TITLE: Open problems in Mathematical Tomography?
QUESTION [9 upvotes]: Since I feel that Tomography can be applied to a wide range of sciences, I was wondering what the current open problems in Tomographic Reconstruction are.
Furthermore, I am curious as to how these conjectures/open problems might help Tomography, once they're solved. Do they allow researchers in biomedical imaging to have a better view of the human body? Do they allow researchers in geophysical engineering to predict the exact location of upcoming earthquakes more quickly? Etc.

REPLY [3 votes]: One open topic I can think of in medical image reconstruction is comressive sensing (or compressed sensing, compressive sampling, etc). This topic becomes popular recently because it has practical meanings.
For example, in computed tomography (CT), the regular case is to let the scanner gantry rotate around the patients for, say 360 degrees, to collect sufficient data (360 projections) for reconstruction. However, if possible, you certainly don't want the patient exposed too much to the radiation. One way to handle this is to reduce the samplings, meaning the CT scanner gantry will acquire data much less then it used to do (ex. only acquire perhaps 30 projections, compared to 360); This will significantly reduce the radiation dose for the patients.
The other example is in MRI scan, since patients have to stay in the chamber for like 30 mins, it would be really nice to sample less if compressive sensing can work perfect.
It is a interesting topic, and you can find many resources online by simply googling compressive sensing if you are interested.