John's equation explained

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\colon R ^{n} → R with compact support the X-ray transform is the integral over all lines in R ^{n} . We will parameterise the lines by pairs of points x,y \in R ^{n} , x \ne y on each line and define u as the ray transform whereu(x,y) = \int\limits ^{inf ty} _{-inf ty} f( x + t(y - x) ) dt. Such functions u are characterized by John's equations(\partial ^{2u})/(\partial x _{i} \partial y _{j}) - (\partial ^{2u})/(\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

\sum\limits ^{2 n} _{i,j =1} a _{ij} (\partial ^{2} u)/(\partial x _{i} \partial x _{j}) + \sum\limits ^{2 n} _{i =1} b _{i (\partial u)/(\partial x _{i})} + cu =0 where n \ge 2 , such that the quadratic form\sum\limits ^{2 n} _{i,j =1} a _{ij} \xi _{i} \xi _{j} can be reduced by a linear change of variables to the form \sum\limits ^{n} _{i =1} \xi ^{2} _{i} - \sum\limits ^{2 n} _{i = n +1} \xi ^{2 .} _{i} 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

• • Á. Kurusa, A characterization of the Radon transform's range by a system of PDEs, J. Math. Anal. Appl., 161(1991), 218--226.
• S K Patch, Consistency conditions upon 3D CT data and the wave equation, Phys. Med. Biol. 47 No 15 (7 August 2002) 2637-2650