View source for Radon transform

[[File:Radon transform.png|thumb|Radon transform. Maps ''f'' on the (''x,{{nbsp}}y'')-domain to ''Rf'' on the (''α,{{nbsp}}s'')-domain.]]
[[File:Sinogram - Two Square Indicator Phantom.svg|thumb|right|Radon transform of the [[indicator function]] of two squares shown in the image below.  Lighter regions indicate larger function values.  Black indicates zero.]]
[[File:Sinogram Source - Two Squares Phantom.png|thumb|right|Original function is equal to one on the white region and zero on the dark region.]]
In [[mathematics]], the '''Radon transform''' is the [[integral transform]] which takes a function ''f'' defined on the plane to a function ''Rf'' defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the [[line integral]] of the function over that line. The transform was introduced in 1917 by [[Johann Radon]],{{sfn|Radon|1917}} who also provided a formula for the inverse transform. Radon further included formulas for the transform in [[three dimensions]], in which the integral is taken over planes (integrating over lines is known as the [[X-ray transform]]).  It was later generalized to higher-dimensional [[Euclidean space]]s, and more broadly in the context of [[integral geometry]].  The [[complex number|complex]] analog of the Radon transform is known as the [[Penrose transform]]. The Radon transform is widely applicable to [[tomography]], the creation of an image from the projection data associated with cross-sectional scans of an object.
{{multiple image
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 | image1    = SheppLogan_Phantom.svg
 | width1    = 170
 | caption1  = [[Shepp-Logan Phantom|Shepp Logan phantom]]
 | image2    = Shepp logan radon.png 
 | width2    = 42
 | caption2  = Radon transform
 | image3    = Shepp logan iradon.png
 | width3    = 170
 | caption3  = Inverse Radon transform
}}

==Explanation==
If a function <math>f</math> 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 wave]]s with different amplitudes and phases.

The Radon transform is useful in [[computed axial tomography]] (CAT scan), [[barcode]] scanners, [[electron microscopy]] of [[macromolecular assemblies]] like [[virus]]es and [[protein complex]]es, [[reflection seismology]] and in the solution of hyperbolic [[partial differential equations]].

==Definition==
Let ''&fnof;''('''x''')&nbsp;=&nbsp;''&fnof;''(''x'',''y'') be a function that satisfies the three regularity conditions:<ref>{{Cite journal|last=Radon|first=J.|date=December 1986|title=On the determination of functions from their integral values along certain manifolds|journal=IEEE Transactions on Medical Imaging|volume=5|issue=4|pages=170–176|doi=10.1109/TMI.1986.4307775|pmid=18244009}}</ref>

# ''&fnof;''(''x'',''y'') is continuous
# the double integral <math>\int\int\frac{|f(x,y)|}{\sqrt{x^2+y^2}}dxdy</math>, extending over the whole plane, converges
# for any arbitrary point <math>(x,y)</math> on the plane it holds that <math>\lim_{r\to\infty}\int_0^{2\pi}f(x+r\cos\phi,y+r\sin\phi)d\phi=0</math>

The Radon transform, ''R&fnof;'', is a function defined on the space of straight lines ''L'' in '''R'''<sup>2</sup> by the [[Line integral#Definition|line integral]] along each such line as

:<math>Rf(L) = \int_L f(\mathbf{x})\,|d\mathbf{x}|.</math>
Concretely, the parametrization of any straight line ''L'' with respect to arc length ''z'' can always be written
:<math>(x(z),y(z)) = \Big( (z\sin\alpha+s\cos\alpha), (-z\cos\alpha+s\sin\alpha)  \Big) \,</math>
where ''s'' is the distance of ''L'' from the origin and <math>\alpha</math> is the angle the normal vector to ''L'' makes with the ''x'' axis.  It follows that the quantities (α,''s'') can be considered as coordinates on the space of all lines in '''R'''<sup>2</sup>, and the Radon transform can be expressed in these coordinates by
:<math>\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}</math>

