Accuracy of fractal and multifractal measures for signal analysis

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Chen, Hongjing
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This thesis is concerned with accurate approximations of multifractal meas res of strange attractors and analysis of the spatial signals (images) from the point of view of strange attractors through multifractal measures. The approximations of multifractal measures of strange attractors is studied through the Renyi dimension, singularity spectrum, and the Mandelbrot dimension. It is based on the probability of each volume element (vel) intersected by the points on the strange attractor. Since the complete strange attractor consists of an infinite number of points, we cannot obtain the theoretical value of the probability; instead, we consider a finite number of points in the vels. Therefore, this study reduces to a finite number of points and finite size of vels. We have shown that, for a given vel size, the Renyi dimension is sensitive to the number of points used in the attractor, and that for a given number of points in the strange attractor, it is also sensitive to the vel size. We also find that for a given vel size, there is a minimum bound on the number of points required. The smaller the vel size, the larger the minimum bound. Furthermore, the Renyi dimension converges when the number of points increases above the minimum bounds. The convergence can be a guideline to determine the number of points required to compute the dimension. The results of the study of the strange attractors can be applied to spatial signals. In this thesis, spatial signals such as images are modelled as strange attractors, and multi-fractals are used to characterize complicated distributions of the grey levels in images. (Abstract shortened by UMI.)