3DGA: Topology-Aware 3D Gaussian Head Avatars with Barycentric Parameterization and Hierarchical Densification

3D Gaussian Splatting has demonstrated remarkable performance in novel view synthesis tasks. However, when applied to controllable avatar reconstruction, existing mesh-embedded approaches suffer from a fundamental yet overlooked challenge: geometric consistency during optimization. Current methods exhibit severe binding instabilities where Gaussian points detach from their assigned mesh patches, leading to visual artifacts. To address this challenge, this paper proposes 3DGA, a principled framework that establishes theoretical guarantees for geometric consistency through constrained barycentric parameterization and introduces semantic-aware resource allocation for efficient avatar generation. Specifically, we design a geometric parameterization mechanism that combines barycentric coordinates and normal displacement, enabling each Gaussian point to be precisely anchored to mesh patches via barycentric coordinates while supporting controlled displacement along the normal direction. This design not only achieves flexible surface-like deformation but also allows Gaussian points to perform shell adjustments while maintaining homomorphic motion with the driving mesh. As a result, the controllability of geometric deformation is significantly enhanced. Furthermore, we propose a hierarchical densification strategy based on facial semantic awareness, which adaptively adjusts the densification thresholds of Gaussian points in different facial regions. This ensures high detail quality in visually critical areas while effectively suppressing the generation of redundant points. The strategy not only improves rendering efficiency but also further optimizes the representation accuracy of facial regions. The experimental results indicate that our method achieves comparable or even superior reconstruction quality compared to state-of-the-art approaches.