=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 ===
[[File:Line_parametrisation.png |thumb|right| '''Figure 1:''' Parametrisation <math>(t,\Phi)</math> 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 <math>t</math> and <math>\Phi</math>, where <math>(t,\Phi)\in \mathbb{R}\times [0;2\pi[</math>, as 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>
Show that for each given line there exist two possible pairs of values <math>(t,\Phi)</math>.

'''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>
Show that for each pair <math>(t,\Phi)</math> there exists one unique ''oriented'' line.

'''Q4:''' A natural pair of coordinates, associated to the family of lines obtained from a given <math>\Phi</math>, is the pair <math>(t,s)\in \mathbb{R}^2</math> of coordinates of a point in the basis <math>(\mathbf{u_t},\mathbf{u_\Phi})</math> related to the line that passes through that point. Provide the expression for <math>(x,y)</math> as a function of <math>(t,s)</math>, as well as the expression for <math>(t,s)</math> as a function of <math>(x,y)</math>. Deduce the relation between the surface elements <math>\textrm{d}x \,\textrm{d}y</math> and <math>\textrm{d}s \,\textrm{d}t</math>.

=== Definition of Radon transform ===
''' Definition: ''' The Radon transform of a function <math> f : \mathbb{R}^2\rightarrow \mathbb{R}</math> is the function <math>\hat f : \mathbb{R}^2\rightarrow \mathbb{R}</math> defined by the following expression

<center><math>
\hat f(t,\mathbf{u_t})=\int_{-\infty}^{+\infty}f(t\mathbf{u_t}+s\mathbf{u_\Phi})\,\mathrm{d}s.
</math></center>

''' Use of the <math>\delta-</math>distribution: ''' The definition of Radon transform can be elegantly written by means of the Dirac-delta distribution. This formulation provides the advantage of generalising the definition of Radon transform to arbitrary dimension.

'''Q5:''' Using the relation
<center><math>
f(t\mathbf{u_t}+s\mathbf{u_\Phi})=\int_{-\infty}^{+\infty}f(t' \mathbf{u_{t'}}+s\mathbf{u_\Phi})\delta(t'-t)\mathrm{d}t',
</math></center>
propose a definition of Radon transform in the form of a surface integral.

'''Q6:''' Propose a definition of the Radon transform of a function <math>f:\mathbb{R}^n\rightarrow \mathbb{R}</math> (<math>n\geq 2</math>). Give a geometrical interpretation of the Radon transform in dimension <math>n=3</math>.

From now on, we will always consider the Radon transform in the two-dimensional case <math>n=2</math>.

=== Some examples ===

==== No calculation needed ====
''' Q7: '''In Figure 2, match each image with the corresponding Radon transform. Images represent functions <math>f: \mathbb{R}^2\rightarrow \mathbb{R}^+</math>. Specify the meaning of the grey scales in the different figures. Draw the axes <math>t,\Phi</math> on the Radon transforms, knowing that in the images in the direct space (plane <math>x,y</math>), the origin of axes <math>x,y</math> is located at the center of the figure.
<center> Some binary images...</center>
[[File:binary_images.png | 800px|thumb|center]]
<center> ... and their Radon transform (in random order):</center>
[[File:radon_transform.png | 800px| thumb|center| ''' Figure 2:''' Qualitative illustration of the Radon transform.]]

'''Q8:''' Indicate (without doing any calculation) the profile of the Radon transform of a constant function on a quasi-point-like support. Similarly, indicate (without any calculation) the profile of the Radon transform of a constant function on a line, and on a line segment.

==== Hand calculations ====
'''Q9:''' Compute the Radon transform of a constant function on a disc of radius <math>R</math>, and null outside.

'''Q10:''' Compute the Radon transform of the function <math>f(x,y)=a\,\exp\left(-\frac{x^2+y^2}{\sigma^2}\right)</math>.

=== Expression for radial functions ===
'''Q11:''' Show that in the case of radial function <math>f(x,y)=F(r)=F(\sqrt{x^2+y^2})</math> the Radon transform can be written as a simple integral transform of the function <math>F(r)</math>.

== Principles of X-ray tomography==
What are the physical principles underlying X-ray tomography? In particular, we will specify which physical quantity is measured, and which quantity we try to reconstruct in form of an image. We will verify that we can express the measured quantities as a Radon transform of the parameter that we try to image. X-ray tomography is based on the inversion of the Radon transform, knowing how to deduce <math>f</math> from a measure of <math>\hat f</math>. The following section addresses the principle at the basis of this inversion.

== Inversion of the Radon transform==
=== Back-projection formula===
'''Q12:''' Explain why the following function
<center><math>
g(\mathbf{r})=\frac{1}{2\pi}\int_0^{2\pi}\hat f (\mathbf{r}\cdot \mathbf{u_t},\mathbf{u_t})\,\mathrm{d}t,
</math></center>
obtained via an angular mean of the Radon transform of a function <math>f</math>, is likely to resemble to the function <math>f</math>. This formula is called ''back-projection formula''.
<center>Some images...</center>
[[File:images_radon.png | 800px| thumb|center]]
<center>... and the back-projection of their Radon transform.</center>
[[File:back_projection.png | 800px| thumb|center| ''' Figure 3:''' Illustration of the back-projection formula.]]

