John's equation

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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 \rightarrow \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 \ne y on each line and define u as the ray transform where

u(x,y) = \int\limits_{-\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

\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 n \ge 2, such that the quadratic form

\sum\limits_{i,j=1}^{2n} a_{ij} \xi_i \xi_j

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

\sum\limits_{i=1}^{n} \xi_i^2 - \sum\limits_{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.


[edit] 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