Mathematical Tools For Imaging Reconstruction

9971151 Grunbaum首席研究员将继续他在几个松散相关领域的工作，包括扩散层析成像、X射线结晶学中的位相问题、应用于正交多项式的达布过程与其零点的静电解释之间的一些关系，以及一些与双谱问题有关的具体问题。最后一个领域是他大约十年前从医学成像中的一个具体问题开始创立的。这一领域已经接触到了数学中几个不相关的领域，从波传播、微分算子的交换环等。主要研究人员的工作继续集中在生物医学领域的反问题与数学物理和信号处理等领域的一些数学发展之间的十字路口。该研究计划是一种双管齐下的方法，旨在确定医学成像中的重要领域，即适合改进数学治疗的成熟领域。在某些情况下，这导致了新的数学发展，最终应该会找到有用的应用。这项工作在雷达检测、非侵入性评估等领域的其他逆问题中也得到了应用。

9971151GrunbaumThe principal investigator will continue his work on a few loosely related areas, including diffuse tomography, the phase problem in X-ray crystallography, some relations between the Darboux process applied to orthogonal polynomials and the electrostatic interpretation of their zeros, and a number of specific problems related to the bispectral problem. This last one is an area that he initiated about a decade ago starting from a concrete problem in medical imaging. This field has made contact with several unrelated fields in mathematics, ranging from wave propagation, commutative rings of differential operators, etc.The principal investigator's work continues to be centered at the crossroads between "inverse problems" in areas of biomedical interest including X-ray, Magnetic Resonance and Optical tomography on the one hand and a number of mathematical developments in areas like mathematical physics and signal processing. The research program is a two-proned approach aimed at identifying important areas in medical imaging that are ripe for improved mathematical treatment. In certain cases this has given rise to new mathematical developments that should eventually find useful applications. This work has proved useful in other inverse problems arising for instance in radar detection, noninvasive evaluation, etc.