Multifractal Properties of a Closed Contour: A Peek beyond the Shape Analysis

Abstract

In recent decades multifractal analysis has been successfully applied to characterize the complex temporal and spatial organization of such diverse natural phenomena as heartbeat dynamics, the dendritic shape of neurons, retinal vessels, rock fractures, and intricately shaped volcanic ash particles. The characterization of multifractal properties of closed contours has remained elusive because applying traditional methods to their quasi-one-dimensional nature yields ambiguous answers. Here we show that multifractal analysis can reveal meaningful and sometimes unexpected information about natural structures with a perimeter well-defined by a closed contour. To this end, we demonstrate how to apply multifractal detrended fluctuation analysis, originally developed for the analysis of time series, to an arbitrary shape of a given study object. In particular, we show the application of the method to fish otoliths, calcareous concretions located in fish's inner ear. Frequently referred to as the fish's “black box", they contain a wealth of information about the fish's life history and thus have recently attracted increasing attention. As an illustrative example, we show that a multifractal approach can uncover unexpected relationships between otolith contours and size and age of fish at maturity.

Responses of small mammals to habitat characteristics in Southern Carpathian forests

“…Neither species richness nor species abundance changed along the elevational gradient in our research area when also considering yearly fluctuations and habitat characteristics (Table 2 ), and our result is in contradiction with the pattern frequently described for mountains worldwide 58 , 59 , including the Eastern Tatras 31 , which shows a reduced species richness with the increase in elevation. But on the other hand, we found species composition to be affected by elevation, with A. flavicollis responding negatively and S. araneus positively.…”

Multifractality and Network Analysis of Phase Transition

“…The MF-DFA method is the generalization of detrended fluctuation analysis (DFA) [5] and has been widely applied to characterize the properties of various non-stationary time series in different fields such as financial market [23–32], physiology [33], biology [34], traffic jamming [35], geophysics [36] and neuroscience [37]. …”

Interplay of model ingredients affecting aggregate shape plasticity in diffusion-limited aggregation

“…We analyzed these fluctuations using the recently proposed traveling observer contour decomposition [25]. The idea behind this technique is to map the contour of the observed structure onto a time series of radial distances from the actual contour, as seen by a virtual observer traveling along the zeroth harmonic (best fit circle), at constant angular speed.…”

“…One interesting possibility to extract information from the contour fluctuations is to deploy a multifractal analysis of the above time series (see, e.g., [25] and references therein). Although that method may reveal a plethora of information, the link between the observed physical phenomena and the multifractal parameters is often nontrivial to make.…”