Collaborative Research: FRG: Inverse Problems in Transport Theory

该奖项支持一组应用和计算数学家以及生物医学工程师的研究，他们将在光学断层扫描和光学分子成像等新兴领域分析、实现和测试图像重建算法。这些成像技术的基础是使用光子探测组织并推断其光学特性，以及使用分子标记，这些标记寻找细胞内部的变化，这些变化是疾病发展的前兆，并发出可在体外检测到的辐射。在数学上，我们将分析辐射传递理论中只有角平均信息时的线性和非线性逆问题。然后，我们将设计和实现鲁棒和精确的算法，重建组织的光学特性和源项的位置从实际可用的测量。我们将基于适用于简单几何的分析方法开发快速图像重建算法，并将使用来自宾夕法尼亚大学团队操作的非接触式光学断层扫描系统的大量实验数据集对其进行测试。brbr该项目汇集了纯数学和应用数学、理论物理和医学领域的跨学科研究团队，以解决生物系统光学成像中的数学问题。这项研究处于努力实现对基本生物学过程和疾病状态的分子理解的前沿。研究人员将把他们在光学物理学和光在高散射介质（如生物组织）中传播的建模方面的专业知识与反问题的数学研究结合起来。长期目标是了解所谓的辐射输运方程的反问题的数学结构，并利用这些知识来开发光学断层成像中图像重建的计算效率方法。

This award supports research by a group of applied and computational mathematicians and biomedical engineers who will analyze, implement, and test image reconstruction algorithms in the emerging fields of optical tomography and optical molecular imaging. These imaging techniques are based on using photons to probe tissues and deduce their optical properties and on using molecular markers, which seek out changes inside cells that are precursors for disease development, and emit radiation that is detectable outside the body. Mathematically, we will analyze linear and non-linear inverse problems in radiative transfer theory when only angularly averaged information is available at the domain boundary. We will then devise and implement robust and accurate algorithms that reconstruct optical properties of tissues and locations of source terms from practically available measurements. We will develop fast image reconstruction algorithms, based on analytic methods applicable in simple geometries, and will test them using large data sets of experimental data from a non-contact optical tomography system operated by the University of Pennsylvania group.brbrThis project brings together an interdisciplinary team of researchers at the interface of pure and applied mathematics, theoretical physics, and medicine in order to attack mathematical problems in optical imaging of biological systems. This research is at the forefront of efforts to achieve a molecular understanding of both basic biological processes and disease states. The investigators will combine their expertise in optical physics and the modeling of the propagation of light in highly-scattering media, such as biological tissue, with mathematical studies of inverse problems. The long term goal is to understand the mathematical structure of inverse problems for the so-called radiative transport equation, and to use this knowledge to develop computationally efficient methods for image reconstruction in optical tomography.