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

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.