Quad Mesh Research Articles

On the flexibility and symmetry of overconstrained mechanisms

In kinematics, a framework is called overconstrained if its continuous flexibility is caused by particular dimensions; in the generic case, a framework of this type is rigid. Famous examples of overconstrained structures are the Bricard octahedra, the Bennett isogram, the Grünbaum framework, Bottema's 16-bar mechanism, Chasles' body-bar framework, Burmester's focal mechanism or flexible quad meshes. The aim of this paper is to present some examples in detail and to focus on their symmetry properties. It turns out that only for a few is a global symmetry a necessary condition for flexibility. Sometimes, there is a hidden symmetry, and in some cases, for example, at the flexible type-3 octahedra or at discrete Voss surfaces, there is only a local symmetry. However, there remain overconstrained frameworks where the underlying algebraic conditions for flexibility have no relation to symmetry at all.

Construction and Application of Subdivision Surface Scheme Using Lagrange Interpolation Polynomial

This paper offers a general formula for surface subdivision rules for quad meshes by using 2-D Lagrange interpolating polynomial [1]. We also see that the result obtained is equivalent to the tensor product of (2N + 4)-point n-ary interpolating curve scheme for N ≥ 0 and n ≥ 2. The simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented by L. Kobbelt [2], which can be directly calculated from the proposed formula. Furthermore, some characteristics and applications of the proposed work are also discussed.

A frontal approach to hex-dominant mesh generation

Indirect quad mesh generation methods rely on an initial triangular mesh. So called triangle-merge techniques are then used to recombine the triangles of the initial mesh into quadrilaterals. This way, high-quality full-quad meshes suitable for finite element calculations can be generated for arbitrary two-dimensional geometries. In this paper, a similar indirect approach is applied to the three-dimensional case, i.e., a method to recombine tetrahedra into hexahedra. Contrary to the 2D case, a 100% recombination rate is seldom attained in 3D. Instead, part of the remaining tetrahedra are combined into prisms and pyramids, eventually yielding a mixed mesh. We show that the percentage of recombined hexahedra strongly depends on the location of the vertices in the initial 3D mesh. If the vertices are placed at random, less than 50% of the tetrahedra will be combined into hexahedra. In order to reach larger ratios, the vertices of the initial mesh need to be anticipatively organized into a lattice-like structure. This can be achieved with a frontal algorithm, which is applicable to both the two- and three-dimensional cases. The quality of the vertex alignment inside the volumes relies on the quality of the alignment on the surfaces. Once the vertex placement process is completed, the region is tetrahedralized with a Delaunay kernel. A maximum number of tetrahedra are then merged into hexahedra using the algorithm of Yamakawa-Shimada. Non-uniform mixed meshes obtained following our approach show a volumic percentage of hexahedra that usually exceeds 80%. The execution times are reasonable. However, non-conformal quadrilateral faces adjacent to triangular faces are present in the final meshes.

Towards FEA over Tangled Quads

In finite element analysis (FEA), a mesh is said to be tangled if it contains an element with negative Jacobian-determinant. Tangling can occur during mesh optimization and mesh morphing. Modern FEA unfortunately cannot handle such tangled meshes, i.e., it will lead to erroneous results. While significant progress has been made on untangling, there are no definitive untangling algorithms.Danczyk and Suresh recently proposed a theoretical extension to FEA such that one could accurately solve boundary value problems over such tangled meshes. However, their investigation was limited to simplicial meshes.In this paper we consider the extension of the above framework to tangled quad meshes that pose additional challenges compared to simplicial meshes. These challenges are identified, and a tangling-framework is developed to address explicit quad tangling where quads are allowed to overlap, but are required to be geometrically convex. Numerical examples illustrate the correctness of the proposed framework, opening new opportunities for meshing algorithms.

An effective quad-dominant meshing method for unorganized point clouds

To perform quad meshing on raw point clouds, existing algorithms usually require a time-consuming parameterization or Voronoi space partition process. In this paper, we propose an effective method to generate quad-dominant meshes directly from unorganized point clouds. In the proposed method, we first apply Marinov’s curvature tensor optimization to the input point cloud to reduce the umbilical regions in order to obtain a smooth curvature tensor. We then propose an efficient marching scheme to extract the curvature lines with controllable density from the point cloud. Finally, we apply a specialized K-Dimension (KD) tree structure, which converts the nearest neighbor searching problem into a sorting problem, to efficiently estimate the intersections of curvature lines and recover the topology of the quad-dominant meshes. We have tested the proposed method on different point clouds. Our results show that the proposed method produces good quality meshes with high computational efficiency and low memory requirement.

