In this thesis we study the general problem of reconstructing a function, defined on a finite lattice from a set of incomplete, noisy and/or ambiguous observations. The goal of this work is to demonstrate the generality and practical value of a probabilistic (in particular, Bayesian) approach to this problem, particularly in the context of Computer Vision. In this approach, the prior knowledge about the solution is expressed in the form of a Gibbsian probability distribution on the space of all possible functions, so that the reconstruction task is formulated as an estimation problem. Our main contributions are the following: (1) We introduce the use of specific error criteria for the design of the optimal Bayesian estimators for several classes of problems, and propose a general (Monte Carlo) procedure for approximating them. This new approach leads to a substantial improvement over the existing schemes, both regarding the quality of the results (particularly for low signal to noise ratios) and the computational efficiency. (2) We apply the Bayesian appraoch to the solution of several problems, some of which are formulated and solved in these terms for the first time. Specifically, these applications are: teh reconstruction of piecewise constant surfaces from sparse and noisy observationsl; the reconstruction of depth from stereoscopic pairs of images and the formation of perceptual clusters. (3) For each one of these applications, we develop fast, deterministic algorithms that approximate the optimal estimators, and illustrate their performance on both synthetic and real data. (4) We propose a new method, based on the analysis of the residual process, for estimating the parameters of the probabilistic models directly from the noisy observations. This scheme leads to an algorithm, which has no free parameters, for the restoration of piecewise uniform images. (5) We analyze the implementation of the algorithms that we develop in non-conventional hardware, such as massively parallel digital machines, and analog and hybrid networks.

The volume of lava flows provide important information on the magnitude of volcanic eruptions, and accurate volumes are necessary to produce reliable models of lava flow emplacement or constrain the internal structure of volcanoes. The most accurate lava flow volumes are obtainable when the topography before and after an eruption are both known, but information for the topography before lava flow emplacement is absent in non-historic lava flows. To calculate the volume of non-historic lava flows, this pre-emplacement topography needs to be reconstructed. Common methods for this include using inverse distance-weighted averages or global polynomial interpolation methods, but these can still underestimate the volume of the flow, and the surface of the flow itself is not considered in these interpolations. A new calculation method seems necessary to better constrain the volume of lava flows, and including the lava flow surface in the volume calculation, given that it is generally excluded during interpolation of pre-emplacement topography, may be the solution to improving lava flow volume calculation for flows where the base surface is unknown. The 2012-2013 Tolbachik lava flow is used to look at potential relationships due to the availability of elevation data before and after the eruption. A quantitative analysis on the relationships between the slope of topography before and after lava flow emplacement and on the relationship between the slope and thickness of lava flows is performed. In addition to this, the slope of the topography calculated over local and regional scales is used as a new interpolation method, and the calculated thickness from the interpolated surface is compared to the known thickness for the lava flow.

In recent years the science revolving image matching algorithms has gotten an upswing mostly due to its benefits in computer vision. This has led to new opportunities for photogrammetric methods to compete with LiDAR data when it comes to 3D-point clouds and generating surface models. In Sweden a project to create a high resolution national height model started in 2009 and today almost the entirety of Sweden has been scanned with LiDAR sensors. The objective for this project is to achieve a height model with high spatial resolution and high accuracy in height. As for today no update of this model is planned in the project so it’s up to each municipality or company who needs a recent height model to update themselves. This thesis aims to investigate the benefits and shortcomings of using photogrammetric measures for generating and updating surface models. Two image matching software are used, ERDAS photogrammetry and Spacemetric Keystone, to generate a 3D point cloud of a rural area in Botkyrka municipality. The point clouds are interpolated into surface models using different interpolation percentiles and different resolutions. The photogrammetric point clouds are evaluated on how well they fit a reference point cloud, the surfaces are evaluated on how they are affected by the different interpolation percentiles and image resolutions. An analysis to see if the accuracy improves when the point cloud is interpolated into a surface. The result shows that photogrammetric point clouds follows the profile of the ground well but contains a lot of noise in the forest covered areas. A lower image resolution improves the accuracy for the forest feature in the surfaces. The results also show that noise-reduction is essential to generate a surface with decent accuracy. Furthermore, the results identify problem areas in dry deciduous forest where the photogrammetric method fails to capture the forest.

