This dissertation addresses three basic issues that arise in the use of Gaussian Markov random fields (GMRFs) in image modelling: the multi-resolution properties, the valid parameter space, and the existence of Maximum Likelihood (ML) and Maximum Entropy (ME) parameter estimates. For the multi-resolution properties, we study GMRFs under two types of resolution transformations, Sampling and Block-to-Point. We show that under both these transformations the coarser level fields are non-Markov, and obtain exact descriptions for their covariances and power spectra. To approximate the coarser level non-Markov fields as GMRFs, we propose a new methodology called the Covariance Invariance Approximation (CIA) and study its measure-theoretic properties. We argue that CIA is better suited to image processing than the free-energy based approximations used in renormalization group studies. On the valid parameter space issue, for both 1-D infinite-length GM processes and 2-D infinite-lattice GMRFs, we present a complete procedure for verifying the validity of a given set of parameters. We illustrate this result by applying it to second-order fields in both 1-D and 2-D, and obtain an explicit and simple description of the respective parameter spaces. We observe that in both these examples, the valid parameter space is considerably larger than the space implied by the previously known sufficient condition. For both 1-D and 2-D finite-lattice fields, we show that the valid parameter space does not admit a simple description. The infinite-lattice conditions, however, provide a tight lower-bound approximation to the valid parameter space of finite-lattice fields. Finally, we consider the existence of the ML and the ME estimates for GMRF parameters. The existence of ME estimates is closely related to the extendibility of covariance sequences. Using this fact in conjunction with our results on the valid parameter space of GMRFs, we obtain analytical and computational solutions to the existence problem. For several examples, we obtain an explicit set of conditions that ascertain extendibility and hence existence. For the general case, we propose a cutting-plane algorithm as an alternative to the two numerical procedures that already exist for determining extendibility, namely, the linear programming algorithm and expanding-hull algorithm. Next, we explore the duality between the valid parameter space of GMRFs and the space of extendible covariances, and their relationships with the space of admissible covariances for finite-size data sequences. Using duality, we also relate the existence of ML estimates to extendibility and show that the existence of ML estimates would have to be ascertained through a computationally intensive linear programming procedure. Finally, we present some results regarding the extendibility of covariances over increasing window sizes.

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In order to autonomously navigate through a complex environment, a mobile robot requires sensory feedback. This feedback will typically include the 3D motion and location of the robot and the 3D structure and motion of obstacles and other environmental features. The general problem considered in this thesis is how this 3D information may be obtained from a sequence of images generated by a camera mounted on a mobile robot. The first set of algorithms developed in this thesis are for robust determination of the 3D pose of the mobile robot from a matched set of model and image landmark features. Least-squares techniques for point and line tokens, which minimize both rotation and translation simultaneously are developed and shown to be far superior to the earlier techniques which solved for rotation first and then translation. However, least-squares techniques fail catastrophically when outliers (or gross errors) are present in the match data. Outliers arise frequently due to incorrect correspondences or gross errors in the 3D model. Robust techniques for pose determination are developed to handle data contaminated by fewer than 50.0% outliers. To make the model based approach widely applicable, it is necessary to be able to automatically build the landmark models. The approach adopted in this thesis is one of model extension and refinement. A partial model of the environment is assumed to exist and this model is extended over a sequence of frames. As will be shown in the experiments, the prior knowledge of the small partial model greatly enhances the robustness of the 3D structure computations. The initial 3D model may have errors and these too are refined over the sequence of frames. Finally, the sensitivity of pose determination and model extension to incorrect estimates of camera parameters is analyzed. It is shown that for small field of view systems, offsets in the image center do not significantly affect the location of the camera and the location of new 3D points in a world coordinate system. Errors in the focal length significantly affect only the component of translation along the optical axis in the pose computation.

This thesis presents methods for the inference of system state and the learning of model structure for a number of hidden-state time series models, within a Bayesian probabilistic framework. Motivating examples are taken from application areas including finance, physical object tracking and audio restoration. The work in this thesis can be broadly divided into three themes: system and parameter estimation in linear jump-diffusion systems, non-parametric model (system) estimation and batch audio restoration. For linear jump-diffusion systems, efficient state estimation methods based on the variable rate particle filter are presented for the general linear case (chapter 3) and a new method of parameter estimation based on Particle MCMC methods is introduced and tested against an alternative method using reversible-jump MCMC (chapter 4). Non-parametric model estimation is examined in two settings: the estimation of non-parametric environment models in a SLAM-style problem, and the estimation of the network structure and forms of linkage between multiple objects. In the former case, a non-parametric Gaussian process prior model is used to learn a potential field model of the environment in which a target moves. Efficient solution methods based on Rao-Blackwellized particle filters are given (chapter 5). In the latter case, a new way of learning non-linear inter-object relationships in multi-object systems is developed, allowing complicated inter-object dynamics to be learnt and causality between objects to be inferred. Again based on Gaussian process prior assumptions, the method allows the identification of a wide range of relationships between objects with minimal assumptions and admits efficient solution, albeit in batch form at present (chapter 6). Finally, the thesis presents some new results in the restoration of audio signals, in particular the removal of impulse noise (pops and clicks) from audio recordings (chapter 7).

The diploma thesis Cyclists speed analysis at the age bracket of 4 to 10 deals with history and origin of bicycle, description and types of the contruction of bicycles for children at the age bracket mentioned above. The thesis includes the statistics of accidents involving cyclists and regulations providing for the ride on a bicycle on a carriageway. The practical part includes measuring cyclists speed when passing a measured section and the analysis of these data. The analysis of data is processed in terms of gear systems as the equipment. Complementary measuring was made in a slight rising. In the second part of the practical part is measured braking of cyclists at a particular age bracket and the evaluation of their deceleration. The conclusion includes comparison with measuring of other authors made so far.

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