John 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 $f\colon\mathbb^n \rightarrow \mathbb$ with compact support the ''X-ray transform'' is the integral over all lines in $\mathbb^n$. We will parameterise the lines by pairs of points $x,y \in \mathbb^n$, $x \ne y$ on each line and define ''$u$'' as the ray transform where
:$u(x,y) = \int\limits_^ f( x + t(y-x) ) dt.$
Such functions ''$u$'' are characterized by John's equations
:$\frac - \frac=0$
which is proved by Fritz John for dimension three and by 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 (a term coined by Richard Courant) is a second order partial differential equation of the form
:$\sum\limits_^ a_\frac + \sum\limits_^ b_i\frac + cu =0$
where $n \ge 2$, such that the quadratic form
:$\sum\limits_^ a_ \xi_i \xi_j$
can be reduced by a linear change of variables to the form
:$\sum\limits_^ \xi_i^2 - \sum\limits_^ \xi_i^2.$
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.

References

*
* Á. Kurusa, A characterization of the Radon transform's range by a system of PDEs, J. Math. Anal. Appl., 161(1991), 218--226.
* 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

Partial differential equations
X-ray computed tomography