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 , its X-ray transform is a function defined on the family of all straight lines in as follows: let the unit vector represent the direction of and let be its signed distance to the origin, so that is represented by the pair (as well as ); then

where is an arbitrary point on the line . 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 of the over the family of all lines which are at a (signed) distance from the point , namely,

where is the Euclidean inner product between and . Radon then showed that the function can be recovered by the formula

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 , dropped at random on a plane, intersects one of the lines of a family of parallel lines located at a distance (cf. also Buffon problem). As explained in [a3], Chapt. 5, the solution leads to the consideration of a measure on the space of all lines in the plane and of invariance under all rigid motions. This measure induces a functional on the family of compact sets by

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