Study On Variational Models And Fast Minimization Methods In Image Segmentation

Image segmentation is not only a fundamental and challenging task in image processing, but also plays an important role in image analysis. With a well-developed mathematical foundation, the variational models have become a hot topic in image segmentation. As the research on image segmentation advances, the investigation on solving segmentation problems becomes popular. This thesis proposes several efficient models and algorithms which overcome the shortcomings of the previous works. The main work and innovation are as follows.1. Inspired by the characteristic of human vision that we observe the objects first in an image, a non-convex background removed model is proposed. The basic idea of the model is to find a curve, out of which the intensity of each pixel is almost the same, and also maximizes this outer area. By using the level set method and the gradient descent method,the energy functional and the evolution equation of the model are obtained. In order to accelerate the evolution of the evolving curve, we use 1 to replace the Dirac function. The theory and experiments both show that the proposed model can segment the images with multi-objects.2. We analyze the limitations of the non-convex models and proposes a convex variational model. The Heaviside function is widely used in image segmentation as the characteristic function of a region. The traditional approximation to Heaviside function often produces a non-convex model which may get stuck in local minima. In order to overcome this limitation, we propose another simple approximation of the Heaviside function,then a constraint convex energy functional is obtained. In this dissertation, we apply three methods, the split Bregman, the gradient descent and the proximity, to minimize this energy functional. For the last two methods, we get two unconstraint functionals by adding different penalties, then we prove that these cannot affect the segmentation result. The numerical experiments demonstrate the high efficiency and accuracy of these methods.3. We discuss the segmentation ability of the common models on intensity inhomogeneous images and propose a new model based on local information of an image. Since a small piece of the intensity inhomogeneous image can be treat as a homogeneous image which is easier to segment, we propose a local intensity average by using the Gauss kernel.Then we take the local average into the constraint convex functional proposed previously and obtain a new model which can successfully segment images with intensity inhomogeneity. At the computational level, an accelerating split Bregman approach is proposed.In contrast, the proposed model performs better in efficiency. Moreover, the model is robust to noise.4. The image can be seen as a surface in 3D space, thus it can be studied by using tools in differential geometry. Based on the knowledge of differential geometry, a feature extraction method is proposed. The feature can be segmented by the previous unconstraint convex model as a new image. At the computational level, we propose a fixed-point algorithm based on the proximity operator and give the convergence proof. Experimental results on synthetic and real images show that the proposed method is efficient to texture images, even the image is nondescript due to the texture.5. For the complex and color images, we propose two variational segmentation models. Since most of the nature images have more than one object or background, the traditional models cannot give complete segmentation results. In order to improve the segmentation ability, we propose a hierarchical segmentation model. The idea of hierarchical segmentation is to segment image iteratively. In each segmentation hierarchy, or equivalently, in each iterative segmentation by the convex model, the segmentation is carried out on the region detected as objects region in the last segmentation hierarchy rather than on the whole image domain. Color image often appears in our life, it has three channels which give more visual information. These three channels are similar, thus the differences among them can show more details. To use the differences more, we propose a transformation of these three channels and treat the new channels as a 3D matrix to be used in segmentation. In this color segmentation model, the split Bregman method is chosen as the minimization approach.