The scope of the present dissertation is a deep lossy compression of still and moving grayscale pictures while maintaining their fidelity, with a specific goal of creating a working prototype of a software system for use in low bandwidth transmission of still satellite imagery and weather briefings with the best preservation of features considered important by the end user.

In this dissertation, we study fractal sets and their properties, especially the open set condition, Hausdorff dimensions and Hausdorff measures for certain fractal constructions.

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Thesis advisor: John E. Ebel / Power-law scaling relationships concerning the earthquake frequency-magnitude distribution and the fractal geometry of spatial seismicity patterns may provide applications to earthquake forecasting and earthquake hazard studies. Past studies on the fractal characteristics of seismic phenomena have observed spatial and temporal differences in earthquake clustering and b value in relation to fractal dimension value. In this thesis, an investigation of the spatiotemporal seismicity patterns in southern California for the years 1982 to 2020 was conducted. The range and temporospatial distribution of b and D2 values for earthquake hypocenters contained in the Southern California Earthquake Data Center catalogue were calculated and shown in time series and spatial distribution maps. b values were calculated using both the Least SquaresMethod and the Maximum Likelihood Method while D2 values were calculated for length scales between 1 km to 10 km. A set of b and D2 values were calculated after declustering for foreshocks and aftershocks using Gardner and Knopoff’s declustering algorithm. b values decreased while D2 values increased on the dates of M > 6.0 earthquakes, whereas b values increased and D2 values decreased on the dates after M > 6.0 earthquakes. Declustering results suggest an influence of earthquake aftershocks to increase D2 values while decreasing b values. The role for b values and D2 values to delineate both the temporal and spatial extent of aftershock sequences for large earthquakes may prove to have an application in earthquake hazard studies. / Thesis (MS) — Boston College, 2022. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Earth and Environmental Sciences.

Yes / Sedimentation processes are fundamental to solids/liquid separation in water and wastewater treatment, and therefore a robust understanding of the settlement characteristics of mass fractal aggregates (flocs) formed in the flocculation stage is fundamental to optimized settlement tank design and operation. However, the use of settling as a technique to determine aggregates’ traits is limited by current understanding of permeability. In this paper, we combine experimental and numerical approaches to assess settling velocities of fractal aggregates. Using a non-intrusive in situ digital image-based method, three- and two-dimensional fractal dimensions were calculated for kaolin-based flocs. By considering shape and fractal dimension, the porosity, density and settling velocities of the flocs were calculated individually, and settling velocities compared with those of spheres of the same density using Stokes’ law. Shape analysis shows that the settling velocities for fractal aggregates may be greater or less than those for perfect spheres. For example, fractal aggregates with floc fractal dimension, Df ¼ 2.61, floc size, df > 320 μm and dp ¼ 7.5 μm settle with lower velocities than those predicted by Stokes’ law; whilst, for Df ¼ 2.33, all aggregates of df > 70 μm and dp ¼ 7.5 μm settled below the velocity calculated by Stokes’ law for spheres. Conversely, fractal settling velocities were higher than spheres for all the range of sizes, when Df of 2.83 was simulated. The ratio of fractal aggregate to sphere settling velocity (the former being obtained from fractal porosity and density considerations), varied from 0.16 to 4.11 for aggregates in the range of 10 and 1,000 μm, primary particle size of 7.5 μm and a three-dimensional fractal dimension between 2.33 and 2.83. However, the ratio decreases to the range of 0.04–2.92 when primary particle size changes to 1.0 μm for the same fractal dimensions. Using the floc analysis technique developed here, the results demonstrate the difference in settlement behaviour between the approach developed here and the traditional Stokes’ law approach using solid spheres. The technique and results demonstrate the improvements in understanding, and hence value to be derived, from an analysis based on fractal, rather than Euclidean, geometry when considering flocculation and subsequent clarification performance / Rodrigo B. Moruzzi is grateful to São Paulo Research Foundation (Fundação de Amparo à Pesquisa do Estado de São Paulo – FAPESP) Grant 2017/19195-7 for financial support and to CNPq for the fellowship Grant 301210/2018-7.

This paper identifies a range of elements and principles useful for the development of an urban theoretical architecture. Acceptance of nature as a design element and in particular the use of nature to bound nodes of high density development are explored. The use of fractal geometry to distribute the urban footprint upon the landscape is introduced along with a tacit development of methodology making the application of fractal geometry useful. Building height restrictions are suggested as usefull to create urban walls and maintain views for tall buildings. It is proposed that the basic unit of urban design and development is a high intensity urban cell. Elements crucial to the life of urban cells are identified. The importance of architectural character in developing the identity of urban space is reinforced and explored. / Master of Architecture

