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In 1963, A.M. Cormack introduced a powerful diagnostic tool in radiology, computerized tomography, which is based on the mathematical properties of the X-ray transform in the Euclidean plane [[#References|[a1]]] (cf. also [[Tomography|Tomography]]). For a compactly supported [[Continuous function|continuous function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x1200301.png" />, its X-ray transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x1200302.png" /> is a function defined on the family of all straight lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x1200303.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x1200304.png" /> as follows: let the unit vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x1200305.png" /> represent the direction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x1200306.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x1200307.png" /> be its signed distance to the origin, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x1200308.png" /> is represented by the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x1200309.png" /> (as well as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003010.png" />); then
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<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/x/x120/x120030/x12003011.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003012.png" /> is an arbitrary point on the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003013.png" />. This transform had already been considered in 1917 by J. Radon, who found its inverse with the help of its adjoint, given by the average value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003014.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003015.png" /> over the family of all lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003016.png" /> which are at a (signed) distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003017.png" /> from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003018.png" />, namely,
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In 1963, A.M. Cormack introduced a powerful diagnostic tool in radiology, computerized tomography, which is based on the mathematical properties of the X-ray transform in the Euclidean plane [[#References|[a1]]] (cf. also [[Tomography|Tomography]]). For a compactly supported [[Continuous function|continuous function]] $f$, its X-ray transform $X f$ is a function defined on the family of all straight lines $\operatorname{l}$ in $\mathbf{R} ^ { 2 }$ as follows: let the unit vector $\theta$ represent the direction of $\operatorname{l}$ and let $p$ be its signed distance to the origin, so that $\operatorname{l}$ is represented by the pair $( \theta , p )$ (as well as $( - \theta , - p )$); then
  
<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/x/x120/x120030/x12003019.png" /></td> </tr></table>
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\begin{equation*} X f ( \text{l} ) = X f ( \theta , p ) = \int _ { - \infty } ^ { \infty } f ( x + t \theta ) d t, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003020.png" /> is the Euclidean [[Inner product|inner product]] between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003022.png" />. Radon then showed that the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003023.png" /> can be recovered by the formula
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where $x$ is an arbitrary point on the line $\operatorname{l}$. This transform had already been considered in 1917 by J. Radon, who found its inverse with the help of its adjoint, given by the average value $F _ { x } ( q )$ of the $X f ( \text{l} )$ over the family of all lines $\operatorname{l}$ which are at a (signed) distance $q$ from the point $x$, namely,
  
<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/x/x120/x120030/x12003024.png" /></td> </tr></table>
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\begin{equation*} F _ { x } ( q ) = \frac { 1 } { 2 \pi } \int _ { S ^ { 1 } } X f ( \theta , x \cdot \theta + q ) d \theta \end{equation*}
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where $x . \theta$ is the Euclidean [[Inner product|inner product]] between $x$ and $\theta$. Radon then showed that the function $f$ can be recovered by the formula
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\begin{equation*} f ( x ) = - \frac { 1 } { \pi } \int _ { 0 } ^ { \infty } \frac { d F _ { x } ( q ) } { q }. \end{equation*}
  
 
The generalization of the X-ray transform to Euclidean spaces of arbitrary dimension and replacing the family of all lines by the family of all affine subspaces of a fixed dimension is known as the [[Radon transform|Radon transform]] [[#References|[a1]]]. For the Radon transform in the broader context of symmetric spaces, see also [[#References|[a2]]].
 
The generalization of the X-ray transform to Euclidean spaces of arbitrary dimension and replacing the family of all lines by the family of all affine subspaces of a fixed dimension is known as the [[Radon transform|Radon transform]] [[#References|[a1]]]. For the Radon transform in the broader context of symmetric spaces, see also [[#References|[a2]]].
  
Note that the adjoint of the X-ray transform can be traced back to the Buffon needle problem (1777): find the average number of times that a needle of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003025.png" />, dropped at random on a plane, intersects one of the lines of a family of parallel lines located at a distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003026.png" /> (cf. also [[Buffon problem|Buffon problem]]). As explained in [[#References|[a3]]], Chapt. 5, the solution leads to the consideration of a [[Measure|measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003027.png" /> on the space of all lines in the plane and of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003028.png" /> invariance under all rigid motions. This measure induces a functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003029.png" /> on the family of compact sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/x/x120/x120030/x12003030.png" /> by
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Note that the adjoint of the X-ray transform can be traced back to the Buffon needle problem (1777): find the average number of times that a needle of length $\operatorname{l}$, dropped at random on a plane, intersects one of the lines of a family of parallel lines located at a distance $D \geq \text{l}$ (cf. also [[Buffon problem|Buffon problem]]). As explained in [[#References|[a3]]], Chapt. 5, the solution leads to the consideration of a [[Measure|measure]] $\omega$ on the space of all lines in the plane and of $\omega$ invariance under all rigid motions. This measure induces a functional $K$ on the family of compact sets $\Omega$ by
  
<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/x/x120/x120030/x12003031.png" /></td> </tr></table>
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\begin{equation*} K ( \Omega ) = \int _ { \lambda \bigcap \Omega \neq \phi } d \omega ( \lambda ), \end{equation*}
  
 
which is basically the adjoint of the X-ray transform. Thus, among the generalizations of the X-ray transform and its adjoint, one also finds basic links to [[Integral geometry|integral geometry]] [[#References|[a3]]], [[#References|[a6]]], combinatorial geometry [[#References|[a4]]], convex geometry [[#References|[a5]]], as well as the [[Pompeiu problem|Pompeiu problem]].
 
