This thesis consists of an introduction and a summary, followed by two papers, both of them on the topic of function spaces on fractals. Paper I: Andreas Brodin, Pointwise Convergence of Haar type Wavelets on Self-Similar Sets, Manuscript. Paper II: Andreas Brodin, Regularization of Wavelet Expansion characterizes Besov Spaces on Fractals, Manuscript. Properties of wavelets, originally constructed by A. Jonsson, are studied in both papers. The wavelets are piecewise polynomial functions on self-similar fractal sets. In Paper I, pointwise convergence of partial sums of the wavelet expansion is investigated. On a specific fractal set, the Sierpinski gasket, pointwise convergence of the partial sums is shown by calculating the kernel explicitly, when the wavelets are piecewise constant functions. For more general self-similar fractals, pointwise convergence of the partial sums and their derivatives, in case the expanded function has higher regularity, is shown using a different technique based on Markov's inequality. A. Jonsson has shown that on a class of totally disconnected self-similar sets it is possible to characterize Besov spaces by means of the magnitude of the coefficients in the wavelet expansion of a function. M. Bodin has extended his results to a class of graph directed self-similar sets introduced by Mauldin and Williams. Unfortunately, these results only holds for fractals such that the sets in the first generation of the fractal are disjoint. In Paper II we are able to characterize Besov spaces on a class of fractals not necessarily sharing this condition by making the wavelet expansion smooth. We create continuous regularizations of the partial sums of the wavelet expansion and show that properties of these regularizations can be used to characterize Besov spaces.

