Mahalanobis-Constrained Center-Based Iterative Method for Gaussian Mixture Estimation

This paper studies fixed-point iteration methods within the contraction-mapping framework and develops a center-based iterative algorithm constrained by Mahalanobis-distance rejection. Building on Banach’s fixed-point theorem, we relate contractivity to the Jacobian norm and derive estimators for the order of convergence, showing the method exhibits linear convergence under the stated conditions. The algorithm modifies standard k-means by using conditional expectations for parameter updates and discarding low-probability tail points of Gaussian components via a p-quantile criterion of the chi-squared Mahalanobis distribution. The procedure is analyzed in both continuous (marginal Gaussian) and discrete noisy settings, with numerical experiments that quantify convergence behavior and robustness to outliers. A spherical clustering validity index is introduced to optimize the p-quantile for pattern detection. Applications to industrial-scene, RGB segmented images demonstrate effective detection of spherical patterns in highly noisy structures, illustrating the method’s practical potential for robust Gaussian mixture estimation and pattern recognition.