Analytical Methods For Certain Inverse Problems In Medical Imaging

各种医学成像技术都是基于解决一类数学问题，称为逆问题。特别是，在正电子发射断层成像(PET)和单光子发射计算机断层成像(SPECT)的医学成像技术中，与之相关的逆问题分别涉及Radon变换和所谓的衰减Radon变换的逆问题。虽然前一种变换的解析公式是众所周知的，但后一种变换的类似公式是最近才得到的。我们将在上述公式和非光滑范数去噪策略的基础上进一步开发分析算法。此外，通过使用各种临床数据，我们将新算法与商业软件进行比较。脑磁图(MEG)和脑电(EEG)等医学技术也涉及到某些逆问题的求解。在最近的一项突破中，PI提出了这些反问题的全面解决方案。我们将进一步开发相关算法，我们将使用复制核技术进行数值实现，并使用各种真实数据将它们与商业软件进行比较。

A variety of medical imaging techniques are based on the solution of a class of mathematical problems, called inverse problems. In particular, in the medical imaging techniques of Positron Emission Tomography (PET) and Single Photon Emission Computed Tomography (SPECT), the associated inverse problems involve the inversion of the Radon transform and of the so-called Attenuated Radon transform, respectively. Although an analytical formula for the former transform is well known, an analogous formula for the latter transform was obtained only recently. We will develop further analytical algorithms based on the above formulae and on non-smooth-norm de-noising strategies. Furthermore, by employing a variety of clinical data, we will compare the new algorithms with commercial software.The medical techniques of Magnetoencephalography (MEG) and Electroencephalography (EEG), also involve the solution of certain inverse problems. In a recent breakthrough, the PI has presented a comprehensive solution of these inverse problems. We will develop further the relevant algorithms, we will use the techniques of reproducing kernels for their numerical implementation, and we will compare them with commercial software using a variety of real data.