The Hausdorff measure functions: A new way to characterize fractal sets

A practical method to experimentally evaluate the Hausdorff dimension: An alternative phase-transition-based methodology

“…The HMSF method has been introduced in our previous papers, [25][26][27] motivated by our concern to distinguish between sets having the same fractal dimension. HMSF can be considered as a combination of two very useful tools: the autocorrelation function, which is dedicated to measure the similarity of the samples' distribution, and the Hausdorff measure and dimension, which are used for complexity quantification.…”

“…This method is based on our previously introduced Hausdorff measure spectrum function (HMSF). [25][26][27] It has the main advantage of providing a very accurate estimate of the Hausdorff dimension without resorting to classical linear regression fit procedure in logarithmic representations commonly used for fractal (box) dimension measurement. This paper is organized as follows.…”

Various Mathematical Approaches to Extract Information from Textures of Increasing Complexities

“…We have considered eight fat fractals having the following values for the parameter a : 3, 4, 5, 6, 8, 10, 12 and 15. The autocorrelation function of these fat fractals is a more irregular curve, preventing thus a valuable application of the Fourier transform as was the case above [15,9]. The irregularity of those curves suggests the utility of the fractal dimension usually used to quantify complexity.…”

“…The method we propose here does not rely on the exact Hausdorff measure definition but substitutes for it, an appropriate approximative quantity. Indeed, except for uniform Cantor-type sets whose HMSF can theoretically be determined [6,7,9,16], it is necessary to use as a substitute, the measure spectrum function, which can be defined in the following way: given a fractal set F , we associate it with a family of covering sets {F i,n }, where n = 0, 1, 2, · · · and i ∈ I n . The set i ∈ I n is the set of indexes whose corresponding intervals F i,n are nonempty.…”

“…The major use of fractal analysis is to measure this complexity through fractal dimensions and other related indexes. However, inadequacy of traditional methods when processing complex systems and the known limitations of popular fractal methods led us to investigate, in parallel, these two different ways of information processing, and to combine them in order to create more powerful and less degenerate methods [6,7,8,9]. This resulted in an extension of classical methods, making them applicable within a nowadays context knowledge and in a benefit of fractal analysis from these classical and very popular methods [7].…”