneous function of its variables of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r07719015.png" />:
+
$$
 +
F ( \xi _ {1} \dots \xi _ {n} ;  C ) ,\ \
 +
( \xi _ {1} \dots x _ {n} , C ) \in \mathbf R  ^ {n+} 1 ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r07719016.png" /></td> </tr></table>
+
is called the Radon transform of the function  $  f $.
 +
It is a homogeneous function of its variables of degree  $  - 1 $:
  
and is related to the Fourier transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r07719017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r07719018.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r07719019.png" /> by
+
$$
 +
F ( \alpha \xi _ {1} \dots \alpha \xi _ {n} ;  \alpha C )  = \
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r07719020.png" /></td> </tr></table>
+
\frac{1}{| \alpha | }
 +
F ( \xi _ {1} \dots \xi _ {n} ; C ) ,
 +
$$
  
The Radon transform is immediately associated with the problem, going back to Radon, of the recovery of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r07719021.png" /> from the values of its integrals calculated over all hyperplanes of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r07719022.png" /> (that is, the problem of the inversion of the Radon transform).
+
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Radon,  "Ueber die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten"  ''Ber. Verh. Sächs. Akad.'' , '''69'''  (1917)  pp. 262–277</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,  M.I. Graev,  N.Ya. Vilenkin,  "Generalized functions" , '''5. Integral geometry and representation theory''' , Acad. Press  (1966)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Radon,  "Ueber die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten"  ''Ber. Verh. Sächs. Akad.'' , '''69'''  (1917)  pp. 262–277</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,  M.I. Graev,  N.Ya. Vilenkin,  "Generalized functions" , '''5. Integral geometry and representation theory''' , Acad. Press  (1966)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
For the far-reaching generalizations of the Radon transform to homogeneous spaces see [[#References|[a3]]].
 
For the far-reaching generalizations of the Radon transform to homogeneous spaces see [[#References|[a3]]].
  
The Radon transform and, in particular, the corresponding inversion formula (i.e. the formula recovering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077190/r07719023.png" /> from its Radon transform) is of central importance in [[Tomography|tomography]].
+
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|tomography]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.R. Deans,  "The Radon transform and some of its applications" , Wiley  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Helgason,  "The Radon transform" , Birkhäuser  (1980)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Helgason,  "Groups and geometric analysis" , Acad. Press  (1984)  pp. Chapt. II, Sect. 4</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.R. Deans,  "The Radon transform and some of its applications" , Wiley  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Helgason,  "The Radon transform" , Birkhäuser  (1980)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Helgason,  "Groups and geometric analysis" , Acad. Press  (1984)  pp. Chapt. II, Sect. 4</TD></TR></table>

Latest revision as of 16:09, 6 January 2024


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
How to Cite This Entry:
Radon transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Radon_transform&oldid=14894
This article was adapted from an original article by R.A. Minlos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article