Este projeto descreve o desenvolvimento, estudo e aplicação de descritores fractais em análise de texturas. Nos últimos anos, a literatura vem apresentando a geometria fractal como uma ferramenta poderosa para a análise de imagens, com aplicações em variados campos da ciência. A maior parte destes trabalhos faz uso direto da dimensão fractal como um descritor do objeto representado na imagem. Entretanto, em função da complexidade de muitos problemas nesta área, algumas soluções foram propostas para melhorar essa análise, usando não apenas o valor da dimensão fractal, mas um conjunto de medidas que pudessem ser extraídas pela geometria fractal e que descrevessem as texturas com maior riqueza e precisão. Entre essas técnicas, destacam-se a metodologia de multifractais, de dimensão fractal multiescala e, mais recentemente, os descritores fractais. Esta última técnica tem se mostrado eficiente na solução de problemas relacionados à discriminação de imagens de texturas e formas, uma vez que os descritores gerados fornecem uma representação direta do padrão de complexidade (distribuição dos detalhes ao longo das escalas de observação) da imagem. Assim, essa solução permite que se tenha uma descrição rica da imagem estudada pela análise da distribuição espacial e/ou espectral dos pixels e intensidade de cores/tons de cinza, com uma modelagem que pode se aproximar da percepção visual humana para a geração de um método automático e preciso. Ocorre, entretanto, que os trabalhos apresentados até o momento sobre descritores fractais focam em métodos de estimativa de dimensão fractal mais conhecidos como Bouligand-Minkowski e Box-counting. Este projeto visa estudar mais a fundo o conceito, generalizando para outras abordagens de dimensão fractal, bem como explorando diferentes formas de se extraírem os descritores a partir da curva logarítmica associada à dimensão. Os métodos desenvolvidos são aplicados à análise de texturas, em problemas de classificação de bases públicas, cujos resultados podem ser comparados com métodos da literatura, bem como a segmentação de imagens de satélite e à identificação automática de amostras obtidas em estudos de nanotecnologia. Os resultados alcançados demonstram o potencial da metodologia desenvolvida para a solução destes problemas, mostrando tratar-se de uma nova fronteira a ser usada e explorada em análise de imagens e visão computacional como um todo. / This project describes the development, study and application of fractal descriptors to texture analysis. Recently, the literature has shown fractal geometry as a powerful tool for image analysis, with applications to several areas of science. Most of these works use fractal dimension as a descriptor of the object depicted in the image. However, due to the complexity of many problems in this context, some solutions have been proposed to improve this analysis. These proposed methods use not only the value of fractal dimension, but a set of measures which could be extracted by fractal geometry to describe the textures with greater richness and accuracy. Among such techniques, we emphasize the multifractal methodology, multiscale fractal dimension and, more recently, fractal descriptors. This latter technique has demonstrated to be efficient in solving problems related to the discrimination of texture and shape images. This is possible as the extracted descriptors provide a direct representation of the complexity (the details distribution along the scales of observation) in the image. Thus, this solution allows for a rich description of the image studied by analyzing the spatial/spectral distribution of pixels and intensity of colors/gray-levels, with a model which can approximate the human visual perception, generating an automatic and precise method. However, the works about fractal descriptors presented in the literature focus on classical methods to estimate fractal dimension, such as Bouligand-Minkowski and Box-counting. This project aims at studying more deeply the concept, generalizing to other approaches in fractal dimension, as well as exploring different ways of extracting the key features from the logarithmic curve associated with the dimension. The developed methods are applied to texture analysis, in classification problems over public databases, whose results can be compared with literature methods, as well as to the segmentation of satellite images and automatically identifying samples obtained from studies on nanotechnology. The results demonstrate the potential of the methodology developed to solve such problems, showing that this is a new frontier to be explored and used in image analysis and computer vision at all.

