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.
Highlights
The objects of classical shape analysis, such as Fourier analysis, wavelet analysis, curvature-based analysis, and geodesic curve analysis [1, 2], are composed of compact differentiable manifolds, smooth curves or surfaces that include their boundaries
In the case of fractal theory, roughness is considered as the main feature evaluated, since it captures the complexity of the shape in terms of the level of protrusions and cavities at different scales, rather than shape in the sense of morphometry. This characteristics is important because variations in the boundary of a natural structure during growth is a response to (i) external boundary conditions and (ii) the internal mechanisms of the growth process
Tel et al [11] proposed an alternative method to solve the first problem, the second issue still remains problematic. These methods assume the contour is a geometrical fractal, and important fine fluctuations around the quasi-one-dimensional structure of the contour perimeter may be ignored. To overcome these technical problems, we propose here a new technique to investigate whether fluctuations of the contour can reveal more information than its bare morphological appearance, combining Regular Fourier Analysis (RFA)
Summary
The objects of classical shape analysis, such as Fourier analysis, wavelet analysis, curvature-based analysis, and geodesic curve analysis [1, 2], are composed of compact differentiable manifolds, smooth curves or surfaces that include their boundaries. In the case of fractal theory, roughness is considered as the main feature evaluated, since it captures the complexity of the shape in terms of the level of protrusions and cavities at different scales, rather than shape in the sense of morphometry This characteristics is important because variations in the boundary of a natural structure during growth is a response to (i) external boundary conditions (surface interaction) and (ii) the internal mechanisms of the growth process. Tel et al [11] proposed an alternative method (the ‘‘Generalized Sand Box Method’’) to solve the first problem, the second issue still remains problematic These methods assume the contour is a geometrical fractal, and important fine fluctuations around the quasi-one-dimensional structure of the contour perimeter may be ignored. These changes are reflected in the formation of micro- and macro-structures around the primordium [17], the chemical composition, thickness, and periodicity of formation, which are correlated with historic events and with the age of the fish [16]
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