Timeline for Partial recovery from Radon transform
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
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| S Apr 23, 2014 at 12:39 | history | suggested | Tommi |
corrected tagging
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| Apr 23, 2014 at 12:30 | review | Suggested edits | |||
| S Apr 23, 2014 at 12:39 | |||||
| Apr 16, 2014 at 10:26 | vote | accept | CommunityBot | ||
| Jan 30, 2014 at 14:39 | comment | added | user45183 | @alvarezpaiva: It seems that you are confused with the notion of radiographs. I did not say that knowing $\int_L f dS$ for an infinite number of distinct lines L is sufficient. Rather, I said that knowing $\int_{p+L} f dS$ for all displacements p and an infinite number of distinct lines L. For brevity one calls $R_L f$ defined by $(R_L f)(p) = (Rf)(L,p) = \int_{p+L} f dS$ a radiograph. Clearly for your specific example, one can reconstruct the function on all slices and by putting the slices together one gets the whole function. | |
| Jan 30, 2014 at 14:36 | comment | added | alvarezpaiva | Take all lines on a plane inside three space, they are infinite in number and they comprise an infinite number of directions. However, knowing the value of the transform for only these lines does not allow you to reconstruct the function. | |
| Jan 30, 2014 at 14:30 | answer | added | user45183 | timeline score: 2 | |
| Jan 30, 2014 at 14:27 | comment | added | user45183 | Sigurd Helgason: Integral Geometry and Radon Transforms, Proposition 7.8 and Andrew Markoe: Analytic Tomography, Theorem 3.144. | |
| Jan 30, 2014 at 11:25 | comment | added | alvarezpaiva | what's your reference for this? | |
| Jan 30, 2014 at 7:12 | history | edited | user45183 | CC BY-SA 3.0 |
changed typos in the variables
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| Jan 29, 2014 at 22:22 | comment | added | user45183 | It is well-known for the X-ray transform in n-dimensional space, that if one knows the radiographs (!) for infinitely many directions, then one can uniquely reconstruct the function under scrutiny. | |
| Jan 29, 2014 at 20:45 | comment | added | alvarezpaiva | It is not true that if you know the line transform for infinitely many lines in three-space then you can recover the function. The best you can do is to know the value of the transform over certain three-dimensional manifolds in the space of lines. | |
| Jan 29, 2014 at 19:51 | history | asked | user45183 | CC BY-SA 3.0 |