Reljefo atvaizdavimas naudojamas geodezijoje, geoinformacinėse sistemose, kompiuteriniuose žaidimuose ir kitur. Dažniausiai išmatuoti reljefo duomenys nėra pakankamai tikslūs, ir su jais atvaizduojami paviršiai atrodo nenatūraliai. Šiame darbe aptariami trys pagrindiniai, viešai prieinami viso Žemės paviršiaus aukščių duomenų šaltiniai: GTOP30, ASTER GDEM ir SRTM. Nagrinėjami populiariausi reljefo interpoliavimo metodai tokie kaip bitiesinis, bikubinis, atvirkščiai proporcingo atstumo ir natūralių kaimynų. SRTM duomenų interaktyviai peržiūrai buvo sukurtas ArcGrid Viewer įrankis, leidžiantis peržiūrėti ArcGrid formatu išsaugotus reljefus. Interpoliavimo metodų tyrimui buvo sukurtas atskiras GeoSG įrankis. Su juo vienu metu galima stebėti kelis paviršius ir atlikti paviršių interpoliavimą bitiesiniu, bikubiniu, atvirkščiai proporcingo atstumo ir natūralių kaimynų metodais. Su šiuo įrankiu atliktas interpoliavimo metodų tikslumo įvertinimas. Nustatyta, kad geriausiai interpoliuoja bikubinis metodas, o blogiausiai – atvirkščiai proporcingo atstumo metodas. / Terrain rendering is used in geodesy, geographic information systems, computer games and other fields. Usually measured terrain data isn’t sufficiently precise, and if used for rendering, the result looks unnatural. This paper analyzes three main topographic data sources of planet Earth, that are publicly available: GTOP30, ASTER GDEM and SRTM. It compares the most often used methods for terrain data: bilinear, bicubic, inverse distance weighted and natural neighbor interpolation. To monitor SRTM data interactively an ArcGrid Viewer tool was created. It can be used to see terrain files in ArcGrid format. To study interpolation methods, a separate GeoSG tool was created. It can display several terrains at once and interpolate terrain data using bilinear, bicubic, inverse distance weighted and natural neighbor methods. This tool is used to evaluate interpolation methods. Results showed, that the best method for interpolation is the bicubic interpolation method, the worst – inverse distance weighted interpolation method.

The work for data analysis of the main methods, fullfilled in Computer mathematic systems (CMS), analysis. Also analysing and comparison of the data analysis of CMS – MAPLE, MATLAB and MATHCAD. There‘s briefly described enter and reading of data, characteristics of statistic data, analysis of variance, regression, interpolation and correlation. In the last section of data system analysis possibilities according to its sophistication, comfortable usage and variety of data function fullfillment. You will finde the examples of solved tasks with CMS after each description of data analysis method.