which is basically the adjoint of the X-ray transform. Thus, among the generalizations of the X-ray transform and its adjoint, one also finds basic links to [[Integral geometry|integral geometry]] [[#References|[a3]]], [[#References|[a6]]], combinatorial geometry [[#References|[a4]]], convex geometry [[#References|[a5]]], as well as the [[Pompeiu problem|Pompeiu problem]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Natterer,  "The mathematics of computerized tomography" , Wiley  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Helgason,  "Geometric analysis on symmetric spaces" , Amer. Math. Soc.  (1994)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.A. Santaló,  "Integral geometry and geometric probability" , ''Encycl. Math. Appl.'' , Addison-Wesley  (1976)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.V. Ambartzumian,  "Combinatorial integral geometry" , Wiley  (1982)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  "Handbook of convex geometry"  P.M. Gruber (ed.)  J.M. Wills (ed.) , '''1; 2''' , North-Holland  (1993)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C.A. Berenstein,  E.L. Grinberg,  "A short bibliography on integral geometry"  ''Gaceta Matematica (R. Acad. Sci. Spain)'' , '''1'''  (1998)  pp. 189–194</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  F. Natterer,  "The mathematics of computerized tomography" , Wiley  (1986)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  S. Helgason,  "Geometric analysis on symmetric spaces" , Amer. Math. Soc.  (1994)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  L.A. Santaló,  "Integral geometry and geometric probability" , ''Encycl. Math. Appl.'' , Addison-Wesley  (1976)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  R.V. Ambartzumian,  "Combinatorial integral geometry" , Wiley  (1982)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  "Handbook of convex geometry"  P.M. Gruber (ed.)  J.M. Wills (ed.) , '''1; 2''' , North-Holland  (1993)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  C.A. Berenstein,  E.L. Grinberg,  "A short bibliography on integral geometry"  ''Gaceta Matematica (R. Acad. Sci. Spain)'' , '''1'''  (1998)  pp. 189–194</td></tr></table>

Revision as of 16:45, 1 July 2020

In 1963, A.M. Cormack introduced a powerful diagnostic tool in radiology, computerized tomography, which is based on the mathematical properties of the X-ray transform in the Euclidean plane [a1] (cf. also Tomography). For a compactly supported continuous function $f$, its X-ray transform $X f$ is a function defined on the family of all straight lines $\operatorname{l}$ in $\mathbf{R} ^ { 2 }$ as follows: let the unit vector $\theta$ represent the direction of $\operatorname{l}$ and let $p$ be its signed distance to the origin, so that $\operatorname{l}$ is represented by the pair $( \theta , p )$ (as well as $( - \theta , - p )$); then

\begin{equation*} X f ( \text{l} ) = X f ( \theta , p ) = \int _ { - \infty } ^ { \infty } f ( x + t \theta ) d t, \end{equation*}

where $x$ is an arbitrary point on the line $\operatorname{l}$. This transform had already been considered in 1917 by J. Radon, who found its inverse with the help of its adjoint, given by the average value $F _ { x } ( q )$ of the $X f ( \text{l} )$ over the family of all lines $\operatorname{l}$ which are at a (signed) distance $q$ from the point $x$, namely,

\begin{equation*} F _ { x } ( q ) = \frac { 1 } { 2 \pi } \int _ { S ^ { 1 } } X f ( \theta , x \cdot \theta + q ) d \theta \end{equation*}

where $x . \theta$ is the Euclidean inner product between $x$ and $\theta$. Radon then showed that the function $f$ can be recovered by the formula

\begin{equation*} f ( x ) = - \frac { 1 } { \pi } \int _ { 0 } ^ { \infty } \frac { d F _ { x } ( q ) } { q }. \end{equation*}

The generalization of the X-ray transform to Euclidean spaces of arbitrary dimension and replacing the family of all lines by the family of all affine subspaces of a fixed dimension is known as the Radon transform [a1]. For the Radon transform in the broader context of symmetric spaces, see also [a2].

Note that the adjoint of the X-ray transform can be traced back to the Buffon needle problem (1777): find the average number of times that a needle of length $\operatorname{l}$, dropped at random on a plane, intersects one of the lines of a family of parallel lines located at a distance $D \geq \text{l}$ (cf. also Buffon problem). As explained in [a3], Chapt. 5, the solution leads to the consideration of a measure $\omega$ on the space of all lines in the plane and of $\omega$ invariance under all rigid motions. This measure induces a functional $K$ on the family of compact sets $\Omega$ by

\begin{equation*} K ( \Omega ) = \int _ { \lambda \bigcap \Omega \neq \phi } d \omega ( \lambda ), \end{equation*}

which is basically the adjoint of the X-ray transform. Thus, among the generalizations of the X-ray transform and its adjoint, one also finds basic links to integral geometry [a3], [a6], combinatorial geometry [a4], convex geometry [a5], as well as the Pompeiu problem.

References

[a1] F. Natterer, "The mathematics of computerized tomography" , Wiley (1986)
[a2] S. Helgason, "Geometric analysis on symmetric spaces" , Amer. Math. Soc. (1994)
[a3] L.A. Santaló, "Integral geometry and geometric probability" , Encycl. Math. Appl. , Addison-Wesley (1976)
[a4] R.V. Ambartzumian, "Combinatorial integral geometry" , Wiley (1982)
[a5] "Handbook of convex geometry" P.M. Gruber (ed.) J.M. Wills (ed.) , 1; 2 , North-Holland (1993)
[a6] C.A. Berenstein, E.L. Grinberg, "A short bibliography on integral geometry" Gaceta Matematica (R. Acad. Sci. Spain) , 1 (1998) pp. 189–194
How to Cite This Entry:
X-ray transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=X-ray_transform&oldid=49951
This article was adapted from an original article by Carlos A. Berenstein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article