We advocate the use of point sets to represent shapes. We provide a definition of a smooth manifold surface from a set of points close to the original surface. The definition is based on local maps from differential geometry, which are approximated by the method of moving least squares (MLS). The computation of points on the surface is local, which results in an out-of-core technique that can handle any point set. We show that the approximation error is bounded and present tools to increase or decrease the density of the points, thus allowing an adjustment of the spacing among the points to control the error. To display the point set surface, we introduce a novel point rendering technique. The idea is to evaluate the local maps according to the image resolution. This results in high quality shading effects and smooth silhouettes at interactive frame rates

We introduce a robust moving least-squares technique for reconstructing a piecewise smooth surface from a potentially noisy point cloud. We use techniques from robust statistics to guide the creation of the neighborhoods used by the moving least squares (MLS) computation. This leads to a conceptually simple approach that provides a unified framework for not only dealing with noise, but also for enabling the modeling of surfaces with sharp features.Our technique is based on a new robust statistics method for outlier detection: the forward-search paradigm. Using this powerful technique, we locally classify regions of a point-set to multiple outlier-free smooth regions. This classification allows us to project points on a locally smooth region rather than a surface that is smooth everywhere, thus defining a piecewise smooth surface and increasing the numerical stability of the projection operator. Furthermore, by treating the points across the discontinuities as outliers, we are able to define sharp features. One of the nice features of our approach is that it automatically disregards outliers during the surface-fitting phase.

We present an anisotropic mesh denoising algorithm that is effective, simple and fast. This is accomplished by filtering vertices of the mesh in the normal direction using local neighborhoods. Motivated by the impressive results of bilateral filtering for image denoising, we adopt it to denoise 3D meshes; addressing the specific issues required in the transition from two-dimensions to manifolds in three dimensions. We show that the proposed method successfully removes noise from meshes while preserving features. Furthermore, the presented algorithm excels in its simplicity both in concept and implementation.

Polygonal meshes provide an efficient representation for 3D shapes. They explicitly captureboth shape surface and topology, and leverage non-uniformity to represent large flat regions as well as sharp, intricate features. This non-uniformity and irregularity, however, inhibits mesh analysis efforts using neural networks that combine convolution and pooling operations. In this paper, we utilize the unique properties of the mesh for a direct analysis of 3D shapes using MeshCNN , a convolutional neural network designed specifically for triangular meshes. Analogous to classic CNNs, MeshCNN combines specialized convolution and pooling layers that operate on the mesh edges, by leveraging their intrinsic geodesic connections. Convolutions are applied on edges and the four edges of their incident triangles, and pooling is applied via an edge collapse operation that retains surface topology, thereby, generating new mesh connectivity for the subsequent convolutions. MeshCNN learns which edges to collapse, thus forming a task-driven process where the network exposes and expands the important features while discarding the redundant ones. We demonstrate the effectiveness of MeshCNN on various learning tasks applied to 3D meshes.

Progressive point set surfaces (PPSS) are a multilevel point-based surface representation. They combine the usability of multilevel scalar displacement maps (e.g. compression, filtering, geometric modeling) with the generality of point-based surface representations (i.e. no fixed homology group or continuity class). The multiscale nature of PPSS fosters the idea of point-based modeling. The basic building block for the construction of PPSS is a projection operator, which maps points in the proximity of the shape onto local polynomial surface approximations. The projection operator allows the computing of displacements from smoother to more detailed levels. Based on the properties of the projection operator we derive an algorithm to construct a base point set.Starting from this base point set, a refinement rule using the projection operator constructs a PPSS from any given manifold surface.