A probabilistic model of X-ray computed tomography

TL;DR

This paper develops a probabilistic model for X-ray CT scans, proving laws of large numbers and central limit theorems that describe the asymptotic behavior of photon count observations, including convergence rates.

Contribution

It introduces a stochastic process model for X-ray measurements and establishes new limit theorems and convergence rates for the observed photon counts.

Findings

Convergence of photon count process to the X-ray transform in $L^2$.

Central limit theorems showing convergence to white noise.

Explicit Berry-Esseen bounds for the rate of convergence.

Abstract

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.