Results¶
Images of different computer-generated Lindenmayer systems (i.e., fractal trees) were analyzed using the open-source image analysis software FracLac. First, the fractal dimension was calculated using regular box-counting, and then with the differential box-count technique. The box-count analysis estimated the objects to have fractal dimensions between \( 1.81 \pm 0.056 < D_B < 1.90 \pm 0.66 \); however, the differential box-count mass dimension was found to be lower on average, with \( D_M = 1.60 \pm 0.10 \) (Table 1).
In Tables 1-X and Figures X-X, the observed mass dimensions of x-ray leaves, branches, and roots are statistically indistinguishable from the MST and fractional Brownian motion (fBm) predictions of \( \alpha = 3/2 \).
From Equation 18, a differential mass dimension for such an image is expected to equal \( 4/3 \) rather than \( 3/2 \) (see Supplementary Information).
Table 1: Fractal mass dimension \( d_m \pm \mu \text{SE} \) and coefficient of variation (CV)¶
where \( \sigma \) is the standard deviation over the mean number of pixels per box. The associated lacunarity \( \Lambda \) and CV are also reported.
| Fractal Type | Pixels | \( d_m \pm \mu \text{SE} \) | \( \mu r^2 \) | \( d_m (\frac{\sigma}{\mu}) \) | \( \Lambda \) | \( \Lambda (\frac{\sigma}{\mu}) \) |
|---|---|---|---|---|---|---|
| Peano Curve 1 (square) | 756,030 | 1.846 ± 0.100 | 0.9966 | 0.0111 | 0.0133 | 0.2128 |
| Peano Curve 2 (rounded) | 1,440,000 | 1.803 ± 0.109 | 0.9959 | 0.0161 | 0.0713 | 0.1348 |
| H-fractal | 3,932,289 | 1.760 ± 0.124 | 0.9945 | 0.0102 | 0.1142 | 0.0746 |
| Pythagoras Tree 1 | 5,000,000 | 1.607 ± 0.106 | 0.9953 | 0.0030 | 0.7003 | 0.0871 |
| Pythagoras Tree 2 | 393,216 | 1.655 ± 0.075 | 0.9976 | 0.0108 | 0.2267 | 0.0899 |
| Barnsley’s Fern | 180,000 | 1.576 ± 0.073 | 0.9973 | 0.0116 | 0.3787 | 0.0680 |
| Fibonacci Tree | 348,140 | 1.470 ± 0.074 | 0.9969 | 0.0118 | 0.8680 | 0.0657 |
Table 2: Observed local mass fractal dimension of five different types of leaves¶
The results were obtained using the FracLac Differential Box Count for a power series with an exponentially increasing box size factor of 0.1.
| Image | Pixels | \( d_M = \ln(\mu_\varepsilon) / \ln \varepsilon \) | \( \mu r^2 \) | \( \mu \text{SE} \) | \( CV (\frac{\sigma}{\mu}) \) |
|---|---|---|---|---|---|
| Coleus | 144,316 | 1.5384 | 0.9938 | 0.1008 | 0.0038 |
| Fig | 315,495 | 1.4844 | 0.9918 | 0.1123 | 0.0026 |
| Nasturtium | 1,115,114 | 1.5525 | 0.9969 | 0.0880 | 0.0010 |
| Ginkgo | 1,086,596 | 1.5135 | 0.9978 | 0.0705 | 0.0020 |
| Fern | 581,196 | 1.5083 | 0.9968 | 0.1268 | 0.0064 |
Table 3: Observed local mass fractal dimension of three branching networks¶
Results were obtained using FracLac Differential Box Count for a power series with an exponentially increasing box size factor of 0.1.
| Image | Pixels | \( d_M = \ln(\mu_\varepsilon) / \ln \varepsilon \) | \( \mu r^2 \) | \( \mu \text{SE} \) | \( CV (\frac{\sigma}{\mu}) \) |
|---|---|---|---|---|---|
| Single branch | 255,285 | 1.4946 | 0.9953 | 0.0850 | 0.0024 |
| Maple branches | 473,450 | 1.4775 | 0.9944 | 0.0920 | 0.0057 |
| Maple root | 617,312 | 1.4549 | 0.9970 | 0.0668 | 0.0016 |
Table 4: Aerial LiDAR Canopy Height Models (CHM) over various forest types¶
Results were obtained using FracLac Differential Box Count for a power series with an exponentially increasing box size factor of 0.1. The predicted mass fractal dimension is \( \frac{3}{2} \) or 1.5.
| Image | Pixels | Mass Dimension \( d_M \) | \( \mu r^2 \) | \( \mu \text{SE} \) | \( CV (\frac{\sigma}{\mu}) \) |
|---|---|---|---|---|---|
| Lowland Rainforest | 361,201 | 1.3313 | 0.9860 | 0.1969 | 0.0128 |
| Pine/Hardwood (South Carolina) | 362,403 | 1.5223 | 0.9898 | 0.1919 | 0.0135 |
| Sierra Madre Oaks (Arizona) | 362,404 | 1.4973 | 0.9886 | 0.1988 | 0.0116 |
| Western Ponderosa Pine (New Mexico) | 361,802 | 1.4566 | 0.9847 | 0.2252 | 0.0144 |
| Southwest Mixed Conifer (New Mexico) | 361,802 | 1.5319 | 0.9869 | 0.2190 | 0.0177 |
| Southwest Spruce-Fir (New Mexico) | 361,802 | 1.5355 | 0.9862 | 0.2255 | 0.0139 |