QC 351 A7 no. 49 / The use of the two -dimensional Fourier transform to compute the form of the optical image has led to new insights into the form of images. The techniques involved are discussed in some detail, and the application to tolerancing of surface errors is used as an ex- ample of the results.

This paper presents a theory of Fourier transforms of complex-valued functions on a finite abelian group and investigates two applications of this theory. Chapter I is an introduction with remarks on notation. Basic theory, including Pontrvagin duality and the Poisson Summation formula, is the subject of Chapter II. In Chapter III the Fourier transform is viewed as an intertwining operator for certain unitary group representations. The solution of the eigenvalue problem of the Fourier transform of functions on the group Z/n of integers module n leads to a proof of the quadratic reciprocity law in Chapter IV. Chapter V addresses the, use of the Fourier transform in computing.

In this thesis a number of problems in harmonic analysis of a geometric flavour are discussed and, in particular, the Lebesgue space mapping properties of certain averaging and Fourier restriction operators are studied. The first three chapters focus on the perspective afforded by affine-geometrical considerations whilst the remaining chapter considers some discrete variants of these problems. In Chapter 1 there is an overview of the basic affine theory of the aforementioned operators and, in particular, the affine arc-length and surface measures are introduced. Chapter 2 presents work of the author, submitted for publication, concerning an operator which takes averages of functions on Euclidean space over both translates and dilates of a fixed polynomial curve. Moreover, the averages are taken with respect to the affine arc-length; this allows one to prove Lebesgue space estimates with a substantial degree of uniformity in the constants. The sharp range of uniform estimates is obtained in all dimensions except for an endpoint. Chapter 3 presents some work of the author, published in Mathematika, concerning a family of Fourier restriction operators closely related to the averaging operators discussed in Chapter 2. Specifically, a Fourier restriction estimate is obtained for a broad class of conic surfaces by introducing a certain measure which exhibits a special kind of affine invariance. Again, the sharp range of estimates is obtained, but the results are limited to the case of 2-dimensional cones. Finally, Chapter 4 discusses some recent joint work of the author and Jim Wright considering the restriction problem over rings of integers modulo a prime power. The sharp range of estimates is obtained for Fourier restriction to the moment curve in finitely-generated free modules over such rings. This is achieved by lifting the problem to the p-adics and applying a classical argument of Drury in this setting. This work aims to demonstrate that rings of integers offer a simplified model for the Euclidean restriction problem.

Regression Wavelet Analysis (RWA) is a novel wavelet-based scheme for coding hyperspectral images that employs multiple regression analysis to exploit the relationships among spectral wavelet transformed components. The scheme is based on a pyramidal prediction, using different regression models, to increase the statistical independence in the wavelet domain For lossless coding, RWA has proven to be superior to other spectral transform like PCA and to the best and most recent coding standard in remote sensing, CCSDS-123.0. In this paper we show that RWA also allows progressive lossy-to-lossless (PLL) coding and that it attains a rate-distortion performance superior to those obtained with state-of-the-art schemes. To take into account the predictive significance of the spectral components, we propose a Prediction Weighting scheme for JPEG2000 that captures the contribution of each transformed component to the prediction process.

