Wavelet transform provides an alternative to the classical Short-Time Fourier Transform (STFT). In contrast to the STFT, which uses a single analysis window, the Wavelet Transform uses shorter windows at higher frequencies and longer windows at lower frequencies. For some particular wavelet functions, the local maxima of the wavelet transform correspond to the sharp variation points of the signal. As an application, wavelet transform is introduced to the character recognition. Local maximum of wavelet transform is used as a local feature to describe character boundary. The wavelet method performs well in the presence of noise. The maximum of wavelet transform is also an important feature for analyzing the properties of brain wave. In our study, we found the maximum of wavelet transform was related to the P300 latency. It provides an easy and efficient way to measure P300 latency.

Event related potentials (ERP's) carry very important information that relates to the performance of the brain functions of the human being. Further studies have identified that one component, in particular, P 300, is affected by the memory process. Matched filter is used to improved the SNR of signal ERP' s. We use the output of the matched filter to distinguish the difference of the waveforms between normal subjects and memory-impaired subjects. In our study, we found that the peak values of the matched filtering output were different between normal subjects and memoryimpaired subjects. Also, as an application, wavelet transform is introduced to the ERP analysis. Local maximum of wavelet transform was used as a local feature to find the relationship between the sharp variation points and the memory process. A comparison between matched filtering and wavelet transform was made and also the correlation coefficients of the peaks and sharp variation points are calculated to find the relationship between the important moments in a memory process.

Abstract not available

Filamentous fungal colonies show a remarkable diversity of different mycelial branching patterns. To date, the characterization of this biostructural complexity has been based on subjective descriptions. Here, computerized image analysis in conjunction with video microscopy has been used to quantify several aspects of fungal growth and differentiation. This was accomplished by applying the new branch of mathematics called Fractal Geometry to this biological system, to provide an objective description of morphological and biochemical complexity. The fractal dimension is useful for describing irregularity and shape complexity in systems that appear to display scaling correlations (between structural units) over several orders of length or size. The branching dynamics of Pycnoporus cinnabarinus have been evaluated using fractals in order to determine whether there was a correlation between branching complexity and the amount of extracellular phenol-oxidase that accumulated during growth. A non-linear branching response was observed when colonies were grown in the presence of the aminoanthraquinone dye, Remazol Brilliant Blue R. Branching complexity could be used to predict the generalized yield of phenol-oxidase that accumulated in submerged culture, or identify paramorphogens that could be used to improve yield. A method to optimize growth of discrete fungal colonies for microscopy and image analysis on microporous membranes revealed secretion sites of the phenoloxidase, laccase as well as the intracellular enzyme, acid phosphatase. This method was further improved using microwave-accelerated heating to detect tip and sheath bound enzyme. The spatial deposition of secreted laccase and acid phosphatase displayed antipersistent scaling in deposition and/or secretion pattern. To overcome inherent statistical limitations of existing methods, a new signal processing tool, called wavelets were applied to analyze both one and two-dimensional data to measure fractal scaling. Two-dimensional wavelet packet analysis (2-d WPA) measured the (i) mass fractal dimension of binary images, or the (ii) self-affine dimension of grey-scale images. Both 1- and 2-d WPA showed comparative accuracy with existing methods yet offered improvements in computational efficiency that were inherent with this multiresolution technique. The fractal dimension was shown to be a sensitive indicator of shape complexity. The discovery of power law scaling was a hallmark of fractal geometry and in many cases returned values that were indicative of a self-organized critical state. This meant that the dynamics of fungal colony branching equilibrium. Hence there was potential for biostructural changes of all sizes, which would allow the system to efficiently adapt to environmental change at both the macro and micro levels.

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 115-121).

