Image super-resolution aims to produce a higher resolution representation of a scene from an ensemble of low-resolution images that may be warped, aliased, blurred and degraded by noise. There are a variety of methods for performing super-resolution described in the literature, and in general they consist of three major steps: image registration, fusion and deblurring. This thesis proposes a novel method of performing the first two of these steps. The ultimate aim of image super-resolution is to produce a higher-quality image that is visually clearer, sharper and contains more detail than the individual input images. Machine algorithms can not assess images qualitatively and typically use a quantitative error criterion, often least-squares. This thesis aims to optimise leastsquares directly using a fast method, in particular one that can be implemented using linear filters; hence, a closed-form solution is required. The concepts of optimal interpolation and resampling are derived and demonstrated in practice. Optimal filters optimised on one image are shown to perform nearoptimally on other images, suggesting that common image features, such as stepedges, can be used to optimise a near-optimal filter without requiring the knowledge of the ground-truth output. This leads to the construction of a pulse model, which is used to derive filters for resampling non-uniformly sampled images that result from the fusion of registered input images. An experimental comparison shows that a 10th order pulse model-based filter outperforms a number of methods common in the literature. The use of optimal interpolation for image registration linearises an otherwise nonlinear problem, resulting in a direct solution. Experimental analysis is used to show that optimal interpolation-based registration outperforms a number of existing methods, both iterative and direct, at a range of noise levels and for both heavily aliased images and images with a limited degree of aliasing. The proposed method offers flexibility in terms of the size of the region of support, offering a good trade-off in terms of computational complexity and accuracy of registration. Together, optimal interpolation-based registration and fusion are shown to perform fast, direct and effective super-resolution.

Darbe nagrinėjamas lygiavertės dozės galios (LDG) laukų struktūros virš mažų teritorijų nustatymas optimalios interpoliacijos metodu, lygiavertės dozės galios laukų struktūros pokyčiai nuo meteorologinių sąlygų. Baltijos jūros priekrantėje prie Juodkrantės atlikti LDG matavimai prie žemės paviršiaus ir vieno metro aukštyje virš jos. Matavimai atlikti 2008 m. liepos 2, 7, 10 ir 12 dienomis 63 taškuose (2,00,2) km teritorijoje esant šiaurės, pietų, rytų ir vakarų vėjo kryptims. Vidutinių dydžių ekstremalios reikšmės buvo 50 nSv/h ir 130 nSv/h. LDG laukų struktūrai nustatyti ir įvertinti buvo panaudotas optimalios interpoliacijos metodas trim atvejams – kai tyrimų teritorijoje atlikta 63, 33 ir 18 matavimų. Gautos identiškos LDG laukų struktūros. Rasta, kad naudojant tik 18 matavimo rezultatų modelyje gaunamas patenkinamas eksperimentinių ir teorinių rezultatų sutapimas, t.y. skirtumas tarp išmatuotų ir sumodeliuotų rezultatų neviršija 20  paklaidos. Nustatyta, kad esant šiaurės, pietų ir vakarų vėjo kryptims, padidėjus vėjo greičiui vidutiniškai 3 kartus, LDG sumažėja apie 8 nSv/h. Esant rytų vėjui, padidėjus vėjo greičiui 3 kartus, LDG padidėja 7 nSv/h. Darbą sudaro 3 dalys: įvadas, darbo metodai ir priemonės, išvados, literatūra. Darbo apimtis – 58 p. teksto be priedų, 46 iliustr., 1 lent., 36 bibliografiniai šaltiniai. Atskirai pridedami darbo priedai ir publikacija, parengta spaudai. / In this work equivalent dose rate (EDR) field structures under small area was estimated by applying optimal interpolation method and EDR fields structure differences was depended on meteorological conditions. EDR measurements were made to the ground surface and one meter above it beside Juodkrante near Baltic seaside. Measurements carried out in 2008 July 2, 7, 10 and 12 days, in 63 points (2.0 x 0.2) km area then was the north, south, east and west wind direction. Extreme values of the average was 50 nSv/h ir 130 nSv/h. To determine EDR fields structure was used the optimal interpolation method in three cases – then in research area conducted 63, 33 and 18 measurements. Received identical EDR fields structure. Found that the use of only 18 measurements in the model obtained satisfactory experimental and theoretical results coincide, i.e., the difference between measured and model results in less than 20  errors. Found that, in the north, south and west wind direction, wind speed increased by an average of 3 times, EDR decreases about 8 nSv/h. In the east wind, the wind speed increased by 3 times, EDR increased 7 nSv/h / h. Structure: introduction, methods and tools of work, conclusions and suggestions, references. Thesis consist of: 58 p. text without appendixes, 46 pictures, 1 tables, 36 bibliographical entries. Appendixes and publication included.

Nous proposons une approche spectrale permettant d'aborder des problèmes d'interpolation à noyaux dont la résolution numérique n'est pas directement envisageable. Un tel cas de figure se produit en particulier lorsque le nombre de données est infini. Nous considérons dans un premier temps le cadre de l'interpolation optimale dans les sous-espaces hilbertiens. Pour un problème donné, un opérateur intégral est défini à partir du noyau sous-jacent et d'une paramétrisation de l'ensemble des données basée sur un espace mesuré. La décomposition spectrale de l'opérateur est utilisée afin d'obtenir une formule de représentation pour l'interpolateur optimal et son approximation est alors rendu possible par troncature du spectre. Le choix de la mesure induit une fonction d'importance sur l'ensemble des données qui se traduit, en cas d'approximation, par une plus ou moins grande précision dans le rendu des données. Nous montrons à titre d'exemple comment cette approche peut être utilisée afin de rendre compte de contraintes de type "conditions aux limites" dans les modèles d'interpolation à noyaux. Le problème du conditionnement des processus gaussiens est également étudié dans ce contexte. Nous abordons enfin dans la dernière partie de notre manuscrit la notion de noyaux conditionnellement positifs. Nous proposons la définition générale de noyaux symétriques conditionnellement positifs relatifs à une espace de référence donné et développons la théorie des sous-espaces semi-hilbertiens leur étant associés. Nous étudions finalement la théorie de l'interpolation optimale dans cette classe d'espaces. / We propose a spectral approach for the resolution of kernel-based interpolation problems of which numerical solution can not be directly computed. Such a situation occurs in particular when the number of data is infinite. We first consider optimal interpolation in Hilbert subspaces. For a given problem, an integral operator is defined from the underlying kernel and a parameterization of the data set based on a measurable space. The spectral decomposition of the operator is used in order to obtain a representation formula for the optimal interpolator and spectral truncation allows its approximation. The choice of the measure on the parameters space introduces a hierarchy onto the data set which allows a tunable precision of the approximation. As an example, we show how this methodology can be used in order to enforce boundary conditions in kernel-based interpolation models. The Gaussian processes conditioning problem is also studied in this context. The last part of this thesis is devoted to the notion of conditionally positive kernels. We propose a general definition of symmetric conditionally positive kernels relative to a given space and exposed the associated theory of semi-Hilbert subspaces. We finally study the optimal interpolation problem in such spaces.