In this thesis, the author studies recursIve subdivision algorithms for curves and surfaces. Several subdivision algorithms are constructed and investigated. Some graphic examples are also presented. Inspired by the Chaikin's algorithm and the Catmull-Clark's algorithm, some non-uniform schemes, the non-uniform corner cutting scheme and the recursive subdivision algorithm for non-uniform B-spline curves, are constructed and analysed. The adapted parametrization is introduced to analyse these non-uniform algorithms. In order to solve the surface interpolation problem, the Dyn-Gregory-Levin's 4-point interpolatory scheme is generalized to surfaces and the 10-point interpolatory subdivision scheme for surfaces is formulated. The so-called Butterfly Scheme, which was firstly introduced by Dyn, Gregory Levin in 1988, is just a special case of the scheme. By studying the Cross-Differences of Directional Divided Differences, a matrix approach for analysing uniform subdivision algorithms for surfaces is established and the convergence of the 10-point scheme over both uniform and non-uniform triangular networks is studied. Another algorithm, the subdivision algorithm for uniform bi-quartic B-spline surfaces over arbitrary topology is introduced and investigated. This algorithm is a generalization of Doo-Sabin's and Catmull-Clark's algorithms. It produces uniform Bi-quartic B-spline patches over uniform data. By studying the local subdivision matrix, which is a circulant, the tangent plane and curvature properties of the limit surfaces at the so-called Extraordinary Points are studied in detail.

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Understanding the events and processes responsible for patterns of within species diversity, provides insight into major evolutionary themes like adaptation, species distributions, and ultimately speciation itself. Here, I combine ecological, genetic and spatial perspectives to evaluate the roles that both historical and contemporary factors have played in shaping the population structure and genetic variation of foxsnakes (Pantherophis gloydi). First, I determine the likely impact of habitat loss on population distribution, through radio-telemetry (32 individuals) at two locations varying in habitat patch size. As predicted, individuals had similar habitat use patterns, but restricted movements to patches of suitable habitat at the more disturbed site. Also, occurrence records spread across a fragmented region were non-randomly distributed and located close to patches of usable habitat, suggesting habitat distribution limits population distribution. Next, I combined habitat suitability modeling with population genetics (589 individuals, 12 microsatellite loci) to infer how foxsnakes disperse through a mosaic of natural and altered landscape features. Boundary regions between genetic clusters were comprised of low suitability habitat (e.g. agricultural fields). Island populations were grouped into a single genetic cluster suggesting open water presents less of a barrier than non-suitable terrestrial habitat. Isolation by distance models had a stronger correlation with genetic data when including resistance values derived from habitat suitability maps, suggesting habitat degradation limits dispersal for foxsnakes. At larger temporal and spatial scales I quantified patterns of genetic diversity and population structure using mitochondrial (101 cytochrome b sequences) and microsatellite (816 individuals, 12 loci) DNA and used Approximate Bayesian computation to test competing models of demographic history. Supporting my predictions, I found models with populations which have undergone population size drops and splitting events continually had more support than models with small founding populations expanding to stable populations. Based on timing, the most likely cause was the cooling of temperatures and infilling of deciduous forest since the Hypisthermal. On a smaller scale, evidence suggested anthropogenic habitat loss has caused further decline and fragmentation. Mitochondrial DNA structure did not correspond to fragmented populations and the majority of foxsnakes had an identical haplotype, suggesting a past bottleneck or selective sweep. / Thesis (Ph.D, Biology) -- Queen's University, 2011-01-11 10:40:52.476