Characterization and Measurement of Random Fractals

with variance Mandelbrot's fractal geometry provides both a description and a mathematical model for many of the seemingly complex shapes found in nature.Such shapes often possess a remarkable invariance under changes of magnification.This statistical self-similarity may be characterized by a fractal dimension D, a number that agrees with our intuitive notion of dimension but need not be an integer.A brief mathematical characterization of random fractals is presented with emphasis on variations of Mandelbrot's fractional Brownian motion.The important concepts of fractal dimension and exact and statisical self-similarity and self-affinity will be reviewed.The various methods and difficulties of estimating the fractal dimension and lacunarity from experimental images or point sets are summarized.