Robust Image Representation of Cultural Heritage Patterns Using Lipschitz-Stable Quaternion Fractional Moments

Abstract

Quaternion Fractional Moment (QFM) descriptors are widely used in geometric pattern recognition due to their ability to encode multi-channel image information and exhibit invariance properties. However, their robustness under real-world acquisition variability, particularly photometric noise, remains insufficiently understood. Based on the Lipschitz stability theorem, which defines a strong, linear form of stability for dynamical systems, applied to one of our previous works, this article improves upon it by introducing a robustness-driven analysis framework that models feature extraction as a stochastic process, where bounded spatio-temporal perturbations generate multiple descriptor realizations for each pattern. Descriptor robustness is directly quantified in feature space using a novel normalized dispersion stability metric. Furthermore, a Lipschitz stability theorem is formally established and proved, providing theoretical guarantees of descriptor robustness under bounded perturbations. Experiments conducted on Moroccan–Andalusian geometric patterns with p4m and p6m symmetry groups demonstrate that the proposed framework achieves high intrinsic stability (σnorm = 0.042 ± 0.010), while preserving state-of-the-art classification performance (Macro-F1 = 0.589 vs. 0.570 under σ = 0.05 noise). These results confirm that robustness is an intrinsic and measurable property of the descriptor, independent of classifier performance. The proposed framework provides both theoretical and methodological support for reliable geometric pattern recognition in cultural heritage imaging under real-world conditions.