CAREER:Development and Application of Compressive Sensing Based Interior Tomography

虽然经典的CT理论的目标是从完整的投影精确重建整个横截面或整个体积，但生物医学应用通常集中在相对较小的内部感兴趣区(ROI)。然而，传统的CT理论不能准确地重建内部ROI，因为这个内部问题在不受约束的情况下不存在唯一解。2007年，PI和他的合作者证明了如果在ROI内已知子区域，则内部问题可以精确而稳定地解决。受压缩传感(CS)理论的启发，PI在2009年提出了基于CS的内部层析成像的概念，并证明了如果ROI是分段常数，则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。该职业提案的目的是推进基于CS的内部层析成像理论和算法的发展，并使传统的全局滤波反投影(FBP)向现代内部重建转变。这三个目标是：1)对一般稀缺性约束模型进行数学分析，以建立唯一性、准确性和稳定性，以及相应离散格式的性质；2)在结合Split-Bregman和统计重建方法的一般POCS框架下，开发和优化新的内部重建算法；3)验证理论结果，通过数值模拟验证所提出的算法，并通过解决大患者问题来验证其有效性。这项研究将与教育和推广活动紧密结合在一起，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)开设研究生和本科生水平的医学图像重建课程。

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).