Tasks: 1. To determine a soil sampling method most suitable for pH tests in soils on flat, rolling and hilly relief using regular grid sampling, soil database (Dirv_DB10LT) and soil agrochemical properties database (DirvAgroch_DB10LT). 2. To investigate the suitability of geostatistical methods for spatial distribution of pH data using different soil sampling methods. 3. To determine an impact of different soil sampling methods on spatial distribution of areas to be limed and on the needed amount of lime. Propositions to be defended: 1. Soil sampling plots for detailed pH tests should be shaped using soil database (Dirv_DB10LT) and soil agrochemical properties database (DirvAgroch_DB10LT); in case of high variability of pH values soil sampling plots should be smaller and in case of lower variability of pH values – larger. 2. Interpolation of pH data using IDW, Simple Kriging and Simple Cokriging methods results in decreased share of determined areas of conditionally acid soils when compared to that obtained using not interpolated pH data. 3. The needed amount of lime depends on soil sampling method. Larger needed amount of lime is calculated when soil samples are collected using databases (Dirv_DB10LT) and (DirvAgroch_DB10LT). / Uždaviniai: 1. Nustatyti dirvoţemio pH tyrimams tinkamiausią ėminių paėmimo metodą lyguminio, banguoto ir kalvoto reljefo plotuose, taikant taisyklingą tinklelį, dirvoţemio (Dirv_DB10LT) ir agrocheminių savybių (DirvAgroch_DB10LT) duomenų bazes. 2. Ištirti geostatistinių metodų tinkamumą pH duomenų erdviniam pasiskirstymui, taikant skirtingus ėminių paėmimo metodus. 3. Nustatyti ėminių paėmimo metodų įtaką kalkintinų plotų erdviniam pasiskirstymui ir kalkių reikmei. 32 Ginami disertacijos teiginiai: 1. Išsamiam dirvoţemio pH tyrimui ėminio paėmimo laukelius tikslinga formuoti naudojant dirvoţemių (Dirv_DB10LT) ir agrocheminių savybių (DirvAgroch_DB10LT) duomenų bazes, esant dideliam pH įvairavimui, dirvoţemio ėminius reikėtų imti tankiau, kai įvairavimas maţesnis – rečiau. 2. Dirvoţemio pH duomenis interpoliuojant IDW, paprastojo krigingo ir paprastojo kokrigingo metodais, sąlygiškai rūgščių plotų gaunama maţiau, palyginti su neinterpoliuotais. 3. Priklausomai nuo dirvoţemio ėminio paėmimo metodo, apskaičiuojama skirtinga kalkinių trąšų reikmė, ji didesnė plotuose, kuriuose dirvoţemio ėminiai imami naudojantis (Dirv_DB10LT) ir (DirvAgroch_DB10LT) duomenų bazėmis.

This thesis presents a new technique for the reconstruction of a smooth surface from a set of 3D data points. The reconstructed surface is represented by an everywhere -continuous subdivision surface which interpolates all the given data points. And the topological structure of the reconstructed surface is exactly the same as that of the data points. The new technique consists of two major steps. First, use an efficient surface reconstruction method to produce a polyhedral approximation to the given data points. Second, construct a Doo-Sabin subdivision surface that smoothly passes through all the data points in the given data set. A new technique is presented for the second step in this thesis. The new technique iteratively modifies the vertices of the polyhedral approximation 1CM until a new control meshM, whose Doo-Sabin subdivision surface interpolatesM, is reached. It is proved that, for any mesh M with any size and any topology, the iterative process is always convergent with Doo-Sabin subdivision scheme. The new technique has the advantages of both a local method and a global method, and the surface reconstruction process can reproduce special features such as edges and corners faithfully.

This thesis presents a new system that reconstructs the 3D representation of dental casts. To maintain the integrity of the 3D representation, a standard model is built to cover the blind spots that the camera cannot reach. The standard model is obtained by scanning a real human mouth model with a laser scanner. Then the model is simplified by an algorithm which is based on iterative contraction of vertex pairs. The simplified standard model uses a local parametrization method to obtain the curvature information. The system uses a digital camera and a square tube mirror in front of the camera to capture multi-view images. The mirror is made of stainless steel in order to avoid double reflections. The reflected areas of the image are considered as images taken by the virtual cameras. Only one camera calibration is needed since the virtual cameras have the same intrinsic parameters as the real camera. Depth is computed by a simple and accurate geometry based method once the corresponding points are identified. Correspondences are selected using a feature point based stereo matching process, including fast normalized cross-correlation and simulated annealing.