More generally, in the ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup>, the Radon transform of a function ''&fnof;'' satisfying the regularity conditions is a function ''R&fnof;'' on the space Σ<sub>''n''</sub> of all [[hyperplane]]s in '''R'''<sup>''n''</sup>.  It is defined by
:<math>Rf(\xi) = \int_\xi f(\mathbf{x})\, d\sigma(\mathbf{x})</math>
for ξ&nbsp;∈Σ<sub>''n''</sub>, where the integral is taken with respect to the natural [[hypersurface]] [[measure (mathematics)|measure]], ''d''σ (generalizing the |''d'''''x'''| term from the 2-dimensional case).  Observe that any element of &Sigma;<sub>''n''</sub> is characterized as the solution locus of an equation
:<math>\mathbf{x}\cdot\alpha = s</math>
where &alpha;&nbsp;&isin;&nbsp;''S''<sup>''n''&minus;1</sup> is a [[unit vector]] and ''s''&nbsp;&isin;&nbsp;'''R'''.  Thus the ''n''-dimensional Radon transform may be rewritten as a function on ''S''<sup>''n''&minus;1</sup>&times;'''R''' via
:<math>Rf(\alpha,s) = \int_{\mathbf{x}\cdot\alpha = s} f(\mathbf{x})\, d\sigma(\mathbf{x}).</math>

It is also possible to generalize the Radon transform still further by integrating instead over ''k''-dimensional affine subspaces of '''R'''<sup>''n''</sup>.  The [[X-ray transform]] is the most widely used special case of this construction, and is obtained by integrating over straight lines.

==Relationship with the Fourier transform==
{{main|Projection-slice theorem}}

[[File:Radon transform via Fourier transform.png|thumb|Computing the 2-dimensional Radon transform in terms of two Fourier transforms.]]

The Radon transform is closely related to the [[Fourier transform]]. We define the univariate Fourier transform here as

:<math>\hat{f}(\omega)=\int_{-\infty}^\infty f(x)e^{-2\pi ix\omega }\,dx.
</math>

and for a function of a 2-vector <math>\mathbf{x}=(x,y)</math>,

:<math>
\hat{f}(\mathbf{w})=\int\limits_{-\infty}^{\infty}
\int\limits_{-\infty }^{\infty}  f(\mathbf{x})e^{-2\pi i\mathbf{x}\cdot\mathbf{w}}\,dx\, dy.
</math>

For convenience, denote <math>\mathcal{R}_\alpha[f](s)= \mathcal{R}[f](\alpha,s)</math>. The [[Fourier slice theorem]] then states

:<math>
\widehat{\mathcal{R}_{\alpha}[f]}(\sigma)=\hat{f}(\sigma\mathbf{n}(\alpha))
</math>

where

:<math>\mathbf{n}(\alpha)= (\cos \alpha,\sin\alpha).</math>

Thus the two-dimensional Fourier transform of the initial function along a line at the inclination angle <math>\alpha</math> is the one variable Fourier transform of the Radon transform (acquired at angle <math>\alpha</math>) 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

:<math>\hat{f}(r\alpha) = \int_{-\infty}^\infty \mathcal{R}f(\alpha,s)e^{-2\pi i sr}\, ds.</math>

==Dual transform==
The dual Radon transform is a kind of [[Hermitian adjoint|adjoint]] to the Radon transform.  Beginning with a function ''g'' on the space Σ<sub>''n''</sub>, the dual Radon transform is the function <math>\mathcal{R}^*g</math> on '''R'''<sup>''n''</sup> defined by
:<math>\mathcal{R}^*g(x) = \int_{x\in\xi} g(\xi)\,d\mu(\xi).</math>
The integral here is taken over the set of all hyperplanes incident with the point ''x''&nbsp;∈&nbsp;'''R'''<sup>''n''</sup>, and the measure ''d''μ is the unique [[probability measure]] on the set <math>\{\xi | x\in\xi\}</math> invariant under rotations about the point ''x''.

Concretely, for the two-dimensional Radon transform, the dual transform is given by
:<math>\mathcal{R}^*g(x) = \frac{1}{2\pi}\int_{\alpha=0}^{2\pi}g(\alpha,\mathbf{n}(\alpha)\cdot\mathbf{x})\,d\alpha.</math>
In the context of image processing, the dual transform is commonly called ''backprojection''{{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.