The usefulness of watershed hydrological process models is considerably increased when they can be extrapolated across spatial and temporal scales. This scale transfer problem, meaning the description and prediction of characteristics and processes at a scale different from the one at which observations and measurements are made, and has become the subject of much current research in hydrology and other areas. Quantitative description of fractal scaling behavior of runoff and stream network morphometry in agricultural watersheds has not been previously reported. In the present study, fractal and multifractal scaling of daily runoff rate in four experimental agricultural watersheds and their associated sub-watersheds (32 in total) were investigated. The time series of daily runoff rate were obtained from the database (comprising about 16,600 station years of rainfall and runoff data for small agricultural watersheds across the U.S.) developed by the Hydrological and Remote Sensing Laboratory, Agricultural Research Service, US Department of Agriculture (HRSL/ARS/USDA). Fractal scaling patterns of the Digital Elevation Model (DEM)-extracted stream network morphometry for these four watersheds were also examined. The morphometry of stream networks of four watersheds were obtained by Geographic Information System (GIS) manipulation of digital elevation data downloaded from the most recent (July 2004) U.S. Geological Survey (USGS) National Elevation Dataset (NED). Several threshold values of contribution area for stream initiation were used to extract stream networks for each of the four watersheds. The principal measures of fractal scaling determined for the runoff series were the Hurst exponent obtained by rescaled range (R/S) analysis, the fractal dimension estimated by the shifted box-counting method, and the multifractal scaling function parameters (a and C1) of the Universal Multifractal Model (UMM). Corresponding measures for the DEM-extracted stream networks at each threshold value were the fractal dimension estimated using the box-counting technique and the Horton ratios of the network. Daily runoff rate exhibited strong long-term dependence and scale invariance over certain time scales. The same fractal dimensions and Hurst exponents were obtained for the sub-watersheds within each watershed. Runoff exhibited multifractal behavior that was well described by UMM. The multifractal parameters a (quantifies how far the process is from monofractality) and C1 (characterizes the sparseness or inhomogeneity of the mean of the process) were reasonably close to each other for sub-watersheds within a watershed and were generally similar among four watersheds. For the DEM-extracted networks, the morphometric attributes and Horton ratios as well as their fractal dimensions were dependent on the threshold values of contribution area used in the extraction process. The fractal dimensions were almost identical for DEM-extracted stream networks of the four watersheds. The DEM-extracted stream network displayed a single scaling pattern, rather than multifractal behavior. Explanation of the physical significance of fractal characteristics of the stream network in relation to runoff time series would require more data than were available in this study. / Ph. D.

A análise de texturas tem atraído um crescente interesse em visão computacional devido a sua importância na caracterização de imagens. Basicamente, as pesquisas em texturas podem ser divididas em duas categorias: texturas estáticas e texturas dinâmicas. As texturas estáticas são caracterizadas por variações de intensidades que formam um determinado padrão repetido espacialmente na imagem. Por outro lado, as texturas dinâmicas são padrões de texturas presentes em uma sequência de imagens. Embora muitas pesquisas tenham sido realizadas, essa área ainda se encontra aberta a estudos, principalmente em texturas dinâmicas por se tratar de um assunto recente e pouco explorado. Este trabalho tem como objetivo o desenvolvimento de pesquisas que abrangem ambos os tipos de texturas nos âmbitos teórico e prático. Em texturas estáticas, foram propostos dois métodos: (i) baseado em caminhadas determinísticas parcialmente auto-repulsivas e dimensão fractal - (ii) baseado em atividade em redes direcionadas. Em texturas dinâmicas, as caminhadas determinísticas parcialmente auto-repulsivas foram estendidas para sequências de imagens e obtiveram resultados interessantes em reconhecimento e segmentação. Os métodos propostos foram aplicados em problemas da biologia e nanotecnologia, apresentando resultados interessantes para o desenvolvimento de ambas as áreas. / Texture analysis has attracted an increasing interest in computer vision due to its importance in describing images. Basically, research on textures can be divided into two categories: static and dynamic textures. Static textures are characterized by intensity variations which form a pattern repeated in the image spatially. On the other hand, dynamic textures are patterns of textures present in a sequence of images. Although many studies have been carried out, this area is still open to study, especially in dynamic textures since it is a recent and little-explored subject. This study aims to develop research covering both types of textures in theoretical and practical fields. In static textures, two methods were proposed: (i) based on deterministic partially self-avoiding walks and fractal dimension - (ii) based on activity in directed networks. In dynamic textures, deterministic partially self-avoiding walks were extended to sequences of images and obtained interesting results in recognition and segmentation. The proposed methods were applied to problems of biology and nanotechnology, presenting interesting results in the development of both areas.

