G 2 -SLAM: Geometrically-guided 3D Gaussian Splatting SLAM via robust loop validation and anisotropic shaping

Monocular 3D Gaussian Splatting (3DGS) SLAM estimates camera motion and dense scene appearance from a single RGB stream, but sparse views make both estimates sensitive to depth uncertainty and false loop closures. We present G²-SLAM, a geometry-guided monocular 3DGS SLAM system that combines depth-Hessian-based loop validation with Hessian-direction-conditioned, image-edge-aware depth regularization. The loop module adds a second-order geometric consistency test before Sim(3) optimization, while the regularizer suppresses intra-region depth fluctuations without strongly smoothing across structural boundaries or requiring pseudo-camera rendering. On Replica, G²-SLAM remains competitive with strong dense SLAM baselines in a high-quality synthetic setting. Across six selected LLFF scenes, it improves the average PSNR from 26.05 to 26.55 dB over HI-SLAM2; on the 27-image room scene, the maximum per-scene gain is 0.86 dB. A short-duration RTK-GPS outdoor study provides a proof-of-concept evaluation under real-world motion. The online tracking component reaches workstation-level frame rates, whereas full global refinement remains offline and is not intended for real-time edge deployment. The project page and public code snapshot are available at https://github.com/galahies/G--SLAM.