Adaptive triangular-mesh reconstruction by mean-curvature-based refinement from point clouds using a moving parabolic approximation

We use a moving parabolic approximation (MPA) to reconstruct a triangular mesh that approximates the underlying surface of a point cloud from closed objects. First, an efficient strategy is presented for constructing a hierarchical grid with adaptive resolution and generating an initial mesh from point clouds. By implementing the MPA algorithm, we can estimate the differential quantities of the underlying surface, and subsequently, we can obtain the local quadratic approximants of the squared distance function for any point in the vicinity of the target shape. Thus, second, we adapt the mesh to the target shape by an optimization procedure that minimizes a quadratic function at each step. With the objective of determining the geometrical features of the target surface, we refine the approximating mesh selectively for the non-flat regions by comparing the estimated curvature from the point clouds and the estimated curvatures computed from the current mesh. Finally, we present various examples that demonstrate the robustness of our method and show that the resulting reconstructions preserve geometric details.