SSD: Smooth Signed Distance Surface Reconstruction

We introduce a new variational formulation for the problem of reconstructing a watertight surface defined by an implicit equation, from a finite set of oriented points; a problem which has attracted a lot of attention for more than two decades. As in the Poisson Surface Reconstruction approach, discretizations of the continuous formulation reduce to the solution of sparse linear systems of equations. But rather than forcing the implicit function to approximate the indicator function of the volume bounded by the implicit surface, in our formulation the implicit function is forced to be a smooth approximation of the signed distance function to the surface. Since an indicator function is discontinuous, its gradient does not exist exactly where it needs to be compared with the normal vector data. The smooth signed distance has approximate unit slope in the neighborhood of the data points. As a result, the normal vector data can be incorporated directly into the energy function without implicit function smoothing. In addition, rather than first extending the oriented points to a vector field within the bounding volume, and then approximating the vector field by a gradient field in the least squares sense, here the vector field is constrained to be the gradient of the implicit function, and a single variational problem is solved directly in one step. The formulation allows for a number of different efficient discretizations, reduces to a finite least squares problem for all linearly parameterized families of functions, and does not require boundary conditions. The resulting algorithms are significantly simpler and easier to implement, and produce results of quality comparable with state‐of‐the‐art algorithms. An efficient implementation based on a primal‐graph octree‐based hybrid finite element‐finite difference discretization, and the Dual Marching Cubes isosurface extraction algorithm, is shown to produce high quality crack‐free adaptive manifold polygon meshes.     

Polyhedral Finite Elements Using Harmonic Basis Functions

Finite element simulations in computer graphics are typically based on tetrahedral or hexahedral elements, which enables simple and efficient implementations, but in turn requires complicated remeshing in case of topological changes or adaptive refinement. We propose a flexible finite element method for arbitrary polyhedral elements, thereby effectively avoiding the need for remeshing. Our polyhedral finite elements are based on harmonic basis functions, which satisfy all necessary conditions for FEM simulations and seamlessly generalize both linear tetrahedral and trilinear hexahedral elements. We discretize harmonic basis functions using the method of fundamental solutions, which enables their flexible computation and efficient evaluation. The versatility of our approach is demonstrated on cutting and adaptive refinement within a simulation framework for corotated linear elasticity.     

Polygonal mesh watermarking using Laplacian coordinates

We propose a watermarking algorithm for polygonal meshes based on the modification of the Laplacian coordinates. More specifically, we first compute the Laplacian coordinates (x,y,z) of the mesh vertices, then construct the histogram of the lengths of the (x,y,z) vectors, and finally, insert the watermark by altering the shape of that histogram. The watermark extraction is carried out blindly, with no reference to the host model. The proposed method is more robust than several existing high capacity watermarking algorithms. In particular, it is able to resist attacks such as translations, rotations, uniform scaling and vertex reordering, due to the invariance of the histogram of the Laplacian vector lengths under such transformations. Compared to the existing robust watermarking methods, our experiments show that the proposed method can better resist common mesh editing attacks, due to the good behaviour of the Laplacian coordinates under such operations.     

Isosurfaces Over Simplicial Partitions of Multiresolution Grids

We provide a simple method that extracts an isosurface that is manifold and intersection‐free from a function over an arbitrary octree. Our method samples the function dual to minimal edges, faces, and cells, and we show how to position those samples to reconstruct sharp and thin features of the surface. Moreover, we describe an error metric designed to guide octree expansion such that flat regions of the function are tiled with fewer polygons than curved regions to create an adaptive polygonalization of the isosurface. We then show how to improve the quality of the triangulation by moving dual vertices to the isosurface and provide a topological test that guarantees we maintain the topology of the surface. While we describe our algorithm in terms of extracting surfaces from volumetric functions, we also show that our algorithm extends to generating manifold level sets of co‐dimension 1 of functions of arbitrary dimension.     

Least Squares Subdivision Surfaces

The usual approach to design subdivision schemes for curves and surfaces basically consists in combining proper rules for regular configurations, with some specific heuristics to handle extraordinary vertices. In this paper, we introduce an alternative approach, called Least Squares Subdivision Surfaces (LS), where the key idea is to iteratively project each vertex onto a local approximation of the current polygonal mesh. While the resulting procedure haves the same complexity as simpler subdivision schemes, our method offers much higher visual quality, especially in the vicinity of extraordinary vertices. Moreover, we show it can be easily generalized to support boundaries and creases. The fitting procedure allows for a local control of the surface from the normals, making LS³ very well suited for interactive freeform modeling applications. We demonstrate our approach on diadic triangular and quadrangular refinement schemes, though it can be applied to any splitting strategies.     

Improving the Parameterization of Approximate Subdivision Surfaces

We provide a method for improving the parameterization of patching schemes that approximate Catmull‐Clark subdivision surfaces, such that the new parameterization conforms better to that of the original subdivision surface. We create this reparameterization in real‐time using a method that only depends on the topology of the surface and is independent of the surface's geometry. Our method can handle patches with more than one extraordinary vertex and avoids the combinatorial increase in both complexity and storage associated with multiple extraordinary vertices. Moreover, the reparameterization function is easy to implement and fast.     

