實驗使用橢球面、環面(torus)資料集，以及 David、Bunny、Hand 等複雜點雲進行測試。在橢球面實驗中，本文將所提出的神經網路方法與三角網格曲率估計方法進行比較，並以高斯曲率與平均曲率的 RMSE 作為評估指標。結果顯示，三角網格方法會受到點數與局部網格品質影響，而所提出的自增廣模型在不同點雲解析度下皆能得到較穩定且較準確的曲率結果。訓練結果也顯示，適當的懲罰參數能改善隱函數微分的穩定性，進而提升曲率計算品質。;Three-dimensional point clouds are widely used in geometric modeling， 3D scanning， and computer vision. In addition to surface reconstruction， curvature is an important quantity for describing local geometric features. However， traditional curvature estimation methods often depend on triangle meshes and local vertex neighborhoods. When the point cloud is sparse or the mesh quality is poor， the estimated curvature may contain large errors or local oscillations.
This thesis uses neural networks to approximate smooth and differentiable surfaces from 3D point clouds. The input coordinates are mapped to a more stable representation， and multi-stage training is used to improve fitting accuracy. MLP and Gaussian RBF models with variable projection are first studied. A Self-Augmented Model is then introduced by adding the surface coordinate to the network input. This augmented input allows implicit differentiation to compute the derivatives required for curvature estimation. A penalty term is also added to improve derivative stability when the Jacobian is close to singular.
Experiments are performed on an ellipsoid， torus datasets， and several complex point clouds， including David， Bunny， and Hand. For the ellipsoid experiment， the proposed neural method is compared with a triangle-mesh-based curvature estimation method. The root mean squared errors of Gaussian curvature and mean curvature are used as evaluation metrics. The results show that the triangle mesh method is sensitive to point density and local mesh quality， while the proposed Self-Augmented Model gives more stable and accurate curvature results under different point-cloud resolutions. The study of the penalty parameters also shows that a suitable penalty setting improves the stability of implicit differentiation and the quality of curvature computation.