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Meniere's disease is a common inner ear disorder that affects balance and hearing. Electrovestibulography (EVestG) is a relatively new vestibular driven test that measures spontaneous and driven field potential activity recorded in the external ear canal in response to various vestibular stimuli. The main objectives of this thesis were to record and analyze EVestG signals in order to 1) testify whether the EVestG technology is capable of classifying individuals with Meniere’s from healthy ones, and if it is, then 2) identify the EVestG tilt stimulus providing the most informative response in relation to identifying Meniere’s symptoms; thus, optimizing the EVestG experimental protocol as a Meniere’s disease diagnostic aid. EVestG signals of two groups of Meniere’s and control individuals during seven different EVestG tilt stimuli were recorded and analyzed by linear and nonlinear signal processing techniques. Data of 14 with Meniere’s disease and 16 healthy individuals were used as the training set, while additional data of 21 individuals with vertiginous disorders (and suspected of Meniere’s disease) and 10 controls were used as the test set. An ad-hoc voting classifier built upon single-feature linear classifiers was designed, and used for classification of the two groups of both training and test datasets. The results showed an overall accuracy of 87% and 84% for training and test datasets, respectively. Among the seven different tilts that each evokes a specific part of the inner ear organ, the side tilt which stimulates most of the labyrinth and particularly the utricle, was found to generate the best characteristic features for identifying Meniere’s disease from controls. Thus, one may simplify the EVestG protocol to only the side tilt stimulus for a quick screening of Meniere’s disease. The proposed method encourages the use of EVestG technology as a non-invasive and potentially reliable diagnostic/screening tool to aid clinical diagnosis of Meniere’s diseases. / October 2016

The main objective of the thesis is to analyse impact of wavelet covariance estimation in the context of Markowitz mean-variance portfolio selection. We use a rolling window to apply maximum overlap discrete wavelet transform to daily returns of 28 companies from DJIA 30 index. In each step, we compute portfolio weights of global minimum variance portfolio and use those weights in the out-of- sample forecasts of portfolio returns. We let rebalancing period to vary in order to test influence of long-term and short-term traders. Moreover, we test impact of different wavelet filters including Haar, D4 and LA8. Results reveal that only portfolios based on the first scale wavelet covariance produce significantly higher returns than portfolios based on the whole sample covariance. The disadvantage of those portfolios is higher riskiness of returns represented by higher Value at Risk and Expected Shortfall, as well as higher instability of portfolio weights represented by shorter period that is required for portfolio weights to significantly differ. The impact of different wavelet filters is rather minor. The results suggest that all relevant information about the financial market is contained in the first wavelet scale and that the dynamics of this scale is more intense than the dynamics of the whole market.

A Dimensão Fractal pode ser utilizada para medir algumas características ligadas a complexidade da imagem, permitindo seu uso em análise de formas e texturas e reconhecimento de padrões. Neste trabalho é apresentado um estudo comparativo entre alguns dos principais métodos de estimativa da Dimensão Fractal. Foi realizada uma análise experimental e um estudo de casos para cada uma das técnicas, levando em consideração aspectos de implementação, precisão, variação de resultados segundo ajuste de parâmetros e tolerância a ruídos. Neste trabalho também foi desenvolvido um estudo sobre a Dimensão Fractal Multiescala, visando seu emprego como metodologia de assinatura de complexidade. Na literatura a técnica de multiescala é limitada ao método de Bouligand-Minkowski, sendo aqui ela estendida para outras metodologias de estimativa de Dimensão Fractal. Por meio de análise experimental as metodologias propostas foram comparadas e os resultados discutidos, enfatizando as vantagens e desvantagens destas técnicas. / Fractal Dimension can be used to measure some characteristics related to the image complexity, allowing its use on shape and texture analysis and pattern recognition. In this work is presented a comparative study among some of the most important methods to estimate Fractal Dimension. It was performed a experimental analysis and a case study for each one of the techniques, considering implementation aspects, precision, variation of results under parameters adjustments and noise tolerance. In this work is also performed a study about MultiScale Fractal Dimension, aiming at its use as a methodology of complexity signature. In the literature the multiscale technique is limited to Bouligand-Minkowski method, being here it extended to other methodologies of estimative of Fractal Dimension. By experimental analysis the proposed methodologies were compared and the results argued, emphasizing the advantages and disadvantages of those techniques.

