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 
with compact support the x-ray transform is the integral over all lines in 
. We will parameterise the lines by pairs of points 
on each line and define u as the ray transform where
then u satisfies John's equation
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
where 
, such that the quadratic form
can be reduced by a linear change of variables to the form
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
