radon transform

File:Sinogram - Two Square Indicator Phantom.svg of two squares shown in the image below. Lighter regions indicate larger function values. Black indicates zero.]]

File:Sinogram Source - Two Squares Phantom.svg

If a function$f$ represents an unknown density, then the Radon transform represents the projection data obtained as the output of a tomographic scan. Hence the inverse of the Radon transform can be used to reconstruct the original density from the projection data, and thus it forms the mathematical underpinning for tomographic reconstruction, also known as iterative reconstruction.

The Radon transform data is often called a sinogram because the Radon transform of an off-center point source is a sinusoid. Consequently, the Radon transform of a number of small objects appears graphically as a number of blurred sine waves with different amplitudes and phases.

The Radon transform is useful in computed axial tomography (CAT scan), barcode scanners, electron microscopy of macromolecular assemblies like viruses and protein complexes, reflection seismology and in the solution of hyperbolic partial differential equations.File:Radon transform sinogram.gifFile:Radon transform example.jpg

Let $f(\backslash textbf\; x)\; =\; f(x,y)$ be a function that satisfies the three regularity conditions:{{sfn|Radon|1986}}

1. $f(\backslash textbf\; x)$ is continuous;
2. the double integral $\backslash iint\backslash dfrac\{\backslash vert\; f(\backslash textbf\; x)\backslash vert\; \}\{\backslash sqrt\{x^2+y^2\}\}\; \backslash ,\; dx\; \backslash ,\; dy$, extending over the whole plane, converges;
3. for any arbitrary point $(x,y)$ on the plane it holds that $\backslash lim\_\{r\backslash to\backslash infty\}\backslash int\_0^\{2\backslash pi\}\; f(x+r\backslash cos\backslash varphi,y+r\backslash sin\backslash varphi)\; \backslash ,\; d\backslash varphi=0.$

The Radon transform, $Rf$, is a function defined on the space of straight lines $L\; \backslash subset\; \backslash mathbb\; R^2$ by the line integral along each such line as:

$$Rf(L)\; =\; \backslash int\_L\; f(\backslash mathbf\{x\})\; \backslash vert\; d\backslash mathbf\{x\}\backslash vert\; .$$Concretely, the parametrization of any straight line $L$ with respect to arc length $z$ can always be written:$$(x(z),y(z))\; =\; \backslash Big(\; (z\backslash sin\backslash alpha+s\backslash cos\backslash alpha),\; (-z\; \backslash cos\backslash alpha\; +\; s\backslash sin\backslash alpha)\; \backslash Big)\; \backslash ,$$where $s$ is the distance of $L$ from the origin and $\backslash alpha$ is the angle the normal vector to $L$ makes with the $X$-axis. It follows that the quantities $(\backslash alpha,s)$ can be considered as coordinates on the space of all lines in $\backslash mathbb\; R^2$, and the Radon transform can be expressed in these coordinates by: $$\backslash begin\{align\}$$

Rf(\alpha,s)

&= \int_{-\infty}^\infty f(x(z),y(z)) \, dz\\

&= \int_{-\infty}^\infty f\big( (z\sin\alpha+s\cos\alpha), (-z\cos\alpha+s\sin\alpha) \big) \, dz.

\end{align}More generally, in the $n$-dimensional Euclidean space $\backslash mathbb\; R^n$, the Radon transform of a function $f$ satisfying the regularity conditions is a function $Rf$ on the space $\backslash Sigma\_n$ of all hyperplanes in $\backslash mathbb\; R^n$. It is defined by:

