Variational, level set and PDE based methods and their applications in digital image processing have been well developed and studied for the past twenty years. These methods were soon applied to some medical image processing problems. However, the study for biological shapes, e.g. surfaces of brains or other human organs, are still in its early stage. The bulk of this dissertation explores some applications of variational, level set and PDE based methods in biological shape processing and analysis. This dissertation also covers some aspects of compressive sensing, ℓ1-minimizations and fast numerical solvers. Their applications in medical image analysis are also studied.;The first topic is on surface restoration using nonlocal means [1], where we extend nonlocal smoothing techniques for image regularization in [12] to surface regularization, with surfaces represented by level set functions. Numerical results show that our extension of nonlocal smoothing to surface regularization is very effective in removing spurious oscillations while preserving and even restoring sharp features. Furthermore, topology corrections are also made by our algorithms for some of the surfaces.;The second topic is on 3D brain aneurysm capturing using level set based method. Inspired by the illusory contour techniques proposed by [36, 37], we present a level set based surface capturing algorithm to capture the aneurysms from the vascular tree. Numerical results are presented to show the accuracy, consistency and robustness of our method in capturing brain aneurysms and volume quantification.;The third topic is on multiscale representations (MSR) of 3D shapes. We introduce a new level set and PDE based MSR for shapes, which is intrinsic to the shape itself, does not need any parametrization, and the details of the MSR reveal important geometric information. Based on the MSR, we then design a surface inpainting algorithm to recover 3D geometry of blood vessels. Because of the nature of irregular morphology in vessels and organs, both phantom and real inpainting scenarios were tested using our new algorithm Numerical results show that the inpainting regions are nicely filled in according to the neighboring geometry of the vessels and allow us to accurately estimate the volume loss of vessels.;The last, but definitely not the least, topic is on Bregman iteration as a fast solver for ℓ1-minimizations in compressive sensing and medical image analysis. We analyzed the convergence properties of linearized Bregman and then improve its convergence speed. We further observe that a general TV-based model can be converted to an ℓ1-minimization which can then be solved efficiently using Bregman iterations. Finally, an application of ℓ1-minimization is considered for needle tracking in ultrasound images.
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