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 $f:mathbb\{R\}^n\; ightarrow\; mathbb\{R\}$ with compact support the "X-ray transform" is the integral over all lines in $mathbb\{R\}^n$. We will parameterise the lines by pairs of points $x,y\; in\; mathbb\{R\}^n,\; x\; e\; y$ on each line and define "u" as the ray transform where:$u(x,y)\; =\; intlimits\_\{-infty\}^\{infty\}\; f(\; x\; +\; t(x-y)\; )\; dt$then "u" satisfies John's equation:$frac\{partial^2u\}\{partial\; x\_i\; partial\; y\_j\}\; -\; frac\{partial^2u\}\{partial\; y\_i\; partial\; x\_j\}=0$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:$sumlimits\_\{i,j=1\}^\{2n\}\; a\_\{ij\}frac\{partial^2\; u\}\{partial\; x\_i\; partial\; x\_j\}\; +\; sumlimits\_\{i=1\}^\{2n\}\; b\_ifrac\{partial\; u\}\{partial\; x\_i\}\; +\; cu\; =0$where $n\; ge\; 2$, such that the quadratic form:$sumlimits\_\{i,j=1\}^\{2n\}\; a\_\{ij\}\; xi\_i\; xi\_j$can be reduced by a linear change of variables to the form :$sumlimits\_\{i=1\}^\{n\}\; xi\_i^2\; -\; sumlimits\_\{i=n+1\}^\{2n\}\; xi\_i^2$Unlike the wave equation and other hyperbolic partial differential equations, 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

* Fritz John, The ultrahyperbolic differential equation with four independent variables, Duke Math. J. 4, no. 2 (1938), 300–322
* 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