Localized Delaunay Refinement for Sampling and Meshing

The technique of Delaunay refinement has been recognized as a versatile tool to generate Delaunay meshes of a variety of geometries. Despite its usefulness, it suffers from one lacuna that limits its application. It does not scale well with the mesh size. As the sample point set grows, the Delaunay triangulation starts stressing the available memory space which ultimately stalls any effective progress. A natural solution to the problem is to maintain the point set in clusters and run the refinement on each individual cluster. However, this needs a careful point insertion strategy and a balanced coordination among the neighboring clusters to ensure consistency across individual meshes. We design an octtree based localized Delaunay refinement method for meshing surfaces in three dimensions which meets these goals. We prove that the algorithm terminates and provide guarantees about structural properties of the output mesh. Experimental results show that the method can avoid memory thrashing while computing large meshes and thus scales much better than the standard Delaunay refinement method.     

Animation‐Aware Quadrangulation

Geometric meshes that model animated characters must be designed while taking into account the deformations that the shape will undergo during animation. We analyze an input sequence of meshes with point‐to‐point correspondence, and we automatically produce a quadrangular mesh that fits well the input animation. We first analyze the local deformation that the surface undergoes at each point, and we initialize a cross field that remains as aligned as possible to the principal directions of deformation throughout the sequence. We then smooth this cross field based on an energy that uses a weighted combination of the initial field and the local amount of stretch. Finally, we compute a field‐aligned quadrangulation with an off‐the‐shelf method. Our technique is fast and very simple to implement, and it significantly improves the quality of the output quad mesh and its suitability for character animation, compared to creating the quad mesh based on a single pose. We present experimental results and comparisons with a state‐of‐the‐art quadrangulation method, on both sequences from 3D scanning and synthetic sequences obtained by a rough animation of a triangulated model.     

SD Models: Super‐Deformed Character Models

Super‐deformed, SD, is a specific artistic style for Japanese manga and anime which exaggerates characters in the goal of appearing cute and funny. The SD style characters are widely used, and can be seen in many anime, CG movies, or games. However, to create an SD model often requires professional skills and considerable time and effort. In this paper, we present a novel technique to generate an SD style counterpart of a normal 3D character model. Our approach uses an optimization guided by a number of constraints that can capture the properties of the SD style. Users can also customize the results by specifying a small set of parameters related to the body proportions and the emphasis of the signature characteristics. With our technique, even a novel user can generate visually pleasing SD models in seconds.     

Stream Surface Parametrization by Flow‐Orthogonal Front Lines

The generation of discrete stream surfaces is an important and challenging task in scientific visualization, which can be considered a particular instance of geometric modeling. The quality of numerically integrated stream surfaces depends on a number of parameters that can be controlled locally, such as time step or distance of adjacent vertices on the front line. In addition there is a parameter that cannot be controlled locally: stream surface meshes tend to show high quality, well‐shaped elements only if the current front line is “globally” approximately perpendicular to the flow direction. We analyze the impact of this geometric property and present a novel solution – a stream surface integrator that forces the front line to be perpendicular to the flow and that generates quad‐dominant meshes with well‐shaped and well‐aligned elements. It is based on the integration of a scaled version of the flow field, and requires repeated minimization of an error functional along the current front line. We show that this leads to computing the 1‐dimensional kernel of a bidiagonal matrix: a linear problem that can be solved efficiently. We compare our method with existing stream surface integrators and apply it to a number of synthetic and real world data sets.     

Dirichlet Harmonic Shape Compression with Feature Preservation for Parameterized Surfaces

With the rapid advancement of 3D scanning devices, large and complicated 3D shapes are becoming ubiquitous, and require large amount of resources to store and transmit them efficiently. This makes shape compression a demanding technique in order for the user to reduce the data transmission latency. Existing shape compression methods could achieve very low bit‐rates by sacrificing shape quality. But none of them guarantees the preservation of salient feature lines that users care. In addition, many 3D shapes come with parametric information for texture mapping purposes. In this paper we describe a spectral method to compress the geometric shapes equipped with arbitrary valid parametric information. It guarantees to preserve user‐specified feature lines while achieving a high compression ratio. By applying the spectral shape analysis – Dirichlet Manifold Harmonics, in the 2D parametric domain, this method provides a progressive compression mechanism to trade‐off between bit‐rate and shape quality. Experiments show that this method provides very low bit‐rate with high shape‐quality and still guarantees the preservation of user‐specified feature lines.     

