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'''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 | '''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 \ | <center><math> | ||
\hat f(t,\mathbf{u_t})=\int_{-\infty}^{+\infty}f(t\mathbf{u_t}+s\mathbf{u_s})\,\mathrm{d}s. | |||
</math></center></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. | ''' 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:''' | '''Q5:''' | ||
Revision as of 22:01, 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (t,\Phi)\in \mathbb{R}\times [0;2\pi[} , displayed in Figure 1. Each angle Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi} is associated to a unique unit vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u_t}} :
Show that for each given line there exist two possible pairs of values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (t,\Phi)} .
Q3: We choose to orient the line positively along the unit vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{u_{\Phi}}} , defined by:
Show that for each pair Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (t,\Phi)} there exists a unique oriented line.
Q4: A natural pair of coordinates, associated to the family of lines obtained from a given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi}
, is the pair Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (t,s)\in \mathbb{R}^2}
of coordinates of a point in the basis Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbf{u_t},\mathbf{u_\Phi})}
related to the line that passes through that point. Provide the expression for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y)}
as a function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (t,s)}
, as well as the expression for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (t,s)}
as a function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y)}
. Deduce the relation between the surface elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{d}x \,\textrm{d}y}
and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{d}s \,\textrm{d}t}
.
Definition of Radon transform
Definition: The Radon transform of a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : \mathbb{R}^2\rightarrow \mathbb{R}} is the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat f : \mathbb{R}^2\rightarrow \mathbb{R}} defined by the following expression
</math>
Use of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta-} 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: