Research On Image Segmentation Based On Variation And Partial Differential Equations

Recently, image segmentation based on variational methods and partial di?erentialequations has become an important branch in the field of image processing. And it is alsoincreasingly becoming a hot project that many researchers pay close attention to. Its aimis to partition an image into several important regions. This thesis mainly studies modelsfor image segmentation, which are based on variational methods and partial di?erentialequations. The main results of this dissertation are listed as follows:1. In this part, this thesis introduced the basic knowledge of images. Besides thisthesis mainly summarized and analyzed advantages and disadvantages of some classicalimage segmentation methods (Geodesic Active Contour model and Chan-Vese model) intheory. And according to a lot of numerical experiments, the advantages and disadvan-tages of these models were further demonstrated.2. From the point of variational methods, the author proposed a new model for seg-mentation of gray images , which is based on the weighted bounded variation model,Geodesic Active Contour model and Region Scalable Fitting model. By using variationalmethods and some properties about the bounded variation space (for example, lower semi-continuity and weak compactness), the author proved the existence of the solutions for thisnew model. According to the optimization knowledge some properties of this new modelwere also obtained. That is, the property that a character function is also a solution ofthis model the author can be obtained. In order to show this model is e?ective, this authorused the dual algorithm that can improve the computational e?ciency to solve this model.And a lot of numerical experiments were done. According to experimental results, it isknown that this model is e?ective for both simple and complex gray images and com-pared with other active contour methods this new model can obtain more accurate edges.Since color images are vector-valued functions, the above model can not be extended tothe color image space simply. Therefore, based on the above model, a new variationalmodel for segmentation of both gray and color images has been proposed. According tosimilar methods, a theorem which is about the existence of solutions of this model can beobtained. And by using the dual algorithm, a solution of this model can be gotten quickly.Moreover, this author did some experiments in order to show this model is e?ective. Ac-cording to experimental results it is shown that this new model is e?ective and compared with other active contour models this new model can segment images more accurately.3. Inspired by the object-detecting function which is used to describe desired ob-jects in images and the above models, a new variational model for segmenting desiredobjects in color images has been proposed. Unlike the above model, both the coe?cientsin the energy functional of the new model and the edge detection function contain anobject-detecting function that is useful for segmenting desired objects. Since this object-detecting function has many good properties such as boundedness, continuity and so on,the author can obtain a theorem that is about the existence for the solution of this modeland can prove a property of this solution that a character function is also a solution of thismodel. In order to show this new model is very e?ective, many numerical experimentswere done. According to experimental results, it is further shown that this new model ise?ective and compared with other active contour methods, this new model can segmentdesired objects and can not be in?uenced by the noise and other objects in images.4. In the nonlocal framework, a new weighted bounded variation semi-norm wasdefined. And this semi-norm was analyzed. Thus, some properties such as lower semi-continuity, equivalence and so on can be obtained. Based on this semi-norm and theclassical Ginzburg-Landau functional, a new variational model for image segmentationwas proposed. In order to show this model is e?ective, the author did a lot of experiments.According to numerical experiments, it is known that this new variational model is verye?ective and compared with other active contour models, it can segment more details ofobjects in images.