Automated Box-Counting Fractal Dimension Analysis: Sliding Window Optimization and Multi-Fractal Validation

This paper presents a systematic methodology for identifying optimal scaling regions in segment-based box-counting fractal dimension calculations through a three-phase algorithmic framework combining grid offset optimization, boundary artifact detection, and sliding window optimization. Unlike traditional pixelated approaches that suffer from rasterization artifacts, the method used directly analyzes geometric line segments, providing superior accuracy for mathematical fractals and other computational applications. The three-phase optimization algorithm automatically determines optimal scaling regions and minimizes discretization bias without manual parameter tuning, achieving significant error reduction compared to traditional methods. Validation across the Koch curve, Sierpinski triangle, Minkowski sausage, Hilbert curve, and Dragon curve demonstrates substantial improvements: excellent accuracy for the Koch curve (0.11% error) and significant error reduction for the Hilbert curve. All optimized results achieve R2≥0.9988. Iteration analysis establishes minimum requirements for reliable measurement, with convergence by level 6+ for the Koch curve and level 3+ for the Sierpinski triangle. Each fractal type exhibits optimal iteration ranges where authentic scaling behavior emerges before discretization artifacts dominate, challenging the assumption that higher iteration levels imply more accurate results. Application to a Rayleigh–Taylor instability interface (D = 1.835 ± 0.0037) demonstrates effectiveness for physical fractal systems where theoretical dimensions are unknown. This work provides objective, automated fractal dimension measurement with comprehensive validation establishing practical guidelines for mathematical and real-world fractal analysis. The sliding window approach eliminates subjective scaling region selection through systematic evaluation of all possible linear regression windows, enabling measurements suitable for automated analysis workflows.