Jar tests are performed to simulate full-scale pre-treatment and particle removal processes. Operators typically conduct them in an effort to attempt alternative treatment doses and strategies without altering the performance of the full-scale drinking water treatment plant. However, information obtained from these tests must be evaluated judiciously, as they currently focus on reduction of specific water quality parameters (i.e., ultraviolet absorption at 254 nm (UV254) and turbidity), and measuring and understanding the significance of coagulant dose on floc size. Consideration of aggregate structure has been less explored due mainly to a lack of appropriate theories to describe the complex random floc structure. Improving the predictive capacity of bench-scale protocols commonly used for optimizing conventional chemical pre-treatment in full-scale drinking water treatment plants is required. Results from settling tests indicated that the production of larger and more settleable flocs could not be described by floc settling velocities and floc sizes. Settling velocities were not directly related to either UV254 or turbidity reductions. Results of the floc characterization tests indicated that measured values of UV254 and turbidity of the supernatant were generally inversely proportional to aggregate D90; that is, the residual UV254 and/or turbidity decreased as the value of D90 increased, which may have been indicative of flocculent settling. No direct relationship could be discerned between fractal dimension D1 (i.e., floc shape) and the UV254 and turbidity of the supernatant; however, the turbidity after flocculation and a period of settling appeared to be inversely proportional to fractal dimension D2 (i.e., porosity). Overall, the results of the experiments have demonstrated that grain size distributions and fractal dimensions might be used to assess and/or predict pre-treatment and/or particle removal performance. Specifically, the relationship between D90 values calculated from samples of flocculated water prior to settling and UV254 and turbidity values of that water after a period of settling may be a simple tool that can be utilized to describe and potentially better predict flocculent settling performance. At present, this appears to be the first such tool of its kind that has been reported.

Jar tests are performed to simulate full-scale pre-treatment and particle removal processes. Operators typically conduct them in an effort to attempt alternative treatment doses and strategies without altering the performance of the full-scale drinking water treatment plant. However, information obtained from these tests must be evaluated judiciously, as they currently focus on reduction of specific water quality parameters (i.e., ultraviolet absorption at 254 nm (UV254) and turbidity), and measuring and understanding the significance of coagulant dose on floc size. Consideration of aggregate structure has been less explored due mainly to a lack of appropriate theories to describe the complex random floc structure. Improving the predictive capacity of bench-scale protocols commonly used for optimizing conventional chemical pre-treatment in full-scale drinking water treatment plants is required. Results from settling tests indicated that the production of larger and more settleable flocs could not be described by floc settling velocities and floc sizes. Settling velocities were not directly related to either UV254 or turbidity reductions. Results of the floc characterization tests indicated that measured values of UV254 and turbidity of the supernatant were generally inversely proportional to aggregate D90; that is, the residual UV254 and/or turbidity decreased as the value of D90 increased, which may have been indicative of flocculent settling. No direct relationship could be discerned between fractal dimension D1 (i.e., floc shape) and the UV254 and turbidity of the supernatant; however, the turbidity after flocculation and a period of settling appeared to be inversely proportional to fractal dimension D2 (i.e., porosity). Overall, the results of the experiments have demonstrated that grain size distributions and fractal dimensions might be used to assess and/or predict pre-treatment and/or particle removal performance. Specifically, the relationship between D90 values calculated from samples of flocculated water prior to settling and UV254 and turbidity values of that water after a period of settling may be a simple tool that can be utilized to describe and potentially better predict flocculent settling performance. At present, this appears to be the first such tool of its kind that has been reported.

