Statistical microlocal analysis in two-dimensional X-ray CT

TL;DR

This paper introduces Statistical Microlocal Analysis (SMA), a novel method for assessing edge detectability in 2D X-ray CT images by incorporating noise and sampling effects into the classical microlocal analysis framework.

Contribution

We develop SMA, a statistical hypothesis testing approach that quantifies edge detectability in noisy, discretely sampled CT data, extending classical microlocal analysis to practical measurement conditions.

Findings

SMA accurately predicts edge detectability in simulated CT data.

The method quantifies uncertainty in edge magnitude and direction.

Strong agreement observed between theory and experiments.

Abstract

In many imaging applications it is important to assess how well the edges of the original object, $f$, are resolved in an image, $f^\text{rec}$, reconstructed from the measured data, $g$. In this paper we consider the case of image reconstruction in 2D X-ray Computed Tomography (CT). Let $f$ be a function describing the object being scanned, and $g=Rf + \eta$ be the Radon transform data in $\mathbb{R}^2$ corrupted by noise, $\eta$, and sampled with step size $\sim\epsilon$. Conventional microlocal analysis provides conditions for edge detectability based on the scanner geometry in the case of continuous, noiseless data (when $\eta = 0$), but does not account for noise and finite sampling step size. We develop a novel technique called Statistical Microlocal Analysis (SMA), which uses a statistical hypothesis testing framework to determine if an image edge (singularity) of $f$ is detectable from $f^\text{rec}$, and we quantify edge detectability using the statistical power of the test. Our approach is based on the theory we developed in previous work, which provides a characterization of $f^\text{rec}$ in local $O(\epsilon)$-size neighborhoods when $\eta \neq 0$. We derive a statistical test for the presence and direction of an edge microlocally given the magnitude of $\eta$ and data sampling step size. Using the properties of the null distribution of the test, we quantify the uncertainty of the edge magnitude and direction. We validate our theory using simulations, which show strong agreement between our predictions and experimental observations. Our work is not only of practical value, but of theoretical value as well. SMA is a natural extension of classical microlocal analysis theory which accounts for practical measurement imperfections, such as noise and finite step size, at the highest possible resolution compatible with the data.