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 [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