In this study, we address the challenge of constructing continuous three-dimensional (3D) models that accurately represent uncertain surfaces, derived from noisy and incomplete LiDAR scanning data. Building upon our prior work, which utilized the Gaussian Process (GP) and Gaussian Mixture Model (GMM) for structured building models, we introduce a more generalized approach tailored for complex surfaces in urban scenes, where four-dimensional (4D) GMM Regression and GP with derivative observations are applied. A Hierarchical GMM (HGMM) is employed to optimize the number of GMM components and speed up the GMM training. With the prior map obtained from HGMM, GP inference is followed for the refinement of the final map. Our approach models the implicit surface of the geo-object and enables the inference of the regions that are not completely covered by measurements. The integration of GMM and GP yields well-calibrated uncertainty estimates alongside the surface model, enhancing both accuracy and reliability. The proposed method is evaluated on the real data collected by a mobile mapping system. Compared to the performance in mapping accuracy and uncertainty quantification of other methods such as Gaussian Process Implicit Surface map (GPIS) and log-Gaussian Process Implicit Surface map (Log-GPIS), the proposed method achieves lower RMSEs, higher log-likelihood values and fewer computational costs for the evaluated datasets.

翻译：本研究旨在解决从含噪且不完整的LiDAR扫描数据中，构建精准表征不确定曲面的连续三维模型的挑战。在前期利用高斯过程（GP）和高斯混合模型（GMM）构建结构化建筑模型工作的基础上，我们提出了一种更为通用的方法，专门适用于城市场景中的复杂曲面，其中采用了四维GMM回归及含导数观测的GP。通过分层GMM（HGMM）优化GMM分量数量并加速训练过程。在通过HGMM获得先验地图后，进一步采用GP推理对最终地图进行细化。我们的方法对地理物体的隐式曲面进行建模，并能够推断未被测量完全覆盖的区域。GMM与GP的集成在曲面模型生成的同时，提供了校准良好的不确定性估计，从而提升了精度与可靠性。该方法在移动测绘系统采集的真实数据上进行了评估。与高斯过程隐式曲面地图（GPIS）及对数高斯过程隐式曲面地图（Log-GPIS）等其他方法在制图精度和不确定性量化方面的性能相比，所提方法在所评估的数据集上实现了更低的均方根误差、更高的对数似然值以及更低的计算成本。