Timeline for Inversion of Radon transform by incomplete data: specific case
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| S Apr 9, 2014 at 11:13 | history | suggested | Tommi |
replaced inverse tag with inverse problems tag
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| Apr 9, 2014 at 11:01 | review | Suggested edits | |||
| S Apr 9, 2014 at 11:13 | |||||
| S Apr 8, 2014 at 13:02 | history | suggested | Tommi | CC BY-SA 3.0 |
added the inverse tag, since there seems to be no inverse problems tag; also fiex latex code
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| Apr 8, 2014 at 12:57 | review | Suggested edits | |||
| S Apr 8, 2014 at 13:02 | |||||
| Oct 24, 2012 at 15:29 | vote | accept | Appliqué | ||
| Oct 22, 2012 at 1:27 | answer | added | an12 | timeline score: 4 | |
| Oct 21, 2012 at 17:01 | comment | added | Nick Alger | The Radon transform smooths $H^s \mapsto H^{s+1/2}$, with eigenvalues $\lambda_i \rightarrow 0$ corresponding to increasingly oscillatory eigenvectors. If you add white noise to a radon transformed image then try to invert it, the $i'th$ components of the noise will be amplified by $1/\lambda_i$, which can be arbitrarily large. To overcome this one usually needs regularization or a prior, depending on whether you take a deterministic or probabilistic approach to the problem. The $1/2$ power smoothing is pretty weak compared to other problems. I don't know any more specifics. | |
| Oct 21, 2012 at 13:16 | comment | added | Appliqué | Thank you Nick Alger, but can you be more exact please? I listened to a course on computer tomography (with applictions in medicine) and in this course we were speaking about reconstruction of function by incomplete data of another character: we know only integrals over any linear manifold, that doesn't intersect some convex body (Cormack-type theorems (1963-1964)). | |
| Oct 21, 2012 at 10:21 | comment | added | Nick Alger | This is well studied in the mathematics of medical imaging, you might look around there. The Radon transform is a smoothing operator with the degree of smoothing being exactly $1/2$ of a derivative, so inversion is mildly ill-posed. | |
| Oct 21, 2012 at 9:19 | comment | added | Appliqué | Of course, I modified a message. | |
| Oct 21, 2012 at 9:18 | history | edited | Appliqué | CC BY-SA 3.0 |
added 1560 characters in body
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| Oct 20, 2012 at 18:31 | comment | added | Dirk | Could you expand on the relation of the Radon transform and economics? (Out of curiosity.) | |
| Oct 20, 2012 at 14:15 | history | asked | Appliqué | CC BY-SA 3.0 |