Discrete Critical Values: a General Framework for Silhouettes Computation

Many shapes resulting from important geometric operations in industrial applications such as Minkowski sums or volume swept by a moving object can be seen as the projection of higher dimensional objects. When such a higher dimensional object is a smooth manifold, the boundary of the projected shape can be computed from the critical points of the projection. In this paper, using the notion of polyhedral chains introduced by Whitney, we introduce a new general framework to define an analogous of the set of critical points of piecewise linear maps defined over discrete objects that can be easily computed. We illustrate our results by showing how they can be used to compute Minkowski sums of polyhedra and volumes swept by moving polyhedra.     

Enhancing Urban Façades via LiDAR‐Based Sculpting

Buildings with symmetrical façades are ubiquitous in urban landscapes and detailed models of these buildings enhance the visual realism of digital urban scenes. However, a vast majority of the existing urban building models in web‐based 3D maps such as Google earth are either less detailed or heavily rely on texturing to render the details. We present a new framework for enhancing the details of such coarse models, using the geometry and symmetry inferred from the light detection and ranging (LiDAR) scans and 2D templates. The user‐defined 2D templates, referred to as coded planar meshes (CPMs), encodes the geometry of the smallest repeating 3D structures of the façades via face codes. Our encoding scheme, take into account the directions, type as well as the offset distance of the sculpting to be applied at the respective locations on the coarse model. In our approach, LiDAR scan is registered with the coarse models taken from Google earth 3D or Bing maps 3D and decomposed into dominant planar segments (each representing the frontal or lateral walls of the building). The façade segments are then split into horizontal and vertical tiles using a weighted point count function defined over the window or door boundaries. This is followed by an automatic identification of CPM locations with the help of a template fitting algorithm that respects the alignment regularity as well as the inter‐element spacing on the façade layout. Finally, 3D boolean sculpting operations are applied over the boxes induced by CPMs and the coarse model, and a detailed 3D model is generated. The proposed framework is capable of modelling details even with occluded scans and enhances not only the frontal façades (facing to the streets) but also the lateral façades of the buildings. We demonstrate the potentials of the proposed framework by providing several examples of enhanced Google earth models and highlight the advantages of our method when designing photo‐realistic urban façades.     

Mesh Segmentation via Spectral Embedding and Contour Analysis

We propose a mesh segmentation algorithm via recursive bisection where at each step, a sub-mesh embedded in 3D is first spectrally projected into the plane and then a contour is extracted from the planar embedding. We rely on two operators to compute the projection: the well-known graph Laplacian and a geometric operator designed to emphasize concavity. The two embeddings reveal distinctive shape semantics of the 3D model and complement each other in capturing the structural or geometrical aspect of a segmentation. Transforming the shape analysis problem to the 2D domain also facilitates our segmentability analysis and sampling tasks. We propose a novel measure of the segmentability of a shape, which is used as the stopping criterionfor our segmentation. The measure is derived from simple area- and perimeter-based convexity measures. We achieve invariance to shape bending through multi-dimensional scaling (MDS) based on the notion of inner distance. We also utilize inner distances to develop a novel sampling scheme to extract two samples along a contour which correspond to two vertices residing on different parts of the sub-mesh. The two samples are used to derive a spectral linear ordering of the mesh faces. We obtain a final cut via a linear search over the face sequence based on part salience, where a choice of weights for different factors of part salience is guided by the result from segmentability analysis.     

Selective Padding for Polycube‐Based Hexahedral Meshing

Hexahedral meshes generated from polycube mapping often exhibit a low number of singularities but also poor‐quality elements located near the surface. It is thus necessary to improve the overall mesh quality, in terms of the minimum scaled Jacobian (MSJ) or average SJ (ASJ). Improving the quality may be obtained via global padding (or pillowing), which pushes the singularities inside by adding an extra layer of hexahedra on the entire domain boundary. Such a global padding operation suffers from a large increase of complexity, with unnecessary hexahedra added. In addition, the quality of elements near the boundary may decrease. We propose a novel optimization method which inserts sheets of hexahedra so as to perform selective padding, where it is most needed for improving the mesh quality. A sheet can pad part of the domain boundary, traverse the domain and form singularities. Our global formulation, based on solving a binary problem, enables us to control the balance between quality improvement, increase of complexity and number of singularities. We show in a series of experiments that our approach increases the MSJ value and preserves (or even improves) the ASJ, while adding fewer hexahedra than global padding.     

