T-I-3draft - Revision history

Frmignacco at 23:14, 30 November 2021

2021-11-30T23:14:54Z

← Older revision		Revision as of 01:14, 1 December 2021
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	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''.		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>Some images...</center>
	[[File:~~images_radon~~.png \| 800px\| thumb\|center]]		[[File:images radon.png \| 800px\| thumb\|center]]
	<center>... and the back-projection of their Radon transform.</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.]]		[[File:back_projection.png \| 800px\| thumb\|center\| ''' Figure 3:''' Illustration of the back-projection formula.]]

Frmignacco at 23:14, 30 November 2021

2021-11-30T23:14:37Z

← Older revision		Revision as of 01:14, 1 December 2021
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	== Radon transform ==		== Radon transform ==
	=== Preliminaries: parametrisation of a line in the plane ===		=== 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.]]		[[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.		'''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.

Frmignacco: /* Radon transform and X-ray tomography */

2021-11-30T22:31:07Z

Radon transform and X-ray tomography

← Older revision		Revision as of 00:31, 1 December 2021
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	=Radon transform and X-ray tomography=		=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 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 ==		== Radon transform ==

Alberto: /* Inversion of the Radon transform */

2021-11-10T16:38:10Z

Inversion of the Radon transform

← Older revision		Revision as of 18:38, 10 November 2021
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	=== Projection-slice theorem ===		=== Projection-slice theorem ===
	Show that, for a given <math>\Phi</math>, the one-dimensional Fourier transform of <math>\hat f (t,\mathbf{u_t})</math> over the variable <math>t</math> is equal to the two-dimensional transform of <math>f(\mathbf{r})</math>:		'''Q13:''' Show that, for a given <math>\Phi</math>, the one-dimensional Fourier transform of <math>\hat f (t,\mathbf{u_t})</math> over the variable <math>t</math> is equal to the two-dimensional transform of <math>f(\mathbf{r})</math>:
	<center><math>		<center><math>
	\int_{-\infty}^{+\infty}\hat f (t,\mathbf{u_t})\, e^{-ikt}\,\mathrm{d}t =\int \int f(\mathbf{r})\, e^{-i(k\mathbf{u_t})\cdot \mathbf{r}}\mathrm{d}\mathbf{r}		\int_{-\infty}^{+\infty}\hat f (t,\mathbf{u_t})\, e^{-ikt}\,\mathrm{d}t =\int \int f(\mathbf{r})\, e^{-i(k\mathbf{u_t})\cdot \mathbf{r}}\mathrm{d}\mathbf{r}
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	=== Inversion formula ===		=== Inversion formula ===
	For brevity, we choose to denote the one-dimensional Fourier transform (notation: <math>\tilde f</math>) of the Radon transform (notation: <math>\hat f</math>) over the variable <math>t</math> as follows:		'''Q14: (to do in class only)''' For brevity, we choose to denote the one-dimensional Fourier transform (notation: <math>\tilde f</math>) of the Radon transform (notation: <math>\hat f</math>) over the variable <math>t</math> as follows:
	<center><math>		<center><math>
	\widetilde{(\hat f)}\,(k,\mathbf{u_t})=\mathrm{TF}_{1D(t)}\left(\hat f (t,\mathbf{u_t})\right)[k].		\widetilde{(\hat f)}\,(k,\mathbf{u_t})=\mathrm{TF}_{1D(t)}\left(\hat f (t,\mathbf{u_t})\right)[k].

Alberto at 16:35, 10 November 2021

2021-11-10T16:35:59Z

← Older revision		Revision as of 18:35, 10 November 2021
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	=== Expression for radial functions ===		=== 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>.~~		'''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==~~
	~~This section is about a documentation project that should be done in pairs. Each pair of students will present their work at the blackboard in ~5 minutes.~~


	What are the physical principles underlying X-ray tomography? In particular, you should specify which physical quantity is measured, and which quantity we try to reconstruct in form of an image. You should 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.		The following section addresses the principle at the basis of this inversion.

Frmignacco at 07:29, 16 October 2021

2021-10-16T07:29:13Z

← Older revision		Revision as of 09:29, 16 October 2021
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	== Principles of X-ray tomography==		== 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.		This section is about a documentation project that should be done in pairs. Each pair of students will present their work at the blackboard in ~5 minutes.


			What are the physical principles underlying X-ray tomography? In particular, you should specify which physical quantity is measured, and which quantity we try to reconstruct in form of an image. You should 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==		== Inversion of the Radon transform==

Frmignacco: /* Hand calculations */

2021-10-16T05:16:38Z

Hand calculations

← Older revision		Revision as of 07:16, 16 October 2021
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	==== Hand calculations ====		==== Hand calculations ====
	'''Q9:''' Compute the Radon transform of a constant ~~function~~ on a disc of radius <math>R</math>, and null outside.		'''Q9:''' Compute the Radon transform of a function that is constant 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>.		'''Q10:''' Compute the Radon transform of the function <math>f(x,y)=a\,\exp\left(-\frac{x^2+y^2}{\sigma^2}\right)</math>.

Frmignacco at 05:12, 16 October 2021

2021-10-16T05:12:10Z

← Older revision		Revision as of 07:12, 16 October 2021
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	== Radon transform ==		== Radon transform ==
	=== Preliminaries: parametrisation of a line in the plane ===		=== 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.)]]		[[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.		'''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.

Frmignacco: /* Inversion formula */

2021-10-15T21:40:06Z

Inversion formula

← Older revision		Revision as of 23:40, 15 October 2021
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	<center><math>		<center><math>
	\widetilde{(\hat f)}\,(k,\mathbf{u_t})=\mathrm{TF}_{1D(t)}\left(\hat f (t,\mathbf{u_t})\right)[k].		\widetilde{(\hat f)}\,(k,\mathbf{u_t})=\mathrm{TF}_{1D(t)}\left(\hat f (t,\mathbf{u_t})\right)[k].
			</math></center>
			From the projection-slice theorem, show that the inversion formula of the Radon transform is:
			<center><math>
			f(\mathbf{r})=\frac{1}{(2\pi)^2}\int_0^\pi\int_{-\infty}^{+\infty}\|k\|\widetilde{(\hat f)}\,(k,\mathbf{u_t})\,e^{+i(\mathbf{r}\cdot\mathbf{u_t})k}\,\mathrm{d}k\mathrm{d}\Phi.
	</math></center>		</math></center>

Frmignacco: /* Inversion formula */

2021-10-15T21:37:03Z

Inversion formula

← Older revision		Revision as of 23:37, 15 October 2021
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	For brevity, we choose to denote the one-dimensional Fourier transform (notation: <math>\tilde f</math>) of the Radon transform (notation: <math>\hat f</math>) over the variable <math>t</math> as follows:		For brevity, we choose to denote the one-dimensional Fourier transform (notation: <math>\tilde f</math>) of the Radon transform (notation: <math>\hat f</math>) over the variable <math>t</math> as follows:
	<center><math>		<center><math>
	\widetilde{(\~~widehat~~ f)}\,(k,\mathbf{u_t})=\mathrm{TF}_{1D(t)}\left(\hat f (t,\mathbf{u_t})\right)[k].		\widetilde{(\hat f)}\,(k,\mathbf{u_t})=\mathrm{TF}_{1D(t)}\left(\hat f (t,\mathbf{u_t})\right)[k].
	</math></center>		</math></center>