AI 中文总结
本文对源在一维光滑曲线上的锥束CT束硬化伪影开展微局部分析，采用FBP型重建，证明伪影为余法型且弱于目标奇异性，还扩展到非光滑曲面并通过金属模拟重建验证理论。
AI 中文摘要
本文针对X射线源被限制在三维空间中一条一维光滑曲线γ上的锥束X射线CT中产生的束硬化伪影，开展了一项新颖的微局部分析。我们表明，当采用标准的比尔-朗伯定律对CT数据进行建模时，模型中的指数项会在数据中产生线性X射线变换不存在的奇异性。我们假设作为重建目标的衰减系数μ在三维空间中的曲面S上存在跳跃间断，其余部分光滑。我们证明，当X射线束同时与S在两个点处相切时，数据中会出现这些额外的奇异性。为了研究数据中的奇异性如何传播到重建空间，我们应用了滤波反投影（FBP）型重建方法。我们证明，由束硬化产生的伪影在局部属于余法型，且位于一个二维曲面上，该曲面是所有与γ相交的双切线射线的并集。这些伪影明显弱于μ的重建跳跃（即所需的奇异性），我们通过相应余法分布的阶数对这一点进行量化。虽然我们的主要理论适用于S的光滑区域，但我们也将其扩展到具有“脊”的非光滑S，这些脊出现在S局部为两个光滑曲面片沿一条曲线横截相交的位置（例如长方体的边缘）。此外，我们给出了圆形锥束CT中金属物体的模拟重建结果，以验证我们的理论。
英文摘要
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 $ γ\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 $μ$ (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 $γ$. The artifacts are notably weaker than the reconstructed jumps of $μ$ (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.
Comments31 pages, 4 figures
