POST-DOCTORAL SCHOLAR IN MEDICAL IMAGING

Fourier transform v. X-ray transform

Incoherence with the Fourier transform

Assume an image is composed of two delta functions of unknown location and unknown amplitude.  According to CS, the image can be reconstructed from four randomly located samples of the Fourier transform.

sparse FT

The important point from CS is that the Fourier sampling is incoherent with the expansion elements (dirac deltas) in which the image is sparse.  Looking at the FT figure, it is obvious that almost any FT sample will give meaningful information on the image.
The situation is different for the x-ray transform.

Incoherence with the X-ray (or fan-beam) transform

The X-ray transform is the data model in computed tomography. 

fan-beam configuration
The X-ray source travels on a circle, and it's location is specified by the angle theta.  A projection image is captured on a detector with the detector bins being identified by the coordinate xi. The measured X-ray transmission of each ray can be converted to the line-integral of the X-ray attenuation map along that ray. A single projection yields line-integrals for all xi at a single theta. Projections are taken from a set of thetas covering the circle, resulting in a 2D data set called a sinogram. The sinogram for our sparse image is shown below.sparse fan-beam projection
Note that there X-ray transform is incoherent with the dirac deltas only in theta, not in xi. This means that a random sample (the line-integral along a single ray) is likely to yield no information on the image.  Four random samples, as was used in the FT case, will not do the job.  Due to the structure of the X-ray transform the xi-direction sampling has to satisfy the Nyquist condition.  But at least the view-angle theta-direction permits sparse sampling. For the 2-point-image-case only two views are necessary.  Random sampling in theta tends to allow sparser sampling than regularly spaced angles between projections.
In practice, random ray sampling doesn't make a whole lot of sense anyway. The patient is being exposed to the X-rays for a single view, so you might as well measure all those rays.