Les codes correcteurs d’effacements sont aujourd’hui une solution bien connueutilisée pour fiabiliser les protocoles de communication ou le stockage distribué desdonnées. La plupart de ces codes sont basés sur l’arithmétique des corps finis, définissantl’addition et la multiplication sur un ensemble fini d’éléments, nécessitantsouvent des opérations complexes à réaliser. En raison de besoins en performancetoujours plus importants, ces codes ont fait l’objet de nombreuses recherches dans lebut d’obtenir de meilleures vitesses d’exécution, tout en ayant la meilleure capacitéde correction possible. Nous proposons une méthode permettant de transformer les éléments de certains corps finis en éléments d’un anneau afin d’y effectuer toutes les opérations dans lebut de simplifier à la fois le processus de codage et de décodage des codes correcteursd’effacements, sans aucun compromis sur les capacités de correction. Nous présentonségalement une technique de réordonnancement des opérations, permettant deréduire davantage le nombre d’opérations nécessaires au codage grâce à certainespropriétés propres aux anneaux utilisés. Enfin, nous analysons les performances decette méthode sur plusieurs architectures matérielles, et détaillons une implémentationsimple, basée uniquement sur des instructions xor et s’adaptant beaucoupplus efficacement que les autres implémentations à un environnement d’exécutionmassivement parallèle. / Erasure codes are widely used to cope with failures for nearly all of today’snetworks communications and storage systems. Most of these codes are based onfinite field arithmetic, defining the addition and the multiplication over a set offinite elements. These operations can be very complex to perform. As a matter offact, codes performance improvements are still an up to date topic considering thecurrent data growth explosion. We propose a method to transform the elements of some finite fields into ring elements and perform the operations in this ring to simplify both coding and decoding of erasure codes, without any threshold on the correction capacities.We also present a scheduling technique allowing to reduce the number of operations thanks to some particular properties of the ring structure. Finally, we analyse the performance ofsuch a method considering several hardware architectures and detail a simple implementation, using only xor operations, fully scalable over a multicore environment.

Student Number : 9708525G - PhD thesis - School of Mathematics - Faculty of Science / Sequences of numbers have important applications in the field of Computer Science. As a result they have become increasingly regarded in Mathematics, since analysis can be instrumental in investigating algorithms. Three concepts are discussed in this thesis, all of which are concerned with ‘words’ or ‘sequences’ of natural numbers where repeated letters are allowed: • The number of distinct values in a sequence with geometric distri- bution In Part I, a sample which is geometrically distributed is considered, with the objective of counting how many different letters occur at least once in the sample. It is concluded that the number of distinct letters grows like log n as n → ∞. This is then generalised to the question of how many letters occur at least b times in a word. • The position of the maximum (and/or minimum) in a sequence with geometric distribution Part II involves many variations on the central theme which addresses the question: “What is the probability that the maximum in a geometrically distributed sample occurs in the first d letters of a word of length n?” (assuming d ≤ n). Initially, d is considered fixed, but in later chapters d is allowed to grow with n. It is found that for 1 ≤ d = o(n), the results are the same as when d is fixed. • The average depth of a key in a binary search tree formed from a sequence with repeated entries Lastly, in Part III, random sequences are examined where repeated letters are allowed. First, the average left-going depth of the first one is found, and later the right-going path to the first r if the alphabet is {1, . . . , r} is examined. The final chapter uses a merge (or ‘shuffle’) operator to obtain the average depth of an arbitrary node, which can be expressed in terms of the left-going and right-going depths.

[pt] A evolução natural dos modelos de interface com o usuário ocorrida nas últimas décadas popularizou o padrão baseado em metáforas puramente visuais. Este processo impediu o acesso de deficientes visuais a computadores e a novas tecnologias. Algumas propostas foram feitas para reverter esta realidade. Entretanto, a maioria delas estava baseada em adaptações dos modelos já existentes, e não em modelos específicos para deficientes visuais. O desenvolvimento de aplicações para tais usuários é uma tarefa que requer a utilização de novas tecnologias, outras ferramentas e outras mídias de comunicação. Esta tese de doutorado propõe a utilização de dispositivos reativos no projeto e implementação de interfaces gráficas capazes de assistir usuários cegos na concretização de tarefas simples de interação em duas dimensões. Tais dispositivos permitem explorar a percepção tátil, geralmente muito apurada em cegos, aumentando a eficiência da comunicação entre o usuário e a interface. Além disto, este trabalho investiga a utilização de transformadas de distância como um poderoso mecanismo de suporte a diversas tarefas de interação bidimensionais. / [en] The natural evolution of user interface models that occurred in the last few decades ended up popularizing a standard model based almost exclusively on visual metaphors. This process has left visually impaired users unable to use computers and to access new technologies. Some actions have been made to revert this scenario. Most of them were based on adapting the existing models instead of creating specific solutions for the visually impaired community. The development of applications for such users requires the use of new technologies, tools and communication media. This thesis proposes the use of force feedback devices in the project and implementation of assistive user interfaces, helping blind users in simple 2D interaction tasks. By exploring the sense of touch, such devices can be used to improve the efficiency of the communication between the user and the interface. Also, this work investigates the use of distance transforms as a powerful mechanism to support many 2D interaction tasks.

