A Fractional-Order Spatiotemporal Unified Energy Framework for Non-Repetitive LiDAR Point Cloud Registration

Non-repetitive scanning LiDARs provide high coverage yet exhibit irregular sampling patterns, which destabilize local features and correspondences. To address this, we propose a novel spatiotemporal unified energy framework that integrates fractional calculus into rigid pose estimation. Spatially, we introduce a Riesz fractional regularization term to impose non-local smoothness constraints on the residual field, mitigating structural inconsistencies. Temporally, we design a Grünwald–Letnikov fractional dynamics solver that leverages long-memory effects of historical gradients to reduce the risk of being trapped in local minima. Comparative experiments on the Stanford 3D, MVTec ITODD, and HomebrewedDB (HB) datasets demonstrate that our method significantly outperforms state-of-the-art geometric and learning-based approaches. Specifically, it maintains a success rate exceeding 90% even under severe sampling perturbations where traditional methods fail. Ablation studies further validate that the introduction of non-local spatial constraints and historical gradient memory significantly reshapes the energy landscape, ensuring robust convergence. This work provides a rigorous theoretical foundation for applying fractional operators to point cloud processing.