El análisis de texturas es un área de estudio interesante con suficiente peso específico dentro de los diferentes campos que componen la visión por ordenador. En este trabajo hemos desarrollado métodos específicos para resolver aspectos importantes de dicha área. El primer acercamiento al tema viene de la mano de un problema de segmentación de un tipo de texturas muy concreto como son las imágenes microscópicas de láminas de mármol. Este primer tipo de imágenes se componen de un conjunto de granos cuyas formas y tamaños sirven a los especialistas para identificar, catalogar y determinar el origen de dichas muestras. Identificar y analizar los granos que componen tales imágenes de manera individual necesita de una etapa de segmentación. En esencia, esto implica la localización de las fronteras representadas en este caso por valles que separan zonas planas asociadas a los granos. De los diferentes métodos estudiados para la detección de dichos valles y para el caso concreto de imágenes petrográficas son los basados en técnicas de morfología matemática los que han dado mejores resultados. Además, la segmentación requiere un filtrado previo para el que se han estudiado nuevamente un conjunto de posibilidades entre las que cabe destacar los algoritmos multirresolución basados en wavelets.El segundo problema que hemos atacado en este trabajo es la clasificación de imágenes de textura. En él también hemos utilizado técnicas multirresolución como base para su resolución. A diferencia de otros enfoques de carácter global que encontramos extensamente en la literatura sobre texturas, nos hemos centrado en problemas donde las diferencias visuales entre las clases de dichas texturas son muy pequeñas. Y puesto que no hemos establecido restricciones fuertes en este análisis, las estrategias desarrolladas son aplicables a un extenso espectro de texturas, como pueden ser las baldosas cerámicas, las imágenes microscópicas de pigmentos de efecto, etc.El enfoque que hemos seguido para la clasificación de texturas implica la consecución de una serie de pasos. Hemos centrado nuestra atención en aquellos pasos asociados con las primeras etapas del proceso requeridas para identificar las características importantes que definen la textura, mientras que la clasificación final de las muestras ha sido realizada mediante métodos de clasificación generales. Para abordar estos primeros pasos dentro del análisis hemos desarrollado una estrategia mediante la cual las características de una imagen se ajustan a un modelo que previamente hemos definido, uno de entre varios modelos que están ordenados por complejidad. Estos modelos están asociados a algoritmos específicos y sus parámetros así como a los cálculos que de ellos se derivan. Eligiendo el modelo adecuado, por tanto, evitamos realizar cálculos que no nos aportan información útil para la clasificación.En un tercer enfoque hemos querido llegar a una descripción de textura que nos permita de forma sencilla su clasificación y su síntesis. Para conseguir este objetivo hemos adoptado por un modelo probabilístico. Dicha descripción de la textura nos permitirá la clasificación a través de la comparación directa de modelos, y también podremos, a partir del modelo probabilístico, sintetizar nuevas imágenes.Para finalizar, comentar que en las dos líneas de trabajo que hemos expuesto, la segmentación y la clasificación de texturas, hemos llegado a soluciones prácticas que han sido evaluadas sobre problemas reales con éxito y además las metodologías propuestas permiten una fácil extensión o adaptación a nuevos casos. Como líneas futuras asociadas a estos temas trataremos por un lado de adaptar la segmentación a imágenes que poco o nada tienen que ver con las texturas, en las que se perseguirá la detección de sujetos y objetos dentro de escenas, como apuntamos más adelante en esta misma memoria. Por otro lado, y relacionado con la clasificación, abordaremos un problema todavía sin solución como es el de la ingeniería inversa en pigmentos de efecto, en otras palabras la determinación de los constituyentes en pinturas metalizadas, y en el que utilizaremos los estudios aquí presentados como base para llegar a una posible solución. / An interesting problem in computer vision is the analysis of texture images. In this work, we have developed specific methods to solve important aspects of this problem. The first approach involves segmentation of a specific type of textures, i.e. those of microscopy images of thin marble sections. These images comprise a pattern of grains whose sizes and shapes help specialists to identify the origin and quality of marble samples. To identify and analyze individual grains in these images represents a problem of image segmentation. In essence, this involves identifying boundary lines represented by valleys which separate flat areas corresponding to grains. Of several methods tested, we found those based on mathematical morphology particularly successful for segmentation of petrographical images. This involves a pre-filtering step for which again several approaches have been explored, including multiresolution algorithms based on wavelets. In the second approach we have also used multiresolution analyses to address the problem of classifying texture images. In contrast to more global approaches found in the literature, we have explored situations where visual differences between textures are rather subtle. Since we have tried to impose relatively few restrictions on these analyses, we have developed strategies that are applicable to a wide range of related texture images, such as images of ceramic tiles, microscopic images of effect pigments, etc.The approach we have used for the classification of texture images involves several technical steps. We have focused our attention in the initial low-level analyses required to identify the general features of the image, whereas the final classification of samples has been performed using generic classification methods. To address the early steps of image analysis, we have developed a strategy whereby the general features of the image fit one of several pre-defined models with increasing levels of complexity. These models are associated to specific algorithms, parameters and calculations for the analysis of the image, thus avoiding calculations that do not provide useful information. Finally, in a third approach we want to arrive to a description of textures in such a way that it should be able to classify and synthesize textures. To reach this goal we adopt a probabilistic model of the texture. This description of the texture allows us to compare textures through comparison of probabilistic models, and also use those probabilities to generate new similar images.In conclusion, we have developed strategies of segmentation and classification of textures that provide solutions to practical problems and are potentially applicable with minor modifications to a wide range of situations. Future research will explore (i) the possibility of adapting segmentation to the analysis of images that do not necessarily involve textures, e.g. localization of subjects in scenes, and (ii) classification of effect pigment images to help identify their components.

This thesis presents two novel coding schemes and applications to both two- and three-dimensional image compression. Image compression can be viewed as methods of functional approximation under a constraint on the amount of information allowable in specifying the approximation. Two methods of approximating functions are discussed: Iterated function systems (IFS) and wavelet-based approximations. IFS methods approximate a function by the fixed point of an iterated operator, using consequences of the Banach contraction mapping principle. Natural images under a wavelet basis have characteristic coefficient magnitude decays which may be used to aid approximation. The relationship between quantization, modelling, and encoding in a compression scheme is examined. Context based adaptive arithmetic coding is described. This encoding method is used in the coding schemes developed. A coder with explicit separation of the modelling and encoding roles is presented: an embedded wavelet bitplane coder based on hierarchical context in the wavelet coefficient trees. Fractal (spatial IFSM) and fractal-wavelet (coefficient tree), or IFSW, coders are discussed. A second coder is proposed, merging the IFSW approaches with the embedded bitplane coder. Performance of the coders, and applications to two- and three-dimensional images are discussed. Applications include two-dimensional still images in greyscale and colour, and three-dimensional streams (video).