"Certain physical problems in electrostatics, magnetostatics, and heat transfer give rise to elliptic boundary value problems with transmission conditions on a layer. We focus on a particular problem with a second order transmission condition, representing an infinitely conductive layer. To approximate irregular layers that may naturally arise, a sequence of layers that converge to the fractal von Koch curve is considered. The solution to this transmission problem with a prefractal layer exhibits singularities due to the transmission condition across the layer as well as the reentrant corners introduced in the domain by the prefractal curve. To solve this problem numerically using a finite element method, the mesh must be adjusted to account for these singularities. We exhibit a general algorithm for creating a finite element discretization of the domain that results in linear convergence of the numerical solution to the true solution in a suitable norm."

The purpose of this paper is to construct a Quasi-volume-filling surface and study its properties. We start with the construction of a volume-filling surface, the Pólya surface, based on Pólya's curve, by rotating the Pólya's curve in 3-dimensional space. Then we construct a Quasi-space-filling curve in 2-dimensions, the Quasi- Pólya curve, which approximates the Pólya's curve and fills a triangle up to a residual small surface of arbitrary size. We prove that the Quasi-Pólya curve satisfies the open set condition, and there exists a unique invariant (self-similar) measure consistent with the normalized Hausdorff measure on it. Moreover, the energy form constructed on Quasi-Pólya curve is proved to be a closed & regular form, and we prove that the Quasi-Pólya curve is a variational fractal in the end. Next, we use the same idea, by rotating the Quasi-Pólya curve in 3-dimensional space, to construct the Quasi-Pólya surface, which is a Quasi-volume-filling surface and approximates to Pólya surface in some sense.

Tese de mestrado. Engenharia Biomédica. Faculdade de Engenharia. Universidade do Porto. 2008