The inversion method for generating non-uniform random variates has some advantages compared to other generation methods, since it monotonically transforms uniform random numbers into non-uniform random variates. Hence it is the method of choice in the simulation literature. However, except for some simple cases where the inverse of the cumulative distribution function is a simple function we need numerical methods. Often inversion by ``brute force" is used, applying either very slow iterative methods or linear interpolation of the CDF and huge tables. But then the user has to accept unnecessarily large errors or excessive memory requirements, that slow down the algorithm. In this paper we demonstrate that with Hermite interpolation of the inverse CDF we can obtain very small error bounds close to machine precision. Using our adaptive interval splitting method this accuracy is reached with moderately sized tables that allow for a fast and simple generation procedure. The algorithms described in this paper have been implemented in ANSI C in a library called UNURAN which is available via anonymous ftp. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

L'arrivée massive d'écrans LCD dits de haute-définition sur le marché, entraîne un besoin acru d'algorithmes d'augmentation de résolution pour l'affichage d'images ou de vidéos dont la résolution est inférieure à celle de l'écran. Nous proposons un schéma novateur d'interpolation d'images, basée sur une analyse multirésolution de la direction des contours. Le but de cette approche est de corriger les artefacts classiques d'interpolation qui apparaissent lorsque des méthodes habituelles sont utilisées (bilinéaire, bicubique), tout en évitant l'apparition des artefacts engendrés par la plupart des méthodes d'interpolation directionnelle. Notre estimation d'orientation de contours, basée sur une division de l'image originale en quadtree et une étude fréquentielle des contours est comparée à deux méthodes faisant référence dans l'état de l'art (transformée de Radon et algorithme de projection utilisé pour la création des bandelettes). Cette comparaison permet d'étudier les comportements de chaque méthode en vue d'une application à des images naturelles. Par la suite, l'interpolation en elle-même est introduite. Elle est basée sur l'utilisation d'un noyau d'interpolation isotrope (cubic-spline), qui est corrigée grâce à un filtrage Gaussien localement orienté dans la direction des contours. Les régions ne contenant pas de contour sont préservées grâce à la création d'un masque construit à partir de filtres de Gabor. Enfin, les résultats de notre interpolation sont comparés à des méthodes d'interpolation directionnelle récentes, afin d'illustrer les bonnes performances de notre algorithme sur des images naturelles de natures variées.

A challenge in hydrological studies in the Canadian Prairie region is to find good-quality meteorological data because many basins are located in remote regions where few stations are available, and existing stations typically have short records and often contain a high number of missing data. The recently released North American Regional Reanalysis (NARR) data set appears to have potential for hydrological studies in data-scarce central Canada. The main objectives of this study are: (1) to evaluate and utilize NARR data for hydrologic modelling and statistical downscaling, (2) to develop methods for estimating missing precipitation data using NARR data, and (3) to investigate and correct NARR precipitation bias in the Canadian Prairie region. Prior to applying NARR for hydrological modelling, the NARR surface data were evaluated by comparison with observed meteorological data over the Canadian Prairie region. The comparison results indicated that NARR is a suitable alternative to observed surface meteorological data and thus useful for hydrological modelling. After evaluation of NARR surface climate data, the SLURP model was set up with input data from NARR and calibrated for several watersheds. The results indicated that the hydrological model can be reasonably calibrated using NARR data as input. The relatively good agreement between precipitation from NARR and observed station data suggests that NARR information may be used in the estimation of missing precipitation records at weather stations. Several traditional methods for estimating missing data were compared with three NARR-based estimation methods. The results show that NARR-based methods significantly improved the estimation of precipitation compared to the traditional methods. The existence of NARR bias is a critical issue that must be addressed prior to the use of the data. Using observed weather station data, a statistical interpolation technique (also known as Optimum Interpolation) was employed to correct gridded NARR precipitation for bias. The results suggest that the method significantly reduces NARR bias over the selected study area.

