T-I-3draft: Difference between revisions
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\hat f(t,\mathbf{u_t})=\int_{-\infty}^{+\infty}f(t\mathbf{u_t}+s\mathbf{u_\Phi})\,\mathrm{d}s. | \hat f(t,\mathbf{u_t})=\int_{-\infty}^{+\infty}f(t\mathbf{u_t}+s\mathbf{u_\Phi})\,\mathrm{d}s. | ||
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''' 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:''' Using the relation | ||
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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', | |||
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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>). |
Revision as of 22:06, 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.
Q4: A natural pair of coordinates, associated to the family of lines obtained from a given , is the pair of coordinates of a point in the basis related to the line that passes through that point. Provide the expression for as a function of , as well as the expression for as a function of . Deduce the relation between the surface elements and .
Definition of Radon transform
Definition: The Radon transform of a function is the function defined by the following expression
Use of the 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
propose a definition of Radon transform in the form of a surface integral.
Q6: Propose a definition of the Radon transform of a function ().