Collaborative research: Mathematics of emerging imaging methods in medicine and homeland security

计算机断层扫描在生物医学成像中起着核心作用。层析成像技术也广泛用于工业无损检测、地质学和地球物理学、天文学以及其他领域。最近，它们在国土安全领域得到了重要应用。在过去的几十年里，已经开发了几种模式并已成为标准。例如，传统的x射线CT扫描、SPECT、MRI、光学、超声和电阻抗断层扫描，仅举几例。然而，诸如检测软组织中的小癌性肿瘤或在大型容器中发现少量核材料等任务仍然是一个重大挑战。近年来，革命性的“混合”或“多物理场”医学成像方法已经出现。通过结合两种或三种不同类型的波（或物理场），这些方法克服了传统断层扫描技术的局限性，以更低的成本和更小的健康危害，提供了否则无法获得的、可能挽救生命的诊断信息。通常，这些模式中的图像是通过复杂的数学程序获得的，而不是通过直接获取。相应的数学，大多还处于非常早期的发展阶段。因此，该项目的第一部分解决了几个新型混合技术中的核心数学和数值问题，这些技术基于磁波、电波、声波和光波的组合，以及所有这些技术共同的一般数学问题。项目的第二部分是针对国土安全问题改进已知方法和开发新的层析成像技术。特别是，我们将重点关注在过境点和港口使用的货物中非法武器级核材料的探测技术。在第三部分中，这些技术将用于解决合成孔径雷达应用、射电断层成像、超声反射测量等领域的一些突出问题。该项目的目标是开发新的医学成像技术，以及在过境点和港口发现非法核材料的有效方法。该项目将对开发和实施几种新的灵敏、廉价和安全的生物医学成像方法、有效的国土安全核威胁探测技术，以及在成像和非破坏性工业测试（例如合成孔径雷达）等其他几个领域的应用产生重大影响。研究结果将通过在高质量的研究期刊上发表的出版物、在国家和国际会议上的发言以及在不同地点举行的一系列讲座来传播。研究生将在该项目中发挥重要作用，这将为他们在精确科学、医学、生物学和国土安全领域的工作做好准备。该项目的部分研究成果将在研究生班、国内和国际学校和会议的系列讲座以及两本专著中发表。

Computerized tomography plays a central role in biomedical imaging. Tomography techniques are also extensively used for industrial non-destructive testing, in geology and geophysics, astronomy, and in other areas. Lately, they have found important applications in homeland security. Over the last decades, several modalities have been developed and have became standard. Such are, for instance, the traditional X-ray CT scan, SPECT, MRI, Optical-, Ultrasound-, and Electrical Impedance Tomography, to name a few. However, such tasks as detection of small cancerous tumors in soft tissues or finding a small amount of nuclear material in a large container still present a significant challenge. In recent years, revolutionary "hybrid" or "multi-physics" methods of medical imaging have been emerging. By combining two or three different types of waves (or physical fields) these methods overcome limitations of classical tomography techniques and deliver otherwise unavailable, potentially life-saving diagnostic information - at a lesser cost and with less health hazard to a patient. As a rule, the images in these modalities are obtained by complex mathematical procedures, rather than through direct acquisition. The corresponding mathematics is, mostly, at very early stages of development. Thus, the first part of the project addresses the central mathematical and numerical issues in several of the novel hybrid techniques, based on combinations of magnetic, electric, acoustic, and optical waves, and the general mathematical issues common to all of these techniques. The second part of the projects is directed toward the improvement of the known methods and development of new tomographic techniques for homeland security problems. In particular, we will focus on the detection techniques for illicit weapons-grade nuclear materials in cargo, to be used at border crossings and harbors. In the third part these techniques will be used to resolve some outstanding problems in Synthetic Aperture Radar applications, radio tomography, ultrasound reflectometry, and other areas.The goal of the project is to develop new techniques of medical imaging, as well as efficient methods of detecting illicit nuclear materials at border crossings and in harbors. The project will have a significant impact on the development and implementation of several new sensitive, inexpensive, and safe methods of biomedical imaging, efficient techniques of nuclear threat detection in homeland security, with applications in several other areas of imaging and non-destructive industrial testing (e.g., synthetic aperture radar). The results will be disseminated through publications in high quality research journals, presentations at national and international conferences, and series of lectures at various venues. Graduate students will play a significant role in the project, which will prepare them for work in the exciting area at the junction of exact sciences, medicine, biology, and homeland security. Parts of the study resulted from the project will be delivered in graduate level classes, lecture series at national and international schools and conferences, and in two monographs.