<p>Predmet istraživanja ove doktorske disertacije je problem<br />interpolacije slike. Glavni fokus disertacije je interpolacija slike<br />uz očuvanje prirodnosti teksture i očuvanje ivica (oštrine)<br />interpolirane slike. Dodatni izazov je da algoritam za interpolaciju<br />slike bude pogodan za primenu u uređajima sa ograničenim resursima.<br />Kvalitet rešenja se ocenjuje poređenjem sa algoritmima poznatim u<br />dostupnoj literaturi korišćenjem odgovarajućih metrika.</p> / <p>This PhD dissertation addresses the problem of image interpolation. The main<br />focus of the dissertation is image interpolation algorithm which preserves<br />edges and keeps a natural texture of interpolated images. Additional challenge<br />for image interpolation algorithm is to be suitable for application on resourcelimited<br />platforms. The quality of the proposed solution is benchmarked against<br />known image interpolation algorithms using appropriate metrics.</p>

Interpolation error estimates in terms of geometric quality measures are established for harmonic coordinates on polytopes in two and three dimensions. First we derive interpolation error estimates over convex polygons that depend on the geometric quality of the triangles in the constrained Delaunay triangulation of the polygon. This characterization is sharp in the sense that families of polygons with poor quality triangles in their constrained Delaunay triangulations are shown to produce large error when interpolating a basic quadratic function. Non-convex polygons exhibit a similar limitation: large constrained Delaunay triangles caused by vertices approaching a non-adjacent edge also lead to large interpolation error. While this relationship is generalized to convex polyhedra in three dimensions, the possibility of sliver tetrahedra in the constrained Delaunay triangulation prevent the analogous estimate from sharply reflecting the actual interpolation error. Non-convex polyhedra are shown to be fundamentally different through an example of a family of polyhedra containing vertices which are arbitrarily close to non-adjacent faces yet the interpolation error remains bounded.

A numerical implementation of the direct Boundary-Domain Integral Equation (BDIE)/ Boundary-Domain Integro-Differential Equations (BDIDEs) and Localized Boundary-Domain Integral Equation (LBDIE)/Localized Boundary-Domain Integro-Differential Equations (LBDIDEs) related to the Neumann and Dirichlet boundary value problem for a scalar elliptic PDE with variable coefficient is discussed in this thesis. The BDIE and LBDIE related to Neumann problem are reduced to a uniquely solvable one by adding an appropriate perturbation operator. The mesh-based discretisation of the BDIE/BDIDEs and LBDIE/LBDIDEs with quadrilateral domain elements leads to systems of linear algebraic equations (discretised BDIE/BDIDEs/LBDIE/BDIDEs). Then the systems obtained from BDIE/BDIDE (discretised BDIE/BDIDE) are solved by the LU decomposition method and Neumann iterations. Convergence of the iterative method is analyzed in relation with the eigen-values of the corresponding discrete BDIE/BDIDE operators obtained numerically. The systems obtained from LBDIE/LBDIDE (discretised LBDIE/LBDIDE) are solved by the LU decomposition method as the Neumann iteration method diverges.

Robust algorithms for the approximation of functions are studied and developed in this thesis. Novel results and algorithms on piecewise polynomial interpolation, rational interpolation and best polynomial and rational approximations are presented. Algorithms for the extension of Chebfun, a software system for the numerical computation with functions, are described. These algorithms allow the construction and manipulation of piecewise smooth functions numerically with machine precision. Breakpoints delimiting subintervals are introduced explicitly, implicitly or automatically, the latter method combining recursive subdivision and edge detection techniques. For interpolation by rational functions with free poles, a novel method is presented. When the interpolation nodes are roots of unity or Chebyshev points the algorithm is particularly simple and relies on discrete Fourier transform matrices, which results in a fast implementation using the Fast Fourier Transform. The method is generalised for arbitrary grids, which requires the construction of polynomials orthogonal on the set of interpolation nodes. The new algorithm has connections with other methods, particularly the work of Jacobi and Kronecker, Berrut and Mittelmann, and Egecioglu and Koc. Computed rational interpolants are compared with the behaviour expected from the theory of convergence of these approximants, and the difficulties due to truncated arithmetic are explained. The appearance of common factors in the numerator and denominator due to finite precision arithmetic is characterised by the behaviour of the singular values of the linear system associated with the rational interpolation problem. Finally, new Remez algorithms for the computation of best polynomial and rational approximations are presented. These algorithms rely on interpolation, for the computation of trial functions, and on Chebfun, for the location of trial references. For polynomials, the algorithm is particularly robust and efficient, and we report experiments with degrees in the thousands. For rational functions, we clarify the numerical issues that affect its application.

