Measuring the Fractal Dimension of Signals: Morphological Covers and Iterative Optimization

Fractals can model many classes of time-series data. The fractal dimension is an important characteristic of fractals that contains information about their geometrical structure at multiple scales. The covering methods are a class of efficient approaches to measure the fractal dimension of an arbitrary fractal signal by creating multiscale covers around the signal’s graph. In this paper we develop a general method that uses multiscale morphological operations with varying structuring elements to unify and extend the theory and digital implementations of covering methods. It is theoretically established that, for the fractal dimension computation, covering one-dimensional signals with planar sets is equivalent to morphologically transforming the signal by one-dimensional functions, which reduces the computational complexity from quadratic in the signal’s length to linear. Then a morphological covering algorithm is developed and applied to discrete-time signals synthesized from Weierstrass functions, fractal interpolation functions, and fractional Brownian motion. Further, for deterministic parametric fractals depending on a single parameter related to their dimension, we develop an optimization method that starts from an initial estimate and iteratively con- verges to the true fractal dimension by searching in the parameter space and minimizing a distance between the original signal and all such signals from the same class. Experimental results are also provided to demonstrate the good performance of the developed methods.