This thesis addresses the problem of representing few group, homogenised neutron cross sections as a function of state parameters (e.g. burn-up, fuel and moderator temperature, etc.) that describe the conditions in the reactor. The problem is multi-dimensional and the cross section samples, required for building the representation, are the result of expensive transport calculations. At the same time, practical applications require high accuracy. The representation method must therefore be efficient in terms of the number of samples needed for constructing the representation, storage requirements and cross section reconstruction time. Sparse grid methods are proposed for constructing such an efficient representation. Approximation through quasi-regression as well as polynomial interpolation, both based on sparse grids, were investigated. These methods have built-in error estimation capabilities and methods for optimising the representation, and scale well with the number of state parameters. An anisotropic sparse grid integrator based on Clenshaw-Curtis quadrature was implemented, verified and coupled to a pre-existing cross section representation system. Some ways to improve the integrator’s performance were also explored. The sparse grid methods were used to construct cross section representations for various Light Water Reactor fuel assemblies. These reactors have different operating conditions, enrichments and state parameters and therefore pose different challenges to a representation method. Additionally, an example where the cross sections have a different group structure, and were calculated using a different transport code, was used to test the representation method. The built-in error measures were tested on independent, uniformly distributed, quasi-random sample points. In all the cases studied, interpolation proved to be more accurate than approximation for the same number of samples. The primary source of error was found to be the Xenon transient at the beginning of an element’s life (BOL). To address this, the domain was split along the burn-up dimension into “start-up” and “operating” representations. As an alternative, the Xenon concentration was set to its equilibrium value for the whole burn-up range. The representations were also improved by applying anisotropic sampling. It was concluded that interpolation on a sparse grid shows promise as a method for building a cross section representation of sufficient accuracy to be used for practical reactor calculations with a reasonable number of samples. / Thesis (MSc Engineering Sciences (Nuclear Engineering))--North-West University, Potchefstroom Campus, 2013.

In this dissertation, the aim is to investigate the empirical relationship between the partial pressure of CO2 (pCO2) and other ocean variables in the Southern Ocean, by using a small percentage of the available data. CO2 is one of the main greenhouse gases that contributes to global warming and climate change. The concentration of anthropogenic CO2 in the atmosphere, however, would have been much higher if some of it was not absorbed by oceanic and terrestrial sinks. The oceans absorb and release CO2 from and to the atmosphere. Large regions in the Southern Ocean are expected to be a CO2 sink. However, the measurements of CO2 concentrations in the ocean are sparse in the Southern Ocean, and accurate values for the sinks and sources cannot be determined. In addition, it is difficult to develop accurate oceanic and ocean-atmosphere models of the Southern Ocean with the sparse observations of CO2 concentrations in this part of the ocean. In this dissertation classical techniques are investigated to determine the empirical relationship between pCO2 and other oceanic variables using in situ measurements. Additionally, sampling techniques are investigated in order to make a judicious selection of a small percentage of the total available data points in order to develop an accurate empirical relationship. Data from the SANAE49 cruise stretching between Antarctica and Cape Town are used in this dissertation. The complete data set contains 6103 data points. The maximum pCO2 value in this stretch is 436.0 μatm, the minimum is 251.2 μatm and the mean is 360.2 μatm. An empirical relationship is investigated between pCO2 and the variables Temperature (T), chlorophyll-a concentration (Chl), Mixed Layer Depth (MLD) and latitude (Lat). The methods are repeated with latitude included and excluded as variable respectively. D-optimal sampling is used to select a small percentage of the available data for determining the empirical relationship. Least squares optimization is used as one method to determine the empirical relationship. For 200 D-optimally sampled points, the pCO2 prediction with the fourth order equation yields a Root Mean Square (RMS) error of 15.39 μatm (on the estimation of pCO2) with latitude excluded as variable and a RMS error of 8.797 μatm with latitude included as variable. Radial basis function (RBF) interpolation is another method that is used to determine the empirical relationship between the variables. The RBF interpolation with 200 D-optimally sampled points yields a RMS error of 9.617 μatm with latitude excluded as variable and a RMS error of 6.716 μatm with latitude included as variable. Optimal scaling is applied to the variables in the RBF interpolation, yielding a RMS error of 9.012 μatm with latitude excluded as variable and a RMS error of 4.065 μatm with latitude included as variable for 200 D-optimally sampled points. / Thesis (MSc (Applied Mathematics))--North-West University, Potchefstroom Campus, 2012