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Abstract Cardiovascular disease is one of the leading causes of death in Canada. Atherosclerosis is considered the primary cause for cardiovascular disease. Optical coherence tomography (OCT) provides a means to minimally invasive imaging and assessment of textural features of atherosclerotic plaque. However, detecting atherosclerotic plaque by visual inspection from Optical Coherence Tomography (OCT) images is usually difficult. Therefore we developed unsupervised segmentation algorithms to automatically detect atherosclerosis plaque from OCT images. We used three different clustering methods to identify atherosclerotic plaque automatically from OCT images. Our method involves data preprocessing of raw OCT images, feature selection and texture feature extraction using the Spatial Gray Level Dependence Matrix method (SGLDM), and the application of three different clustering techniques: K-means, Fuzzy C-means and Gustafson-Kessel algorithms to segment the plaque regions from OCT images and to map the cluster regions (background, vascular tissue, OCT degraded signal region and Atherosclerosis plaque) from the feature-space back to the original preprocessed OCT image. We validated our results by comparing our segmented OCT images with actual photographic images of vascular tissue with plaque. / October 2015

Variational methods and Partial Differential Equations (PDEs) have been extensively employed for the mathematical formulation of a myriad of problems describing physical phenomena such as heat propagation, thermodynamic transformations and many more. In imaging, PDEs following variational principles are often considered. In their general form these models combine a regularisation and a data fitting term, balancing one against the other appropriately. Total variation (TV) regularisation is often used due to its edgepreserving and smoothing properties. In this thesis, we focus on the design of TV-based models for several different applications. We start considering PDE models encoding higher-order derivatives to overcome wellknown TV reconstruction drawbacks. Due to their high differential order and nonlinear nature, the computation of the numerical solution of these equations is often challenging. In this thesis, we propose directional splitting techniques and use Newton-type methods that despite these numerical hurdles render reliable and efficient computational schemes. Next, we discuss the problem of choosing the appropriate data fitting term in the case when multiple noise statistics in the data are present due, for instance, to different acquisition and transmission problems. We propose a novel variational model which encodes appropriately and consistently the different noise distributions in this case. Balancing the effect of the regularisation against the data fitting is also crucial. For this sake, we consider a learning approach which estimates the optimal ratio between the two by using training sets of examples via bilevel optimisation. Numerically, we use a combination of SemiSmooth (SSN) and quasi-Newton methods to solve the problem efficiently. Finally, we consider TV-based models in the framework of graphs for image segmentation problems. Here, spectral properties combined with matrix completion techniques are needed to overcome the computational limitations due to the large amount of image data. Further, a semi-supervised technique for the measurement of the segmented region by means of the Hough transform is proposed.