Polar NURBS Surface with Curvature Continuity

Polar NURBS surface is a kind of periodic NURBS surface, one boundary of which shrinks to a degenerate polar point. The specific topology of its control‐point mesh offers the ability to represent a cap‐like surface, which is common in geometric modeling. However, there is a critical and challenging problem that hinders its application: curvature continuity at the extraordinary singular pole. We first propose a sufficient and necessary condition of curvature continuity at the pole. Then, we present constructive methods for the two key problems respectively: how to construct a polar NURBS surface with curvature continuity and how to reform an ordinary polar NURBS surface to curvature continuous. The algorithms only depend on the symbolic representation and operations of NURBS, and they introduce no restrictions on the degree or the knot vectors. Examples and comparisons demonstrate the applications of the curvature‐continuous polar NURBS surface in hole‐filling and free‐shape modeling.     

Anisotropic Point Set Surfaces

Point Set Surfaces define smooth surfaces from regular samples based on weighted averaging of the points. Because weighting is done based on a spatial scale parameter, point set surfaces apply basically only to regular samples. We suggest to attach individual weight functions to each sample rather than to the location in space. This extends Point Set Surfaces to irregular settings, including anisotropic sampling adjusting to the principal curvatures of the surface. In particular, we describe how to represent surfaces with ellipsoidal weight functions per sample. Details of deriving such a representation from typical inputs and computing points on the surface are discussed.     

Persistent Heat Signature for Pose‐oblivious Matching of Incomplete Models

Although understanding of shape features in the context of shape matching and retrieval has made considerable progress in recent years, the case for partial and incomplete models in presence of pose variations still begs a robust and efficient solution. A signature that encodes features at multi‐scales in a pose invariant manner is more appropriate for this case. The Heat Kernel Signature function from spectral theory exhibits this multi‐scale property. We show how this concept can be merged with the persistent homology to design a novel efficient pose‐oblivious matching algorithm for all models, be they partial, incomplete, or complete. We make the algorithm scalable so that it can handle large data sets. Several test results show the robustness of our approach.     

Scales and Scale‐like Structures

We present a method for generating scales and scale‐like structures on a polygonal mesh through surface replacement. As input, we require a triangular mesh that will be covered with scales and one or more proxy‐models to be used as the scale's shape. A user begins scale generation by drawing a lateral line on the model to control the distribution and orientation of scales on the surface. We then create a vector field over the surface to control an anisotropic Voronoi tessellation, which represents the region occupied by each scale. Next we replace these regions by cutting the proxy model to match the boundary of the Voronoi region and deform the cut model onto the surface. The result is a fully connected 2‐manifold that is suitable for subsequent post‐processing applications like surface subdivision.     

Noise‐Adaptive Shape Reconstruction from Raw Point Sets

We propose a noise‐adaptive shape reconstruction method specialized to smooth, closed shapes. Our algorithm takes as input a defect‐laden point set with variable noise and outliers, and comprises three main steps. First, we compute a novel noise‐adaptive distance function to the inferred shape, which relies on the assumption that the inferred shape is a smooth submanifold of known dimension. Second, we estimate the sign and confidence of the function at a set of seed points, through minimizing a quadratic energy expressed on the edges of a uniform random graph. Third, we compute a signed implicit function through a random walker approach with soft constraints chosen as the most confident seed points computed in previous step.     

Moving Least Squares Coordinates

We propose a new family of barycentric coordinates that have closed‐forms for arbitrary 2D polygons. These coordinates are easy to compute and have linear precision even for open polygons. Not only do these coordinates have linear precision, but we can create coordinates that reproduce polynomials of a set degree m as long as degree m polynomials are specified along the boundary of the polygon. We also show how to extend these coordinates to interpolate derivatives specified on the boundary.     

Contouring Discrete Indicator Functions

We present a method for calculating the boundary of objects from Discrete Indicator Functions that store 2‐material volume fractions with a high degree of accuracy. Although Marching Cubes and its derivatives are effective methods for calculating contours of functions sampled over discrete grids, these methods perform poorly when contouring non‐smooth functions such as Discrete Indicator Functions. In particular, Marching Cubes will generate surfaces that exhibit aliasing and oscillations around the exact surface. We derive a simple solution to remove these problems by using a new function to calculate the positions of vertices along cell edges that is efficient, easy to implement, and does not require any optimization or iteration. Finally, we provide empirical evidence that the error introduced by our contouring method is significantly less than is introduced by Marching Cubes.     

On the shape of a set of points and lines in the plane

Detailed geometric models of the real world are in increasing demand. LiDAR data is appropriate to reconstruct urban models. In urban scenes, the individual surfaces can be reconstructed and connected to form the scene geometry. There are various methods for reconstructing the free‐form shape of a point sample on a single surface. However, these methods do not take the context of the surface into account. We present the guided α‐shape: an extension of the well known α‐shape that uses lines (guides) to indicate preferred locations for the boundary of the shape. The guided α‐shape uses (parts of) these lines as boundary where the points suggest that this is appropriate. We prove that the guided α‐shape can be constructed in O((n + m) log (n + m)) time, from an input of n points and m guides. We apply guided α‐shapes to urban reconstruction from LiDAR, where neighboring surfaces can be connected conveniently along their intersection lines into adjacent surfaces of a 3D model. We analyze guided α‐shapes of both synthetic and real data and show they are consistently better than α‐shapes for this application.