基于截断锥束投影的感兴趣区域重建
Region-of-Interest reconstruction from truncated cone-beam projections
摘要：
感兴趣区域（ROI）断层摄影旨在仅使用与$C$相交的X射线投影来重建身体内的感兴趣区域$C$，目的是当仅需要检查身体的小的特定区域时减少总体辐射暴露。我们考虑从位于光滑曲线$Γ$上的源获取X射线，在$\mathbb{R}^3$中验证经典的Tuy条件。在这种情况下，平滑密度$f$的{ \it non-trucated}锥束变换$D f$允许显式逆$Z$ ;然而$Z$不能从ROI截断投影直接重建$f$。为了处理ROI层析成像问题，我们引入了一种新的重建方法。对于$L^{\infty}(B)$中的密度$f$，其中$B$是$\mathbb{R}^3$中的有界球，我们的方法迭代结合ROI截断投影的算子$U$，由算子$Z$和适当的正则化算子反演。假设只有知识的投影对应于一个球形的ROI $C \subset B$，给定$ε>0$，我们证明，如果$C$是足够大的，我们的迭代重建算法收敛均匀的$ε$ -准确的近似$f$，其中的准确性取决于在Sobolev规范$W^5(B)$量化的$f$的规律性。该结果表明存在临界ROI半径，确保ROI重建算法收敛于$ε$-$f$的精确近似。我们数值验证了这些理论结果，使用模拟采集的ROI截断锥束投影数据的多个采集几何形状。数值实验表明，临界ROI半径相对于支持区域$B$是相当小的。
英文摘要：
Region-of-Interest (ROI) tomography aims at reconstructing a region of interest $C$ inside a body using only x-ray projections intersecting $C$ with the goal to reduce overall radiation exposure when only a small specific region of the body needs to be examined. We consider x-ray acquisition from sources located on a smooth curve $Γ$ in $\mathbb{R}^3$ verifying classical Tuy's condition. In this situation, the {\it non-trucated} cone-beam transform $D f$ of smooth densities $f$ admits an explicit inverse $Z$; however $Z$ cannot directly reconstruct $f$ from ROI-truncated projections. To deal with the ROI tomography problem, we introduce a novel reconstruction approach. For densities $f$ in $L^{\infty}(B)$ where $B$ is a bounded ball in $\mathbb{R}^3$, our method iterates an operator $U$ combining ROI-truncated projections, inversion by the operator $Z$ and appropriate regularization operators. Assuming only knowledge of projections corresponding to a spherical ROI $C \subset B$, given $ε>0$, we prove that if $C$ is sufficiently large our iterative reconstruction algorithm converges uniformly to an $ε$-accurate approximation of $f$, where the accuracy depends on the regularity of $f$ quantified in the Sobolev norm $W^5(B)$. This result shows the existence of a critical ROI radius ensuring the convergence of the ROI reconstruction algorithm to $ε$-accurate approximations of $f$. We numerically verified these theoretical results using simulated acquisition of ROI-truncated cone-beam projection data for multiple acquisition geometries. Numerical experiments indicate that the critical ROI radius is fairly small with respect to the support region~$B$.
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