We propose a data-driven method for geometric feature detection in 3D scattered point clouds and its integration into an adaptive spline-approximation framework. For each point, we compute geometric quantities derived from the covariance matrix of its local neighborhood; these features are then used as inputs to a multi-layer perceptron applied independently to every point. This results in an effective technique for identifying points near edge features in the input model. The detected features guide an adaptive reconstruction scheme based on Locally Refined B-splines (LR B-splines), exploiting their anisotropic refinement capabilities while avoiding the cost of traditional error-estimation-driven adaptive loops. Numerical experiments demonstrate that the proposed approach improves the robustness of the feature detection and the quality of the reconstruction compared with other local spline refinement strategies.

We propose a new paradigm for scattered data fitting with adaptive spline constructions based on the key interplay between parameterization and adaptivity. Specifically, we introduce two novel adaptive fitting schemes that combine moving parameterizations with adaptive spline refinement for highly accurate CAD models reconstruction from real-world scattered point clouds. The first scheme alternates surface fitting and data parameter optimization. The second scheme jointly optimizes the parameters and the surface control points. To combine the proposed fitting methods with adaptive spline constructions, we present a key treatment of boundary points. Industrial examples show that updating the parameterization, within an adaptive spline approximation framework, significantly reduces the number of degrees of freedom needed for a certain accuracy, especially if spline adaptivity is driven by suitably graded hierarchical meshes. The numerical experiments employ THB-splines, thus exploiting the existing CAD integration within the considered industrial setting, nevertheless, any adaptive spline construction can be chosen.

Abstract Surface reconstruction from scattered point clouds is the process of generating surfaces from unstructured data configurations retrieved using an acquisition device such as a laser scanner. Smooth surfaces are possible with the use of spline representations, an established mathematical tool in computer-aided design and related application areas. One key step in the surface reconstruction process is the parameterization of the points, that is, the construction of a proper mapping of the 3D point cloud to a planar domain that preserves surface boundary and interior points. Despite achieving a remarkable progress, existing heuristics for generating a suitable parameterization face challenges related to the accuracy, the robustness with respect to noise, and the computational efficiency of the results. In this work, we propose a boundary-informed dynamic graph convolutional network (BIDGCN) characterized by a novel boundary-informed input layer, with special focus on applications related to adaptive spline approximation of scattered data. The newly introduced layer propagates given boundary information to the interior of the point cloud, in order to let the input data be suitably processed by successive graph convolutional network layers. We apply our BIDGCN model to the problem of parameterizing three-dimensional unstructured data sets over a planar domain. A selection of numerical examples shows the effectiveness of the proposed approach for adaptive spline fitting with (truncated) hierarchical B-spline constructions. In our experiments, improved accuracy is obtained, e.g., from 60% up to 80% for noisy data, while speedups ranging from 4 up to 180 times are observed with respect to classical algorithms. Moreover, our method automatically predicts the local neighborhood graph, leading to much more robust results without the need for delicate free parameter selection.

Abstract The use of rotation‐minimizing directed frames (RMDFs) for defining smoothly varying camera orientations along given spatial paths, in real or virtual environments, is proposed. A directed frame on a space curve${\bf r}(\xi)$ is a varying orthonormal basis$({\bf o},{\bf p},{\bf q})$ for ℝ3such that${\bf o}(\xi)={\bf r}(\xi)/|{\bf r}(\xi)|$ coincides with the unit polar vector from the origin to each curve point, and such a frame is rotation‐minimizing if its angular velocity vector${\bf \omega}$ maintains a vanishing component alongo. To facilitate computation of rotation‐minimizing directed frames, it is shown that the basic theory is equivalent to the established theory for rotation‐minimizing adapted frames—for which one frame vector coincides with the tangent${\bf t}(\xi)={\bf r}'(\xi)/|{\bf r}'(\xi)|$ at each curve point—if one replaces the given space curve by its anti‐hodograph (i.e., indefinite integral). A family of polynomial curves on which RMDFs can be computed exactly by a rational function integration, the Pythagorean (P) curves, is also introduced, together with algorithms for their construction. Copyright © 2009 John Wiley & Sons, Ltd.