Persistent Homology Analysis of AI-Generated Fractal Patterns: A Mathematical Framework for Evaluating Geometric Authenticity

Abstract

We present a mathematical framework for analyzing fractal patterns in AI-generated images using persistent homology. Given a text-to-image mapping (Formula presented.), we demonstrate that the persistent homology groups (Formula presented.) of sublevel set filtrations (Formula presented.) characterize multi-scale geometric structures, where (Formula presented.) is the grayscale intensity function of a generated image. The primary challenge lies in quantifying self-similarity in scales, which we address by analyzing birth–death pairs (Formula presented.) in the persistence diagram (Formula presented.). Our contribution extends beyond applying the stability theorem to AI-generated fractals; we establish how the self-similarity inherent in fractal patterns manifests in the persistence diagrams of generated images. We validate our approach using the Stable Diffusion 3.5 model for four fractal categories: ferns, trees, spirals, and crystals. An analysis of guidance scale effects (Formula presented.) reveals monotonic relationships between model parameters and topological features. Stability testing confirms robustness under noise perturbations (Formula presented.), with feature count variations (Formula presented.). Our framework provides a foundation for enhancing generative models and evaluating their geometric fidelity in fractal pattern synthesis. © 2024 by the authors.