Editing Radon transform (section)

===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}}