===Intertwining property===
Let Δ denote the [[Laplacian]] on '''R'''<sup>''n''</sup>: 
:<math>\Delta = \frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2}.</math>
This is a natural rotationally invariant second-order [[differential operator]].  On Σ<sub>''n''</sub>, the "radial" second derivative
:<math>Lf(\alpha,s) \equiv \frac{\partial^2}{\partial s^2} f(\alpha,s)</math>
is also rotationally invariant. The Radon transform and its dual are [[intertwining operator]]s for these two differential operators in the sense that{{sfn|Helgason|1984|loc=Lemma I.2.1}}
:<math>\mathcal{R}(\Delta f) = L (\mathcal{R}f),\quad \mathcal{R}^* (Lg) = \Delta(\mathcal{R}^*g).</math>
In analyzing the solutions to the wave equation in multiple spatial dimensions, the intertwining property leads to the translational representation of Lax and Philips.<ref>{{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}}</ref> In imaging<ref>
{{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|url= http://hal.archives-ouvertes.fr/hal-00881872
}}</ref> and numerical analysis<ref>{{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}}</ref> this is exploited to reduce multi-dimensional problems into single-dimensional ones, as a dimensional splitting method.

==Reconstruction approaches==

The process of  ''reconstruction'' produces the image (or function <math>f</math> in the previous section) from its projection data. ''Reconstruction'' is an [[inverse problem]].

===Radon inversion formula===
In the 2D case, the most commonly used analytical formula to recover <math>f</math> from its Radon transform is the ''Filtered Backprojection Formula'' or ''Radon Inversion Formula'':

:<math>f(\mathbf{x})=\int^{\pi}_{0}(\mathcal{R}f(\cdot,\theta)*h)(\left\langle\mathbf{x},\mathbf{n}_{\theta} \right\rangle) d\theta</math>{{sfn|Candès|2016a}}

where <math>h</math> is such that <math>\hat{h}(k)=|k|</math>.{{sfn|Candès|2016b}}

The convolution kernel <math>h</math> is referred to as [[Ramp filter]] in some literature.

===Ill-posedness===
Intuitively, in the ''filtered backprojection'' formula, by analogy with differentiation, for which <math>\left(\widehat{\frac{d}{dx}f}\right)\!(k)=ik\widehat{f}(k)</math>, 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:

We have <math>\widehat{\mathcal{R}^*\mathcal{R}g}(k)=\frac{1}{||\mathbf{k}||}\hat{g}(\mathbf{k})</math>

where <math>\mathcal{R}^*</math> is the previously defined [[adjoint]] to the Radon Transform.

Thus for <math>g(\mathbf{x})=e^{i \left\langle\mathbf{k}_0,\mathbf{x}\right\rangle}</math>, 
:<math> \mathcal{R}^*\mathcal{R}g=\frac{1}{||\mathbf{k_0}||}e^{i \left\langle\mathbf{k}_0,\mathbf{x}\right\rangle}</math>.

The complex exponential <math>e^{i \left\langle\mathbf{k}_0,\mathbf{x}\right\rangle}</math> is thus an eigenfunction of <math>\mathcal{R}^*\mathcal{R}</math> with eigenvalue <math>\frac{1}{||\mathbf{k_0}||}</math>. Thus the singular values of <math>\mathcal{R}</math> are <math>\sqrt{\frac{1}{||\mathbf{k}||}}</math>. Since these singular values tend to 0, <math>\mathcal{R}^{-1}</math> is unbounded.{{sfn|Candès|2016b}}

===Iterative reconstruction methods===
{{main|Iterative reconstruction}}
Compared with the ''Filtered Backprojection'' method, iterative reconstruction costs large computation time, limiting its practical use. However, due to the ill-posedness of Radon Inversion, the ''Filtered Backprojection'' method may be infeasible in the presence of discontinuity or noise. Iterative reconstruction methods (''e.g.'', [[SAMV (algorithm)|iterative Sparse Asymptotic Minimum Variance]]<ref name=AbeidaZhang>{{cite journal | last=Abeida | first=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 }}</ref>) could provide metal artifact reduction, noise and dose reduction for the reconstructed result that attract much research interest around the world.

