Voxel counting and fractal geometry

Background

The Minkowski–Bouligand dimension, or commonly known as the box-counting method, is a way of sampling complex patterns in a given object by breaking the object into smaller and smaller boxes while extracting the rate of change. Here because we are dealing with point clouds with three dimensions, voxels of a given volume are used instead of boxes.

In practice, a given point cloud can be covered using a single large voxel (N₁ = 1) of size S₁, but as S is reduced in volume (S₁ > S₂ … > S_n), the number of voxels required (N > 1) to cover it will increase. Since N increases as a power function, it is positive to estimate the Minkowski-Boulingand dimension by a log-log linear model. This can be done by rTLS following the next steps.

Steps

1. Read the file

First, we need to read the point clouds in R. This can be done using the function fread of data.table for a fast reading or any other traditional approach for reading tables.

For this example specifically, we will use a point cloud already embedded in rTLS, pc_tree. This point cloud was created from a group of scans of a tree. It has a coarse resolution of 0.05 m for TLS standards in order to integrate it as an example into the package. The file can be load following:

library(rTLS)
data(pc_tree)

#Visualize the point cloud
rgl::plot3d(pc_tree)

2. Run voxel_counting

The following step is running voxel_counting. In order to estimate the fractal geometry we need to consider a series of requirements. Specifically we need to consider the minimum and maximum voxel edge length (lower and upper cut-off), and the number of scales to perform. These are important steps since if the voxel edge length is lower than the minimum distance between points or higher than the largest range of XYZ we will loss the power-law relationship between the number of voxels and the voxel size. Using voxel_counting we can define both requirements using voxel.sizes manually, or if this is null, we can define the number of scales to use by length.out and selecting the minimum voxel size by min.size. If voxel.sizes = NULL, the function selects the maximum voxel size based on the largest range of XYZ. Since the resolution of the point cloud is 0.05 m we can use this as a value value of minimum voxel size. This can be performed following:

fractals <- voxels_counting(pc_tree, min.size = 0.05)

Other ways to select the min.size is estimating the minimum distance between points. This can be done also using ‘min_distance’ function also from rTLS.

#Regular processing
low_size <- min_distance(pc_tree)

#or using parallel processing
low_size <- min_distance(pc_tree, threads = 4)

3. Estimation of the fractal geometry by voxel counting

Using the outputs of voxel_counting is then possible to estimate the fractal geometry. As mentioned before, this can be done by fitting a log-log regression between N (fractals$N_voxels) and S (fractals$Voxel.size^3). In practice, the Minkowski–Bouligand dimension is estimated using 1/S in order to obtain positive slopes. This can be done following:

dimention <- lm(log10(N_voxels) ~ log10(1/(Edge.X*Edge.Y*Edge.Z)), data = fractals)

The coefficients from this relationship represent the fractal dimension (d_MB;dimention$coefficients[2]) and the intercept (intercept_MB;, dimention$coefficients[1]). In general, the resulting d_MB may have a value between 0 and 1, where values close to 1 represents a tree that uniformly occupies their 3D space (such as a Menger sponge with the greatest surface‐to‐volume ratio), while values close to 0 represents a cylindrical tree (such as a pole‐like object) or a point cloud with two points. On the other hand, the intercept_MB may have positive and negative values, where high values tend to be associated with large objects that require several voxels to fill at different scales.