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

Abstract

We introduce a methodology to estimate numerically the Hausdorff dimension of a geometric set. This practical method has been conceived as a subsequent tool of another context study, associated to our concern to distinguish between various fractal sets. Its conception is natural since it can be related to the original idea involved in the definitions of Hausdorff measure and Hausdorff dimension. It is based on the critical behavior of the measure spectrum functions of the set around its Hausdorff dimension value. We illustrate on several well-known examples, the ability of this method to accurately estimate the Hausdorff dimension. Also, we show how the transition property, exhibited by the quantities used as substitutes of the Hausdorff measure in the corresponding fractal dimension relationships, can be used to accurately estimate the fractal dimension. To show the potential of our method, we also report the results of Hausdorff dimension measurements on some typical examples, compared to a direct application of the scaling relation involved in the box-counting dimension definition.

Various Mathematical Approaches to Extract Information from Textures of Increasing Complexities

“…A first approximating quantity is similar to the one involved in the box-counting dimension: Q 1 (n, s) = N (n)ǫ s n , where N (n) is the number of nonempty boxes of coarse-graining elements of size ǫ n . A better approximating quantity is what we have previously named the adaptive coverings, noted Q 2 [16]. Indeed, we have successfully used this quantity to practically evaluate the Hausdorff dimension.…”

“…where s H is the Hausdorff dimension of the set F [16]. Fig.5 shows the HMSF of the Cantor fractal set as well as of a geometrical multifractal set.…”

“…Q 2 gathers into one element each group of joined coarse-graining elements, and redefines in this way another covering family (adaptive). It is usable when the condition that the maximum length of its adaptive covering goes to zero when the coarsegraining level n goes to infinity [16]. We believe that for advanced needs of practical problems, this approximation exercise should be continued in order to reach better estimations.…”

“…When dealing with a given fractal set, one uses a coarse-graining process to practically represent the set at different resolutions. If we use a coarse-graining whose elements are all of size ǫ n at the coarse-graining level n, we can define different quantities to approximate the Hausdorff measure [16]. A first approximating quantity is similar to the one involved in the box-counting dimension: Q 1 (n, s) = N (n)ǫ s n , where N (n) is the number of nonempty boxes of coarse-graining elements of size ǫ n .…”

“…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.…”