==Inversion formulas==

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}}
:<math>c_n f = (-\Delta)^{(n-1)/2}R^*Rf\,</math>
where 
:<math>c_n = (4\pi)^{(n-1)/2}\frac{\Gamma(n/2)}{\Gamma(1/2)}.</math>
and the power of the Laplacian (&minus;Δ)<sup>(''n''&minus;1)/2</sup> is defined as a [[pseudodifferential operator]] if necessary by the [[Fourier transform]]
:<math>\left[\mathcal{F}(-\Delta)^{(n-1)/2}\phi\right](\xi) = |2\pi\xi|^{n-1}(\mathcal{F}\phi)(\xi).</math>

For computational purposes, the power of the Laplacian is commuted with the dual transform ''R''<sup>*</sup> to give{{sfn|Helgason|1984|loc=Theorem I.2.16}}
:<math>c_nf = \begin{cases}
R^*\frac{d^{n-1}}{ds^{n-1}}Rf & n \rm{\ odd}\\
R^*H_s\frac{d^{n-1}}{ds^{n-1}}Rf & n \rm{\ even}
\end{cases}
</math>
where ''H''<sub>''s''</sub> is the [[Hilbert transform]] with respect to the ''s'' variable.  In two dimensions, the operator ''H''<sub>''s''</sub>''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 ƒ of two variables
:<math>f = \frac{1}{2}R^{*}H_s\frac{d}{ds}Rf.</math>
Thus in an image processing context the original image ƒ can be recovered from the 'sinogram' data ''R''ƒ by applying a ramp filter (in the <math>s</math>  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}}
:<math>f(x) = \frac{1}{2}(2\pi)^{1-n}(-1)^{(n-1)/2}\int_{S^{n-1}}\frac{\partial^{n-1}}{\partial s^{n-1}}Rf(\alpha,\alpha\cdot x)\,d\alpha</math>
if ''n'' is odd, and
:<math>f(x) = (2\pi)^{-n}(-1)^{n/2}\int_{-\infty}^\infty \frac{1}{q}\int_{S^{n-1}}\frac{\partial^{n-1}}{\partial s^{n-1}}Rf(\alpha,\alpha\cdot x + q)\,d\alpha\,dq</math>
if ''n'' is even.

The dual transform can also be inverted by an analogous formula:
:<math>c_n g = (-L)^{(n-1)/2}R(R^*g).\,</math>

==Radon transform in algebraic geometry==

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

Write

:<math>\mathbf P^d \stackrel{p_1} \gets H \stackrel{p_2}\to \mathbf P^{\vee, d}</math>

for the [[incidence relation|universal hyperplane]], i.e., ''H'' consists of pairs (''x'', ''h'') where ''x'' is a point in ''d''-dimensional [[projective space]] <math>\mathbf P^d</math> 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 category|derived categories]] of [[étale sheaf|étale sheaves]]

:<math>Rad := Rp_{2,*} p_1^* : D(\mathbf P^d) \to D(\mathbf P^{\vee, d}).</math>

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

== See also ==
* [[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}}
* [[Crofton formula|Cauchy-Crofton theorem]] is a closely related formula for computing the length of curves in space.
* [[Fast Fourier transform]]

