Microlocal analysis of non-linear artifacts in cone beam CT

Abstract

We present a novel microlocal analysis of beam hardening artifacts arising in cone-beam X-ray CT, where the set of X-ray sources is restricted to a 1D smooth curve $ \gamma\subset \mathbb{R}^3$. We show that, when the CT data is modeled in the standard way using the Beer Lambert law, the exponential term in the model creates singularities in the data that are not present in the linear X-ray transform. We assume that the attenuation coefficient $\mu$ (the reconstruction target) has a jump discontinuity across a surface $\mathcal{S}\subset\mathbb{R}^3$, and is smooth otherwise. We prove that these additional singularities in the data occur when the X-ray beam is tangent to $\mathcal{S}$ at two points simultaneously. To investigate how the singularities in the data propagate to the reconstruction space, we apply Filtered Back Projection (FBP) type reconstruction. We prove that the artifacts due to beam hardening are locally of conormal type, and lie on a 2-D surface which is the union of all double tangent rays which intersect $\gamma$. The artifacts are notably weaker than the reconstructed jumps of $\mu$ (i.e., the desired singularities), and we quantify this using the order of the corresponding conormal distributions. While our primary theory applies to the regions of $\mathcal{S}$ that are smooth, we also extend our theory to non-smooth $\mathcal{S}$ with "ridges." These arise where $\mathcal{S}$ is locally the intersection of two smooth surface patches meeting transversely along a curve (e.g., the edge of a cuboid). In addition, we present simulated reconstructions of metal objects in circular cone-beam CT to validate our theory.