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

Figure 1: Parametrisation $(t,\Phi )$ 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 $\mathbf {u_{t}}$:

$\mathbf {u_{t}} =\cos(\Phi )\mathbf {u_{x}} +\sin(\Phi )\mathbf {u_{y}} .$

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 $\mathbf {u_{\Phi }}$, defined by:

$\mathbf {u_{\Phi }} =-\sin(\Phi )\mathbf {u_{x}} +\cos(\Phi )\mathbf {u_{y}} .$

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 $\Phi$, is the pair $(t,s)\in \mathbb {R} ^{2}$ 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 $(x,y)$ as a function of $(t,s)$, as well as the expression for $(t,s)$ as a function of $(x,y)$. Deduce the relation between the surface elements ${\textrm {d}}x\,{\textrm {d}}y$ and ${\textrm {d}}s\,{\textrm {d}}t$.

Definition of Radon transform

Definition: The Radon transform of a function $f:\mathbb {R} ^{2}\rightarrow \mathbb {R}$ is the function ${\hat {f}}:\mathbb {R} ^{2}\rightarrow \mathbb {R}$ 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 $\delta -$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

$f(t\mathbf {u_{t}} +s\mathbf {u_{\Phi }} )=\int _{-\infty }^{+\infty }f(t'\mathbf {u_{t}} +s\mathbf {u_{\Phi }} )\delta (t'-t)\mathrm {d} t',$

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

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

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 $f:\mathbb {R} ^{2}\rightarrow \mathbb {R} ^{+}$. Specify the meaning of the grey scales in the different figures. Draw the axes $t,\Phi$ on the Radon transforms, knowing that in the images in the direct space (plane $x,y$), the origin of axes $x,y$ 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 $R$, 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 $f(x,y)=F(r)=F({\sqrt {x^{2}+y^{2}}})$ the Radon transform can be written as a simple integral transform of the function $F(r)$.

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 $f$, is likely to resemble to the function $f$. 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 ${\hat {f}}(t,\mathbf {u_{t}} )$ over the variable $t$ is equal to the two-dimensional transform of $f(\mathbf {r} )$:

$\int _{-\infty }^{+\infty }{\hat {f}}(t,\mathbf {u_{t}} )\,e^{-ikt}\,\mathrm {d} t=\int \int f(\mathbf {r} )\,e^{-i(k\mathbf {u_{t}} )\cdot \mathbf {r} }\mathrm {d} \mathbf {r}$

or, similarly,

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

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: ${\hat {f}}$) over the variable $t$ 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:

$f(\mathbf {r} )={\frac {1}{(2\pi )^{2}}}\int _{0}^{\pi }\int _{-\infty }^{+\infty }|k|{\widetilde {({\hat {f}})}}\,(k,\mathbf {u_{t}} )\,e^{+i(\mathbf {r} \cdot \mathbf {u_{t}} )k}\,\mathrm {d} k\mathrm {d} \Phi .$