Collaborative Research: Direct Reconstruction Methods for Optical Tomography and Related Inverse Problems

本提案的目的是发展直接图像重建方法的光学层析成像。光学断层扫描是一种新兴的生物医学成像方式，它使用多次散射光作为组织结构和功能的探针。该项目将包括三个组成部分。(i)我们将开发基于逆玻恩级数的直接重建方法，从边界测量中恢复辐射输运方程和扩散方程的吸收和扩散系数。我们将描述所提出的方法的收敛性、近似误差和稳定性。（ii）开发基于逆玻恩级数的快速重构算法。这些算法将被设计用于多达10^8个测量值的数据集。算法将使用数值模拟正演数据和从非接触式光学层析成像系统获得的实验数据进行测试。（iii）我们将探索与光学层析成像具有共同数学结构的相关逆问题的直接反演方法。特别是，我们将开发和分析基于Born序列反演的电阻抗层析成像和近场麦克斯韦方程的重建方法。生物医学成像的重大挑战之一是开发高分辨率的生理和分子功能成像方法。这项研究的长期目标是通过开发必要的数学工具来帮助解决这个问题。在比目前可能的更早的阶段检测和表征疾病的能力将具有变革性，并且与开发用于功能成像的稳健和准确的图像重建算法至关重要。两名研究生和一名本科生将接受培训，以适应这种跨学科的环境。研究结果将广泛传播，计算机代码将公开供非商业用途。

The objective of this proposal is to develop direct image reconstruction methods for optical tomography. Optical tomography is an emerging biomedical imaging modality that uses multiply-scattered light as a probe of tissue structure and function. The project will involve three components. (i) We will develop direct reconstruction methods, based on the inverse Born series, for recovering the absorption and diffusion coefficients of the radiative transport equation and the diffusion equation from boundary measurements. We will characterize the convergence, approximation error and stability of the proposed methods. (ii) We will develop fast reconstruction algorithms based on the inverse Born series. The algorithms will be designed to be used with data sets as large as 10^8 measurements. The algorithms will be tested using numerically simulated forward data and experimental data obtained from a noncontact optical tomography system. (iii) We will explore direct inversion methods for related inverse problems that share a common mathematical structure with optical tomography. In particular, we will develop and analyze reconstruction methods, based on inversion of the Born series, for electrical impedance tomography and for the Maxwell equations in the near-field.One of the grand challenges in biomedical imaging is to develop high-resolution methods for imaging of physiological and molecular function. The long-term goal of this research is to contribute to the solution to this problem by developing the necessary mathematical tools. The ability to detect and characterize disease at much earlier stages than is currently possible would be transformative and is critically linked to the development of robust and accurate image reconstruction algorithms for functional imaging. Two graduate students and an undergraduate will be trained to function in this interdisciplinary environment. The results of the research will be broadly disseminated and computer codes will be made publicly available for noncommercial use.