Mathematical Sciences: Mathematical Tools for Imaging Reconstruction

9400097 Grunbaum这个项目由应用数学计划、计算生物学计划和计算数学计划共同资助，将考虑一些由生物医学成像问题引发的逆问题，包括现有的X射线和磁共振等模式，以及扩散断层扫描和在X射线结晶学直接方法中使用高阶不变量等新兴模式。一些具体的研究领域包括：X射线层析成像中的有限角度重建问题，磁共振成像中用于切片选择的优化脉冲的设计，扩散层析成像中非线性反转算法的发展，以及在X射线晶体学直接方法中使用高阶不变量来解决相位问题的新方法的发展。该项目由应用数学计划、计算生物学计划和计算数学计划共同资助，围绕着生物医学成像中出现的反问题，包括X射线和磁共振成像。特别是，它将探索数学物理的发展如何在漫反射断层成像中产生洞察力，即使用极低能量的探测器(如红外激光)进行医学成像。虽然这项研究可能在医学诊断的某些特定领域具有相当大的实用价值，如乳房X光照相和新生儿诊断，但其主要思想和概念的适用性超出了医学成像，从结晶学到雷达目标估计和半导体等领域。

9400097 Grunbaum This project, which is jointly funded by the Applied Mathematics Program, the Computational Biology Program, and the Computational Mathematics Program, will consider a number of inverse problems motivated by problems in biomedical imaging, including existing modalities such as X-ray and magnetic resonance, as well as emerging ones such as diffuse tomography and the use of higher order invariants in the direct method of X-ray crystallography. Some specific areas of investigation include work on aspects of the limited angle reconstruction problem in X-ray tomography, the design of optimnized pulses for slice selection in magnetic resonance imaging, the development of nonlinear inversion algorithms in diffuse tomography, and the development of new ways of using higher order invariants within the direct method of X-ray crystallography to solve the phase problem. This project, which is jointly funded by the Applied Mathematics Program, the Computational Biology Program, and the Computational Mathematics Program, centers around inverse problems arising in biomedical imaging, including X-ray and magnetic resonance imaging. In particular, it will explore ways in which developments in mathematical physics can yield insights in diffuse tomography, that is, medical imaging using very low energy probes such as infra-red lasers. While this research is likely to be of considerable practical value in certain specific areas of medical diagnostics such as mammographies and neonatal diagnostics, the principal ideas and concepts have applicability beyond medical imaging, from areas such as crystallography to radar target estimation and semi-conductors.