The particular case of some internal region being illuminated from all directions has also been thoroughly explored - in the mathematical literature it is known as the 'interior problem.' Perhaps surprisingly, no part of this internal region can be accurately reconstructed in general. However, when examining the FBP algorithm closely one notes that the reconstruction at a specific point uses _all_ the x-ray data, even from rays that do not pass through that point. So, until the turn of the century, the understanding was that classical computed tomography was 'all or nothing' - either all rays through the sample are measured, in which case the sample can be reconstructed; or some rays are missing (typically, some projections are truncated) in which case nothing can be recovered.

In this talk, it will be shown that even with truncated projections, certain regions of interest (ROIs) can be accurately recovered. Over the last 5 years, new mathematical results have shown that (nearly) any ROI that is illuminated from all angles can be reconstructed provided it is not internal. For some cases, relatively simple new algorithms have been devised that verify the theory and can be readily implemented for direct image reconstruction. It turns out that the Hilbert transform is the relevant operator, and not the Fourier transform.
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The speaker

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