Study On Variational Models For Image Segmentation And Numerical Implementations

Image segmentation is a fundamental processing step and key technology, which bridges image processing and image analysis. Its goal is to partition a given image into several non-overlapped regions according to some similarity criteria. Recently, variational methods for image segmentation have been widely paid attention by many researchers, by virtue of their flexible structure, diverse form and excellent performance. Generally speaking, variational models consist of several energy functionals, each of which reflects to some of the characteristics(intensity, edge, color and texture, etc.) of image to be processed, and can also be subject to a priori knowledge about the shape and characteristics of the target to be segmented. Energy functional is a flexible framework that can integrate both low level vision property from image and various informations(prior knowledges) about the object to be extracted seamlessly. Minimizing the energy functional makes the evolution curve(or surface) move so as to obtain a complete representation of the segmentation regions.This thesis concentrates on some theoretical issues of variational models for image segmentation as well as concrete problems to be solved in practical application. The author constructs some new variational models, analyzes its theoretical essence, and puts forward effective numerical implementation schemes.The main contents can be summarized as follows:1. We propose a convex variational model for image segmentationThe Mumford-Shah functional, as the most classical variational model for image segmentation, is the groundwork of many of the existing image segmentation model. However, solving the model numerically is difficult when direct implementations are performed. Chan and Vese first study the reduced Mumford-Shah model using the level set method and proposed the well-known Chan-Vese model which creates the research of variational level set models. As a result, thousands of variational level set models have been proposed. However, the energy functional of the existing variational level set models is non-convex, which may leads to local minima. This is a serious problem because the local minima of energy functional often provide poor segmentation results. The segmentation results thus depend on the initializations of contour. To solve this problem, we propose a strictly convex energy functional in a level set formulation for two-phase image segmentation. We prove that the value of the unique global minimizer for the energy functional is within the interval ?-1, 1? for any image. A pointwise convergent numerical scheme is presented to solve the level set evolution equation. The proposed model is allowed for flexible initializations and can set a reasonable termination criterion on the algorithm. The proposed model has been successfully applied to some synthesized and real images with promising results.2. For the regularization problem of the level set function, we proposed an indirectly regularized variational level set modelSince image segmentation is often affected by factors such as noise and weak boundary, it is often necessary to impose some kind of regularity constraint on the level set function, such as length regularization, TV regularization and 1H regularization. Existing variational level set models directly imposed the regular constraints on the level set function. In this paper, we propose an indirectly regularized variational level set model in which the regularization is posed indirectly on the level set function via an auxiliary function, which is used to remove the influence of high noise on level set evolution while ensuring the active contours not to pass through weak object boundaries. Since the proposed energy functional is convex, it thus is robust to the initializations. Finally, the propose model can be solved efficiently by means of the alternating minimization algorithm, which avoids the limitations of traditional gradient descent method. We show that the alternating minimization algorithm is convergent for the proposed model under mild conditions.3. For the problem of intensity inhomogeneity, we proposed a Retinex based piecewise constant variational modelA severe challenge for image segmentation is illumination bias(gain inhomogeneity, shadow) which arises from various imperfect aspects of image acquisition process, such as spatial variations in illumination and imperfection of imaging devices, etc. Intensity inhomogeneity causes overlaps in intensity distributions of different regions and leads to the intensity variation even for the same region within an image, which greatly decrease the accuracy of image segmentation algorithms.To deal with the intensity inhomogeneity, we propose a new variational model for simultaneous image segmentation and bias correction. Based on the Retinex theory, we decompose the original image into a smooth bias component and a structure part, where the structure part is expected to be piecewise constant. We propose a variational model for image segmentation and bias correction by modeling the structure part in the way of piecewise constant and integrating the Retinex decomposition. Based on the alternating minimization algorithm, we present a numerical method to solve the minimization problem efficiently. Experimental results on images from diverse modalities demonstrate the proposed model can achieve the global minimum and is insensitive to the initializations of level set function. In addition, the proposed model outperforms the other models in terms of the efficiency and accuracy.4. We develop a two-stage method to deal with oil spill images from synthetic aperture radar imageryTarget tracking and target detection technology for Synthetic Aperture Radar(SAR) images have wide spread application in the national economic and military field and thus have very important research significance. However, SAR image segmentation is yet a difficult task. This difficulty arises from the fact that most of SAR images are degraded by high noise and low contrast, and characterized by intensity inhomogeneity.A two-stage method is developed for oil spill image segmentation. The first stage of our method is to obtain an enhanced image by suppressing the backscattering effect from the original oil spill image. Once the enhanced image is obtained, then in the second stage, a variational segmentation model is presented for dealing with the obtained enhanced image. The data term of the energy functional is constructed for the enhanced image in a piecewise constant way. In addition, a Cahn-Hilliard type regularization term is introduced into the energy functional. The variational model is numerically solved by alternating minimization scheme. Numerical examples on the oil spill images show that the proposed method outperforms the two representative state-of-the-art methods in terms of the efficiency and accuracy.