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

合作研究：断射线氡变换反演及应用

• 批准号: 1115616
• 负责人: Vadim Markel
• 金额: $ 16.29万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115616&HistoricalAwards=false
• 关键词: Collaborative; Research; Inversion; Broken; Ray

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.

该项目组建了一个由应用和计算数学家和物理学家组成的团队，负责开发、分析和实现断射线氡变换（BRT）及其推广的重建算法。 BRT 描述单散射粒子或波的传播。这种情况是临床能量下的 X 射线成像或近乎透明的组织和模型生物体的光学成像的典型情况。本研究的目的是为 BRT 提供理论上、数值上合理且实际适用的重建算法，并应用于弱散射区域的 X 射线计算机断层扫描和光学断层扫描。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，以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（在索博列夫空间的规模上）也是一个问题。微局域分析技术可用于表征奇点的传播。 用于反演 BRT 的高效数值算法将使用来自辐射传输前向求解器的数据进行实施和测试，这些求解器可解释单散射和多散射，从而将研究与实验制度联系起来。成像的巨大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携带有关它们所穿过的介质的有用信息。调查人员的目的是证明事实并非如此。通过利用利用散射存在的数学方法和计算方法，他们寻求改变各种生物医学和安全相关的 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以及他们与三名研究生的合作。预计通过出版物和公开软件的生成，研究结果将得到广泛传播。