Inverse Problems with Internal Data

计算机断层扫描(CT)、磁共振成像(MRI)和超分辨显微镜等成像技术的进步改变了临床医学和基础生物医学研究的实践。虽然众所周知，这类技术的发展依赖于物理学和工程学的进步，但应用数学和计算数学也发挥了重要作用，这一点却鲜为人知。这项研究项目研究了在新的医学生物医学成像模式中出现的数学问题，其中新的新测量发挥了关键作用。这项研究将研究新的数学算法，这些算法将导致光学成像在分辨率(在较小尺度上可视化结构)和计算速度方面的改进。特别是，该项目的目标是设计稳健和准确的图像重建算法，以便在比目前可能的阶段更早的阶段检测和确定疾病的特征。首席研究员对应用数学和理论物理有研究兴趣。他也是一名内科医生。研究生将被训练在这种跨学科的环境中发挥作用。本项目的目标是研究生物医学光学成像中出现的内部数据的反问题。我们将考虑两类问题。(I)Teh Pi将开发数学上合理的成像方法，用于分辨率低于衍射极限的成像，也称为超分辨率成像。本文的工作包括对有内源的逆散射问题的分析和重建算法的发展。这些算法将使用来自物理上真实的数值模拟的数据进行测试和表征。(Ii)PI将研究声光成像中出现的逆问题。这项研究将集中在相干多次散射的机制上，与非相干成像相比，这将导致相当大的数学简化。特别是，PI将开发重建方法，从相干声光测量中恢复辐射传输方程的吸收和散射系数。最后，将研究声光效应模型的改进在图像重建中的作用。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Advances in imaging technologies such as computed tomography (CT), magnetic resonance imaging (MRI) and super resolution microscopy have transformed the practice of clinical medicine and basic biomedical research. Although the development of such technologies is well known to depend upon progress in physics and engineering, it is less well known that applied and computational mathematics has also played an essential role. This research project studies mathematical questions that arise in new medical biomedical imaging modalities in which new novel measurements play a key role. The research will study novel mathematical algorithms that will lead to improvements in optical imaging both with respect to resolution (visualizing structures at smaller scales) and computational speed. In particular, the project aims to devise robust and accurate image reconstruction algorithms that may lead to the detection and characterization of disease at much earlier stages than is currently possible. The principal investigator has research interests in applied mathematics and theoretical physics. He is also a physician. Graduate students will be trained to function in this interdisciplinary environment.The objective of this project is to investigate inverse problems with internal data that arise in biomedical optical imaging. Two classes of problems will be considered. (i) Teh PI will develop mathematically-justified methods for imaging below the diffraction limit of resolution, also known as superresolution imaging. The proposed work includes both analysis of the inverse scattering problem with internal sources and the development of reconstruction algorithms. The algorithms will be tested and characterized using data from physically realistic numerical simulations. (ii) The PI will study inverse problems that arise in acousto-optic imaging. The research will focus on the regime of coherent multiple scattering which leads to considerable mathematical simplifications compared to incoherent imaging. In particular, the PI will develop reconstruction methods for recovering the absorption and scattering coefficients of the radiative transport equation from coherent acousto-optic measurements. Finally, the role of improvements in modeling of the acousto-optic effect on image reconstruction will be investigated.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.