John's equation
Encyclopedia
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
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.
X-ray transform
In mathematics, the X-ray transform is an integral transform introduced by that is one of the cornerstones of modern integral geometry. It is very closely related to the Radon transform, and coincides with it in two dimensions...
of a function. It is named after Fritz John
Fritz John
Fritz John was a German-born mathematician specialising in partial differential equations and ill-posed problems. His early work was on the Radon transform and he is remembered for John's equation.-Biography:...
.
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 wherethen 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
Richard Courant
Richard Courant was a German American mathematician.- Life :Courant was born in Lublinitz in the German Empire's Prussian Province of Silesia. During his youth, his parents had to move quite often, to Glatz, Breslau, and in 1905 to Berlin. He stayed in Breslau and entered the university there...
) is a second order partial differential equation of the form
where
, such that the quadratic formQuadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
can be reduced by a linear change of variables to the form
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.