Este trabalho utiliza o método da Salsicha de Minkowski usando dilatação exata para estimação da dimensão fractal em imagens de superfícies de SPM (Microscópio de Varredura por Ponta de prova). Descrevemos uma rotina que permite o cálculo de uma série de dilatações da superfície original em relação a vários raios. O método de dilatação exata considera todas as possíveis salsichas envolvendo um pré-cálculo das distâncias (raios) numa grade ortogonal, que são armazenadas em uma lista junto com suas coordenadas relativas. A partir daí, realizamos um estudo multiescala sobre a curva log-log do volume dilatado em termos dos raios a fim de obter o valor da dimensão fractal para a superfície analisada. Para isso aplicamos dois métodos numéricos exatos, os quais são baseados em: diferenciação da curva por diferenças finitas e, por diferenciação usando uma propriedade da Transformada de Fourier. Os valores da derivada do sinal obtido permitem caracterizar a evolução da dimensão fractal da superfície ao longo de várias escalas espaciais, isto é, a dimensão fractal apresenta um comportamento dinâmico em termos de escalas espaciais definida pelos raios. / This work uses the Minkowski Sausage method using exact dilation for estimating fractal dimension to SPM (Scanning Probe Microscopy) surface images. We describe a routine that permits the calculation of a series of dilations of the original surface, with respect to several radii. The exact dilation method considers all the possible sausages, involving pre-calculation of the distances (radiuses) in the orthogonal lattice, which are stored into a list together with the relative coordinates. Afterwards, we did a multiscale analysis of the log-log plot of the dilated volume in terms of radiuses, in order to obtain the dimension fractal of the studied surface. For this we applied two accurate numerical methods, which are based: on the differentiation by Finite Difference and, by differentiation using a Fourier transform property. The derivative values obtained allow to characterize the evolution of the fractal dimension of the surface along several spatial scales, i.e., the fractal dimension presents a dynamic behavior in terms of different spatial scales defined by radiuses.

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No description available.

An investigation into the complex function of muscle using non-invasive imaging and novel analytical approaches. / The human body is inherently complex as seen through the structural organization of muscle in terms of its contractile subunit organization and scaling, innervation patterns, and vascular organization. However, the functional complexity of muscle such as its state of oxygenation, metabolism or blood-flow has been less well explored. Thus in an effort to improve our understanding of muscle, blood oxygenation level dependent (BOLD) magnetic resonance imaging of the lower leg, at rest and during a variety of weighted plantar-flexion paradigms, at 40% maximal voluntary contraction, was employed. Prior to experimentation, on 11 healthy subjects, an ergometer and electromyogram (EMG), suitable for use within the MRI, were constructed to allow for concurrent exercise and image acquisition. After collecting muscle BOLD data, four novel techniques were using to describe muscle function. The first technique used the fractal dimension, a measure of complexity, conveying the rate of variation of muscle blood flow at rest. This technique was able to determine differences between the muscles of lower leg, which have varying distributions of muscle fibre types based on function. The second exploratory technique was the use of the phase space, which provides insight into state/state-transitions of a system over time. The phase space representation of the BOLD signal provided novel insight into the muscle activation state. It demonstrated that muscle has more than the two blood flow states of reduced levels at rest and increased levels when exercising. The third technique involved using a signal saturation (SAT) region, proximal to the imaging region, to mitigate the arterial in-flow effects to more accurately represent muscle activation. By observing the correlation between the ideal reference and recorded signal, the acquisition with the arterial suppression improved the assessment of what regions in the muscle were active in the range borderline activation, which has the highest uncertainty. The final outlook on muscle behaviour involved using measures of fatigue from the collected EMG data to develop novel metrics of fatigue based on the BOLD signal. Concurrent BOLD and EMG of the anterior compartment of the lower leg during a plantar-flexion block design, demonstrated that the change in blood-flow between rest and contracted states is an excellent indicator of muscle fatigue. The primary outlook of this thesis is to provide a unique data collection and analytic framework to describe muscle behaviour, which was achieved using non-invasive measures with a complex outlook. / Thesis / Master of Applied Science (MASc) / The human body is complex, and an incredible amount of research has been done to better understand it. Specifically, muscle is built and works in a complex way to allow us to move and perform everyday tasks. There are many diseases that affect how a muscle works, which is why there is a need to describe muscle performance when it is healthy and unhealthy. In this research, muscle behaviour is explored by taking pictures of the leg. From these pictures the blood flow in the calf and shin was measured both when staying still and when performing exercise. Four new techniques were created to describe the blood flow in the leg. The first technique measured how complex the muscle activity is, while staying still. If blood-flow changes a lot in a short amount of time, it is complex. This showed that the different components of muscle, either used for stamina or power, receive blood differently. The second technique used a different way of looking at the muscle to show that there are many different rates and amounts of blood-flow in the muscle. It revealed that muscle has more than the two blood flow options of 1) the normal level when resting and 2) the increased level when exercising. The third technique involved using an image filter to get a clean image of the muscle without the blood vessels affecting or misrepresenting the image. It was able to show what muscle regions were involved in exercise more accurately than before. The final technique involved measuring muscle electricity and blood flow at the same time, to find out when the muscle was exhausted. It demonstrated that muscle, when exhausted, showed larger changes in blood flow when going from resting to exercising. Overall, this research described how muscle performs in healthy individuals using new techniques. These techniques can now be used to compare healthy muscle to damaged/diseased muscle to determine how the muscle is recovering or to diagnose muscular disease.

