T-I-3

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.

Radon transform

Preliminaries: parametrisation of a line in the plane

thumb|right| Figure 1: Parametrisation 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)} 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 $t$ and $\Phi$, where $(t,\Phi )\in \mathbb {R} \times [0;2\pi [$, as displayed in Figure 1. Each angle $\Phi$ 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}} :

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} = \cos(\Phi) \mathbf{u_x}+ \sin(\Phi)\mathbf{u_y}.}

Show that for each given line there exist two possible pairs of values $(t,\Phi )$.

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:

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}} = -\sin(\Phi)\mathbf{u_x} + \cos(\Phi)\mathbf{u_y}.}

Show that for each pair $(t,\Phi )$ there exists one 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 $(\mathbf {u_{t}} ,\mathbf {u_{\Phi }} )$ 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 $(t,s)$ 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 ${\textrm {d}}s\,{\textrm {d}}t$.

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

${\hat {f}}(t,\mathbf {u_{t}} )=\int _{-\infty }^{+\infty }f(t\mathbf {u_{t}} +s\mathbf {u_{\Phi }} )\,\mathrm {d} s.$

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: Using the relation

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(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', }

propose a definition of Radon transform in the form of a surface integral.

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

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

Some examples

No calculation needed

Q7: In Figure 2, match each image with the corresponding Radon transform. Images represent functions 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}^+} . Specify the meaning of the grey scales in the different figures. Draw the axes 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} on the Radon transforms, knowing that in the images in the direct space (plane $x,y$), the origin of axes 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} is located at the center of the figure.

Some binary images...

... and their Radon transform (in random order):

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 function that is constant on a disc of radius 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 R} , and null outside.

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

Expression for radial functions

Q11: Show that in the case of radial 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(x,y)=F(r)=F(\sqrt{x^2+y^2})} the Radon transform can be written as a simple integral transform of 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 F(r)} .

Inversion of the Radon transform

Back-projection formula

Q12: Explain why the following function

$g(\mathbf {r} )={\frac {1}{2\pi }}\int _{0}^{2\pi }{\hat {f}}(\mathbf {r} \cdot \mathbf {u_{t}} ,\mathbf {u_{t}} )\,\mathrm {d} \Phi ,$

obtained via an angular mean of 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} , is likely to resemble to 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 f} . This formula is called back-projection formula.

Some images...

... and the back-projection of their Radon transform.

Figure 3: Illustration of the back-projection formula.

Figure 3 displays the application of the back-projection formula to the examples that we have previously considered (binary images), as well as to an example from medical imaging context. Does the back-projection formula allow to inverse the Radon transform? Explain.

Projection-slice theorem

Show that, for a given $\Phi$, the one-dimensional Fourier transform 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 \hat f (t,\mathbf{u_t})} over the variable 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} is equal to the two-dimensional transform of $f(\mathbf {r} )$:

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 \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} }

or, similarly,

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{TF}_{1D(t)}\left(\hat f(t,\mathbf{u_t})\right)[k]=\textrm{TF}_{2D}\left( f(x,y)\right)[k\mathbf{u_t}]. }

In the context of medical imaging, this relation is called projection-slice theorem. It relates the two-dimensional Fourier transform of a function to its Radon transform.

Recall the meaning of the Fourier transform of a function of one variable and of two variables. From the projection-slice theorem, deduce an interpretation of the two-dimensional Fourier transform in terms of the one-dimensional Fourier transform.

Inversion formula

For brevity, we choose to denote the one-dimensional Fourier transform (notation: ${\tilde {f}}$) of the Radon transform (notation: 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} ) over the variable 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} as follows:

${\widetilde {({\hat {f}})}}\,(k,\mathbf {u_{t}} )=\mathrm {TF} _{1D(t)}\left({\hat {f}}(t,\mathbf {u_{t}} )\right)[k].$

From the projection-slice theorem, show that the inversion formula of the Radon transform is:

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(\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. }