Collaborative Research: Inversion of the Broken-Ray Radon Transform and Applications

该项目汇集了一组应用和计算数学家和物理学家，以开发、分析和实现破损射线Radon变换(BRT)及其推广的重建算法。BRT描述了单次散射粒子或波的传播。这种情况在临床能量X射线成像或对几乎透明的组织和模型生物的光学成像中是典型的。这项研究的目的是为BRT提供理论上、数值上合理和实用的重建算法，并应用于弱散射区域的X射线计算机层析成像和光学层析成像。特别是，研究人员建议推导和分析基于BRT的扫描协议和相应的反演技术，以重建吸收和散射系数。将开发相关的扫描协议，在空间分辨率和噪声稳定性之间提供最佳平衡。此外，唯一性和稳定性问题(在Sobolev空间的尺度上)也是一个令人关注的问题。微局部分析技术可以用来描述奇点的传播。利用从辐射传输正向解算器获得的数据，将实施和测试有效的数值算法来反演BRT，这些数据既可以解释单次散射，也可以解释多次散射，从而将研究与实验制度联系起来。成像中最大的挑战之一是解决散射问题。人们普遍认为，只有未散射的粒子或波才能携带有关它们所经过的介质的有用信息。调查人员的目的是证明情况并非如此。通过利用数学方法和利用散射存在的计算方法，他们寻求改变各种生物医学和与安全相关的X射线和光学成像技术。这项研究是应用和计算数学家和物理学家之间的合作，他们与三名研究生合作。预计将通过出版物和生成公开可用的软件来广泛传播研究结果。

This project assembles a team of applied and computational mathematicians and physicists to develop, analyze and implement reconstruction algorithms for the broken-ray Radon transform (BRT) and its generalizations. The BRT describes the propagation of single-scattered particles or waves. This situation is typical of x-ray imaging at clinical energies or optical imaging of nearly transparent tissues and model organisms. The intent of the proposed research is to provide theoretically, numerically justified and practically applicable reconstruction algorithms for the BRT with applications to both x-ray computed tomography and optical tomography in the weak-scattering regime. In particular, the investigators propose to derive and analyze BRT-based scanning protocols and corresponding inversion techniques to reconstruct the absorption and scattering coefficients. Associated scanning protocols, which provide the optimum balance between spatial resolution and stability to noise, are to be developed. In addition, questions of uniqueness and stability (in the scale of Sobolev spaces) are a concern. Techniques of microlocal analysis may be used to characterize the propagation of singularities. Efficient numerical algorithms for inverting the BRT are to be implemented and tested using data derived from radiative transport forward solvers that account for both single- and multiple-scattering, hence connecting the research to the experimental regime. One of the grand challenges in imaging is to address the problem of scattering. It is generally believed that only unscattered particles or waves carry useful information about the medium through which they have traveled. The Investigators aim to show that this is not the case. By making use of mathematical methods and computational approaches that exploit the presence of scattering, they seek to transform a variety of biomedical and security-related x-ray and optical imaging technologies. This research is a collaboration between applied and computational mathematicians and physicists and their work with three graduate students. Broad dissemination of the results of the research is anticipated through publications and generation of publicly available software.