Fractal Dimension Estimation

Fractal geometry provides a useful model for describing complex 3-D surfaces typical of natural objects. The fractal dimension is one feature that is useful in describing fractal surfaces. In part I of the dissertation, we develop a new a method for calculating the fractal dimension of 3 -D surfaces using the maximum likelihood estimate (MLE). We formulate a model to describe the fractal surface. We then derive the likelihood function for that model and estimate the fractal dimension by maximizing the likelihood function. We then compare the new method (MLE) with the Box-dimension estimation method (one of the most popular methods) in small and large windows. In small windows the histogram of the fractal dimension using MLE is more consistent than the Box method (i.e., 2.0 < D < 3.0). It has a low variance, and peaks closer to the value used to generate synthetic fractal images. In large windows we obtain a functional relationship between a known fractal dimension and its estimate and the results shows that the MLE has a better linear functional relationship than the Box method. In part II of the dissertation, we analyze additive noise to single variable fractal path. We include the additive noise in the MLE model and estimate the fractal dimension and compare the results with the Box method which cannot handle additive noise. The results show that the MLE gives better results than the Box method.