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 German-American mathematician Fritz John.

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

${\displaystyle u(x,y)=\int \limits _{-\infty }^{\infty }f(x+t(y-x))dt.}$

Such functions ${\displaystyle u}$ are characterized by John's equations

${\displaystyle {\frac {\partial ^{2}u}{\partial x_{i}\partial y_{j}}}-{\frac {\partial ^{2}u}{\partial y_{i}\partial x_{j}}}=0}$

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

${\displaystyle \sum \limits _{i,j=1}^{2n}a_{ij}{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}+\sum \limits _{i=1}^{2n}b_{i}{\frac {\partial u}{\partial x_{i}}}+cu=0}$

where ${\displaystyle n\geq 2}$, such that the quadratic form

${\displaystyle \sum \limits _{i,j=1}^{2n}a_{ij}\xi _{i}\xi _{j}}$

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

${\displaystyle \sum \limits _{i=1}^{n}\xi _{i}^{2}-\sum \limits _{i=n+1}^{2n}\xi _{i}^{2}.}$

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

• 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