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 M:T→I, we demonstrate that the persistent homology groups Hk(t) of sublevel set filtrations {f−1((−∞,t])}t∈R characterize multi-scale geometric structures, where f:M(p)→R 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 (bi,di) in the persistence diagram PD(M(p)). 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 γ∈[4.0,8.0] reveals monotonic relationships between model parameters and topological features. Stability testing confirms robustness under noise perturbations η≤0.2, with feature count variations Δμf<0.5. Our framework provides a foundation for enhancing generative models and evaluating their geometric fidelity in fractal pattern synthesis.