John's equation
John's equation is an ultrahyperbolic partial differential equation satisfied by the X-ray transform of a function. It is named after Fritz John.
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
which is proved by Fritz John for dimension three and by :hu:Kurusa Árpád|Kurusa for higher dimensions.
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 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.