{{multiple image

| align = left

| direction = horizontal

| image1 = SheppLogan_Phantom.svg

| caption1 = Shepp Logan phantom

| image2 = Shepp logan radon.png

| caption2 = Radon transform

| image3 = Shepp logan iradon.png

| caption3 = Inverse Radon transform

| total_width = 400

}}

$$Rf(\backslash xi)\; =\; \backslash int\_\backslash xi\; f(\backslash mathbf\{x\})\backslash ,\; d\backslash sigma(\backslash mathbf\{x\}),\; \backslash quad\; \backslash forall\; \backslash xi\; \backslash in\; \backslash Sigma\_n$$where the integral is taken with respect to the natural hypersurface measure, $d\; \backslash sigma$ (generalizing the $\backslash vert\; d\backslash mathbf\{x\}\backslash vert$ term from the $2$-dimensional case). Observe that any element of $\backslash Sigma\_n$ is characterized as the solution locus of an equation $\backslash mathbf\{x\}\backslash cdot\backslash alpha\; =\; s$, where $\backslash alpha\; \backslash in\; S^\{n-1\}$ is a unit vector and $s\; \backslash in\; \backslash mathbb\; R$. Thus the $n$-dimensional Radon transform may be rewritten as a function on $S^\{n-1\}\; \backslash times\; \backslash mathbb\; R$ via: $$Rf(\backslash alpha,s)\; =\; \backslash int\_\{\backslash mathbf\{x\}\backslash cdot\backslash alpha\; =\; s\}\; f(\backslash mathbf\{x\})\backslash ,\; d\backslash sigma(\backslash mathbf\{x\}).$$It is also possible to generalize the Radon transform still further by integrating instead over $k$-dimensional affine subspaces of $\backslash mathbb\; R^n$. The X-ray transform is the most widely used special case of this construction, and is obtained by integrating over straight lines.

{{main|Projection-slice theorem}}

File:Radon transform via Fourier transform.png

The Radon transform is closely related to the Fourier transform. We define the univariate Fourier transform here as: $$\backslash hat\{f\}(\backslash omega)=\backslash int\_\{-\backslash infty\}^\backslash infty\; f(x)e^\{-2\backslash pi\; ix\backslash omega\; \}\backslash ,dx.$$

For a function of a $2$-vector $\backslash mathbf\{x\}=(x,y)$, the univariate Fourier transform is: $$$$

\hat{f}(\mathbf{w})=\iint_{\mathbb R^2} f(\mathbf{x})e^{-2\pi i\mathbf{x}\cdot\mathbf{w}}\,dx\, dy.

For convenience, denote $\backslash mathcal\{R\}\_\backslash alpha[f](s)=\; \backslash mathcal\{R\}[f](\backslash alpha,s)$. The Fourier slice theorem then states: $$$$

\widehat{\mathcal{R}_{\alpha}[f]}(\sigma)=\hat{f}(\sigma\mathbf{n}(\alpha))

where $\backslash mathbf\{n\}(\backslash alpha)=\; (\backslash cos\; \backslash alpha,\backslash sin\backslash alpha).$

Thus the two-dimensional Fourier transform of the initial function along a line at the inclination angle $\backslash alpha$ is the one variable Fourier transform of the Radon transform (acquired at angle $\backslash alpha$) of that function. This fact can be used to compute both the Radon transform and its inverse. The result can be generalized into n dimensions: $$\backslash hat\{f\}(r\backslash alpha)\; =\; \backslash int\_\{\backslash mathbb\; R\}\backslash mathcal\{R\}f(\backslash alpha,s)e^\{-2\backslash pi\; i\; sr\}\; \backslash ,\; ds.$$

The dual Radon transform is a kind of adjoint to the Radon transform. Beginning with a function g on the space $\backslash Sigma\_n$, the dual Radon transform is the function $\backslash mathcal\{R\}^*g$ on Rⁿ defined by: $$\backslash mathcal\{R\}^*g(\backslash mathbf\{x\})\; =\; \backslash int\_\{\backslash mathbf\{x\}\backslash in\backslash xi\}\; g(\backslash xi)\backslash ,d\backslash mu(\backslash xi).$$The integral here is taken over the set of all hyperplanes incident with the point $\backslash textbf\; x\; \backslash in\; \backslash mathbb\; R^n$, and the measure $d\; \backslash mu$ is the unique probability measure on the set $\backslash \{\backslash xi\; |\; \backslash mathbf\{x\}\backslash in\backslash xi\backslash \}$ invariant under rotations about the point $\backslash mathbf\{x\}$.