In practice, yield curves, i.e. plots of relation between yields and times to maturity for a group of comparable securities, are an important tool for assets and liabilities pricing as well as for financial decision making. The theoretical risk-free yield curve represents the term structure of interest rates that are used e.g. in insurance industry for pricing the liabilities, for which reserves are created, or also as a benchmark for pricing other assets in the market. When constructing the yield curve, it is not possible to observe yields of a group of assets for all maturities. That is why we use various mathematical methods which enable us to construct the yield curve also for unobserved maturities. In this thesis, some of these methods are introduced. The Svensson's method is one of the most important and frequently used ones. We use this method to derive the coupon curve from Czech government bonds aiming to construct the risk-free zero coupon yield curve. Later on, we use different weights for particular bonds trying to improve pricing of all the bonds based on the derived curve. Then, we also look for the curve that minimizes the mean squared error of estimated (compared to observed) prices. Because problems with liquidity can appear especially for long maturities, we apply all of the procedures to a...

The polytopic model (PM) structure is often used in the areas of automatic control and fault detection and isolation (FDI). It is an alternative to the multiple model approach which explicitly allows for interpolation among local models. This thesis proposes a novel approach to PM estimation by modeling the set of PM weights as a random vector with Dirichlet Distribution (DD). A new approximate (adaptive) PM estimator, referred to as a Quasi-Bayesian Adaptive Kalman Filter (QBAKF) is derived and implemented. The model weights and state estimation in the QBAKF is performed adaptively by a simple QB weights' estimator and a single KF on the PM with the estimated weights. Since PM estimation problem is nonlinear and non-Gaussian, a DD marginalized particle filter (DDMPF) is also developed and implemented similar to MPF. The simulation results show that the newly proposed algorithms have better estimation accuracy, design simplicity, and computational requirements for PM estimation.

Soil property variability is a large component of the overall environmental variability that Precision Agriculture practices seek to address. Thus, the creation of accurate field soil maps from field soil samples is of utmost importance to practitioners of Precision Agriculture, as understanding and characterizing variability is the first step in addressing it. Today, growers often interpolate their soil maps in a “black-box” fashion, and there is a need for an easy to use, accurate method of interpolation. In this study, current interpolation practices are examined as a benchmark, a Random Forest (RF) based prediction framework utilizes public data to aid predictions, and the RF framework is exposed via a webtool. A high density (0.20 ha/sample) field soil sample dataset provides 28 training points and 82 validation points to be used as a case study. In the prediction of soil percent organic matter (OM), the grid and ordinary kriging interpolations both had higher Mean Absolute Error (MAE) scores than a field average prediction, though the difference was not statistically significant at a 5\% confidence level. A RF framework interpolation utilizing a high resolution (1.52 m) DEM and distances to known points as the feature set had a significantly lower MAE score than the field average, grid, and ordinary kriging interpolations. The results suggest that for the study site, RF framework performed better compared to a field average, a grid based, and an ordinary kriging interpolation methods.

All financial institutions that provide options to counterparties will in most cases get involved withMonte Carlo simulations. Options with a payoff function that depends on asset’s value at differenttime points over its lifespan are so called path dependent options. This path dependency impli-cates that there exists no parametric solution and the price must hence be estimated, it is hereMonte Carlo methods come into the picture. The problem though with this fundamental optionpricing method is the computational time. Prices fluctuate continuously on the open market withrespect to different risk factors and since it’s impossible to re-evaluate the option for all shifts dueto its computing intensive nature, estimations of the option price must be used. Estimating theprice from known points will of course never produce the same result as a full re-evaluation but anestimation method that produces reliable results and greatly reduces computing time is desirable.This thesis will evaluate different approaches and try to minimize the estimation error with respectto a certain number of risk factors.This is the background for our master thesis at Swedbank. The goal is to create multiple estima-tion methods and compare them to Swedbank’s current estimation model. By doing this we couldpotentially provide Swedbank with improvement ideas regarding some of its option products andrisk measurements. This thesis is primarily based on two estimation methods that estimate optionprices with respect to two variable risk factors, the value of the underlying assets and volatility.The first method is a grid that uses a second order Taylor expansion and the sensitivities delta,gamma and vega. The other method uses a grid of pre-simulated option prices for different shiftsin risk factors. The interpolation technique that is used in this method is calledPiecewise CubicHermiteinterpolation. The methods (or referred to as approaches in the report) are implementedto handle a relative change of 50 percent in the underlying asset’s index value, which is the firstrisk factor. Concerning the second risk factor, volatility, both methods estimate prices for a 50percent relative downward change and an upward change of 400 percent from the initial volatility.Should there emerge even more extreme market conditions both methods use linear extrapolationto estimate a new option price.