==Notes==
{{reflist}}

==References==
{{refbegin}}
* {{citation |title = Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten |last = Radon |first = Johann |authorlink = Johann Radon |journal = Berichte über die Verhandlungen der Königlich-Sächsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse [Reports on the proceedings of the Royal Saxonian Academy of Sciences at Leipzig, mathematical and physical section] |location = Leipzig |publisher = Teubner |year = 1917 |issue = 69 |pages = 262–277 }}; ''Translation:'' {{citation |title = On the determination of functions from their integral values along certain manifolds |journal = IEEE Transactions on Medical Imaging |year = 1986 |volume = 5 |pages = 170–176 |author = Radon, J. |author2 = Parks, P.C. (translator) |doi = 10.1109/TMI.1986.4307775 |pmid = 18244009 |issue = 4 }}.
* {{springer|id=t/t092980|title=Tomography|first=J.B.T.M.|last=Roerdink}}.
* {{citation |first = Sigurdur |last = Helgason |authorlink = Sigurður Helgason (mathematician) |title = Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions |year = 1984 |publisher = Academic Press |isbn = 0-12-338301-3 }}.
* {{cite web |title = Applied Fourier Analysis and Elements of Modern Signal Processing - Lecture 9 |last1 = Candès |first1 = Emmanuel |url = http://statweb.stanford.edu/~candes/math262/Lectures/Lecture09.pdf |date = February 2, 2016a  |ref = harv }}
* {{cite web |title = Applied Fourier Analysis and Elements of Modern Signal Processing - Lecture 10 |last1 = Candès |first1 = Emmanuel |url = http://statweb.stanford.edu/~candes/math262/Lectures/Lecture10.pdf |date = February 4, 2016b  |ref = harv }}
* {{cite web |url = http://www.owlnet.rice.edu/~elec539/Projects97/cult/node2.html |title = Filtered Back Projection |work = Tomographic Reconstruction of SPECT Data |last = Nygren |first = Anders J. |year = 1997 |ref = harv }}
* {{cite web |title = A short introduction to the Radon and Hough transforms and how they relate to each other |last1 = van Ginkel |first1 = M. |last2 = Hendricks |first2 = C.L. Luengo |last3 = van Vliet |first3 = L.J. |year = 2004 |url = http://tnw.home.tudelft.nl/fileadmin/Faculteit/TNW/Over_de_faculteit/Afdelingen/Imaging_Science_and_Technology/Research/Research_Groups/Quantitative_Imaging/Publications/Technical_Reports/doc/mvanginkel_radonandhough_tr2004.pdf |archive-url = https://web.archive.org/web/20160729172119/http://tnw.home.tudelft.nl/fileadmin/Faculteit/TNW/Over_de_faculteit/Afdelingen/Imaging_Science_and_Technology/Research/Research_Groups/Quantitative_Imaging/Publications/Technical_Reports/doc/mvanginkel_radonandhough_tr2004.pdf |archive-date = 2016-07-29 |url-status = live |ref = harv }}
{{refend}}

==Further reading==
* {{cite book|author1=Lokenath Debnath|author2=Dambaru Bhatta|title=Integral Transforms and Their Applications|url=https://books.google.com/?id=WbZcqdvCEfwC&printsec=frontcover&dq=integral+transform#v=snippet&q=Radon&f=false|date=19 April 2016|publisher=CRC Press|isbn=978-1-4200-1091-6}}
* {{citation |last = Deans |first = Stanley R. |title = The Radon Transform and Some of Its Applications |year = 1983 |publisher = John Wiley & Sons |location = New York }}
* {{Citation |last1 = Helgason |first1 = Sigurdur |title = Geometric analysis on symmetric spaces |publisher = [[American Mathematical Society]] |location = Providence, R.I. |edition = 2nd |series = Mathematical Surveys and Monographs |isbn = 978-0-8218-4530-1 |mr = 2463854 |year = 2008 |volume = 39 |doi = 10.1090/surv/039 }}
* {{citation |last = Herman |first = Gabor T. |authorlink = Gabor Herman |title = Fundamentals of Computerized Tomography: Image Reconstruction from Projections |year = 2009 |publisher = Springer |edition = 2nd |isbn = 978-1-85233-617-2 }}
* {{springer |id = r/r077190 |title = Radon transform |first = R.A. |last = Minlos}}
* {{citation |first = Frank |last = Natterer |title = The Mathematics of Computerized Tomography |series = Classics in Applied Mathematics |volume = 32 |publisher = Society for Industrial and Applied Mathematics |isbn = 0-89871-493-1 |date = June 2001 }}
* {{citation |first1 = Frank |last1 = Natterer |first2 = Frank |last2 = Wübbeling |title = Mathematical Methods in Image Reconstruction |publisher = Society for Industrial and Applied Mathematics |isbn = 0-89871-472-9 |year = 2001 }}
* {{Citation|authorlink1=Reinhardt Kiehl|last1=Kiehl|first1=Reinhardt|last2=Weissauer|first2=Rainer|
isbn=3-540-41457-6
|title=Weil conjectures, perverse sheaves and l'adic Fourier transform|publisher=Springer|year=2001|mr=1855066|doi=10.1007/978-3-662-04576-3}}

==External links==
* {{MathWorld |urlname = RadonTransform |pagename = Radon Transform}}
* {{cite AV media |date = September 10, 2015 |title = Analytical projection (the Radon transform) |medium = video |institution = [[University of Antwerp]] |series = Part of the "Computed Tomography and the ASTRA Toolbox" course |url = https://www.youtube.com/watch?v=MA2y_2YySq0 }}

[[Category:Integral geometry]]
[[Category:Integral transforms]]