Concretely, for the two-dimensional Radon transform, the dual transform is given by: $$\backslash mathcal\{R\}^*g(\backslash mathbf\{x\})\; =\; \backslash frac\{1\}\{2\backslash pi\}\backslash int\_\{\backslash alpha=0\}^\{2\backslash pi\}g(\backslash alpha,\backslash mathbf\{n\}(\backslash alpha)\backslash cdot\backslash mathbf\{x\})\backslash ,d\backslash alpha.$$ In the context of image processing, the dual transform is commonly called back-projection{{sfn|Roerdink|2001}} as it takes a function defined on each line in the plane and 'smears' or projects it back over the line to produce an image.

Let $\backslash Delta$ denote the Laplacian on $\backslash mathbb\; R^n$ defined by:$$\backslash Delta\; =\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; x\_1^2\}+\backslash cdots+\backslash frac\{\backslash partial^2\}\{\backslash partial\; x\_n^2\}$$This is a natural rotationally invariant second-order differential operator. On $\backslash Sigma\_n$, the "radial" second derivative $Lf(\backslash alpha,s)\; \backslash equiv\; \backslash frac\{\backslash partial^2\}\{\backslash partial\; s^2\}\; f(\backslash alpha,s)$ is also rotationally invariant. The Radon transform and its dual are intertwining operators for these two differential operators in the sense that:{{sfn|Helgason|1984|loc=Lemma I.2.1}} $$\backslash mathcal\{R\}(\backslash Delta\; f)\; =\; L\; (\backslash mathcal\{R\}f),\backslash quad\; \backslash mathcal\{R\}^*\; (Lg)\; =\; \backslash Delta(\backslash mathcal\{R\}^*g).$$In analysing the solutions to the wave equation in multiple spatial dimensions, the intertwining property leads to the translational representation of Lax and Philips.{{cite journal|last1 = Lax|first1= P. D.|last2 = Philips|first2= R. S.|title=Scattering theory|journal= Bull. Amer. Math. Soc.|date= 1964|volume= 70|number=1|pages=130–142|doi=10.1090/s0002-9904-1964-11051-x|doi-access= free}} In imaging

{{cite journal

|last1 = Bonneel |first1= N.

|last2 = Rabin |first2= J.

|last3 = Peyre|first3= G.

|last4 = Pfister|first4= H.

|title= Sliced and Radon Wasserstein Barycenters of Measures

|journal= Journal of Mathematical Imaging and Vision|date=2015|volume=51|number=1|pages=22–25 |doi=10.1007/s10851-014-0506-3|bibcode= 2015JMIV...51...22B

|s2cid= 1907942

|url= http://hal.archives-ouvertes.fr/hal-00881872

}} and numerical analysis{{cite journal|last = Rim|first= D.|title=Dimensional Splitting of Hyperbolic Partial Differential Equations Using the Radon Transform

|journal= SIAM J. Sci. Comput.|date= 2018|volume= 40|number= 6 |pages=A4184–A4207|doi=10.1137/17m1135633|arxiv= 1705.03609|bibcode= 2018SJSC...40A4184R|s2cid= 115193737}} this is exploited to reduce multi-dimensional problems into single-dimensional ones, as a dimensional splitting method.

The process of reconstruction produces the image (or function $f$ in the previous section) from its projection data. Reconstruction is an inverse problem.

In the two-dimensional case, the most commonly used analytical formula to recover $f$ from its Radon transform is the Filtered Back-projection Formula or Radon Inversion Formula{{sfn|Candès|2016b}}: $$f(\backslash mathbf\{x\})=\backslash int^\backslash pi\_0\; (\backslash mathcal\{R\}f(\backslash cdot,\backslash theta)*h)(\backslash left\backslash langle\backslash mathbf\{x\},\backslash mathbf\{n\}\_\backslash theta\; \backslash right\backslash rangle)\; \backslash ,\; d\backslash theta$$where $h$ is such that $\backslash hat\{h\}(k)=|k|$.{{sfn|Candès|2016b}} The convolution kernel $h$ is referred to as Ramp filter in some literature.

