Editing Radon transform (section)

==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 two-dimensional case, the most commonly used analytical formula to recover <math>f</math> from its Radon transform is the ''Filtered Back-projection Formula'' or ''Radon Inversion Formula{{sfn|Candès|2016b}}'': <math display="block">f(\mathbf{x})=\int^\pi_0 (\mathcal{R}f(\cdot,\theta)*h)(\left\langle\mathbf{x},\mathbf{n}_\theta \right\rangle) \, d\theta</math>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 back-projection'' formula, by analogy with differentiation, for which <math display="inline">\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:<math display="block">\widehat{\mathcal{R}^*\mathcal{R} [g]}(\mathbf{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>, we have: <math display="block"> \mathcal{R}^*\mathcal{R}[g](\mathbf{x}) = \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 display="inline">\frac{1}{\|\mathbf{k}_0\|}</math>. Thus the singular values of <math>\mathcal{R}</math> are <math display="inline">\frac{1}\sqrt{\|\mathbf{k}\|}</math>. Since these singular values tend to <math>0</math>, <math>\mathcal{R}^{-1}</math> is unbounded.{{sfn|Candès|2016b}}

===Iterative reconstruction methods===
{{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.'' [[SAMV (algorithm)|iterative Sparse Asymptotic Minimum Variance]]<ref name=AbeidaZhang>{{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 }}</ref>) could provide metal artefact reduction, noise and dose reduction for the reconstructed result that attract much research interest around the world.