**The 2D Radon transform**

## The 2D Radon transform

### Definition

The Radon transform using the notations in the section on the description of linear structures in 2D with ) for an *f* real function:

Let us follow the convention and , now the function *f* Radon transformed with variables :

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.

### Example

Let us take a "disk" of radius *r*, where the function values is *1* inside radius *r* otherwise 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 . Thus:

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 coordinate system, that is called a sinogram, and it is the topic of the next section.