O uso de imagens é crucial na medicina, e seu uso no diagnóstico de doenças é uma das principais ferramentas clínicas da atualidade. Porém, frequentemente necessitam de pós-processamento para serem úteis. Embora ferramentas clássicas sejam utilizadas para esse fim, elas não dão tratamento específico a certas características de imagens fractais, como as provindas de sistemas biológicos. Nesse enfoque, este trabalho objetivou a criação de novas bases de interpolação utilizando a Q-Estatística para verificar se seriam estas seriam adequadas à representação de objetos com características fractais que as bases clássicas. Foram criados dois tipos de splines: uma unidimensional e outra bidimensional, que permitiram um tipo diferente de interpolação, fundamentado na q-Estatística. Os testes demonstraram a potencialidade dessas ferramentas para uso em sinais e imagens médicas, com acentuada redução do erro de interpolação no caso unidimensional (em até 351,876%) e uma redução sutil no caso bidimensional (0,3%). Como resultado adicional, foram criados filtros de imagens e avaliados seus resultados em imagens médicas, que resultaram em melhorias de até 1.340% de ganho efetivo na remoção de ruídos de natureza fractal (marrom). Os resultados sugerem que as q-bases desenvolvidas foram capazes de representar melhor imagens e sinais médicos, bem como é interessante o uso dos filtros desenvolvidos na remoção de diversos tipos de ruído do tipo 1/f^b. / The use of images is crucial in modern medicine, and diagnostic imaging is a major clinical tools used in detecting, monitoring and completion of many treatments. However, often the images need to be post-processed for display to health professionals or automated analysis, searching for signs of abnormalities. Although classical tools are used for that purpose, they do not give special treatment to certain characteristics of fractal images, such as those coming from biological systems. These characteristics are produced, in general, by complex dynamic systems as a result of internal interactions of sub-system components, giving the system a fractal character. In this context, the main objective of this work was to propose interpolation bases using the Q-statistic, creating bases of Q-interpolation, and verify if such bases would be best suited to the representation of objects with fractal characteristics than classical bases, assumed the premise that such a theory model best this kind of phenomenon than classical theory. Based on this hypothesis, we created two types of splines: one-dimensional and one-dimensional, called Q-splines, which allow a different type of interpolation and they can capture behaviors as super-additive or sub-additive among the constituents of a spline. These models have demonstrated numerically the potential use of this type of interpolation for use in signals and medical images, reducing the interpolation error by up to 351.876 % in the one-dimensional case and 0.3 % in two dimensional. As secondary results, were defined two families of image filters, called anisotropic Q-filters and isotropic Q-filters, and their results were evaluated in real medical images. In virtually all analyzes it was possible to obtain the best results from conventional approaches, sometimes with improvements of 1.340 % in some filters, in removing noise fractal nature (brown). The results were more modest for the interpolation of two-dimensional images, however, generally proved exciting and encouraging, clearly showing that these new approaches are not only viable, but also can produce better results compared to classical approaches. Based on these results, we concluded that the Q-bases developed are best able to represent not only signs but medical imaging (1D and 2D) although its use can be improved by the adoption of approaches adapted to the vector representation of information, that favor the use of splines. Similarly, the Q-filters were more suitable for the processing of medical signals when compared to conventional approaches.

The modal logic S5 gives us a simple technical tool to analyze some main notions from philosophy (e.g. metaphysical necessity and epistemological concepts such as knowledge and belief). Although S5 can be axiomatized by some simple rules, this logic shows some puzzling properties. For example, an interpolation result holds for the propositional version, but this same result fails when we add first-order quantifiers to this logic. In this dissertation, we study the failure of the Definability and Interpolation Theorems for first-order S5. At the same time, we combine the results of justification logic and we investigate the quantified justification counterpart of S5 (first-order JT45). In this way we explore the relationship between justification logic and modal logic to see if justification logic can contribute to the literature concerning the restoration of the Interpolation Theorem. / A lógica modal S5 nos oferece um ferramental técnico para analizar algumas noções filosóficas centrais (por exemplo, necessidade metafísica e certos conceitos epistemológicos como conhecimento e crença). Apesar de ser axiomatizada por princípios simples, esta lógica apresenta algumas propriedades peculiares. Uma das mais notórias é a seguinte: podemos provar o Teorema da Interpolação para a versão proposicional, mas esse mesmo teorema não pode ser provado quando adicionamos quantificadores de primeira ordem a essa lógica. Nesta dissertação vamos estudar a falha dos Teoremas da Definibilidade e da Interpolação para a versão quantificada de S5. Ao mesmo tempo, vamos combinar os resultados da lógica da justificação e investigar a contraparte da versão quantificada de S5 na lógica da justificação (a lógica chamada JT45 de primeira ordem). Desse modo, vamos explorar a relação entre lógica modal e lógica da justificação para ver se a lógica da justificação pode contribuir para a restauração do Teorema da Interpolação.