Mathematical models and techniques for medical imaging

Partial differential equations(PDE) and variational methods have been researched vigorously and applied successfully to medical image processing in the last two decades. The acquired medical imaging data such as Magnetic Resonance Imaging have reached high isotropic resolution in 3D, with scalar, vector or matrix at each grid point. Automatic extraction of useful information from these huge high dimensional multi-valued datasets requires advanced computational algorithms, especially in the presence of image noise and blur. In this thesis, we proposed several variational and level set based methods for medical data denoising, deblurring and segmentation. For matrix valued image denoising in Diffusion Tensor Imaging(DTI), we extended the idea of channel coupling in the vector valued total variation paper by Blomgren and Chan and incorporated a term in the constrained optimization to guarantee positive definiteness. For DTI fiber tracking and segmentation, we proposed a new approach based on the level set framework which can automatically capture multiple curves with branching. For image deblurring, we developed a model to segment and recover blurred image where different regions of an image is blurred by different point spread functions. We showed that we can recover the image if the initial guess is tuned appropriately even though the problem is very ill-posed. Finally, we discussed an improved version of the combined wavelet and total variation method and its use in medical image denoising.