Positioning Fractal Dimension and Lacunarity in the IBSI Feature Space: Simulation With and Without Wavelets

Fractal dimension (Frac) and lacunarity (Lac) are frequently proposed as biomarkers of multiscale image complexity, but their incremental value over standardized radiomics remains uncertain. We position both measures within the Image Biomarker Standardisation Initiative (IBSI) feature space by running a fully reproducible comparison in two settings. In a baseline experiment, we analyze N=1000 simulated 64×64 textured ROIs discretized to Ng=64, computing 92 IBSI descriptors together with Frac (box counting) and Lac (gliding box), for 94 features per ROI. In a wavelet-augmented experiment, we analyze N=1000 ROIs and add level-1 wavelet descriptors by recomputing first-order and GLCM features in each sub-band (LL, LH, HL, and HH), contributing 4×(19+19)=152 additional features and yielding 246 features per ROI. Feature similarity is summarized by a consensus score that averages z-scored absolute Pearson and Spearman correlations, distance correlation, maximal information coefficient, and cosine similarity, and is visualized with clustered heatmaps, dendrograms, sparse networks, PCA loadings, and UMAP and t-SNE embeddings. Across both settings a stable two-block organization emerges. Frac co-locates with contrast, difference, and short-run statistics that capture high-frequency variation; when wavelets are included, detail-band terms from LH, HL, and HH join this group. Lac co-locates with measures of large, coherent structure—GLSZM zone size, GLRLM long-run, and high-gray-level emphases—and with GLCM homogeneity and correlation; LL (approximation) wavelet features align with this block. Pairwise associations are modest in the baseline but become very strong with wavelets (for example, Frac versus GLCM difference entropy, which summarizes the randomness of gray-level differences, with |r|≈0.98; and Lac versus GLCM inverse difference normalized (IDN), a homogeneity measure that weights small intensity differences more heavily, with |r|≈0.96). The multimetric consensus and geometric embeddings consistently place Frac and Lac in overlapping yet separable neighborhoods, indicating related but non-duplicative information. Practically, Frac and Lac are most useful when multiscale heterogeneity is central and they add a measurable signal beyond strong IBSI baselines (with or without wavelets); otherwise, closely related variance can be absorbed by standard texture families.