Mathematics for Imaging with Waves

当人们为观察到的结果寻找原因或想要产生期望的结果时，在所有的科学和技术领域都会出现逆问题。计算能力的提高和强大算法的发展使得将反问题技术应用于日益复杂的现实问题成为可能。这项研究将侧重于逆问题的应用，包括一些医疗技术以及成像中的其他问题，例如在地球的子结构中定位石油和矿藏，从望远镜数据创建天体物理图像，寻找材料内的裂缝和界面，形状优化等。计算机断层扫描（CT）、磁共振成像（MRI）和超声等医学成像技术是逆问题发挥重要作用并帮助挽救生命的例子，本研究将发展几个基本逆问题的数学理论。第一个项目涉及一种相对较新的医学成像技术，称为电阻抗断层扫描。在EIT中，人们试图通过在物体的边界处进行电压和电流测量来确定物体的电特性。一个潜在的应用是在医学成像中，特别是在检测肺水肿或癌性肿瘤中，因为这种异常在正常情况下具有非常不同的电特性。第二个项目是走时层析成像。在这种成像技术中，主要研究者（PI）将用不同类型的波（如电磁波或声波）探测物体。 通过测量波通过介质的传播时间，PI将试图确定介质的属性。第三个项目是非线性声学中的反问题。它在超声中有应用，特别是在称为组织谐波成像（THI）的医学成像技术中。THI是超声诊断（US）的常规组成部分，在该方法中，由非线性基波US波传播产生的高频谐波产生的图像包含较少的伪影。该奖项反映了NSF的法定使命，通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Inverse problems arise in all fields of science and technology when one seeks a cause for an observed effect or wants to produce a desired effect. The increase in computing power and the development of powerful algorithms have made it possible to apply the techniques of inverse problems to real-world problems of growing complexity. This research will focus on inverse problems with applications that include a number of medical techniques as well as other problems in imaging, such as locating oil and mineral deposits in the Earth's substructure, creating of astrophysical images from telescope data, finding cracks and interfaces within materials, shape optimization, and many others. The familiar medical imaging technologies of computed tomography (CT) scans, magnetic resonance imaging (MRI) and ultrasound are examples where inverse problems have played a fundamental role and have helped to save lives.This research will develop the mathematical theory of several fundamental inverse problems. The first project deals with a relatively new medical imaging technique called electrical impedance tomography. In EIT, one attempts to determine an object's electrical properties by making voltage and current measurements at the boundary of the object. One potential application is in medical imaging, particularly in detecting pulmonary edema or a cancerous tumor since such anomalies have very different electrical properties in normal situations. The second project is travel time tomography. In this imaging technique, the Principal Investigator (PI) will probe the object with different types of waves, like electromagnetic waves or sound waves. By measuring the travel times of the waves going through the medium, the PI will attempt to determine the properties of the medium. The third project is an inverse problem arising in nonlinear acoustics. It has applications in ultrasound, particularly in a medical imaging technique called tissue harmonic imaging (THI). THI is a routinely used component of diagnostic ultrasonography (US), and higher-frequency harmonic waves produced by nonlinear fundamental US wave propagation in the method generate images containing fewer artifacts.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.