ROI reconstruction from truncated cone-beam projections
ROI reconstruction from truncated cone-beam projections
复制标题
从截断锥束投影重建 ROI
DOI：
10.3934/ipi.2018002
复制
发表时间：
2018
影响因子：
1.3
通讯作者：
Daniel Vera
中科院分区：
文献类型：
--
作者：
R. Azencott;B. Bodmann;Tasadduk Chowdhury;D. Labate;Anando Sen;Daniel Vera
Region-of-Interest (ROI) tomography aims at reconstructing a region of interest \begin{document} $C$ \end{document} inside a body using only x-ray projections intersecting \begin{document} $C$ \end{document} and it is useful to reduce overall radiation exposure when only a small specific region of a body needs to be examined. We consider x-ray acquisition from sources located on a smooth curve \begin{document} $Γ$ \end{document} in \begin{document} $\mathbb R^3$ \end{document} verifying the classical Tuy condition. In this generic situation, the non-trucated cone-beam transform of smooth density functions \begin{document} $f$ \end{document} admits an explicit inverse \begin{document} $Z$ \end{document} as originally shown by Grangeat. However \begin{document} $Z$ \end{document} cannot directly reconstruct \begin{document} $f$ \end{document} from ROI-truncated projections. To deal with the ROI tomography problem, we introduce a novel reconstruction approach. For densities \begin{document} $f$ \end{document} in \begin{document} $L^{∞}(B)$ \end{document} where \begin{document} $B$ \end{document} is a bounded ball in \begin{document} $\mathbb R^3$ \end{document} , our method iterates an operator \begin{document} $U$ \end{document} combining ROI-truncated projections, inversion by the operator \begin{document} $Z$ \end{document} and appropriate regularization operators. Assuming only knowledge of projections corresponding to a spherical ROI \begin{document} $C \subset B$ \end{document} , given \begin{document} $e >0$ \end{document} , we prove that if \begin{document} $C$ \end{document} is sufficiently large our iterative reconstruction algorithm converges at exponential speed to an \begin{document} $e$ \end{document} -accurate approximation of \begin{document} $f$ \end{document} in \begin{document} $L^{∞}$ \end{document} . The accuracy depends on the regularity of \begin{document} $f$ \end{document} quantified by its Sobolev norm in \begin{document} $W^5(B)$ \end{document} . Our result guarantees the existence of a critical ROI radius ensuring the convergence of our ROI reconstruction algorithm to an \begin{document} $e$ \end{document} -accurate approximation of \begin{document} $f$ \end{document} . We have 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 \begin{document} $B$ \end{document} .
Region-of-Interest (ROI) tomography aims at reconstructing a region of interest \begin{document} $C$ \end{document} inside a body using only x-ray projections intersecting \begin{document} $C$ \end{document} and it is useful to reduce overall radiation exposure when only a small specific region of a body needs to be examined. We consider x-ray acquisition from sources located on a smooth curve \begin{document} $Γ$ \end{document} in \begin{document} $\mathbb R^3$ \end{document} verifying the classical Tuy condition. In this generic situation, the non-trucated cone-beam transform of smooth density functions \begin{document} $f$ \end{document} admits an explicit inverse \begin{document} $Z$ \end{document} as originally shown by Grangeat. However \begin{document} $Z$ \end{document} cannot directly reconstruct \begin{document} $f$ \end{document} from ROI-truncated projections. To deal with the ROI tomography problem, we introduce a novel reconstruction approach. For densities \begin{document} $f$ \end{document} in \begin{document} $L^{∞}(B)$ \end{document} where \begin{document} $B$ \end{document} is a bounded ball in \begin{document} $\mathbb R^3$ \end{document} , our method iterates an operator \begin{document} $U$ \end{document} combining ROI-truncated projections, inversion by the operator \begin{document} $Z$ \end{document} and appropriate regularization operators. Assuming only knowledge of projections corresponding to a spherical ROI \begin{document} $C \subset B$ \end{document} , given \begin{document} $e >0$ \end{document} , we prove that if \begin{document} $C$ \end{document} is sufficiently large our iterative reconstruction algorithm converges at exponential speed to an \begin{document} $e$ \end{document} -accurate approximation of \begin{document} $f$ \end{document} in \begin{document} $L^{∞}$ \end{document} . The accuracy depends on the regularity of \begin{document} $f$ \end{document} quantified by its Sobolev norm in \begin{document} $W^5(B)$ \end{document} . Our result guarantees the existence of a critical ROI radius ensuring the convergence of our ROI reconstruction algorithm to an \begin{document} $e$ \end{document} -accurate approximation of \begin{document} $f$ \end{document} . We have 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 \begin{document} $B$ \end{document} .