Editing Radon transform (section)

== 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 <math>\Sigma_n</math>, the dual Radon transform is the function <math>\mathcal{R}^*g</math> on '''R'''<sup>''n''</sup> defined by: <math display="block">\mathcal{R}^*g(\mathbf{x}) = \int_{\mathbf{x}\in\xi} g(\xi)\,d\mu(\xi).</math>The integral here is taken over the set of all hyperplanes incident with the point <math>\textbf x \in \mathbb R^n</math>, and the measure <math>d \mu</math> is the unique [[probability measure]] on the set <math>\{\xi | \mathbf{x}\in\xi\}</math> invariant under rotations about the point <math>\mathbf{x}</math>.

Concretely, for the two-dimensional Radon transform, the dual transform is given by: <math display="block">\mathcal{R}^*g(\mathbf{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 ''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.

===Intertwining property===
Let <math>\Delta</math> denote the [[Laplacian]] on <math>\mathbb R^n</math> defined by:<math display="block">\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 <math>\Sigma_n</math>, 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 display="block">\mathcal{R}(\Delta f) = L (\mathcal{R}f),\quad \mathcal{R}^* (Lg) = \Delta(\mathcal{R}^*g).</math>In analysing 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|doi-access= free}}</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|bibcode= 2015JMIV...51...22B
|s2cid= 1907942
|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|bibcode= 2018SJSC...40A4184R|s2cid= 115193737}}</ref> this is exploited to reduce multi-dimensional problems into single-dimensional ones, as a dimensional splitting method.