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 [math]\displaystyle{ f\colon\mathbb{R}^n \rightarrow \mathbb{R} }[/math] with compact support the X-ray transform is the integral over all lines in [math]\displaystyle{ \mathbb{R}^n }[/math]. We will parameterise the lines by pairs of points [math]\displaystyle{ x,y \in \mathbb{R}^n }[/math], [math]\displaystyle{ x \ne y }[/math] on each line and define [math]\displaystyle{ u }[/math] as the ray transform where

[math]\displaystyle{ u(x,y) = \int\limits_{-\infty}^{\infty} f( x + t(y-x) ) dt. }[/math]

Such functions [math]\displaystyle{ u }[/math] are characterized by John's equations

[math]\displaystyle{ \frac{\partial^2u}{\partial x_i \partial y_j} - \frac{\partial^2u}{\partial y_i \partial x_j}=0 }[/math]

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

[math]\displaystyle{ \sum\limits_{i,j=1}^{2n} a_{ij}\frac{\partial^2 u}{\partial x_i \partial x_j} + \sum\limits_{i=1}^{2n} b_i\frac{\partial u}{\partial x_i} + cu =0 }[/math]

where [math]\displaystyle{ n \ge 2 }[/math], such that the quadratic form

[math]\displaystyle{ \sum\limits_{i,j=1}^{2n} a_{ij} \xi_i \xi_j }[/math]

can be reduced by a linear change of variables to the form

[math]\displaystyle{ \sum\limits_{i=1}^{n} \xi_i^2 - \sum\limits_{i=n+1}^{2n} \xi_i^2. }[/math]

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

• 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, http://projecteuclid.org/euclid.dmj/1077490637 
• 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