| Image segmentation is a very important topic in image processing. It is the process of identifying object outlines within images. There are quite a few efficient algorithms for segmentation such as the model based approaches, pattern recognition techniques, tracking-based approaches, artificial intelligence-based approaches, etc. In this thesis, we mainly study two kinds of image segmentation problems. More precisely, one kind problem is the vessel segmentation problem in medical imaging, the other is the generic image segmentation problem, i.e., two-phase and multiphase image segmentation for very general images, for example medical, noisy, and blurry images, etc.;In Part I of this thesis, we focus on the vessel segmentation problem in medical Images, and our tight-frame based vessel segmentation algorithm will be proposed. Tight-frame, a generalization of orthogonal wavelets, has been used successfully in various problems in image processing, including inpainting, impulse noise removal, super-resolution image restoration, etc. In this part, we propose to apply the tight-frame approach to automatically identify tube-like structures in medical imaging, with the primary application of segmenting blood vessels in magnetic resonance angiography images. Our method iteratively refines a region that encloses the potential boundary of the vessels. At each iteration, we apply the tight-frame algorithm to denoise and smooth the potential boundary and sharpen the region. The cost per iteration is proportional to the number of pixels in the image. We prove that the iteration converges in a finite number of steps to a binary image whereby the segmentation of the vessels can be done straightforwardly. Numerical experiments on synthetic and real 2D/3D images demonstrate that our method is more accuracy when compared with some representative segmentation methods, and it usually converges within a few iterations.;Part II of this thesis focuses on generic image segmentation problem. The Mumford-Shah model is one of the most important image segmentation models, and has been studied extensively in the last twenty years. In this part, based on the Mumford-Shah model, our convex image segmentation model will be proposed. It can be seen as finding a smooth approximation g to the piecewise smooth solution of the Mumford-Shah model. Once g is obtained, the two-phase or multiphase segmentation is done by thresholding g. The thresholds can be given by the users to reveal specific features in the image or they can be obtained automatically using a K-means method. Because of the convexity of our model, g can be solved efficiently by techniques like the split-Bregman algorithm or the Chambolle-Pock method. We prove that our model is convergent and the solution g is always unique. In our method, there is no need to specify the number of segments K (K ≥ 2) before finding g. We can obtain any K-phase segmentations by choosing (K – 1) thresholds after g is found; and there is no need to recompute g if the thresholds are changed. Experimental results show that our method performs better than many standard 2-phase or multi-phase segmentation methods for very general images, including anti-mass, noisy, and blurry images. |
