The sinogram
The sinogram
The 2D Radon transform is usually graphically represented as a sinogram, which means the intensity values in the coordinate system of variables .
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 (x_{0},y_{0}). With that:
Thus


The result is nonzero only in points, where
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 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 . In the (x,y) space the expression describing this line is:
Now let us take its Radon transform:
substituting
Taking the coefficient of s:
if the result of the expression is
that results in a bounded result as it does not contain singularity.
If holds then a Dirac delta is independent of the s integration variable:
Finally we obtain nonzero results at the point of on the sinogram, apart from the finite part.


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 .
Noncentered disk:


Noncentered square:


SheppLogan head phanstom:


Picture of a foal:


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