Variational Tensor-Based Models for Image Diffusion in Non-Linear Domains

This dissertation addresses the problem of adaptive image filtering. Although the topic has a long history in the image processing community, researchers continuously present novel methods to obtain ever better image restoration results. With an expanding market for individuals who wish to share their everyday life on social media, imaging techniques such as compact cameras and smart phones are important factors. Naturally, every producer of imaging equipment desires to exploit cheap camera components while supplying high quality images. One step in this pipeline is to use sophisticated imaging software including, e.g., noise reduction to reduce manufacturing costs, while maintaining image quality. This thesis is based on traditional formulations such as isotropic and tensor-based anisotropic diffusion for image denoising. The difference from main-stream denoising methods is that this thesis explores the effects of introducing contextual information as prior knowledge for image denoising into the filtering schemes. To achieve this, the adaptive filtering theory is formulated from an energy minimization standpoint. The core contributions of this work is the introduction of a novel tensor-based functional which unifies and generalises standard diffusion methods. Additionally, the explicit Euler-Lagrange equation is derived which, if solved, yield the stationary point for the minimization problem. Several aspects of the functional are presented in detail which include, but are not limited to, tensor symmetry constraints and convexity. Also, the classical problem of finding a variational formulation to a given tensor-based partial differential equation is studied. The presented framework is applied in problem formulation that includes non-linear domain transformation, e.g., visualization of medical images. Additionally, the framework is also used to exploit locally estimated probability density functions or the channel representation to drive the filtering process. Furthermore, one of the first truly tensor-based formulations of total variation is presented. The key to the formulation is the gradient energy tensor, which does not require spatial regularization of its tensor components. It is shown empirically in several computer vision applications, such as corner detection and optical flow, that the gradient energy tensor is a viable replacement for the commonly used structure tensor. Moreover, the gradient energy tensor is used in the traditional tensor-based anisotropic diffusion scheme. This approach results in significant improvements in computational speed when the scheme is implemented on a graphical processing unit compared to using the commonly used structure tensor.VIDINACIPGARNICSEMC^