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
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.
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.
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
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.