Localized Quadrilateral Coarsening

In this paper we introduce a coarsening algorithm for quadrilateral meshes that generates quality, quad-only connectivity during level-of- coarsening creation. A novel aspect of this work is development and implementation of a localized adaptation of the polychord collapse operator to better control and preserve important surface components. We describe a novel weighting scheme for automatic deletion selection that considers surface attributes, as well as localized queue updates that allow for improved data structures and computational performance opportunities over previous techniques. Additionally, this work supports optional and intuitive user controls for tailored simplification results.     

Variational Surface Approximation and Model Selection

We consider the problem of approximating an arbitrary generic surface with a given set of simple surface primitives. In contrast to previous approaches based on variational surface approximation, which are primarily concerned with finding an optimal partitioning of the input geometry, we propose to integrate a model selection step into the algorithm in order to also optimize the type of primitive for each proxy. Our method is a joint global optimization of both the partitioning of the input surface as well as the types and number of used shape proxies. Thus, our method performs an automatic trade‐off between representation complexity and approximation error without relying on a user supplied predetermined number of shape proxies. This way concise surface representations are found that better exploit the full approximative power of the employed primitive types.     

Semi-regular Quadrilateral-only Remeshing from Simplified Base Domains

Semi-regular meshes describe surface models that exhibit a structural regularity that facilitates many geometric processing algorithms. We introduce a technique to construct semi-regular, quad-only meshes from input surface meshes of arbitrary polygonal type and genus. The algorithm generates a quad-only model through subdivision of the input polygons, then simplifies to a base domain that is homeomorphic to the original mesh. During the simplification, a novel hierarchical mapping method, keyframe mapping , stores specific levels-of-detail to guide the mapping of the original vertices to the base domain. The algorithm implements a scheme for refinement with adaptive resampling of the base domain and backward projects to the original surface. As a byproduct of the remeshing scheme, a surface parameterization is associated with the remesh vertices to facilitate subsequent geometric processing, i.e. texture mapping, subdivision surfaces and spline-based modeling.     

Particle Level Set Advection for the Interactive Visualization of Unsteady 3D Flow

Typically, flow volumes are visualized by defining their boundary as iso‐surface of a level set function. Grid‐based level sets offer a good global representation but suffer from numerical diffusion of surface detail, whereas particle‐based methods preserve details more accurately but introduce the problem of unequal global representation. The particle level set (PLS) method combines the advantages of both approaches by interchanging the information between the grid and the particles. Our work demonstrates that the PLS technique can be adapted to volumetric dye advection via streak volumes, and to the visualization by time surfaces and path volumes. We achieve this with a modified and extended PLS, including a model for dye injection. A new algorithmic interpretation of PLS is introduced to exploit the efficiency of the GPU, leading to interactive visualization. Finally, we demonstrate the high quality and usefulness of PLS flow visualization by providing quantitative results on volume preservation and by discussing typical applications of 3D flow visualization.     

ARAPLBS: Robust and Efficient Elasticity‐Based Optimization of Weights and Skeleton Joints for Linear Blend Skinning with Parametrized Bones

We present a fast, robust and high‐quality technique to skin a mesh with reference to a skeleton. We consider the space of possible skeleton deformations (based on skeletal constraints, or skeletal animations), and compute skinning weights based on an optimization scheme to obtain as‐rigid‐as‐possible (ARAP) corresponding mesh deformations. We support stretchable‐and‐twistable bones (STBs) and spines by generalizing the ARAP deformations to stretchable deformers. In addition, our approach can optimize joint placements. If wanted, a user can guide and interact with the results, which is facilitated by an interactive feedback, reached via an efficient sparsification scheme. We demonstrate our technique on challenging inputs (STBs and spines, triangle and tetrahedral meshes featuring missing elements, boundaries, self‐intersections or wire edges).     

State of the Art on Stylized Fabrication

Digital fabrication devices are powerful tools for creating tangible reproductions of 3D digital models. Most available printing technologies aim at producing an accurate copy of a tridimensional shape. However, fabrication technologies can also be used to create a stylistic representation of a digital shape. We refer to this class of methods as ‘stylized fabrication methods’. These methods abstract geometric and physical features of a given shape to create an unconventional representation, to produce an optical illusion or to devise a particular interaction with the fabricated model. In this state‐of‐the‐art report, we classify and overview this broad and emerging class of approaches and also propose possible directions for future research.     

