T-I-3draft: Difference between revisions
Frmignacco (talk | contribs) |
Frmignacco (talk | contribs) |
||
Line 9: | Line 9: | ||
'''Q2:''' In the context of Radon transform, we choose to define a line via the parameters <math>t</math> and <math>\Phi</math>, where <math>(t,\Phi)\in \mathbb{R}\times [0;2\pi[</math>, displayed in Figure 1. Each angle <math>\Phi</math> is associated to a unique unit vector <math>\mathbf{u_t}</math>: | '''Q2:''' In the context of Radon transform, we choose to define a line via the parameters <math>t</math> and <math>\Phi</math>, where <math>(t,\Phi)\in \mathbb{R}\times [0;2\pi[</math>, displayed in Figure 1. Each angle <math>\Phi</math> is associated to a unique unit vector <math>\mathbf{u_t}</math>: | ||
<center><math>\mathbf{u_t} = \cos(\Phi) \mathbf{u_x}+ \sin(\Phi)\mathbf{u_y}.</math></center> | |||
<center><math>\mathbf{u_t} = \cos(\Phi) \mathbf{u_x}+ \sin(\Phi)\mathbf{u_y}</math></center> | |||
Show that each given line corresponds to two possible pairs of values <math>(t,\Phi)</math>. | Show that each given line corresponds to two possible pairs of values <math>(t,\Phi)</math>. |
Revision as of 21:34, 15 October 2021
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
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 and , where , displayed in Figure 1. Each angle is associated to a unique unit vector :
Show that each given line corresponds to two possible pairs of values .