Mathematical Tools For Imaging Reconstruction

9971151Grunbaum主要研究者将继续他的工作在一些松散的相关领域，包括扩散断层扫描，在X射线晶体学的相位问题，一些达布过程之间的关系适用于正交多项式和静电解释他们的零，和一些具体的问题有关的bispectral问题。最后一个领域是他大约十年前从医学成像的一个具体问题开始的。这个领域已经与数学中几个不相关的领域取得了联系，从波传播，微分算子的交换环等，主要研究者的工作继续集中在生物医学领域的“逆问题”之间的十字路口，包括X射线，一方面是磁共振和光学断层扫描，另一方面是数学物理和信号处理等领域的一些数学发展。该研究计划是一个双管齐下的方法，旨在确定医学成像中的重要领域，这些领域已经成熟，可以改进数学处理。在某些情况下，这引起了新的数学发展，最终应该找到有用的应用。这项工作已被证明是有用的，在其他反问题所产生的，例如在雷达探测，非侵入性评估等。

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.