Hybrid Imaging with Acoustic and Optics: Efficient Computation via Constructive Analysis

本项目关注医学成像中一些混合方法的数学理论和计算工具的发展。通过混合方法，我们指的是成像技术，其中具有不同底层物理的两个模态耦合在一起，以实现所涉及的每个模态单独无法实现的对比度和分辨率。该项目关注的是一类特殊的混合成像模式，将超声成像与光学成像相结合，以实现对异质介质光学特性的高分辨率成像。我们的目标是(i)为这些混合模式开发快速和鲁棒的图像重建算法以供实际使用，以及（ii）培养该领域数学和计算研究的高级本科生，研究生和博士后研究人员。本项目拟开展的主要研究工作包括但不限于：(i)针对超声特性未知的光声层析成像（PAT）多系数反演问题开发高效的反演算法；（ii）开发用于分子成像的定量荧光PAT （fPAT）和双光子PAT （TP-PAT）反问题的有效重建算法；（iii）对超声调制荧光光学断层扫描（UMfOT）中的反问题进行数学分析，并在相同的设置下制定稳健的图像重建策略；（iv）开发传输方程的快速算法，用于局部测量的PAT、fPAT、TP-PAT和UMfOT基于优化的迭代图像重建策略。在拟议项目中发展的数学思想有望为许多科学和技术领域中出现的偏微分方程的许多类似逆系数问题的数学和数值解提供见解。

This project is concerned with the development of mathematical theory and computational tools for some hybrid methods in medical imaging. By hybrid methods we refer to imaging techniques where two modalities with distinct underlying physics are coupled together to achieve the contrast and resolution not attainable separately by each of the modalities involved. This project is concerned with a specific class of hybrid imaging modalities that couple ultrasound imaging with optical imaging to achieve high-resolution imaging of optical properties of heterogeneous media. Our objectives are (i) to develop fast and robust image reconstruction algorithms for these hybrid modalities for practical use, and (ii) to train advanced undergraduate students, graduate students, and postdoctoral researchers on the mathematical and computational research in this field. The main proposed research efforts in this project include, but not limited to: (i) to develop efficient inversion algorithms for multi-coefficient inverse problems in photoacoustic tomography (PAT) with unknown ultrasound properties; (ii) to develop efficient reconstruction algorithms for inverse problems in quantitative fluorescence PAT (fPAT) and two-photon PAT (TP-PAT) for molecular imaging; (iii) to perform mathematical analysis on inverse problems in ultrasound-modulated fluorescence optical tomography (UMfOT) and to develop robust image reconstruction strategies in the same setting; and (iv) to develop fast algorithms for the transport equation for optimization-based iterative image reconstruction strategies for PAT, fPAT, TP-PAT and UMfOT with partial measurements. The mathematical ideas developed in the proposed projects are expected to provide insights on the mathematical and numerical solutions of many similar inverse coefficient problems to partial differential equations that appear in many fields of science and technology.