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John's equation is an
ultrahyperbolic partial differential equation In the mathematical field of differential equations, the ultrahyperbolic equation is a partial differential equation (PDE) for an unknown scalar function of variables of the form \frac + \cdots + \frac - \frac - \cdots - \frac = 0. More ...
satisfied by the
X-ray transform In mathematics, the X-ray transform (also called ray transform or John transform) is an integral transform introduced by Fritz John in 1938 that is one of the cornerstones of modern integral geometry. It is very closely related to the Radon tra ...
of a function. It is named after Fritz John. Given a function f\colon\mathbb^n \rightarrow \mathbb with compact support the ''X-ray transform'' is the integral over all lines in \mathbb^n. We will parameterise the lines by pairs of points x,y \in \mathbb^n, x \ne y on each line and define ''u'' as the ray transform where : u(x,y) = \int\limits_^ f( x + t(y-x) ) dt. Such functions ''u'' are characterized by John's equations : \frac - \frac=0 which is proved by Fritz John for dimension three and by Kurusa for higher dimensions. In three-dimensional x-ray
computerized tomography A computed tomography scan (CT scan; formerly called computed axial tomography scan or CAT scan) is a medical imaging technique used to obtain detailed internal images of the body. The personnel that perform CT scans are called radiographers ...
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_^ a_\frac + \sum\limits_^ b_i\frac + cu =0 where n \ge 2, such that the
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...
: \sum\limits_^ a_ \xi_i \xi_j can be reduced by a linear change of variables to the form : \sum\limits_^ \xi_i^2 - \sum\limits_^ \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

* * 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 {{doi, 10.1088/0031-9155/47/15/306 Partial differential equations X-ray computed tomography