The Wavelet Method For Several Problems In Fractal Theory

There are many problems to be solved in modern fractal theory, especially focuses on studies of computation on fractal dimension,inverse problem and sigularities. The scale property of wavelet is well suited for analyzing non-stationarity and localization and used to extract microscopic information about the scaling properties of fractals. In this dissertation, based on wavelet theory,the following problems are discussed:1. Traditional computional methods and limitation of dimension of fractal curve being analyzed ,A model of fractal curve having multidimension is proposed. Based on noise reduction for wavelet softhresholding, computional formula and algorithm about point-wise local dimension on the curve are given. Results from simulations show the validity of he algorithm.2. Given a graph of a fractal interpolation function which is the attractor of an unknown IPS with affine constration maps, the maps are found based on the self-similarity of the zero-crossing points of wavelet transform. The effectiveness of method is shown in an example.3. The concept of time-varying dimension are proposed and introduced in two kinds of biased stochastic process with locally self-similarity. The estimation formula and algorithm of Hurst Index are based on Daubechies wavelet analysis of samples dada . Simulation result indicates effectiveness of estimation method and proves that estimation value is a consistent result of true value. Week-index analysis of Shanghaistock market is taken as a real example by time-varying dimension.4. Based on correlation-decaying charecteristics of discrete wavelet coefficients oflf processes and the least-square algorithm estimation for spectral parameter of yfprocesses is presented without knowing the distribution of the parameter.5. A parameterization method for tracing wavelet maxima lines at the finest scale continuously is presented, by means of which the sigularity of multifractal spectrum can be analyzed well. Topologic bifurcation situation of maxima lines is also discussed. In the end ,an estimation for multifractal spectrum is proposed in terms of wavelet module maxima lines.