A probabilistic model of X-ray computed tomography
We consider a discrete stochastic process, indexed by lines through the unit disk in the plane, which models the observed photon counts in a medical X-ray tomography scan. We first prove a functional law of large numbers, showing that this process converges in $L^2$ to the X-ray transform of the underlying attenuation function. We then prove a family of functional central limit theorems, which show that the normalized observations converge to a white noise on the space of lines, provided the growth rate of the mean number of photons per line is greater than a certain power of the number of lines scanned. Using this family of theorems, we can reduce that power arbitrarily close to zero by adding correction terms to the normalization. We also prove a Berry-Esseen inequality that gives a concrete rate of convergence for each functional central limit theorem in our family of theorems.