| Images of various kinds are increasingly important to medical diagnostic processes. Difficult problems are encountered in selecting both the most appropriate image reconstruction methods and post-processing methods. Reconstruction methods involve producing optimal quality images from raw data. Post-processing methods involve acquiring the highest quality information from these images. Medical image processing (i.e., reconstruction and post processing) has created tremendous opportunities for mathematical modeling, analysis, and computation.; I examined various applications of the variational partial differential equation (PDE) method. This method is one of the most recent and successful approaches to medical image processing. I worked on the following important medical image processing problems: clustering, denoising, segmentation, registration, tomography, inpainting, and field tracking. Novel PDE based models and corresponding numerical methods are introduced for each of these problems.; We proposed an original framework for segmentation that may incorporate various forms of prior information. This information can include shape, key points, or intensity profiles. We also proposed the level set formulation and a numerical algorithm for the model. We proposed a novel inpainting model and applied it to magnetic resonance imaging (MRI). This novel inpainting model considers the automatic choice of diffusion type. A specific modification of this model for sensitivity maps was produced. We also introduced a new framework (the modified mumford-shad model by level set) for tomography problems with extremely noisy and limited data. This algorithm is flexible, it can be used for any geometry with or without prior information. This model is very easy to modify for various cases. Finally, we proposed a novel tracking algorithm. The proposed method solves multi-diffusion-direction and branch problems existing in fiber tracking.; We applied those models on segmentation of ultrasound cardiac images, MRI brain images, noise tomography, inpainting of sensitivity maps for MRI surface coils, and fiber tracking. Experimental results and computational costs were compared with conventional methods. |
