Given a function with compact support the X-ray transform is the integral over all lines in . We will parameterise the lines by pairs of points , on each line and define as the ray transform where
Such functions are characterized by John's equations
In three-dimensional x-ray computerized tomography John's equation can be solved to fill in missing data, for example where the data is obtained from a point source traversing a curve, typically a helix.
More generally an ultrahyperbolic partial differential equation (a term coined by Richard Courant) is a second order partial differential equation of the form
where , such that the quadratic form
can be reduced by a linear change of variables to the form
It is not possible to arbitrarily specify the value of the solution on a non-characteristic hypersurface. John's paper however does give examples of manifolds on which an arbitrary specification of u can be extended to a solution.
- John, Fritz (1938), "The ultrahyperbolic differential equation with four independent variables", Duke Mathematical Journal 4 (2): 300–322, doi:10.1215/S0012-7094-38-00423-5, ISSN 0012-7094, MR 1546052, Zbl 0019.02404
- Á. Kurusa, A characterization of the Radon transform's range by a system of PDEs, J. Math. Anal. Appl., 161(1991), 218--226. doi:10.1016/0022-247X(91)90371-6
- S K Patch, Consistency conditions upon 3D CT data and the wave equation, Phys. Med. Biol. 47 No 15 (7 August 2002) 2637-2650 doi:10.1088/0031-9155/47/15/306