Intuitively, in the filtered back-projection formula, by analogy with differentiation, for which $\backslash left(\backslash widehat\{\backslash frac\{d\}\{dx\}f\}\backslash right)\backslash !(k)=ik\backslash widehat\{f\}(k)$, we see that the filter performs an operation similar to a derivative. Roughly speaking, then, the filter makes objects more singular. A quantitive statement of the ill-posedness of Radon inversion goes as follows:$$\backslash widehat\{\backslash mathcal\{R\}^*\backslash mathcal\{R\}\; [g]\}(\backslash mathbf\{k\})\; =\; \backslash frac\{1\}\{\backslash |\backslash mathbf\{k\}\backslash |\}\; \backslash hat\{g\}(\backslash mathbf\{k\})$$

where $\backslash mathcal\{R\}^*$ is the previously defined adjoint to the Radon transform. Thus for $g(\backslash mathbf\{x\})\; =\; e^\{i\; \backslash left\backslash langle\backslash mathbf\{k\}\_0,\backslash mathbf\{x\}\backslash right\backslash rangle\}$, we have: $$\backslash mathcal\{R\}^*\backslash mathcal\{R\}[g](\backslash mathbf\{x\})\; =\; \backslash frac\{1\}\{\backslash |\backslash mathbf\{k\_0\}\backslash |\}\; e^\{i\; \backslash left\backslash langle\backslash mathbf\{k\}\_0,\backslash mathbf\{x\}\backslash right\backslash rangle\}$$

The complex exponential $e^\{i\; \backslash left\backslash langle\backslash mathbf\{k\}\_0,\backslash mathbf\{x\}\backslash right\backslash rangle\}$ is thus an eigenfunction of $\backslash mathcal\{R\}^*\backslash mathcal\{R\}$ with eigenvalue $\backslash frac\{1\}\{\backslash |\backslash mathbf\{k\}\_0\backslash |\}$. Thus the singular values of $\backslash mathcal\{R\}$ are $\backslash frac\{1\}\backslash sqrt\{\backslash |\backslash mathbf\{k\}\backslash |\}$. Since these singular values tend to $0$, $\backslash mathcal\{R\}^\{-1\}$ is unbounded.{{sfn|Candès|2016b}}

{{main|Iterative reconstruction}}

Compared with the Filtered Back-projection method, iterative reconstruction costs large computation time, limiting its practical use. However, due to the ill-posedness of Radon Inversion, the Filtered Back-projection method may be infeasible in the presence of discontinuity or noise. Iterative reconstruction methods (e.g. iterative Sparse Asymptotic Minimum Variance{{cite journal | last1=Abeida | first1=Habti | last2=Zhang | first2=Qilin | last3=Li | first3=Jian | last4=Merabtine | first4=Nadjim | title=Iterative Sparse Asymptotic Minimum Variance Based Approaches for Array Processing | journal=IEEE Transactions on Signal Processing | publisher=IEEE | volume=61 | issue=4 | year=2013 | issn=1053-587X | doi=10.1109/tsp.2012.2231676 | pages=933–944 | url=https://qilin-zhang.github.io/_pages/pdfs/SAMVpaper.pdf | bibcode=2013ITSP...61..933A | arxiv=1802.03070 | s2cid=16276001 }}) could provide metal artefact reduction, noise and dose reduction for the reconstructed result that attract much research interest around the world.

Explicit and computationally efficient inversion formulas for the Radon transform and its dual are available. The Radon transform in $n$ dimensions can be inverted by the formula:{{sfn|Helgason|1984|loc=Theorem I.2.13}} $$c\_n\; f\; =\; (-\backslash Delta)^\{(n-1)/2\}R^*Rf\backslash ,$$where $c\_n\; =\; (4\backslash pi)^\{(n-1)/2\}\backslash frac\{\backslash Gamma(n/2)\}\{\backslash Gamma(1/2)\}$, and the power of the Laplacian $(-\backslash Delta)^\{(n-1)/2\}$ is defined as a pseudo-differential operator if necessary by the Fourier transform: $$\backslash left[\backslash mathcal\{F\}(-\backslash Delta)^\{(n-1)/2\}\; \backslash varphi\backslash right](\backslash xi)\; =\; |2\backslash pi\backslash xi|^\{n-1\}(\backslash mathcal\{F\}\backslash varphi)(\backslash xi).$$For computational purposes, the power of the Laplacian is commuted with the dual transform $R^*$ to give:{{sfn|Helgason|1984|loc=Theorem I.2.16}} $$c\_nf\; =\; \backslash begin\{cases\}$$