QEx

The most popular and actively researched class of quad remeshing techniques is the family of parametrization based quad meshing methods . They all strive to generate an integer-grid map , i.e. a parametrization of the input surface into R 2 such that the canonical grid of integer iso-lines forms a quad mesh when mapped back onto the surface in R 3 . An essential, albeit broadly neglected aspect of these methods is the quad extraction step, i.e. the materialization of an actual quad mesh from the mere "quad texture". Quad (mesh) extraction is often believed to be a trivial matter but quite the opposite is true: numerous special cases, ambiguities induced by numerical inaccuracies and limited solver precision, as well as imperfections in the maps produced by most methods (unless costly countermeasures are taken) pose significant challenges to the quad extractor. We present a method to sanitize a provided parametrization such that it becomes numerically consistent even in a limited precision floating point representation. Based on this we are able to provide a comprehensive and sound description of how to perform quad extraction robustly and without the need for any complex tolerance thresholds or disambiguation rules. On top of that we develop a novel strategy to cope with common local fold-overs in the parametrization. This allows our method, dubbed QEx , to generate all-quadrilateral meshes where otherwise holes, non-quad polygons or no output at all would have been produced. We thus enable the practical use of an entire class of maps that was previously considered defective. Since state of the art quad meshing methods spend a significant share of their run time solely to prevent local fold-overs, using our method it is now possible to obtain quad meshes significantly quicker than before. We also provide libQEx, an open source C++ reference implementation of our method and thus significantly lower the bar to enter the field of quad meshing.

Multiresolution Attributes for Hardware Tessellated Objects

Hardware tessellation is one of the latest GPU features. Triangle or quad meshes are tessellated on-the-fly, where the tessellation level is chosen adaptively in a separate shader. The hardware tessellator only generates topology; attributes such as positions or texture coordinates of the newly generated vertices are determined in a domain shader. Typical applications of hardware tessellation are view dependent tessellation of parametric surfaces and displacement mapping. Often, the attributes for the newly generated vertices are stored in textures, which requires uv unwrapping, chartification, and atlas generation of the input mesh--a process that is time consuming and often requires manual intervention. In this paper, we present an alternative representation that directly stores optimized attribute values for typical hardware tessellation patterns and simply assigns these attributes to the generated vertices at render time. Using a multilevel fitting approach, the attribute values are optimized for several resolutions. Thereby, we require no parameterization, save memory by adapting the density of the samples to the content, and avoid discontinuities by construction. Our representation is optimally suited for displacement mapping: it automatically generates seamless, view-dependent displacement mapped models. The multilevel fitting approach generates better low-resolution displacement maps than simple downfiltering. By properly blending levels, we avoid artifacts such as popping or swimming surfaces. We also show other possible applications such as signal-optimized texturing or light baking. Our representation can be evaluated in a pixel shader, resulting in signal adaptive, parameterization-free texturing, comparable to PTex or Mesh Colors. Performance evaluation shows that our representation is on par with standard texture mapping and can be updated in real time, allowing for application such as interactive sculpting.

Animation‐Aware Quadrangulation

AbstractGeometric 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.

Sketch-based generation and editing of quad meshes

Coarse quad meshes are the preferred representation for animating characters in movies and video games. In these scenarios, artists want explicit control over the edge flows and the singularities of the quad mesh. Despite the significant advances in recent years, existing automatic quad remeshing algorithms are not yet able to achieve the quality of manually created remeshings. We present an interactive system for manual quad remeshing that provides the user with a high degree of control while avoiding the tediousness involved in existing manual tools. With our sketch-based interface the user constructs a quad mesh by defining patches consisting of individual quads. The desired edge flow is intuitively specified by the sketched patch boundaries, and the mesh topology can be adjusted by varying the number of edge subdivisions at patch boundaries. Our system automatically inserts singularities inside patches if necessary, while providing the user with direct control of their topological and geometrical locations. We developed a set of novel user interfaces that assist the user in constructing a curve network representing such patch boundaries. The effectiveness of our system is demonstrated through a user evaluation with professional artists. Our system is also useful for editing automatically generated quad meshes.