Physiological signals, electroencephalogram (EEG) and heart rate variability (HRV), are generated by complex self-regulating systems. These signals are extremely inhomogeneous and nonstationary, and fluctuate in an irregular and highly complex manner. These fluctuations are due to underlying dynamics of the system. The synchronous neural activity measured as scalp EEG indicates underlying neural dynamics of the brain. Hence, quantitative EEG analysis has become a very useful tool in interpreting results from physiological experiments. The analysis of HRV provides valuable information to assess the autonomous nervous system (ANS). The HRV can be significantly affected by physiological state changes and many disease states. Hence, HRV analysis is becoming a major experimental and diagnostic tool. In this thesis, we focus on the study of EEG and HRV time series using tools from nonlinear time series analysis with special emphasis on its implications in detecting physiological state changes such as, in diseases like epileptic seizure and schizophrenia, and in altered states of consciousness as in sleep and meditation. The proposed nonlinear techniques are used in discriminating different physiological states from control states. Artifact processing of EEG signal Interferences (artifacts) from various sources unavoidably contaminate EEG recordings. In quantitative analysis, results can differ significantly by these artifacts, which may lead to wrong interpretation of the results. In this part of the thesis, we have devised methods to minimize ocular and muscle artifacts in EEG. The artifact correction methods are based on blind source separation (BSS) techniques such as singular value decomposition (SVD), algorithm for multiple signal extraction (AMUSE), canonical correlation analysis (CCA), information maximization (INFOMAX) independent component analysis (ICA) and joint approximate diagonalization of eigen-matrices (JADE) ICA. We have proposed a method to simulate clean and artifact corrupted EEG data based on the BSS methods. In order to enhance the performance of BSS methods, a technique called wavelet-filtered component inclusion method has been introduced. In addition, second-order statistics (SOS) and higher-order statistics (HOS) based BSS methods have been studied considering less number of EEG channels; and performance comparison of these methods has also been made. We have also addressed the problem of simultaneous correction of ocular and muscle artifacts in EEG recordings using the BSS methods. Irrespective of the BSS methods, the component elimination method has introduced high spectral error in all the bands after reconstruction of clean EEG. However, the wavelet filtered component inclusion method has retained almost all spectral powers of EEG channels in theta, alpha, and beta bands after ocular artifact minimization. When the number of EEG channels is very less, the enhanced CCA (SOS BSS) has given superior artifact minimization results than HOS BSS methods, especially in delta band. The component elimination method is used in muscle artifact minimization, and hence the SVD method cannot be used for this purpose since it leads to large spectral distortion of reconstructed EEG. The AMUSE and CCA methods have given comparable performance in muscle artifact minimization. In addition, the JADE method has introduced less mean spectral error compared to other methods. The CCA method has shown superior performance in simultaneous minimization of ocular and muscle artifacts, and AMUSE and JADE methods have given comparable results. Furthermore, the less computation time of wavelet enhanced SOS BSS methods make them very useful in real clinical environments. Fractal characterization of time series In biomedical signal analysis, fractal dimension (FD) is used as a quantitative measure to estimate complexity of physiological signals. Such analysis helps to study physiological processes of underlying systems. The FD can also be used to study dynamics of transitions between different states of systems like brain and ANS, in various physiological and pathological states. In this part, we have proposed a method to estimate FD of time series, called multiresolution box-counting (MRBC) method. A modification of this method resulted in multiresolution length (MRL) method. The estimation performance of the proposed methods is compared with that of Katz, Sevcik, and Higuchi methods, by simulating mathematically defined fractal signals, and also the computation time is compared between the methods. The MRBC and MRL methods have given comparable performance to that of Higuchi method, in estimating FD of waveforms, with the advantage of less