R^*\frac{d^{n-1}}{ds^{n-1}}Rf & n \text{ odd}\\

R^* \mathcal H_s\frac{d^{n-1}}{ds^{n-1}}Rf & n \text{ even}

\end{cases}

where $\backslash mathcal\; H\_s$ is the Hilbert transform with respect to the s variable. In two dimensions, the operator $\backslash mathcal\; H\_s\backslash frac\{d\}\{ds\}$

appears in image processing as a ramp filter.{{sfn|Nygren|1997}} One can prove directly from the Fourier slice theorem and change of variables for integration that for a compactly supported continuous function $f$

of two variables: $$f\; =\; \backslash frac\{1\}\{2\}R^\{*\}\backslash mathcal\; H\_s\backslash frac\{d\}\{ds\}Rf.$$Thus in an image processing context the original image $f$

can be recovered from the 'sinogram' data $Rf$

by applying a ramp filter (in the $s$ variable) and then back-projecting. As the filtering step can be performed efficiently (for example using digital signal processing techniques) and the back projection step is simply an accumulation of values in the pixels of the image, this results in a highly efficient, and hence widely used, algorithm.

Explicitly, the inversion formula obtained by the latter method is:{{sfn|Roerdink|2001}} $$f(x)\; =$$

\begin{cases}

\displaystyle - \imath 2\pi (2\pi)^{-n}(-1)^{n/2}\int_{S^{n-1}}\frac{\partial^{n-1}}{2\partial s^{n-1}}Rf(\alpha,\alpha\cdot x)\,d\alpha & n \text{ odd} \\

\displaystyle (2\pi)^{-n}(-1)^{n/2}\iint_{\mathbb R \times S^{n-1}}\frac{\partial^{n-1}}{q\partial s^{n-1}} Rf(\alpha,\alpha\cdot x + q)\,d\alpha\,dq & n \text{ even} \\

\end{cases}The dual transform can also be inverted by an analogous formula: $$c\_n\; g\; =\; (-L)^\{(n-1)/2\}R(R^*g).\backslash ,$$

In algebraic geometry, a Radon transform (also known as the Brylinski–Radon transform) is constructed as follows.

Write

:$\backslash mathbf\; P^d\; \backslash ,\; \backslash stackrel\{p\_1\}\; \backslash gets\; \backslash ,\; H\; \backslash ,\; \backslash stackrel\{p\_2\}\backslash to\; \backslash ,\; \backslash mathbf\; P^\{\backslash vee,\; d\}$

for the universal hyperplane, i.e., H consists of pairs (x, h) where x is a point in d-dimensional projective space $\backslash mathbf\; P^d$ and h is a point in the dual projective space (in other words, x is a line through the origin in (d+1)-dimensional affine space, and h is a hyperplane in that space) such that x is contained in h.

Then the Brylinksi–Radon transform is the functor between appropriate derived categories of étale sheaves

:$\backslash operatorname\{Rad\}\; :=\; Rp\_\{2,*\}\; p\_1^*\; :\; D(\backslash mathbf\; P^d)\; \backslash to\; D(\backslash mathbf\; P^\{\backslash vee,\; d\}).$

The main theorem about this transform is that this transform induces an equivalence of the categories of perverse sheaves on the projective space and its dual projective space, up to constant sheaves.{{harvtxt|Kiehl|Weissauer|2001|loc=Ch. IV, Cor. 2.4}}

• Periodogram
• Matched filter
• Deconvolution
• X-ray transform
• Funk transform
• The Hough transform, when written in a continuous form, is very similar, if not equivalent, to the Radon transform.{{sfn|van Ginkel|Hendricks|van Vliet|2004}}
• Cauchy–Crofton theorem is a closely related formula for computing the length of curves in space.
• Fast Fourier transform

{{reflist}}

{{sfn whitelist |CITEREFRoerdink2001}}

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{{refend}}

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Category:Integral geometry

Category:Integral transforms

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