Towards Globally Optimal Normal Orientations for Large Point Clouds

Various processing algorithms on point set surfaces rely on consistently oriented normals (e.g. Poisson surface reconstruction). While several approaches exist for the calculation of normal directions, in most cases, their orientation has to be determined in a subsequent step. This paper generalizes propagation‐based approaches by reformulating the task as a graph‐based energy minimization problem. By applying global solvers, we can achieve more consistent orientations than simple greedy optimizations. Furthermore, we present a streaming‐based framework for orienting large point clouds. This framework orients patches locally and generates a globally consistent patch orientation on a reduced neighbour graph, which achieves similar quality to orienting the full graph.     

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.     

Uniformization and Density Adaptation for Point Cloud Data Via Graph Laplacian

Point cloud data is one of the most common types of input for geometric processing applications. In this paper, we study the point cloud density adaptation problem that underlies many pre‐processing tasks of points data. Specifically, given a (sparse) set of points Q sampling an unknown surface and a target density function, the goal is to adapt Q to match the target distribution. We propose a simple and robust framework that is effective at achieving both local uniformity and precise global density distribution control. Our approach relies on the Gaussian‐weighted graph Laplacian and works purely in the points setting. While it is well known that graph Laplacian is related to mean‐curvature flow and thus has denoising ability, our algorithm uses certain information encoded in the graph Laplacian that is orthogonal to the mean‐curvature flow. Furthermore, by leveraging the natural scale parameter contained in the Gaussian kernel and combining it with a simulated annealing idea, our algorithm moves points in a multi‐scale manner. The resulting algorithm relies much less on the input points to have a good initial distribution (neither uniform nor close to the target density distribution) than many previous refinement‐based methods. We demonstrate the simplicity and effectiveness of our algorithm with point clouds sampled from different underlying surfaces with various geometric and topological properties.     

A Biorthogonal Wavelet Approach based on Dual Subdivision

In this paper a biorthogonal wavelet approach based on Doo‐Sabin subdivision is presented. In the dual subdivision like Doo‐Sabin scheme, all the old control vertices disappear after one subdivision step, which is a big challenge to the biorthogonal wavelet construction. In our approach, the barycenters of the V‐faces corresponding to the old vertices are selected as the vertices associated with the scaling functions to construct the scaling space. The lifting scheme is used to guarantee the fitting quality of the wavelet transform, and a local orthogonalization is introduced with a discrete inner product operation to improve the computation efficiency. Sharp feature modeling based on extended Doo‐Sabin subdivision rules is also discussed in the framework of our wavelet construction. The presented wavelet construction is proven to be stable and effective by the experimental results.     

Interactive Mesh Smoothing for Medical Applications

Surface models derived from medical image data often exhibit artefacts, such as noise and staircases, which can be reduced by applying mesh smoothing filters. Usually, an iterative adaption of smoothing parameters to the specific data and continuous re‐evaluation of accuracy and curvature is required. Depending on the number of vertices and the filter algorithm, computation time may vary strongly and interfere with an interactive mesh generation procedure. In this paper, we present an approach to improve the handling of mesh smoothing filters. Based on a GPU mesh smoothing implementation of uniform and anisotropic filters, model quality is evaluated in real‐time and provided to the user to support the mental optimization of input parameters. This is achieved by means of quality graphs and quality bars. Moreover, this framework is used to find appropriate smoothing parameters automatically and to provide data‐specific parameter suggestions. These suggestions are employed to generate a preview gallery with different smoothing suggestions. The preview functionality is additionally used for the inspection of specific artefacts and their possible reduction with different parameter sets.     

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.     

Taylor Prediction for Mesh Geometry Compression

In this paper, we introduce a new formalism for mesh geometry prediction. We derive a class of smooth linear predictors from a simple approach based on the Taylor expansion of the mesh geometry function. We use this method as a generic way to compute weights for various linear predictors used for mesh compression and compare them with those of existing methods. We show that our scheme is actually equivalent to the Modified Butterfly subdivision scheme used for wavelet mesh compression. We also build new efficient predictors that can be used for connectivity‐driven compression in place of other schemes like Average/Dual Parallelogram Prediction and High Degree Polygon Prediction. The new predictors use the same neighbourhood, but do not make any assumption on mesh anisotropy. In the case of Average Parallelogram Prediction, our new weights improve compression rates from 3% to 18% on our test meshes. For Dual Parallelogram Prediction, our weights are equivalent to those of the previous Freelence approach, that outperforms traditional schemes by 16% on average. Our method effectively shows that these weights are optimal for the class of smooth meshes. Modifying existing schemes to make use of our method is free because only the prediction weights have to be modified in the code.