Measuring Fractal Dimension: Morphological Estimates and Iterative Optimization.

摘要

Fractals are mathematical sets with a high degree of geometrical complexity that can model many natural phenomena. Examples include physical objects such as clouds, mountains, trees and coastlines, as well as image intensity signals that emanate from them (assuming certain restrictions on the object's reflectance and illumination). Although, the fractal images are the most popularized class of fractals due to their fantastic resemblance with natural scenes, there are also numerous natural processes described by time-series measurements that are fractals. The one-dimensional signals f(t) representing these measurements are fractals in the sense that their graph G(f) = ((t,y) : y = f(t)) is a fractal set. Thus, modeling fractal signals is of great interest in signal and image analysis. An important characteristic of fractals useful for their description and classification is their fractal dimension D, which exceeds their topological dimension. Intuitively, D measures the degree of their boundary fragmentation or roughness. It makes meaningful the measurement of metric aspects of fractal sets such as their length or area. Keywords: Morphology, Iterations, Optimization, Mathematical models, G Fractal dimensions, Signal processing, Image processing, Reprints. (jg)