# 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 (x0,y0). With that:

Thus

 Pointin the (x,y) space Point in the sinogram space

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.

 line in the (x,y) spacen 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 in the (x,y) space Sinogram (Radon transform) X-axis: angle, Y axis: paramtere t

Non-centered square:

 image in the (x,y) space Sinogram (Radon transform) X-axis: angle, Y axis: paramtere t

 image in the (x,y) space Sinogram (Radon transform) X-axis: angle, Y axis: paramtere t

Picture of a foal:

 image of a foal Radon transform of the picture of a foal

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