computational time. In addition, various properties of the FD are studied and discussed in connection with classical signal processing concepts such as amplitude, frequency, sampling frequency, effect of noise, band width, correlation, etc. The FD value of signals has increased with number of harmonics, noise variance, band-width, and mid-band frequency, and decreased with degree of correlation in AR signal. An analogy between Katz FD and smoothed Teager energy operator has also been made. Application of fractal analysis to EEG and HRV time series The fluctuation of EEG potentials normally depends upon degree of alertness, and varies in amplitude and frequency. Hence, the EEG is an important clinical tool for studying sleep and sleep related disorders, epileptic seizures, schizophrenia, and meditation. In this part of the thesis, we have used FD which gives signal complexity, and detrended fluctuation analysis (DFA) which gives multiscale exponent of time series to quantify EEG. We have extended the concept of FD to multiscale FD to compute complexity of time series at multiple scales. The main applications of the proposed method are epileptic seizure detection, sleep stage detection, schizophrenia EEG analysis, and analysis of heart rate variability during meditation. For seizure detection, we have used intracranial EEG recordings with seizure-free and seizure intervals. In sleep EEG analysis, whole-night sleep EEG is used and results are compared with the manually scored hypnogram. The schizophrenia symptom is further categorized into positive and negative symptoms and complexity is estimated using FD and DFA. We have also analyzed HRV data of Chi and Kundalini meditation using FD and DFA techniques. In all the applications considered, we have tested for statistical significance of the computed parameters, between the case of interest and corresponding control cases, to discriminate between the physiological states. The ocular artifact has reduced FD while muscle artifact increased FD of EEG. The FD of seizure EEG has shown high value compared to that of seizure-free EEG. In addition, the seizure-free EEG has more DFA exponent-1 than seizure EEG. The value of FD of EEG is decreased with deepening of sleep, wake state having high FD value. The FD of REM state sleep EEG showed value between that of wake and state-1. The DFA exponent-1 has increased with deepening of sleep state, having small value for wake state. The REM state has given exponent-1 value between wake and state-1. The schizophrenia subjects have shown lower FD value than healthy controls in all the EEG channels except the bilateral temporal and occipital regions. The positive symptom sub-group has shown comparatively high FD values than healthy controls as well as overall schizophrenia sample in the bilateral tempero-parietal-occipital region. In addition, the positive symptom sub-group has shown significantly higher regional FD values than negative symptom sub-group especially in right temporal region. The overall schizophrenia samples as well as the positive and negative subgroup have shown least FD values in the bilateral frontal region. The values of DFA exponent-2 have shown significant high value in schizophrenia samples. In addition, the schizophrenia group has shown less DFA exponent-1 in bilateral temporal region than healthy control. The FD, multiscale FD, DFA exponents have shown significant performance in discriminating different physiological states from control states. The FD value of HRV time series during meditation is less compared to pre-meditation state in both Chi and Kundalini meditation. Irrespective of the type of meditation, meditation state has shown significantly high DFA exponent-1 than pre-meditation state, and significantly high DFA exponent-2 in pre-meditation state compared to meditation state. Functional connectivity analysis of brain during meditation In functionally related regions of the brain, even in those regions separated by substantial distances, the EEG fluctuations are synchronous, which is termed as functional connectivity. In this part, a novel application of functional connectivity analysis of brain using graph theoretic approach has been made on the EEG recorded from meditation practitioners. We have used 16 channel EEG data from subjects while performing Raja Yoga meditation. The pre-meditation condition is used as control state, against which meditation state is compared. For finding connectivity between EEG of various channels, we have computed pair-wise linear correlation and mutual information between the EEG channels, to form a connection matrix of size 16x16. Then, various graph parameters, such as average connection