View source for X-ray transform

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

<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>

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,

<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>

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

<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>

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

<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>

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====
<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>