Analytic and Numerical Methods for Emerging Tomography Techniques

断层成像技术用于对感兴趣对象的内部结构进行非侵入性可视化，已经改变了许多领域，包括天文学、医学、工业无损检测、国土安全、考古学、生物学、地球物理学等。与技术创新一起，数学方法在推动这些技术方面发挥了重要作用。该项目将解决两种有前景的断层扫描技术所产生的数学问题：多能量计算机断层扫描(MECT)和康普顿相机成像(CCI)。这两种方法在成像应用中都非常有用。MECT是一种X射线透射式成像方法，它使用X射线衰减的能量依赖性来确定感兴趣对象的元素组成。传统的单能量CT使用简化的(线性)X射线传输模型，因此产生的灰度图像仅显示被扫描物体的形态。相比之下，由于建模更准确，MECT可以提供定量信息和彩色可视化。MECT被认为是CT成像的再创造，预计将在未来几年对医学成像产生重大影响。康普顿相机在天文学中被用作探测大气或宇宙伽马射线源的望远镜。初步研究表明，它可用于广泛的医学成像应用，以及放射性去污染。这一结果也将用于国土安全成像，以检测非法核材料。除了它的实际好处，该项目还将为研究生和本科生创造培训、研究经验和职业发展的机会。MECT中的图像重建需要解决一个非线性逆问题，对于这个问题必须使用迭代方法，因为还没有找到解析解。虽然在MECT中发展图像重建算法一直是一个非常活跃的研究领域，但系统地分析反演的唯一性和稳定性的研究一直是有限的。在目前的实践中，产生唯一和稳定重建的扫描协议主要是通过检查从物理模型实验获得的图像中的噪声来确定的。该项目的一个主要目标是根据MECT测量模型的稳定性估计，开发一种系统的方法来设计MECT扫描参数。在医疗应用中使用康普顿相机的一个主要挑战将是克服图像重建的复杂性。研究人员将为康普顿相机开发有效和计算效率高的图像重建方法，以及对允许的探测器几何形状和测量不确定度进行分析。该奖项反映了NSF的法定使命，通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为是值得支持的。

Tomography techniques, which are used for non-invasive visualization of the interior structure of an object of interest, have transformed many fields, including astronomy, medicine, industrial non-destructive testing, homeland security, archaeology, biology, geophysics, and others. Together with technological innovations, mathematical approaches have been instrumental in advancing these techniques. The project will address mathematical problems arising from two promising tomography techniques: Multi-energy Computed Tomography (MECT) and Compton Camera Imaging (CCI). Both methods are highly useful in imaging applications. MECT is an x-ray transmission imaging method that uses the energy dependence of x-ray attenuation to determine the elemental composition of an object of interest. The conventional single-energy CT uses a simplified (linear) model for x-ray transmission, and thus, produces a grayscale image revealing only the morphology of scanned objects. In contrast, MECT can provide quantitative information and visualization in color, as a result of the more accurate modeling. MECT is regarded as a reinvention of CT imaging and is anticipated to have a significant impact on medical imaging in the coming years. Compton cameras have been used in astronomy as a telescope to detect atmospheric or cosmic gamma-ray sources. Pilot studies have shown that it can be used in a wide range of medical imaging applications, as well as radioactive decontamination. The result will also be useful in homeland security imaging for detecting illicit nuclear materials. Besides its practical benefits, the project will create opportunities for training, research experience, and career development for graduate and undergraduate students. It will also facilitate interdisciplinary collaborations.Image reconstruction in MECT requires solving a nonlinear inverse problem, for which iterative approaches must be used because an analytical solution has yet to be discovered. Although developing image reconstruction algorithms in MECT has been a very active research area, the studies systematically analyzing the uniqueness and stability of inversion have been limited. In the current practice, the scan protocol yielding unique and stable reconstructions is primarily determined by examining the noise in images obtained from physical phantom experiments. A major goal of the project is to develop a systematic approach for the design of MECT scan parameters based on the stability estimates for the MECT measurement model. A major challenge for using Compton cameras in medical applications will be overcoming the complexity of image reconstruction. The investigator will develop effective and computationally efficient image reconstruction methods for Compton cameras as well as analyses for admissible detector geometries and measurement uncertainties. Numerical techniques for the assessment of the theoretical results of the project will also be developed.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.