T-I-3draft

Radon transform and X-ray tomography

The goal of this homework is to introduce the Radon transform of a two-dimensional function. We will show that this transform is invertible and the inverse involves the Fourier transform in two dimensions. From a practical point of view, the Radon transform is the basis of X-ray tomography (as well as X-ray scanning), applied in the medical context in order to obtain cross-section images of different organs. The second part of the homework consists of a documentation work to be conducted in pairs: each pair of students should prepare a blackboard presentation of approximately five minutes on this part.

Radon transform

Preliminaries: parametrisation of a line in the plane

Figure 1: Parametrisation $(t,\Phi )$ of a line in the plane.)

Q1: In a two-dimensional space, how many parameters are needed in order to define a line? Provide some examples of equations that define a unique line in the plane.

Q2: In the context of Radon transform, we choose to define a line via the parameters $t$ and $\Phi$, where $(t,\Phi )\in \mathbb {R} \times [0;2\pi [$, displayed in Figure 1. Each angle $\Phi$ is associated to a unique unit vector $\mathbf {u_{t}}$:

$\mathbf {u_{t}} =\cos(\Phi )\mathbf {u_{x}} +\sin(\Phi )\mathbf {u_{y}} .$

Show that for each given line there exist two possible pairs of values $(t,\Phi )$.

Q3: We choose to orient the line positively along the unit vector $\mathbf {u_{\Phi }}$, defined by:

$\mathbf {u_{\Phi }} =-\sin(\Phi )\mathbf {u_{x}} +\cos(\Phi )\mathbf {u_{y}} .$

Show that for each pair <math>(t,\Phi)</math/> there exists a unique oriented line.