Uncertainty Principles and Gabor Framework in Image Segmentation, Analysis and Synthesis

This work revolves around Gabor filters. It involves theoretical aspects such as the uncertainty principle as well as practical issues such as filter bank design and application of Gabor filters to texture segmentation. This thesis is composed of three main parts. In the first part, we explore the role of the metric of the Gabor feature space, and its usefulness in capturing what we refer to as texture-gradients, and address the issue of textured image segmentation. Gabor filters tuned to a set of orientations, scales and frequencies are applied to images to create the Gabor feature space. A two-dimensional Riemannian manifold of local features is extracted via the Beltrami framework. The metric of this surface provides a good indicator of texture changes and is used, therefore, in a Beltrami-based diffusion mechanism and in a geodesic active contours algorithm for texture segmentation. The performance of the proposed algorithm is compared to that of the edgeless active contours algorithm. Moreover, an integrated approach, that combines the geodesic and edgeless active contours algorithms, is applied for texture segmentation. We show that considering both boundary and region information yields more robust and accurate texture segmentation results. In the second part, we consider the local attributes of Gabor filters, in terms of the uncertainty principle, and explore the affine Weyl-Heisenberg group. The uncertainty principle is a fundamental concept in the context of signal and image processing, just as much as it has been in the framework of physics and more recently in harmonic analysis. Uncertainty principles can be derived using a group theoretic approach. This approach also yields formalism for finding functions that minimize the uncertainty. A general theorem that associates an uncertainty principle with a pair of self-adjoint operators is used in finding the minimizers of the uncertainty related to various groups. In this study we address the uncertainty principle in the context of the Weyl-Heisenberg, the SIM(2), the affine and the affine-Weyl-Heisenberg groups. We explore the relationship between the twodimensional affine group and the SIM(2) group in terms of the uncertainty minimizers. The uncertainty principle is also extended to the affine-WeylHeisenberg group in one dimension. Possible minimizers related to these groups are also presented and the scale-space properties of some of the minimizers are explored. In the third part, we move on from local attributes of a single filter to the global attributes of a Gabor filter bank, and consider the issue of tessellation of the combined space, so that the tightness of the frame obtained can be evaluated. The studies of Daugman, Lee and Manjunath provide three completely different treatments of the Gabor space. We provide the link between these studies, and obtain a quantitative framework for the task of filter bank design. The organization of this thesis is as follows: • In chapter 1 we provide an overview and introduce the main issues dealt with in this work. • Chapter 2 deals with the Gabor space geodesic active contours. There, we describe de-noising and segmentation algorithms that are based on the metric of the Gabor feature space. • Chapter 3 discusses the property of the Gabor function as the minimizer of the uncertainty related to time and frequency localization. We study this concept in terms of other groups, and aim at finding the appropriate minimizers. • In chapter 4 we deal with the task of filter bank design strategy. The criterion we use to assess the goodness of a filter bank is the tightness of the frame it forms. We explore several design issues and relate our work to previous studies done in this field. • In the appendix we provide a detailed calculation of the frame bounds when the scaling in the x and y directions is not identical.