Man-made objects usually exhibit descriptive curved features (i.e., curve networks). The curve network of an object conveys its high-level geometric and topological structure. We present a framework for extracting feature curve networks from unstructured point cloud data. Our framework first generates a set of initial curved segments fitting highly curved regions. We then optimize these curved segments to respect both data fitting and structural regularities. Finally, the optimized curved segments are extended and connected into curve networks using a clustering method. To facilitate effectiveness in case of severe missing data and to resolve ambiguities, we develop a user interface for completing the curve networks. Experiments on various imperfect point cloud data validate the effectiveness of our curve network extraction framework. We demonstrate the usefulness of the extracted curve networks for surface reconstruction from incomplete point clouds.

Intrinsic Delaunay triangulation (IDT) naturally generalizes Delaunay triangulation from R^2 to curved surfaces. Due to many favorable properties, the IDT whose vertex set includes all mesh vertices is of particular interest in polygonal mesh processing. To date, the only way for constructing such IDT is the edge-flipping algorithm, which iteratively flips non-Delaunay edges to become locally Delaunay. Although this algorithm is conceptually simple and guarantees to terminate in finite steps, it has no known time complexity and may also produce triangulations containing faces with only two edges. This article develops a new method to obtain proper IDTs on manifold triangle meshes.We first compute a geodesic Voronoi diagram (GVD) by taking all mesh vertices as generators and then find its dual graph. The sufficient condition for the dual graph to be a proper triangulation is that all Voronoi cells satisfy the so-called closed ball property. To guarantee the closed ball property everywhere, a certain sampling criterion is required. For Voronoi cells that violate the closed ball property, we fix them by computing topologically safe regions, inwhich auxiliary sites can be addedwithout changing the topology of theVoronoi diagram beyond them.Given a meshwith n vertices, we prove that by adding at most O(n) auxiliary sites, the computed GVD satisfies the closed ball property, and hence its dual graph is a proper IDT. Our method has a theoretical worst-case time complexity O(n^2 + tn log n), where t is the number of obtuse angles in the mesh. Computational results show that it empirically runs in linear time on real-world models.

Non-rigid registration of 3D shapes is an essential task of increasing importance as commodity depth sensors become more widely available for scanning dynamic scenes. Non-rigid registration is much more challenging than rigid registration as it estimates a set of local transformations instead of a single global transformation, and hence is prone to the overfitting issue due to underdetermination. The common wisdom in previous methods is to impose an l2-norm regularization on the local transformation differences. However, the l2-norm regularization tends to bias the solution towards outliers and noise with heavy-tailed distribution, which is verified by the poor goodness- of-fit of the Gaussian distribution over transformation differences. On the contrary, Laplacian distribution fits well with the transformation differences, suggesting the use of a sparsity prior. We propose a sparse non-rigid registration (SNR) method with an l1-norm regularized model for transformation estimation, which is effectively solved by an alternate direction method (ADM) under the augmented Lagrangian framework. We also devise a multi-resolution scheme for robust and progressive registration. Results on both public datasets and our scanned datasets show the superiority of our method, particularly in handling large-scale deformations as well as outliers and noise.

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