This thesis presents two novel coding schemes and applications to both two- and three-dimensional image compression. Image compression can be viewed as methods of functional approximation under a constraint on the amount of information allowable in specifying the approximation. Two methods of approximating functions are discussed: Iterated function systems (IFS) and wavelet-based approximations. IFS methods approximate a function by the fixed point of an iterated operator, using consequences of the Banach contraction mapping principle. Natural images under a wavelet basis have characteristic coefficient magnitude decays which may be used to aid approximation. The relationship between quantization, modelling, and encoding in a compression scheme is examined. Context based adaptive arithmetic coding is described. This encoding method is used in the coding schemes developed. A coder with explicit separation of the modelling and encoding roles is presented: an embedded wavelet bitplane coder based on hierarchical context in the wavelet coefficient trees. Fractal (spatial IFSM) and fractal-wavelet (coefficient tree), or IFSW, coders are discussed. A second coder is proposed, merging the IFSW approaches with the embedded bitplane coder. Performance of the coders, and applications to two- and three-dimensional images are discussed. Applications include two-dimensional still images in greyscale and colour, and three-dimensional streams (video).

With an increasing emphasis on personal privacy, security, and convenience, the security of identification system is an important issue nowadays. In this thesis, two intelligent identification systems, laser speckle image identification system and laser-based finger biometric system, are proposed to perform superior solutions for identification applications. In laser speckle image identification system, we investigated the characteristics of laser speckle as well as proposed an appropriate algorithm to establish this system. The proposed algorithm is a coarse-to-fine process which identifies laser speckle images systematically. In laser-based finger biometric system, a new biometric approach is described to proceed personal identification using a scanner with a low power laser scans across the surface of the finger and continuously recording the reflected intensity at a fixed position. Experimental results show that the recognition rates of the proposed system are both 100%.

In this dissertation we study the problem of allocating computational resources and managing applications in a data center to serve incoming requests in such a way that the energy usage, reliability and quality of service considerations are balanced. The problem is motivated by the growing energy consumption by data centers in the world and their overall inefficiency. This work is focused on designing flexible and robust strategies to manage the resources in such a way that the system is able to meet the service agreements even when the load conditions change. As a first step, we study the control of a Markovian queueing system with controllable number of servers and service rates (M=Mt=kt ) to minimize effort and holding costs. We present structural properties of the optimal policy and suggest an algorithm to find good performance policies even for large cases. Then we present a reactive/proactive approach, and a tailor-made wavelet-based forecasting procedure to determine the resource allocation in a single application setting; the method is tested by simulation with real web traces. The main feature of this method is its robustness and flexibility to meet QoS goals even when the traffic behavior changes. The system was tested by simulating a system with a time service factor QoS agreement. Finally, we consider the multi-application setting and develop a novel load consolidation strategy (of combining applications that are traditionally hosted on different servers) to reduce the server-load variability and the number of booting cycles in order to obtain a better capacity allocation.

The source and nature, as well as the history of ray-effects, is described. A benchmark code, using piecewise constant functions in angle and diamond differencing in space, is derived in order to analyze four sample problems. The results of this analysis are presented showing the ray effects and how increasing the resolution (number of angles) eliminates them. The theory of wavelets is introduced and the use of wavelets in multiresolution analysis is discussed. This multiresolution analysis is applied to the transport equation, and equations that can be solved to calculate the coefficients in the wavelet expansion for the angular flux are derived. The use of thresholding to eliminate wavelet coefficients that are not required to adequately solve a problem is then discussed. An iterative sweeping algorithm, called the SN-WN method, is derived to solve the wavelet-based equations. The convergence of the SN-WN method is discussed. An algorithm for solving the equations is derived, by solving a matrix within each cell directly for the expansion coefficients. This algorithm is called the CWWN method. The results of applying the CW-WN method to the benchmark problems are presented. These results show that more research is needed to improve the convergence of the SN-WN method, and that the CW-WN method is computationally too costly to be seriously considered.

Thesis (Ph. D.)--University of Missouri-Columbia, 2000. / Typescript. Vita. Includes bibliographical references (leaves 109-112). Also available on the Internet.

Thesis (Ph. D.)--University of Hong Kong, 2001. / Includes bibliographical references.

Thesis (Ph. D.)--University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 135-143). Available also in a digital version from Dissertation Abstracts.