Editing Radon transform (section)

==Definition==
Let <math>f(\textbf x) = f(x,y)</math> be a function that satisfies the three regularity conditions:{{sfn|Radon|1986}}

# <math>f(\textbf x)</math> is continuous;
# the double integral <math>\iint\dfrac{\vert f(\textbf x)\vert }{\sqrt{x^2+y^2}} \, dx \, dy</math>, extending over the whole plane, converges;
# for any arbitrary point <math>(x,y)</math> on the plane it holds that <math>\lim_{r\to\infty}\int_0^{2\pi} f(x+r\cos\varphi,y+r\sin\varphi) \, d\varphi=0.</math>


The Radon transform, <math>Rf</math>, is a function defined on the space of straight lines <math>L \subset \mathbb R^2</math> by the [[Line integral#Definition|line integral]] along each such line as:
<math display="block">Rf(L) = \int_L f(\mathbf{x}) \vert d\mathbf{x}\vert .</math>Concretely, the parametrization of any straight line ''<math>L</math>'' with respect to arc length <math>z</math> can always be written:<math display="block">(x(z),y(z)) = \Big( (z\sin\alpha+s\cos\alpha), (-z \cos\alpha + s\sin\alpha)  \Big) \,</math>where <math>s</math> is the distance of <math>L </math> from the origin and <math>\alpha</math> is the angle the normal vector to ''<math>L</math>'' makes with the <math>X</math>-axis. It follows that the quantities <math>(\alpha,s)</math> can be considered as coordinates on the space of all lines in <math>\mathbb R^2</math>, and the Radon transform can be expressed in these coordinates by: <math display="block">\begin{align}
Rf(\alpha,s)
&= \int_{-\infty}^\infty f(x(z),y(z)) \, dz\\
&= \int_{-\infty}^\infty f\big(  (z\sin\alpha+s\cos\alpha), (-z\cos\alpha+s\sin\alpha) \big) \, dz.
\end{align}</math>More generally, in the <math>n</math>-dimensional [[Euclidean space]] <math>\mathbb R^n</math>, the Radon transform of a function <math>f</math> satisfying the regularity conditions is a function ''<math>Rf</math>'' on the space <math>\Sigma_n</math> of all [[hyperplane]]s in <math>\mathbb R^n</math>.  It is defined by:

{{multiple image
| align = left
| direction = horizontal
| image1 = SheppLogan_Phantom.svg
| caption1 = [[Shepp-Logan Phantom|Shepp Logan phantom]]
| image2 = Shepp logan radon.png
| caption2 = Radon transform
| image3 = Shepp logan iradon.png
| caption3 = Inverse Radon transform
| total_width = 400
}}

<math display="block">Rf(\xi) = \int_\xi f(\mathbf{x})\, d\sigma(\mathbf{x}), \quad \forall \xi \in \Sigma_n</math>where the integral is taken with respect to the natural [[hypersurface]] [[measure (mathematics)|measure]], <math>d \sigma</math> (generalizing the <math>\vert d\mathbf{x}\vert</math> term from the <math>2</math>-dimensional case). Observe that any element of <math>\Sigma_n</math> is characterized as the solution locus of an equation <math>\mathbf{x}\cdot\alpha = s</math>, where <math>\alpha \in S^{n-1}</math> is a [[unit vector]] and <math>s \in \mathbb R</math>.  Thus the <math>n</math>-dimensional Radon transform may be rewritten as a function on <math>S^{n-1} \times \mathbb R</math> via: <math display="block">Rf(\alpha,s) = \int_{\mathbf{x}\cdot\alpha = s} f(\mathbf{x})\, d\sigma(\mathbf{x}).</math>It is also possible to generalize the Radon transform still further by integrating instead over <math>k</math>-dimensional affine subspaces of <math>\mathbb R^n</math>.  The [[X-ray transform]] is the most widely used special case of this construction, and is obtained by integrating over straight lines.