Ao modelar-se depósitos minerais é comum enfrentarmos o problema de estimar múltiplos atributos possivelmente correlacionados, onde algumas variáveis são amostradas menos densamente do que outras. A falta de dados impõe um problema que requer atenção antes de qualquer modelagem subsequente. Precisamos, ao final, de modelos que sejam estatisticamente representativos. A maioria dos conjuntos de dados de problemas práticos são amostrados de maneira heterotópica e, para obter resultados coerentes, é preciso entender os motivos pelos quais alguns dados faltam e quais são os mecanismos que influenciaram a ausência de informações. A teoria de dados faltantes relaciona as amostras ausentes com aquelas medidas através de três mecanismos distintos: Faltante Completamente Aleatório (Missing Completely At Random - MCAR), Faltante Aleatório (Missing At Random - MAR) e Faltante Não Aleatório (Missing Not At Random - MNAR). O último mecanismo é extremamente complexo e a literatura recomenda ser tratado inicialmente como um mecanismo MAR. E após uma transformação fixa deve ser aplicada aos valores complementados para que estes se transformem em valores MNAR Embora existam métodos estatísticos clássicos para lidar com dados faltantes, tais abordagens ignoram a correlação espacial, uma característica que ocorre naturalmente em dados geológicos. A metodologia adequada para tratar com a falta de dados geológicos é a atualização bayesiana, em que se inserem valores sob mecanismo MAR considerando a correlação espacial. No presente estudo, a atualização bayesiana foi combinada com transformações fixas para tratar o mecanismo de falta de dados MNAR em dados geológicos. A transformação fixa aqui empregada é baseada no erro de inserção gerado em um cenário MAR no conjunto de dados. Assim, com o conjunto completo resultante foi utilizado em uma simulação sequencial gaussiana dos teores de uma base de dados multivariada, apresentando resultados satisfatórios, superiores aos obtidos por meio da cossimulação sequencial gaussiana, não inserindo qualquer viés no modelo final. / When modeling mineral deposits, it is common to face the problem of estimating multiple attributes possibly correlated where some variables are more densely sampled then others. Missing data imposes a problem that requires attention prior to any subsequent modeling. The later requires estimation models statistically representative. Most practical data sets are often heterotopically sampled, and to obtain coherent results one must understand the reasons why there are missing data and what are the mechanisms that cause the absence of information. The theory of missing data relates the missing samples to those measured through three different mechanisms: Missing Completely At Random (MCAR), Missing At Random (MAR), and Missing Not At Random (MNAR). The last mechanism is quite complex to deal with, and the literature recommends being treated as a MAR mechanism and after a fixed transform should be applied to the imputed values so that these turn into MNAR imputed values. Even though there are classical statistical methods to deal with missing data, such approaches ignore spatial correlation, a feature that occurs naturally in geological data. The adequate methodology to deal with missing geologic data is Bayesian Updating, which approaches the MAR mechanism and accounts for spatial correlation. In the present study, bayesian updating was used combined with fixed transforms to treat MNAR missing data mechanism in geologic data. The fixed transform herein used is based on the error of MAR imputation on the data set. The resulting complete set was then used on a sequential gaussian simulation of the grades on a multivariate data set, presenting satisfactory results, superior to those obtained through sequential gaussian cossimulation, not inserting any biases on the final model.