This thesis provides contributions to research in Bayesian modeling and shrinkage in the wavelet domain. Wavelets are a powerful tool to describe phenomena rapidly changing in time, and wavelet-based modeling has become a standard technique in many areas of statistics, and more broadly, in sciences and engineering. Bayesian modeling and estimation in the wavelet domain have found useful applications in nonparametric regression, image denoising, and many other areas. In this thesis, we build on the existing techniques and propose new methods for applications in nonparametric regression, image denoising, and partially linear models. The thesis consists of an overview chapter and four main topics. In Chapter 1, we provide an overview of recent developments and the current status of Bayesian wavelet shrinkage research. The chapter contains an extensive literature review consisting of almost 100 references. The main focus of the overview chapter is on nonparametric regression, where the observations come from an unknown function contaminated with Gaussian noise. We present many methods which employ model-based and adaptive shrinkage of the wavelet coefficients through Bayes rules. These includes new developments such as dependence models, complex wavelets, and Markov chain Monte Carlo (MCMC) strategies. Some applications of Bayesian wavelet shrinkage, such as curve classification, are discussed. In Chapter 2, we propose the Gibbs Sampling Wavelet Smoother (GSWS), an adaptive wavelet denoising methodology. We use the traditional mixture prior on the wavelet coefficients, but also formulate a fully Bayesian hierarchical model in the wavelet domain accounting for the uncertainty of the prior parameters by placing hyperpriors on them. Since a closed-form solution to the Bayes estimator does not exist, the procedure is computational, in which the posterior mean is computed via MCMC simulations. We show how to efficiently develop a Gibbs sampling algorithm for the proposed model. The developed procedure is fully Bayesian, is adaptive to the underlying signal, and provides good denoising performance compared to state-of-the-art methods. Application of the method is illustrated on a real data set arising from the analysis of metabolic pathways, where an iterative shrinkage procedure is developed to preserve the mass balance of the metabolites in the system. We also show how the methodology can be extended to complex wavelet bases. In Chapter 3, we propose a wavelet-based denoising methodology based on a Bayesian hierarchical model using a double Weibull prior. The interesting feature is that in contrast to the mixture priors traditionally used by some state-of-the-art methods, the wavelet coefficients are modeled by a single density. Two estimators are developed, one based on the posterior mean and the other based on the larger posterior mode; and we show how to calculate these estimators efficiently. The methodology provides good denoising performance, comparable even to state-of-the-art methods that use a mixture prior and an empirical Bayes setting of hyperparameters; this is demonstrated by simulations on standard test functions. An application to a real-word data set is also considered. In Chapter 4, we propose a wavelet shrinkage method based on a neighborhood of wavelet coefficients, which includes two neighboring coefficients and a parental coefficient. The methodology is called Lambda-neighborhood wavelet shrinkage, motivated by the shape of the considered neighborhood. We propose a Bayesian hierarchical model using a contaminated exponential prior on the total mean energy in the Lambda-neighborhood. The hyperparameters in the model are estimated by the empirical Bayes method, and the posterior mean, median, and Bayes factor are obtained and used in the estimation of the total mean energy. Shrinkage of the neighboring coefficients is based on the ratio of the estimated and observed energy. The proposed methodology is comparable and often superior to several established wavelet denoising methods that utilize neighboring information, which is demonstrated by extensive simulations. An application to a real-world data set from inductance plethysmography is considered, and an extension to image denoising is discussed. In Chapter 5, we propose a wavelet-based methodology for estimation and variable selection in partially linear models. The inference is conducted in the wavelet domain, which provides a sparse and localized decomposition appropriate for nonparametric components with various degrees of smoothness. A hierarchical Bayes model is formulated on the parameters of this representation, where the estimation and variable selection is performed by a Gibbs sampling procedure. For both the parametric and nonparametric part of the model we are using point-mass-at-zero contamination priors with a double exponential spread distribution. In this sense we extend the model of Chapter 2 to partially linear models. Only a few papers in the area of partially linear wavelet models exist, and we show that the proposed methodology is often superior to the existing methods with respect to the task of estimating model parameters. Moreover, the method is able to perform Bayesian variable selection by a stochastic search for the parametric part of the model.

In this dissertation, we address the statistical analysis under the multiscale framework for the self-similar process. Motivated by the problems arising from geophysics and health informatics, we develop a set of statistical measures as discriminative summaries of the self-similar process. These measures include Multiscale Schur Monotone (MSM) measures, Geometric Attributes of Multifractal Spectrum (GAMFS), Quasi-Hurst exponents, Mallat Model and Tsallis Maxent Model. These measures are used as methods to quantify the difference (or similarities) or as input (feature) vectors in the classification model. As the cornstone of GAMFS, we study the estimation of multifractal spectrum and adopt a Weighted Least Squares (WLS) schemes in the wavelet domain to minimize the heteroskedastic effects , which is inherent because the sample variances of the wavelet coefficients depend on the scale. We also propose a Combined K-Nearest-Neighbor classifier (Comb-K-NN) to address the inhomogeneity of the class attributes, which is indicated by the large variations between subsets of input vectors. The Comb-K-NN classifier stabilizes the variations in the sense of reducing the misclassification rates. Bayesian justifications of Comb-K-NN classifier are provided. GAMFS, Quasi-Hurst exponents, Mallat Model and Tsallis Maxent Model are used in the study of assessing the effects of atmospheric stability on the turbulence measurements in the inertial subrange. We also formulate the criteria for success in evaluating how atmospheric stability alters the MFS of a single flow variable time series as a statistical classification model. We use the multifractal discriminate model as the solution of this problem. Also, high frequency pupil-diameter dynamic measurements, which are well documented as measures of mental workload, are summarized using both GAMFS and MSM. These summaries are further used as the feature vector in the Comb-K-NN classifier. The serious inhomogeneity among subjects in the same user group makes classification difficult. These difficulties are overcome by using Comb-K-NN classifier.

This thesis addresses four topics in the area of applied harmonic analysis. First, we show that the affine densities of separable wavelet frames affect the frame properties. In particular, we describe a new relationship between the affine densities, frame bounds and weighted admissibility constants of the mother wavelets of pairs of separable wavelet frames. This result is also extended to wavelet frame sequences. Second, we consider affine pseudodifferential operators, generalizations of pseudodifferential operators that model wideband wireless communication channels. We find two classes of Banach spaces, characterized by wavelet and ridgelet transforms, so that inclusion of the kernel and symbol in appropriate spaces ensures the operator is Schatten p-class. Third, we examine the Schatten class properties of pseudodifferential operators. Using Gabor frame techniques, we show that if the kernel of a pseudodifferential operator lies in a particular mixed modulation space, then the operator is Schatten p-class. This result improves existing theorems and is sharp in the sense that larger mixed modulation spaces yield operators that are not Schatten class. The implications of this result for the Kohn-Nirenberg symbol of a pseudodifferential operator are also described. Lastly, Fourier integral operators are analyzed with Gabor frame techniques. We show that, given a certain smoothness in the phase function of a Fourier integral operator, the inclusion of the symbol in appropriate mixed modulation spaces is sufficient to guarantee that the operator is Schatten p-class.

Thesis (Ph. D.)--University of Missouri-Columbia, 2000. / Typescript. Vita. Includes bibliographical references (leaves 202-221). Also available on the Internet.

Thesis (M.S.)--Mississippi State University. Department of Electrical & Computer Engineering. / Title from title screen. Includes bibliographical references.