In recent years, rapid coral bleaching and extinction have led to numerous conservative efforts geared towards managing endangered reefs, which provides habitat, shelter, and nursing grounds for a wide range of tropical and subtropical marine life. However, these efforts are often marred by the most vital issue of them all: the delineation of different coral species. The current method for identifying corals relies on researchers to first categorize corals into discrete categories based on features of the skeleton known as morphological traits, then identify the correct species within the category. The challenge often comes at the first step, since the morphological trait approach neither accounts for a single species either having more than one discrete morphological trait nor does it assume different morphological traits due to exposure to various environmental factors (phenotypic plasticity).

In bullets, misidentification (which leads to miscommunication later down the line) can be due to:

• High level of phenotypic plasticity: one single species may assume different morphologies due to exposure to various environmental factors, leading to a single species being categorized under different families
• Crossbreeding leads to mixed traits: different ‘families’ can breed with each other, leading to corals with mixed characteristics that do not fit neatly into predefined categories

This article addresses the challenge of identifying and capturing a colony’s structural complexity by proposing a promising continuous trait: fractal dimension. Additionally, we introduce a toolkit to aid in the calculation of fractal dimension, providing researchers with a reliable tool for describing and differentiating coral colonies.

Why Fractal Dimension: The Role of Fractal Geometry in Nature

Fractal geometry is the mathematical concept of repeating patterns that exist within a shape or structure. To quote Benoit Mandelbrot, the father of fractal geometry, a fractal is a “Rough or fragmented geometric shape that can be split into parts each of which is (at least approximately) a reduced-size copy of the whole”. Nature, with its plethora of irregular objects and organisms, surprisingly obeys fractal geometry. In fact, many objects cannot easily be described by classical geometry.

Take trees for example: as a tree grows, it produces branches that split into smaller and smaller branches, creating a repeating pattern of self-similarity. This pattern can be seen at all scales, from the branching of the trunk and main limbs to the tiniest twigs and leaves.

Many industries are beginning to realize the importance of this method, some example uses are highlighted below:

• Describing ecological systems (population, community, landscape)
• Insect movements
• Image process/analysis
• Chaotic trajectories
• Arteries (blood movement path)

This article showcases a novel approach to incorporating fractal geometry in distinguishing coral structures: using a fractal dimension, the foundational measure.

Fractal Dimension

In fractal geometry, D is a basic parameter used to characterize the structural complexity of fractals (Mandelbrot, 1983). We can begin to understand fractal dimension D through the world we’re accustomed to. An ideal topological line has dimension D=1, the ideal plane has D=2, and the ideal space has D=3. Dimension can be defined by the following equation, where N is the number of repetitions an object will undergo, and s is the number of sides to the object:

Find below the illustration of how each dimension is calculated given N and s.

But what about a jagged line? Or a convoluted surface? Fractal geometry dictates that it actually occupies a non-integer dimension. A jagged line, depending on how irregular it is, can occupy between dimensions of 1 (less irregular) or 2 (highly irregular). The same logic applies to a convoluted surface.

There are several advantages in using fractal dimension to describe an object. Fractal dimension, unlike traditional measures, picks up on small pattern changes and is both orientation and size invariant. This makes re-scaling and re-orienting unnecessary.

• Orientation invariant
• Scale invariant
• Highly sensitive to structural changes in the object

Calculation methods & examples:

Currently, the most widely used methods to calculate fractal dimension are the Bouligand-Minkowski method and the Cube Counting method. The Bouligand-Minkowski method is explored in an earlier paper by Reichert et. al in 2017. In this paper, we will apply fractal dimension using the Cube Counting method. The fractal dimension equation is illustrated below:

We can quickly double check this equation by taking the limit and observe that it converges to equation (1). One would expect in the trivial case that s is completely smooth (e.g. a smooth surface).

Now, we are equipped to break this down into executable calculation methods:

1. Measure how many boxes (N) of specified length (s) are required to cover the perimeter
2. Scale down (lower the s, slowly making our way for s to converge to 0) and measure again
3. Repeat steps 1-2 multiple times
4. Create log-log plot of number of boxes (N) vs scale (s)
5. Find D, the slope of the plot

Below is the a pictorial illustration:

Materials & Methods

Materials:

This paper utilizes 3D scans of 289 corals from the NTU Coral Museum. The scans are compiled into .obj files for further analysis.

Methods:

There are three main steps to the methods:

1. .obj file modifications (data cleaning)
2. Cube counting & data collection
3. Data analysis

Post-processing & modifications

Due to the way the corals are obtained (they come in fragmented pieces), there are some areas of the coral that are not facing their environment. In other words, the smooth, broken surfaces are not what we want to include in our data collection because they do not accurately describe the structure of our coral. We have removed those broken surfaces from our data set to ensure an accurate analysis.

Since fractal dimension possesses rotation invariance and scale invariance, prior normalization and reorientation are unnecessary (Reichert et al., 2017).

Cube counting & data collection

We then created a cube counting algorithm in Python (attached in the toolkit below). The program performs the following steps to obtain the fractal dimension:

1. Place the 3D object in a 3D space with box grids of length s. 
2. Count the number of boxes Ns containing the vertices of the object. 
3. Repeat steps 1-2 a number of times with box grids of different lengths s.
4. Plot the curve of log(s) vs log (Ns).
5. Fit a line to the curve.
6. Obtain the fractal dimension D by calculating the slope of the fitted line.

In addition, the program calculates the surface area, volume, bounding length, bounding width, and bounding height of the corals.

For each colony, the 3D Coral Analysis Toolbox provides the sets N(s) and s, as well as the log-log plot along with the fitted line indicating the slope for the fractal dimension.

Other methods: Minkowski Bouligand Method

As mentioned earlier, a 2017 paper has examined the fractal dimension of corals via the Minkowski-Bouligand method. That paper employs a ball-dilation method, where each object’s fractal dimension is calculated by placing a ball of radius r at each vertex, recording the V(r) of the influence volume, then dilating the radii and repeating the same measurement. As the radius increases, interactions between spheres increase, and a characteristic influence volume begins to form. Similar to the box-counting method, one produces a log-log plot of V(r) vs r and determines the fractal dimension through the slope of the fitted line.

Results

The results section is currently under construction. We’ve received some feedback from our reviewers that we’re eager to implement. Please check back later for the updated version!