Editing Radon transform (section)

==Inversion formulas==

Explicit and computationally efficient inversion formulas for the Radon transform and its dual are available. The Radon transform in <math>n</math> dimensions can be inverted by the formula:{{sfn|Helgason|1984|loc=Theorem I.2.13}} <math display="block">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 <math>(-\Delta)^{(n-1)/2}</math> is defined as a [[pseudodifferential operator|pseudo-differential operator]] if necessary by the [[Fourier transform]]: <math display="block">\left[\mathcal{F}(-\Delta)^{(n-1)/2} \varphi\right](\xi) = |2\pi\xi|^{n-1}(\mathcal{F}\varphi)(\xi).</math>For computational purposes, the power of the Laplacian is commuted with the dual transform <math>R^*</math> to give:{{sfn|Helgason|1984|loc=Theorem I.2.16}} <math display="block">c_nf = \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}
</math>where <math>\mathcal H_s</math> is the [[Hilbert transform]] with respect to the ''s'' variable.  In two dimensions, the operator <math>\mathcal H_s\frac{d}{ds}
</math> 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 <math>f
</math> of two variables: <math display="block">f = \frac{1}{2}R^{*}\mathcal H_s\frac{d}{ds}Rf.</math>Thus in an image processing context the original image <math>f
</math> can be recovered from the 'sinogram' data <math>Rf
</math> 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 display="block">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}</math>The dual transform can also be inverted by an analogous formula: <math display="block">c_n g = (-L)^{(n-1)/2}R(R^*g).\,</math>