Fractal Signal Analysis Using Mathematical Morphology

Publisher Summary This chapter provides the definitions of basic morphological transformations for sets and signals—i.e., the erosion, dilation, and opening operations—and surveys the theory of fractal dimensions. There is a proliferation of fractal dimensions, all of which are more or less capable of measuring the degree of fragmentation of a signal's graph; their definitions and interrelationships are also discussed in the chapter. Emphasis is given on the Minkowski–Bouligand dimension, whose analysis is done using morphological operations. The chapter also reviews three classes of parametric fractal signals and related algorithms for their synthesis. The performance of the presented morphological method for measuring fractal dimension is tested by applying it to the above synthetic fractal signals. In the chapter, various covering methods—a class of general and efficient approaches to compute the fractal dimension of arbitrary fractal signals—are discussed. The morphological covering approach to find the fractal dimension of 2D signals are described in the chapter followed by the fractal binary image modeling using collages.