Mathematical Tools for Imaging Reconstruction

该研究计划有一个双管齐下的方法，旨在确定成像中的重要领域，这些领域已经成熟，可以进行增强的数学处理。 PI的主要目标是将新开发的数学工具应用于此类成像问题，并考虑新的成像情况，表明需要开发或采用新的数学技术。 标值球函数的经典理论统一了许多经典的逼近方法。 这些包括雅可比多项式，贝塞尔函数，拉盖尔多项式，厄米多项式和勒让德函数。 PI已经将这个理论扩展到矩阵值的情况。 这将是解决成像问题的主要工具。 自然候选者涉及偏振或各向异性（如在光学层析成像）发挥重要作用的领域。 这些成像方法将应用于生物医学领域，包括X射线，磁共振，X射线晶体学和光学断层扫描。 国家研究理事会报告说，在光学或扩散层析成像的新兴领域有很大的发展潜力，其中人们的目标是使用非常低的能量探针，如红外激光器来获得图像。 从一开始就需要处理衰减和散射。 在这项研究中开发的技术应该提高使用这些技术重建图像的能力。 设想的其他应用是在新生儿诊所进行重复的乳房X光检查和某些脑血流研究（测量氧含量）。 PI将与开发仪器硬件的人员密切合作。

The research program has a two-pronged approach aimed at identifying important areas in imaging that are ripe for enhanced mathematical treatment. The PI's main goal is to apply newly developed mathematical tools to such imaging problems as well as to consider new imaging situations that suggest the need to develop or adapt new mathematical techniques. The classical theory of scalar valued spherical functions unifies a number of classical approximation methods. These include Jacobi polynomials, Bessel functions, Laguerre polynomials, Hermite polynomials, and Legendre functions. The PI has extended this theory to matrix valued situations. This will be the main tool in solving the imaging problems. Natural candidates involve areas where polarization or anisotropy (as in optical tomography) play important roles. These imaging methods will be applied in biomedical areas, including X-ray, magnetic resonance, X-ray crystallography, and optical tomography. The National Research Council rported that there was a potential for great progress in the emerging field of optical or diffuse tomography, where one aims at obtaining images with the use of very low energy probes, like infrared lasers. From the beginning there was a need to deal both with attenuation and scattering. The techniques developed in this research should improve the ability to reconstruct images using these techniques. Other applications envisaged are repeated mammographies and certain brain blood flow studies (measuring oxygen content) in neonatal clinics. The PI will work closely with those who are developing the instrumation hardware.