Performance Assessment of Sampling Methods for the Box-Counting Method

The box-counting method is a fundamental tool for estimating fractal dimensions from 2D binary images, but its accuracy depends heavily on the sampling strategy used to select box sizes. Traditional approaches—such as geometric and arithmetic stepping—either produce sparse data at large scales or bias toward over-sampling at small scales. More recent strategies, including modified geometric and coverage-ratio stepping, improve pixel utilization but still suffer from low utilization rates and uneven data distribution on log–log plots. We propose an enhanced sampling strategy called the group competition step (GCS) method. GCS partitions the sampling range into subgroups and applies a localized competition mechanism based on coverage ratio values to produce a more balanced distribution of step sizes. The proposed method was evaluated alongside five existing sampling schemes across a dataset of 120 deterministic fractal images encompassing both geometric and non-geometric scales paired with fractional box-counting. On geometric benchmark images, GCS achieved superior performance with the lowest mean absolute error (MAE = 0.0062) and mean absolute percentage error (MAPE = 0.3561%), securing the lowest root mean square error (RMSE) in five out of eight benchmark geometries. Across the full 120-image test suite incorporating fractional remainder boundary counting, GCS maintained its performance lead (MAE = 0.0077, MAPE = 0.4569%) and recorded a Wilcoxon signed-rank test p-value of 0.5469, confirming statistical unbiasedness. These results demonstrate that GCS provides a more reliable framework for precise fractal-dimension estimation.