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.

💡 Research Summary

The paper develops a rigorous probabilistic framework for the photon‑count data obtained in X‑ray computed tomography (CT). The authors consider the unit disk D⊂ℝ² and a non‑negative, Lipschitz attenuation function f defined on D. For each line y intersecting D they define the X‑ray transform Xf(y)=∫_{Ly∩D}f dx, which maps f to a function on the space Z of all such lines. In a CT scan the number of photons sent along a line is random; it follows a Poisson distribution with mean N, while each photon independently survives the passage with probability p_y=exp(−Xf(y)). Consequently the number of emerging photons S_y is Poisson(N p_y).

A major technical difficulty is that S_y can be zero, making the logarithm in the usual estimator −log(S_y/N) undefined. The authors therefore introduce three practical regularizations: (N1) add one to every count, (N2) add one only when the count is zero, and (N3) repeat the scan until at least one photon is detected. The main body of the paper adopts (N1) for mathematical simplicity, defining the observable Y_{n,m,N}^f(y)=−log