CAREER:Development and Application of Compressive Sensing Based Interior Tomography

摘要：虽然经典的计算机断层扫描（CT）理论的目标是从完整的投影中精确重建整个截面或整个体积，但生物医学应用通常侧重于相对较小的内部兴趣区域（roi）。然而，传统的CT理论不能仅通过与x射线相关的截断投影精确地重建内部ROI，因为这个内部问题在无约束环境下没有唯一的解决方案。2007年，PI和他的合作者证明，如果ROI内部已知子区域，则可以精确稳定地解决内部问题。受压缩感知（CS）理论的启发，PI于2009年提出了基于CS的内部层摄影术的概念，并证明了当ROI为分段常数时，通过内部扫描可以实现精确的内部重建，随后将其推广到分段多项式ROI的情况。本次CAREER提案的目标是推进基于cs的内部层析成像理论和算法，并实现从传统的全局滤波反投影（FBP）到当代内部重建的范式转变。这三个目标是：1)对一般的稀缺性约束模型进行数学分析，建立唯一性、精确性和稳定性，以及相应的离散格式的性质；2)结合split-Bregman和统计重建方法，在一般POCS框架下开发和优化新的内部重建算法；3)通过数值模拟验证了理论发现和算法的有效性，并通过解决大病人问题证明了算法的实用性。该研究将与教育和推广活动紧密结合，包括在弗吉尼亚理工大学-维克森林大学生物医学工程与科学学院（SBES）为研究生和本科生开设医学图像重建课程。

ABSTRACT-1149679While classic computed tomography (CT) theory targets exact reconstruction of a whole cross-section or entire volume from complete projections, biomedical applications often focus on relatively small internal region-of-interests (ROIs). However, traditional CT theory cannot exactly reconstruct an internal ROI only from truncated projections associated with x-rays through the ROI because this interior problem does not have a unique solution in an unconstrained setting. In 2007, the PI and his collaborators proved that the interior problem can be exactly and stably solved if a sub-region is known inside the ROI. Inspired by the compressive sensing (CS) theory, in 2009 the PI proposed the concept of CS-based interior tomography and proved that exact interior reconstruction is achievable with an interior scan if the ROI is piecewise constant, which is subsequently extended to the case of piecewise polynomial ROI.The goal of this CAREER proposal is to advance the CS-based interior tomography theory and algorithms, and make a paradigm shift from traditional global filtered back-projection (FBP) to contemporary interior reconstruction. The three objectives are to 1) perform mathematical analysis on a general scarcity constraint model to establish uniqueness, exactness and stability, as well as the properties of the corresponding discrete scheme; 2) develop and optimize novel interior reconstruction algorithms in a general POCS framework incorporating the split-Bregman and statistical reconstruction methods; 3) verify the theoretical findings and validate the proposed algorithms via numerical simulation, and demonstrate its utility by solving the big patient problem. The research will be closely integrated with educational and outreach activities including creating a Medical Image Reconstruction course at both graduate and undergraduate levels at the Virginia Tech-Wake Forest University School of Biomedical Engineering and Sciences (SBES).