In this thesis we present new ideas and mathematical insights in the field of Compton Scattering Tomography (CST), an X-ray and gamma ray imaging technique which uses Compton scattered data to reconstruct an electron density of the target. This is an area not considered extensively in the literature, with only two dimensional gamma ray (monochromatic source) CST problems being analysed thus far. The analytic treatment of the polychromatic source case is left untouched and while there are three dimensional acquisition geometries in CST which consider the reconstruction of gamma ray source intensities, an explicit three dimensional electron density reconstruction from Compton scatter data is yet to be obtained. Noting this gap in the literature, we aim to make new and significant advancements in CST, in particular in answering the questions of the three dimensional density reconstruction and polychromatic source problem. Specifically we provide novel and conclusive results on the stability and uniqueness properties of two and three dimensional inverse problems in CST through an analysis of a disc transform and a generalized spindle torus transform. In the final chapter of the thesis we give a novel analysis of the stability of a spindle torus transform from a microlocal perspective. The practical application of our inversion methods to fields in X-ray and gamma ray imaging are also assessed through simulation work.

Based on the construction of Gabor feature space, which is important in applying pixel similarity computations, we formalize the space using rotation-invariant Gabor filter banks and apply optimizations in texture feature space. In image colorizations, the pixels that have similar Gabor features appear similar colors, our approach can colorize natural images globally, without the restriction of the disjoint regions with similar texture-like appearances. Our approach supports the two-pass colorization processes: coloring optimization in Gabor space and color detailing for progressive effects. We further work on the video colorization using the optimized Gabor flow computing, including coloring keyframes, color propagation by Gabor filtering, and optimized parallel computing over the video. Our video colorization is designed in a spatiotemporal manner to keep temporal coherence, and provides simple closed-form solutions in energy optimization that yield fast colonizations. Moreover, we develop parallel surface texturing of geometric models on GPU, generating spatially-varying visual appearances. We incorporate the Gabor feature space for the searching of 2D exemplars, to determine the k-coherence candidate pixels. The multi-pass correction in synthesis is applied to the local neighborhood for parallel processes. The iso/aniso-scale texture synthesis leverages the strengths of GPU computing, so to synthesize the iso/aniso-scale texturing appearance in parallel over arbitrary surfaces. Our experimental results showed that our approach produces simply controllable texturing effects of surface synthesis, generating texture-similar and spatially-varying visual appearances with GPU accelerated performance. / Texture feature similarity has long been crucial and important topic in VR/graphics applications, such as image and video colorizations, surface texture synthesis and geometry image applications. Generally, the image feature is highly subjective, depending on not only the image pixels but also interactive users. Existing colorization and surface texture synthesis pay little attention to the generation of conforming color/textures that accurately reflect exemplar structures or user's intension. Realistic surface synthesis remains a challenging task in VR/graphics researches. In this dissertation, we focus on the encoding of the Gabor filter banks into texture feature similarity computations and GPU-parallel surface rendering faithfully, including image/vodeo colorizations, parallel texturing of geometric surfaces, and multiresolution rendering on sole-cube maps (SCMs). / We further explore the GPU-based multiresolution rendering on solecube maps (SCMs). Our SCMs on GPU generate adaptive mesh surfaces dynamically, and are fully developed in parallelization for large-scale and complex VR environments. We also encapsulate the differential coordinates in SCMs, reflecting the local geometric characteristics for geometric modeling and interactive animation applications. For the future work, we will work on improving the image/ video feature analysis framework in VR/graphics applications. The further work lying in the surface texture synthesis includes the interactive control of texture orientations by surface vector fields using sketch editing, so to widen the gamut of interactive tools available for texturing artists and end users. / Sheng, Bin. / Adviser: Hanqin Sun. / Source: Dissertation Abstracts International, Volume: 73-04, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 128-142). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.