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The sinogram

The sinogram

 
The 2D Radon transform is usually graphically represented as a sinogram, which means the intensity values \mathfrak{R}f\left ( \vartheta,t \right ) in the coordinate system of variables $\left ( \vartheta,t \right )}.

To understand the sinogram ( the result of a Radon transform) let us investigate the sinogram of a point and a line.

The Radon transform of a point

Let us take a point with coordinates (x0,y0). With that:

$ f\left ( x,y)=\delta \left ( x-x_{0} \right )\delta \left ( y-y_{0} \right )
Thus
\mathfrak{R}f\left ( t,\vartheta \right )=\int_{-\infty }^{\infty}\int_{-\infty }^{\infty}\delta \left ( x-x_{0} \right )\delta \left ( y-y_{0} \right ) \delta \left ( t-x\cos \left (\vartheta  \right )-y\sin \left (  \vartheta\right )\right )dxdy=
\delta \left ( t-x_{0}\cos \left (\vartheta  \right )-y_{0}\sin \left (  \vartheta\right )\right )


Image
Pointin the (x,y) space


Image
Point in the sinogram space

 
The result is nonzero only in points, where

$t=x_{0}\cos \left (\vartheta  \right )-y_{0}\sin \left (  \vartheta\right )

Now the result obtained -as the Radon transform of a point- resembles a sine, this is why the graph representation of the Radon transform of variables $\left ( \vartheta,t \right )} is called a sinogram.

Radon transform of a line

Now let us take a line and use the usual parametrization of offset and angle and choose some fixed values for them of $\left ( \vartheta_{0},t_{0} \right ). In the (x,y) space the expression describing this line is:

$  f\left ( x,y \right )=\delta \left ( t_{0}-x\cos \left (\vartheta_{0}  \right )-y\sin \left (  \vartheta_{0}\right )\right )
Now let us take its Radon transform:

$\mathfrak{R}f=\int_{-\infty }^{\infty }f\left ( t\cos \vartheta -s\sin \vartheta ,t\sin \vartheta +s\cos \vartheta  \right )ds
substituting
$\int_{-\infty }^{\infty }\left \delta (t_{0}- \left (t\cos \vartheta -s\sin \vartheta  \right )\cos \vartheta _{0} -\left (t\sin \vartheta +s\cos \vartheta  \right )\sin \vartheta _{0}  \right )  ds=
\int_{-\infty }^{\infty }\left \delta (t_{0}- t\cos\left ( \vartheta -\vartheta_{0} \right ) -s\sin \left (\vartheta-\vartheta_{0}  \right )  \right )  ds
Taking the coefficient of s:
$ \int_{-\infty }^{\infty }\frac{1}{\left | sin \left (\vartheta-\vartheta_{0}  \right ) \right |} \left \delta (\frac{t_{0}- t\cos\left ( \vartheta -\vartheta_{0} \right )}{sin \left (\vartheta-\vartheta_{0}  \right )} -s  \right )  ds

if the result of the expression $\vartheta\neq \vartheta_{0} is
$
\frac{1}{\left | sin \left (\vartheta-\vartheta_{0}  \right ) \right |}
that results in a bounded result as it does not contain singularity.
If $\vartheta= \vartheta_{0} holds then a Dirac delta is independent of the s integration variable:

$ \int_{-\infty }^{\infty }\left \delta (t_{0}- t \right ) ds =\left\{\begin{matrix}
\infty  \mid t_{0}=t\\ 
0 \mid t_{0}\neq t
\end{matrix}\right.

Finally we obtain nonzero results at the point of $\left ( \vartheta_{0},t_{0} \right ) on the sinogram, apart from the finite part.


Image
line in the (x,y) spacen

Image
Radon transform of a line (approximate)

 

General sinograms

 
Interpreting a sinogram is not an easy task, as a starting point, based on the above, points of a sinogram may correspond to lines. Usually, the more recognizable features are sinusoids that belong to compact structures in the (x,y) space.

Now we show some random examples .

Non-centered disk:


Image
image in the (x,y) space

Image
Sinogram (Radon transform) X-axis: angle, Y axis: paramtere t

 
Non-centered square:


Image
image in the (x,y) space

Image
Sinogram (Radon transform) X-axis: angle, Y axis: paramtere t

 
Shepp-Logan head phanstom:


Image
image in the (x,y) space

Image
Sinogram (Radon transform) X-axis: angle, Y axis: paramtere t

 
Picture of a foal:


Image
image of a foal

Image
Radon transform of the picture of a foal

 
In the next section we discuss the general properties of the Radon transform.

 


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