Inverse Problems in Optical Imaging

计算机断层扫描（CT）和磁共振成像（MRI）等成像技术的进步已经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，这些技术的发展取决于物理学和工程学的进步，但人们不太清楚的是，应用数学和计算数学也发挥了重要作用。该研究项目研究了新的医学成像模式中出现的数学问题，这些模式单独使用近红外光或与超声结合使用。该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（在较小尺度下可视化结构）和计算速度方面的改进。特别是，该项目的目的是设计强大的和准确的图像重建算法，可能会导致在更早的阶段比目前可能的疾病的检测和表征。本研究的目的是调查生物医学光学成像中出现的逆问题。该项目包括两个组成部分。(i)漫射光光学层析成像的几何校正图像重建算法的发展。分析将在辐射传输理论的框架内进行。该项目旨在开发基于逆Born级数拓扑约化的重建方法，用于从边界测量恢复辐射输运方程的吸收和散射系数，并表征这些方法的收敛性和近似误差。它计划采用这些结果开发和测试快速重建算法适用于非接触式光学层析成像系统。 (ii)声光成像中的逆问题研究。这种最近开发的成像模式结合了高分辨率的超声成像与生理上重要的对比度的光学成像。工作包括相关的逆散射问题的理论分析，以及相关的图像重建算法的数值测试。特别是，该项目旨在开发重建方法，从声光测量中恢复辐射传输方程的吸收和散射系数。

Advances in imaging technologies such as computed tomography (CT) and magnetic resonance imaging (MRI) 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 imaging modalities that make use of near-infrared light either alone, or in combination with ultrasound. 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 objective of this research is to investigate inverse problems that arise in biomedical optical imaging. The project involves two components. (i) Development of mathematically-justified image reconstruction algorithms for optical tomography with diffuse light. The analysis will be carried out within the framework of radiative transport theory. The project aims to develop reconstruction methods based on topological reduction of the inverse Born series, for recovering the absorption and scattering coefficients of the radiative transport equation from boundary measurements, and to characterize the convergence and approximation error of these methods. It is planned to employ these results to develop and test fast reconstruction algorithms suitable for use with a noncontact optical tomography system. (ii) Study of inverse problems that arise in acousto-optic imaging. This recently developed imaging modality combines the high-resolution of ultrasound imaging with the physiologically important contrast of optical imaging. The work includes theoretical analysis of related inverse scattering problems, together with numerical tests of associated image reconstruction algorithms. In particular, the project aims to develop reconstruction methods for recovering the absorption and scattering coefficients of the radiative transport equation from acousto-optic measurements.