T-I-3draft: Difference between revisions
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'''Q3:''' We choose to ''orient'' the line positively along the unit vector <math>\mathbf{u_{\Phi}}</math>, defined by: | '''Q3:''' We choose to ''orient'' the line positively along the unit vector <math>\mathbf{u_{\Phi}}</math>, defined by: | ||
<center><math> \mathbf{u_{\Phi}} = -\sin(\Phi)\mathbf{u_x} + \cos(\Phi)\mathbf{u_y}.</math></center> | <center><math> \mathbf{u_{\Phi}} = -\sin(\Phi)\mathbf{u_x} + \cos(\Phi)\mathbf{u_y}.</math></center> | ||
Show that for each pair <math>(t,\Phi)</math | Show that for each pair <math>(t,\Phi)</math> there exists a unique ''oriented'' line. |
Revision as of 21:39, 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 for each given line there exist two possible pairs of values .
Q3: We choose to orient the line positively along the unit vector , defined by:
Show that for each pair there exists a unique oriented line.