The principal aim of this thesis was to develop and implement a standardised protocol for the fractal analysis of cancellous bone architecture. Cancellous bone structure from different skeletal sites in groups of osteoporotic, osteoarthritic and normal individuals was analysed. The results of fractal analysis were explained in the context of conventional bone histomorphometry and a priori knowledge to advance the understanding of cancellous bone architecture. There has been much effort devoted to the pursuit of descriptors of cancellous bone complexity. The aim of these endeavours has been to develop morphological descriptors of bone quality that explain the functional properties of the cancellous bone structure for age-related changes, the effect of disease processes or the effect of therapeutic agents on the diseased skeleton. The fractal analysis of the complexity of cancellous bone architecture promises to be an exciting addition to existing analytical techniques. The establishment of a standardised methodology for the fractal analysis of cancellous bone encompassed many components. Knowledge of the stereological and histomorphometric principles that are employed in currently available techniques enabled a comprehensive examination of the factors that effect the measurement of the fractal dimensions. The methodology presented in this thesis has been optimised specifically for measuring sectional fractal dimensions in histological sections of cancellous bone. The sectional fractal dimensions show that, over three ranges of scale, cancellous bone is effectively fractal at multiple sites in the normal skeleton. The three sectional fractal dimensions describe different morphological compartments of the cancellous bone structure. Fractal 1 describes the surface texture of the trabeculae, fractal 2 describes the shape or form of individual trabeculae and fractal 3 describes the spatial arrangement or overall architecture of the cancellous bone. This thesis reports that in the normal skeleton there are differences between skeletal sites for the fractal dimensions, which are dependent on the functional properties of the skeletal sites. Fractal 2 and fractal 3 for subchondral cancellous bone is greater than vertebral body and iliac crest cancellous bone, which indicates greater morphological complexity. Also, fractal 2 and fractal 3 in subchondral cancellous bone show an age-related decrease, which suggests that the cancellous bone structure becomes less complex with age. This interostotic variability in response to ageing is indicative of the heterogeneity in functional properties of cancellous bone in the skeleton. In this thesis, fractal analysis has been shown to detect morphological differences in the cancellous bone between normals, osteoporotics and osteoarthritics in the compressive and tensile trabeculae of the femoral head and the iliac crest. These data have provided new insights into the mechanisms of change to cancellous bone structure in ageing and in disease. Age-related changes in the structural parameters of cancellous bone are seen at all the skeletal sites in the normals but are only present in the compressive trabeculae of the femoral head in the osteoporotics and not at all in the osteoarthritics. These observations indicate that these disease processes are associated with an uncoupling of the control mechanisms that affect cancellous bone structural complexity. In the normals, the fractal dimensions only show age-related change in the tensile trabeculae of the femoral head, suggesting that fractal analysis is not suitable for detecting the age-related changes that are quantified by the structural parameters of cancellous bone in these study groups but the fractal dimensions detect underlying cancellous bone complexity independent of age. For the osteoporotics, fractal 1 is the same at all skeletal sites. This suggests that the relative levels of remodeling activity are the same for both normals and osteoporotics. Fractal 2 for both the compressive and tensile trabeculae in the femur is significantly lower for the osteoporotics than the normals but in the iliac crest, fractal 2 is the same. This implies that in the femoral head the osteoporotics have trabeculae that are significantly less complex in shape than the normals. This phenomenon is not seen in the iliac crest, which is usually the site of diagnostic biopsy. Therefore, biopsies for diagnosis of osteoporosis may not show changes in cancellous bone structural complexity that are evident in disease affected sites. The structural parameters of cancellous bone show that osteoporotics lose whole trabeculae due to perforation of trabeculae, through decreased Tb.N and increased Tb.Sp. This leads to less interconnection between trabeculae, loss of branching and more rounded trabeculae, hence the trabeculae are less complex in shape. For fractal 3, in compressive and tensile regions of the femur the osteoporotics are significantly lower than the normals and in the iliac crest the osteoporotics are the same as the normals. This indicates that in the femoral head the spatial arrangement of the trabeculae within the cancellous structure of the osteoporotics is less complex. The structural parameters of cancellous bone show that there is loss of whole trabeculae, which is associated with increased spatial separation between the trabeculae as bone is lost. For the osteoarthritics, fractal 1 is the same as the normals at all skeletal sites. Fractal 2 for the compressive trabeculae in the femoral head is significantly higher for the osteoarthritics than the normals but in the tensile trabeculae of the femoral head and the iliac crest fractal 2 for the osteoarthritics is the same as the normals. This implies that in the compressive trabeculae of the femoral head the osteoarthritics have trabeculae that are significantly more complex in shape than the normals. The structural parameters of cancellous bone show that the compressive trabeculae of the femoral head are thicker, more numerous and less widely separated with greater BV/TV than the normals. This leads to greater interconnectivity between trabeculae and more complex branching, hence the trabeculae are more complex in shape. For fractal 3, in the compressive and tensile regions of the femoral head the osteoarthritics and the normals are the same but in the iliac crest the osteoarthritics are lower than the normals. This indicates that the spatial arrangement of the trabeculae within the cancellous structure of the osteoarthritics does not change in response to the disease process in subchondral cancellous bone adjacent to the articular lesion but in the iliac crest the spatial arrangement of the trabeculae in osteoarthritics is less complex in shape. The structural parameters of cancellous bone show that BV/TV is increased in the compressive and tensile trabeculae of the femoral head but not in the iliac crest of the osteoarthritics. This indicates that the spatial complexity of the trabecular arrangement within the cancellous structure of osteoarthritics changes independently of changes in cancellous bone structure detected by the structural parameters of cancellous bone. The sectional fractal dimensions have detected differences in morphological complexity between the normals and disease groups and between the skeletal sites. These novel data have been obtained using an innovative technique that is not dependent on assumptions based on conceptual models of cancellous bone structure. A priori knowledge of bone biology is utilised to enable the fractal analysis to measure specific morphological entities within the cancellous bone structure. The fractal dimensions have identified changes in the morphological complexity of specific components of the cancellous structure, which are not identified by existing model-based morphometric techniques. This has enabled new understanding of how change to cancellous bone structure occurs as a result of a disease process. Fractal analysis is a novel tool that will prove useful for the study of changes in cancellous bone structure due to disease and to study the use of therapies to alter or maintain the quality of cancellous bone architecture. / Thesis (Ph.D.)--Medical School, 2002.