O desenvolvimento da tecnologia e a propagação de sistemas computacionais nos mais variados domínios do conhecimento têm contribuído para a geração e o armazenamento de uma quantidade constantemente crescente de dados, em uma velocidade maior da que somos capazes de processar. De um modo geral, a principal razão para o armazenamento dessa enorme quantidade de dados é a utilização deles em benefício da humanidade. Diversas áreas têm se dedicado à pesquisa e a proposta de métodos e processos para tratar esses dados. Um desses processos é a Descoberta de Conhecimento em Bases de Dados, a qual tem como objetivo extrair conhecimento a partir das informações contidas nesses dados. Para alcançar esse objetivo, usualmente são construídos modelos (hipóteses), os quais podem ser gerados com o apoio de diferentes áreas tal como a de Aprendizado de Máquina. A Seleção de Atributos desempenha uma tarefa essencial dentro desse processo, pois representa um problema de fundamental importância em aprendizado de máquina, sendo freqüentemente realizada como uma etapa de pré-processamento. Seu objetivo é selecionar os atributos mais importantes, pois atributos não relevantes e/ou redundantes podem reduzir a precisão e a compreensibilidade das hipóteses induzidas por algoritmos de aprendizado supervisionado. Vários algoritmos para a seleção de atributos relevantes têm sido propostosna literatura. Entretanto, trabalhos recentes têm mostrado que também deve-se levar em conta a redundância para selecionar os atributos importantes, pois os atributos redundantes também afetam a qualidade das hipóteses induzidas. Para selecionar alguns e descartar outros, é preciso determinar a importância dos atributos segundo algum critério. Entre os vários critérios de importância de atributos propostos, alguns estão baseados em medidas de distância, consistência ou informação, enquanto outros são fundamentados em medidas de dependência. Outra questão essencial são as avaliações experimentais, as quais representam um importante instrumento de estimativa de performance de algoritmos de seleção de atributos, visto que não existe análise matemática que permita predizer que algoritmo de seleção de atributos será melhor que outro. Essas comparações entre performance de algoritmos são geralmente realizadas por meio da análise do erro do modelo construído a partir dos subconjuntos de atributos selecionados por esses algoritmos. Contudo, somente a consideração desse parâmetro não é suficiente; outras questões devem ser consideradas, tal como a percentagem de redução da quantidade de atributos desses subconjuntos de atributos selecionados. Neste trabalho é proposto um algoritmo que separa as análises de relevância e de redundância de atributos e introduz a utilização da Dimensão Fractal para tratar atributos redundantes em aprendizado supervisionado. É também proposto um modelo de avaliação de performance de algoritmos de seleção de atributos baseado no erro da hipótese construída e na percentagem de redução da quantidade de atributos selecionados. Resultados experimentais utilizando vários conjuntos de dados e diversos algoritmos consolidados na literatura, que selecionam atributos importantes, mostram que nossa proposta é competitiva com esses algoritmos. Outra questão importante relacionada à extração de conhecimento a partir de bases de dados é o formato no qual os dados estão representados. Usualmente, é necessário que os exemplos estejam descritos no formato atributo-valor. Neste trabalho também propomos um metodologia para dar suporte, por meio de um processo semi-automático, à construção de conjuntos de dados nesse formato, originados de informações de pacientes contidas em laudos médicos que estão descritos em linguagem natural. Esse processo foi aplicado com sucesso a um caso real. / Progress in computer systems and devices applied to a different number of fields, have made it possible to collect and store an increasing amount of data. Moreover, this technological advance enables the storage of a huge amount of data which is difficult to process unless new approaches are used. The main reason to maintain all these data is to use it in a general way for the benefit of humanity. Many areas are engaged in the research and proposal of methods and processes to deal with this growing data. One such process is Knowledge Discovery from Databases, which aims at finding valuable and interesting knowledge which may be hidden inside the data. In order to extract knowledge from data, models (hypothesis) are usually developed supported by many fields such as Machine Learning. Feature Selection plays an important role in this process since it represents a central problem in machine learning and is frequently applied as a data pre-processing step. Its objective is to choose a subset from the original features that describes a data set, according to some importance criterion, by removing irrelevant and/or redundant features, as they may decrease data quality and reduce comprehensibility of hypotheses induced by supervised learning algorithms. Most of the state-of-art feature selection algorithms mainly focus on finding relevant features. However, it has been shown that relevance alone is not sufficient to select important features. Different approaches have been proposed to select features, among them the filter approach. The idea of this approach is to remove features before the model's induction takes place, based on general characteristics from the data set. For the purpose of selecting features and discarding others, it is necessary to measure the features' goodness, and many importance measures have been proposed. Some of them are based on distance measures, consistency of data and information content, while others are founded on dependence measures. As there is no mathematical analysis capable of predicting whether a feature selection algorithm will produce better feature subsets than others, it is important to empirically evaluate the performance of these algorithms. Comparisons among algorithms' performance is usually carried out through the model's error analysis. Nevertheless, this sole parameter is not complete enough, and other issues, such as percentage of the feature's subset reduction should also be taken into account. In this work we propose a filter that decouples features' relevance and redundancy analysis, and introduces the use of Fractal Dimension to deal with redundant features. We also propose a performance evaluation model based on the constructed hypothesis' error and the percentage of reduction obtained from the selected feature subset. Experimental results obtained using well known feature selection algorithms on several data sets show that our proposal is competitive with them. Another important issue related to knowledge extraction from data is the format the data is represented. Usually, it is necessary to describe examples in the so-called attribute-value format. This work also proposes a methodology to support, through a semi-automatic process, the construction of a database in the attribute-value format from patient information contained in medical findings which are described in natural language. This process was successfully applied to a real case.