The power of 3D fractal dimensions for comparative shape and structural complexity analyses of irregularly shaped organisms

Abstract

The increasing interest in 3D morphological approaches for addressing evolutionary and ecological questions has driven the need for reliable, efficient, and easily accessible methods for shape quantification and complexity characterization. This applies in particular to irregularly shaped organisms, such as stony corals, which are challenging to study using traditional morphometric analyses. A potential approach to assess shape and spatial complexity of 3D documented organisms are 3D fractal dimension analyses that combine information from various spatial scales, thus enabling a holistic shape quantification. Therefore, the latest 3D scanning technology and fractal dimension analyses were used to test the performance of this novel combination of methods for quantifying inter‐ and intraspecific differences in shape and spatial complexity of highly irregularly shaped organisms. As model taxa, six widespread species of stony corals were used, belonging to the genera Acropora, Pocillopora and Porites. First, high‐resolution shape models of coral colonies were generated using 3D structured light scanning. Then the univariate fractal dimension D and the multivariate multiscale fractal dimension as well as two traditional univariate morphological measures (surface–volume ratio and rugosity) were calculated. Finally all parameters were tested for significant interspecific and intraspecific differences, and the discriminative power of the two fractal dimension and the two traditional measures were assessed. The results show that 3D shape analyses based on fractal dimensions can be used to quantify both inter‐ and intraspecific variations in irregularly shaped organisms such as corals. Compared to traditional methods, fractal dimensions performed at least as good at the interspecific level and considerably better at the intraspecific level. This study demonstrates that 3D fractal dimension analyses are an efficient and easily applicable method for shape quantification and complexity characterization of irregularly shaped organisms, such as stony corals. Given the specific properties of fractal dimensions, they may either serve as an alternative complexity measure or as an additional characteristic for studying irregularly shaped organisms. Ultimately, fractal dimensions may provide a more integrative understanding of morphological variation within an ecological and evolutionary context, and could open up new opportunities for semi‐automatic or automatic species determination based on 3D morphological images.

Relevance of coral geometry in the outcomes of the coral-algal benthic war

“…Higher fractal dimensions lead to more convoluted surfaces or perimeters with a larger number of wrinkles and textures that increase the effective surface and perimeter of corals (Falconer, 2003; Okie, 2013). Previous studies found fractality among corals at different scales (Basillais, 1997; Bradbury & Reichelt, 1983; Knudby & LeDrew, 2007; Martin-Garin et al, 2007; Mark, 1984; Purkis et al, 2006; Reichert et al, 2017; Zawada & Brock, 2009), but the measurements at the coral colony scale of interest in the present study were inconclusive (Mark 1984).…”

“…Corals with mean fractal dimensions smaller than two, DS < 2, displayed surfaces with holes and large peninsulas, while corals with fractal dimensions larger than two, DS > 2, displayed more compact surfaces with richer and more wrinkled textures (Figure 3c). Additional geometric metrics such as rugosity, vector dispersion, multivariate multiscale fractal dimension, and multifractal analysis (Reichert et al, 2017; Young et al, 2017, Chakraborty et al, 2016) might be necessary to refine the coral geometric analysis presented here.…”

Integrating Three-Dimensional Benthic Habitat Characterization Techniques into Ecological Monitoring of Coral Reefs

“…Fractal dimension has been previously suggested as a matric for quantifying the shapes of corals or structural complexity of coral reefs [15,[34][35][36]. Fractal dimension (D) was calculated as D = 2 − slope logS(δ)/log(δ), in which δ is a resolution of the raster and S(δ) is the 3D surface area at the given resolution [15].…”

Quantifying Coral Morphology

“…However, while these results highlight differences between growth forms across a range of processes, they are unable to directly assess process-based hypotheses for the observed differences, nor can the results be generalised to other growth forms or taxa with similar morphological adaptations but different overall morphology (e.g., sponges, hydrozoans, algae, plants, etc). As such, recent studies have begun to explore techniques for quantifying and comparing the three-dimensional shape of corals (Lavy et al 2015;Reichert et al 2017;Bythell, Pan, and Lee 2001;House et al 2018). We build on this work to develop a quantitative schema for coral morphology via variables that capture shape variation.…”