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 boundary artifact detection, sliding window optimization, and grid offset 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 Koch curves, Sierpinski triangles, Minkowski sausages, Hilbert curves, and Dragon curves demonstrates substantial improvements: excellent accuracy for Koch curves (0.11% error) and significant error reduction for Hilbert curves. All optimized results achieve R2≥0.9988. Iteration analysis establishes minimum requirements for reliable measurement, with convergence by level 6+ for Koch curves and level 3+ for Sierpinski triangles. 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. This work provides objective, automated fractal dimension measurement with comprehensive validation establishing practical guidelines for mathematical 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.