This thesis considers statistical modelling of natural image data. Obtaining advances in this field can have significant impact for both engineering applications, and for the understanding of the human visual system. Several recent advances in natural image modelling have been obtained with the use of unsupervised feature learning. We consider a class of such models, restricted Boltzmann machines (RBMs), used in many recent state-of-the-art image models. We develop extensions of these stochastic artificial neural networks, and use them as a basis for building more effective image models, and tools for computational vision. We first develop a novel framework for obtaining Boltzmann machines, in which the hidden unit activations co-transform with transformed input stimuli in a stable and predictable way throughout the network. We define such models to be transformation equivariant. Such properties have been shown useful for computer vision systems, and have been motivational for example in the development of steerable filters, a widely used classical feature extraction technique. Translation equivariant feature sharing has been the standard method for scaling image models beyond patch-sized data to large images. In our framework we extend shallow and deep models to account for other kinds of transformations as well, focusing on in-plane rotations. Motivated by the unsatisfactory results of current generative natural image models, we take a step back, and evaluate whether they are able to model a subclass of the data, natural image textures. This is a necessary subcomponent of any credible model for visual scenes. We assess the performance of a state- of-the-art model of natural images for texture generation, using a dataset and evaluation techniques from in prior work. We also perform a dissection of the model architecture, uncovering the properties important for good performance. Building on this, we develop structured extensions for more complicated data comprised of textures from multiple classes, using the single-texture model architecture as a basis. These models are shown to be able to produce state-of-the-art texture synthesis results quantitatively, and are also effective qualitatively. It is demonstrated empirically that the developed multiple-texture framework provides a means to generate images of differently textured regions, more generic globally varying textures, and can also be used for texture interpolation, where the approach is radically dfferent from the others in the area. Finally we consider visual boundary prediction from natural images. The work aims to improve understanding of Boltzmann machines in the generation of image segment boundaries, and to investigate deep neural network architectures for learning the boundary detection problem. The developed networks (which avoid several hand-crafted model and feature designs commonly used for the problem), produce the fastest reported inference times in the literature, combined with state-of-the-art performance.

Η παρούσα διατριβή αφορά στη μελέτη του συνόρου υπερβολικών δισδιάστατων πολυέδρων. Οι χώροι οι οποίοι μελετώνται κατασκευάζονται κολλώντας υπερβολικά τρίγωνα τα οποία έχουν 2 τουλάχιστον κορυφές στο άπειρο. Οι συγκολλήσεις γίνονται με ισομετρίες κατά μήκος των πλευρών των τριγώνων και οι χώροι οι οποίοι προκύπτουν εφοδιάζονται φυσιολογικά με μία γεωμετρία η οποία έχει ομοιότητες με την γεωμετρία των υπερβολικών πολλαπλοτήτων. Αρχικά μελετάμε τις βασικές ιδιότητες των δισδιάστατων ιδεωδών πολυέδρων και αποδεικνύουμε ότι: «Για κάθε δύο σημεία του συνόρου του καθολικού καλύμματος του χώρου που κατασκευάζουμε, υπάρχει άπειρο πλήθος υποχώρων του συνόρου ομοιομορφικών με το οι οποίοι περιέχουν τα σημεία αυτά». Στη συνέχεια, για μια ειδική κλάση πολυέδρων που κατασκευάζουμε κολλώντας με ισομετρίες κατά μήκος των πλευρών τους πεπερασμένα υπερβολικά τρίγωνα τα οποία έχουν δύο κορυφές στο άπειρο, αποδεικνύουμε επιπλέον ότι: «το σύνορο του καθολικού καλύμματος του χώρου που κατασκευάζουμε είναι τοπικά συνεκτικό κατά τόξα». Τέλος, στην τρίτη ενότητα δίδουμε μια τοπολογική περιγραφή του συνόρου των ιδεωδών πολυέδρων διάστασης 2. / The present work is related to the study of the visual boundary of hyperbolic two dimensional simplicial complexes. We construct (and study) spaces by gluing hyperbolic triangles with at least two vertices at infinity. We glue the triangles by isometries along their sides and we study the derived spaces. In the first chapter it is proved that for every two points in the visual boundary of the universal covering of a two dimensional ideal polyhedron, there is an infinity of paths joining them. In the second chapter, a class of hyperbolic two dimensional complexes X is defined. Is is shown that the limit set of the action of π1(X) on the universal covering of X, is equal to the visual boundary and also that the visual boundary is path connected and locally path connected. Finally, in the third chapter a kind of Sierpinski set is described which is homeomorphic to the visual boundary of certain ideal polyhedra.