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The 2D Radon transform

The 2D Radon transform



The Radon transform using the notations in the section on the description of linear structures in 2D with $ \mathbf{x}\in \mathbb{R}^{2}, $ t\in \mathbb{R} , \boldsymbol{\omega}\in \mathbb{S}^{1} ) for an f real function:

$ g\left ( L \right )=\mathfrak{R}f =\int_{L}f\left (\textbf{x}  \right )d\textbf{x}

Let us follow the convention $ t\in \left \{ 0,\infty \right \} and $ \vartheta\in \left \{ 0,2\pi \right \}, now the function f Radon transformed with variables $\left ( \vartheta,t \right )} :

$ \mathfrak{R}f\left ( t,\vartheta \right )=\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }
{ f\left ( \mathbf{x}  \right )\delta \left ( t-\boldsymbol{\omega} \mathbf{x} \right )}dxdy=
\int_{-\infty }^{\infty }
{ f\left ( t\boldsymbol{\omega}+s\boldsymbol{\omega}_{\perp}  \right )}ds =

  \int_{-\infty }^{\infty }
{ f\left ( t \cos \left (\vartheta  \right ) - s \sin \left (\vartheta  \right ) ,t \sin \left (\vartheta  \right ) + s \cos \left (\vartheta  \right ) \right )}ds

For the existence of the Radon transform we have to require that the improper integral above exists (more precisely that is integrable in a Lebesgue sense). Johann Radon showed that if f is continuous and has a compact support the Radon transform is unique.


Let us take a "disk" of radius r, where the function values is 1 inside radius r otherwise 0:
$ f\left ( x,y \right ))=\left\{ \begin{matrix} 
 \left \{ \left (x,y  \right ) \mid x^{2}+y^{2} \leq r\right \}   : 1
 \left \{ \left (x,y  \right ) \mid x^{2}+y^{2} >   r\right \}   : 0


As the function value is 1 or 0, the Radon transform will be given by the limits of the support: for an arbitrary angle at a given t the nonzero values are on the domain of $ -\sqrt{r^{2}-t^{2}} \to \sqrt{r^{2}-t^{2}}. Thus:

\mathfrak{R}f\left ( t,\vartheta \right )=\int_{-\sqrt{r^{2}-t^{2}} }^{\sqrt{r^{2}-t^{2}}
{ 1 } ds = 2\sqrt{r^{2}-t^{2}}
if t<r and 0 otherwise. The result is independent of angle, has compact support, but despite the original constant function it is not constant.

A 2D Radon transform can be graphed in a $\left ( \vartheta,t \right )} coordinate system, that is called a sinogram, and it is the topic of the next section.

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