density, degree of nodes, characteristic path length, and cluster index, are computed from the connection matrix. The computed parameters are projected on to the scalp to get topographic head maps that give spatial variation of the parameter, and results are compared between meditation and pre-meditation states. The meditation state has shown low average connection density, less characteristic path length, and high average degree in fronto-central and central regions. Furthermore, high cluster index is shown in frontal and central regions than pre-meditation state. The parameters such as complexity, characteristic path length and average connection density are used as features in quadratic discriminant classifier to classify meditation and pre-meditation state, and have given good accuracy performance. Connectivity analysis using mutual information has given high average connection density in meditation state in theta, alpha and beta bands compared to pre-meditation state. The characteristic path length is high in delta, alpha and beta bands in meditation state. In addition, the meditation state has shown high degree and cluster index in theta and beta bands compared to pre-meditation state. Nonlinear dynamical characterization of HRV during meditation The cardiovascular system is influenced by internal dynamics as well as from various external factors, which makes the system more dynamic and nonlinear. In this part of the thesis, a novel application of using HRV data for studying Chi and Kundalini meditation has been made. The HRV time series are embedded into higher dimensional phase-space using Takens’ embedding theorem to reconstruct the attractor. After estimating the minimum embedding dimension to unfold the attractor dynamics, the complexity of the attractor is computed using correlation dimension, Lyapunov exponent, and nonlinearity scores. In all the analyses, the pre-meditation state is used as control state against which meditation state is compared. The statistical significance of the parameters estimated is tested to discriminate meditation state from control state. The HRV time series of both pre-meditation and meditation have shown similar minimum embedding dimensions in both Chi and Kundalini meditation. Irrespective of the type of meditation, the meditation state has shown high correlation dimension, largest Lyapunov exponent, and low nonlinearity score compared to pre-meditation state. Recurrent quantification analysis of HRV during meditation In this part, a novel application of recurrent quantification analysis (RQA) to HRV during meditation is studied. Here, the time series is embedded into a higher dimensional phase-space and Euclidean distance between the embedded vectors is calculated to form a distance matrix. The matrix is converted into binary matrix by applying a suitable threshold, and plotted as image to get recurrence plot. Various parameters are extracted from the recurrence plot such as percent recurrence rate, diagonal parameters (determinism, divergence, entropy, ratio), and vertical or horizontal parameters (laminarity, trapping time, maximal vertical line length). The procedure is applied to HRV data during meditation and pre-meditation (control) to discriminate between the states. The HRV of meditation state has shown more diagonal line structure whereas more black patches are observed in pre-meditation state. In addition, at low embedding dimensions, the meditation state has shown low recurrence rate, high determinism, low divergence, low entropy, high ratio, high laminarity, high trapping time, and less maximal vertical line length compared to pre-meditation state. These RQA parameters have shown superior performance in discriminating meditation state from control state.

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.

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.

The efficient market hypothesis is one of the most important propositions in finance theory and has been subjected to years of rigorous empirical testing. We examine power of a new tool for evaluating market efficiency, fractal dimension. Characteristics and abilities of fractal dimension measure are explored through extensive Monte Carlo simulations. We prove that it provides an accurate evaluation of market's efficiency and its changes. This approach is highly innovative and creates new possibilities for examination of markets. The uniqueness of fractal dimension is in its ability to assign a numerical ranking to examined series describing the level of (in)efficiency; it is accurate for small samples of observations and quickly reflects changes in market efficiency structure. Powered by TCPDF (www.tcpdf.org)