A probabilistic model of X-ray computed tomography

TL;DR

This paper develops a probabilistic framework for X-ray CT by modeling photon counts along lines through the unit disk and relating the log-count measurements to the X-ray transform $Xf$. It proves a functional law of large numbers showing $Y_{n,m,N}f$ converges to the discretized transform $X_{n,m}f$, and establishes a family of functional central limit theorems with hierarchical correction terms that reduce the required photon count normalization from $N\sim n^2$ toward much smaller scales; a Berry-Esseen inequality provides explicit convergence rates. The main results rigorously connect shot-noise statistics to the deterministic X-ray transform and provide practical guidance on normalization strategies in CT under Poisson noise. Methodologically, the work combines Taylor expansions, Poisson moment bounds, and weak convergence in $\mathcal{D}'(Z)$ to yield sharp limit theorems and finite-sample error controls, with extensions to alternative normalizations. The findings have theoretical significance for the statistical understanding of CT data and can inform photon-counting protocols in imaging applications where log-count measurements approximate line integrals.

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.

Theorems & Definitions (31)

• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Theorem 2.3
• Theorem 2.4
• Corollary 2.5
• Corollary 2.6
• Lemma 3.1
• proof