Benchmark Instability in Fractal Dimension Estimation: Distortion Induced by Gray-Level Mapping in Synthetic FBM Images

Synthetic fractional Brownian motion (FBM) images serve as standard data for assessing fractal dimension (FD) estimation techniques. The synthesis pipeline transforms continuous FBM matrices into 8-bit grayscale images using linear mapping and quantization, a process often regarded as benign and rarely documented. If this procedure distorts FD estimations, algorithm comparisons based on such benchmarks merge performance with preprocessing errors. We demonstrate that grayscale conversion induces systematic distortion in FD estimation. Identical matrices were initially processed using three linear mapping strategies with varying emphases (direct, 3σ statistical, external-coefficient) and subsequently assessed with four FD algorithms (two DBC variants, Higuchi, PSD). The results demonstrate that linear mapping significantly alters FD estimates. In particular, the FD regression slope of the PSD approach notably decreased from 0.9955 (direct mapping) to 0.6400 (external-coefficient mapping), whereas Higuchi displayed negligible sensitivity. The near-perfect log-log linearity ruled out scaling breakdown. The mapping strategies produce distinct grayscale statistical properties that amplify quantization residuals. We present the relative residual to measure this amplification. The relative residual correlates strongly with FD deviations for DBC and PSD methods (r up to 0.97), while showing limited association with the Higuchi estimator. These results violate the assumption of benchmark neutrality in FBM-based FD assessment. FD benchmarking studies should, therefore, report preprocessing strategies and minimize relative residuals to ensure algorithmic comparability.