Analytical Methods For Certain Inverse Problems In Medical Imaging

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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.