T-I-3draft

2021-10-15T21:13:03Z

Frmignacco: /* Back-projection formula */

=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 ===
[[File:Line_parametrisation.png |thumb|right| '''Figure 1:''' Parametrisation <math>(t,\Phi)</math> 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 <math>t</math> and <math>\Phi</math>, where <math>(t,\Phi)\in \mathbb{R}\times [0;2\pi[</math>, as 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>
Show that for each given line there exist two possible pairs of values <math>(t,\Phi)</math>.

'''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>
Show that for each pair <math>(t,\Phi)</math> there exists one unique ''oriented'' line.

'''Q4:''' A natural pair of coordinates, associated to the family of lines obtained from a given <math>\Phi</math>, is the pair <math>(t,s)\in \mathbb{R}^2</math> of coordinates of a point in the basis <math>(\mathbf{u_t},\mathbf{u_\Phi})</math> related to the line that passes through that point. Provide the expression for <math>(x,y)</math> as a function of <math>(t,s)</math>, as well as the expression for <math>(t,s)</math> as a function of <math>(x,y)</math>. Deduce the relation between the surface elements <math>\textrm{d}x \,\textrm{d}y</math> and <math>\textrm{d}s \,\textrm{d}t</math>.

=== Definition of Radon transform ===
''' Definition: ''' The Radon transform of a function <math> f : \mathbb{R}^2\rightarrow \mathbb{R}</math> is the function <math>\hat f : \mathbb{R}^2\rightarrow \mathbb{R}</math> defined by the following expression

<center><math>
\hat f(t,\mathbf{u_t})=\int_{-\infty}^{+\infty}f(t\mathbf{u_t}+s\mathbf{u_\Phi})\,\mathrm{d}s.
</math></center>

''' Use of the <math>\delta-</math>distribution: ''' The definition of Radon transform can be elegantly written by means of the Dirac-delta distribution. This formulation provides the advantage of generalising the definition of Radon transform to arbitrary dimension.

'''Q5:''' Using the relation
<center><math>
f(t\mathbf{u_t}+s\mathbf{u_\Phi})=\int_{-\infty}^{+\infty}f(t' \mathbf{u_{t'}}+s\mathbf{u_\Phi})\delta(t'-t)\mathrm{d}t',
</math></center>
propose a definition of Radon transform in the form of a surface integral.

'''Q6:''' Propose a definition of the Radon transform of a function <math>f:\mathbb{R}^n\rightarrow \mathbb{R}</math> (<math>n\geq 2</math>). Give a geometrical interpretation of the Radon transform in dimension <math>n=3</math>.

From now on, we will always consider the Radon transform in the two-dimensional case <math>n=2</math>.

=== Some examples ===

==== No calculation needed ====
''' Q7: '''In Figure 2, match each image with the corresponding Radon transform. Images represent functions <math>f: \mathbb{R}^2\rightarrow \mathbb{R}^+</math>. Specify the meaning of the grey scales in the different figures. Draw the axes <math>t,\Phi</math> on the Radon transforms, knowing that in the images in the direct space (plane <math>x,y</math>), the origin of axes <math>x,y</math> is located at the center of the figure.
<center> Some binary images...</center>
[[File:binary_images.png | 800px|thumb|center]]
<center> ... and their Radon transform (in random order):</center>
[[File:radon_transform.png | 800px| thumb|center| ''' Figure 2:''' Qualitative illustration of the Radon transform.]]

'''Q8:''' Indicate (without doing any calculation) the profile of the Radon transform of a constant function on a quasi-point-like support. Similarly, indicate (without any calculation) the profile of the Radon transform of a constant function on a line, and on a line segment.

==== Hand calculations ====
'''Q9:''' Compute the Radon transform of a constant function on a disc of radius <math>R</math>, and null outside.

'''Q10:''' Compute the Radon transform of the function <math>f(x,y)=a\,\exp\left(-\frac{x^2+y^2}{\sigma^2}\right)</math>.

=== Expression for radial functions ===
'''Q11:''' Show that in the case of radial function <math>f(x,y)=F(r)=F(\sqrt{x^2+y^2})</math> the Radon transform can be written as a simple integral transform of the function <math>F(r)</math>.

== Principles of X-ray tomography==
What are the physical principles underlying X-ray tomography? In particular, we will specify which physical quantity is measured, and which quantity we try to reconstruct in form of an image. We will verify that we can express the measured quantities as a Radon transform of the parameter that we try to image. X-ray tomography is based on the inversion of the Radon transform, knowing how to deduce <math>f</math> from a measure of <math>\hat f</math>. The following section addresses the principle at the basis of this inversion.

== Inversion of the Radon transform==
=== Back-projection formula===
'''Q12:''' Explain why the following function
<center><math>
g(\mathbf{r})=\frac{1}{2\pi}\int_0^{2\pi}\hat f (\mathbf{r}\cdot \mathbf{u_t},\mathbf{u_t})\,\mathrm{d}t,
</math></center>
obtained via an angular mean of the Radon transform of a function <math>f</math>, is likely to resemble to the function <math>f</math>. This formula is called ''back-projection formula''.
<center>Some images...</center>

<center>... and the back-projection of their Radon transform.</center>

File:Back projection.png

2021-10-15T21:12:05Z

Frmignacco:

File:Images radon.png

2021-10-15T21:11:50Z

Frmignacco:

T-I-3draft

T-I-3draft

2021-10-15T19:39:29Z

Frmignacco: /* Preliminaries: parametrisation of a line in the plane */