Various methods in shape and analysis and image segmentation and registration

Image segmentation and registration are vital to image processing, image analysis, and computer vision. In the last decade, research into these areas has developed at a rapid pace by using various mathematical methods. In this dissertation, variational partial differential equations are proposed to solve image segmentation and registration problems.; A new variational partial differential equation based level set method for simultaneous image segmentation and non-rigid registration is presented. This technique incorporates both prior shape and intensity information. Since global rigid registration has limited applicability when non-rigid shapes are considered, a transformation is created which is the sum of a global rigid function and a local non-rigid deformation. This model is tested against two-chamber end-systolic endocardial ultrasound images of thirteen human patients. The experimental results provide preliminary evidence of the effectiveness of the model in detecting the boundaries of the incompletely resolved objects which were plagued by noise, dropout, and artifact. Develop algorithms for generating the mean shape has a significant role in image segmentation, since the mean shape can be used as a prior shape to acquire better segmentation results. As alternate ways, several statistical algorithms including a shape related energy function, Self-Organizing map with Procrustes methods, and Self-Organizing map with a principal component analysis to generate mean shape and clustering are also presented.; Region based image segmentation and registration models which contains three interrelated sections: image segmentation using a modified Mumford-Shah technique, region based segmentation using a prior shape, and simultaneous segmentation and registration using Mumford-Shah model are also proposed. The first goal is to develop an image segmentation technique using a modified Mumford-Shah model. A variational region intensity based image segmentation model is presented. The boundary of the given image is extracted using a modified Mumford-Shah segmentation technique. Even though the two phase case is performed for image segmentation during the numerical experiments, the suggested model can be applied to more general cases. Another goal is to develop a region based image segmentation technique using prior shape information. The prior shape information is extracted by using a modified Mumford-Shah segmentation technique. The prior shape knowledge supports the segmentation process for a novel image. Finally, a region based model for simultaneous image segmentation and registration is also presented. The purpose of the model is to segment and register given images simultaneously utilizing a modified Mumford-Shah technique and region intensity values. The segmentation is obtained by minimizing a modified Mumford-Shah model. A global rigid registration is assisted by the segmentation information and region intensity values. A segmentation and registration process is obtained simultaneously in this model. In addition, the model can also be applied to the case of non-rigid registration. The numerical experiments of the proposed models are tested against synthetic data and simulated human brain MRI images. The experimental results provide preliminary evidence of the effectiveness of the proposed models.