Integer-grid maps for reliable quad meshing

Quadrilateral remeshing approaches based on global parametrization enable many desirable mesh properties. Two of the most important ones are (1) high regularity due to explicit control over irregular vertices and (2) smooth distribution of distortion achieved by convex variational formulations. Apart from these strengths, state-of-the-art techniques suffer from limited reliability on real-world input data, i.e. the determined map might have degeneracies like (local) non-injectivities and consequently often cannot be used directly to generate a quadrilateral mesh. In this paper we propose a novel convex Mixed-Integer Quadratic Programming (MIQP) formulation which ensures by construction that the resulting map is within the class of so called Integer-Grid Maps that are guaranteed to imply a quad mesh. In order to overcome the NP-hardness of MIQP and to be able to remesh typical input geometries in acceptable time we propose two additional problem specific optimizations: a complexity reduction algorithm and singularity separating conditions. While the former decouples the dimension of the MIQP search space from the input complexity of the triangle mesh and thus is able to dramatically speed up the computation without inducing inaccuracies, the latter improves the continuous relaxation, which is crucial for the success of modern MIQP optimizers. Our experiments show that the reliability of the resulting algorithm does not only annihilate the main drawback of parametrization based quad-remeshing but moreover enables the global search for high-quality coarse quad layouts - a difficult task solely tackled by greedy methodologies before.

A Divide-and-Conquer Approach to Quad Remeshing

Many natural and man-made objects consist of simple primitives, similar components, and various symmetry structures. This paper presents a divide-and-conquer quadrangulation approach that exploits such global structural information. Given a model represented in triangular mesh, we first segment it into a set of submeshes, and compare them with some predefined quad mesh templates. For the submeshes that are similar to a predefined template, we remesh them as the template up to a number of subdivisions. For the others, we adopt the wave-based quadrangulation technique to remesh them with extensions to preserve symmetric structure and generate compatible quad mesh boundary. To ensure that the individually remeshed submeshes can be seamlessly stitched together, we formulate a mixed-integer optimization problem and design a heuristic solver to optimize the subdivision numbers and the size fields on the submesh boundaries. With this divider-and-conquer quadrangulation framework, we are able to process very large models that are very difficult for the previous techniques. Since the submeshes can be remeshed individually in any order, the remeshing procedure can run in parallel. Experimental results showed that the proposed method can preserve the high-level structures, and process large complex surfaces robustly and efficiently.

Fitting polynomial surfaces to triangular meshes with Voronoi squared distance minimization

This paper introduces Voronoi squared distance minimization (VSDM), an algorithm that fits a surface to an input mesh. VSDM minimizes an objective function that corresponds to a Voronoi-based approximation of the overall squared distance function between the surface and the input mesh (SDM). This objective function is a generalization of the one minimized by centroidal Voronoi tessellation, and can be minimized by a quasi-Newton solver. VSDM naturally adapts the orientation of the mesh elements to best approximate the input, without estimating any differential quantities. Therefore, it can be applied to triangle soups or surfaces with degenerate triangles, topological noise and sharp features. Applications of fitting quad meshes and polynomial surfaces to input triangular meshes are demonstrated.

All-hex meshing using singularity-restricted field

Decomposing a volume into high-quality hexahedral cells is a challenging task in geometric modeling and computational geometry. Inspired by the use of cross field in quad meshing and the CubeCover approach in hex meshing, we present a complete all-hex meshing framework based on singularity-restricted field that is essential to induce a valid all-hex structure. Given a volume represented by a tetrahedral mesh, we first compute a boundary-aligned 3D frame field inside it, then convert the frame field to be singularity-restricted by our effective topological operations. In our all-hex meshing framework, we apply the CubeCover method to achieve the volume parametrization. For reducing degenerate elements appearing in the volume parametrization, we also propose novel tetrahedral split operations to preprocess singularity-restricted frame fields. Experimental results show that our algorithm generates high-quality all-hex meshes from a variety of 3D volumes robustly and efficiently.

GPU-optimized indirect scalar displacement mapping

Displacement mapping refers to a technique for rendering a high-frequency surface by adding geometric details encoded in a displacement map to a low-frequency base surface. This paper proposes a method for indirectly accessing the base surface using a special displacement map and then carrying out scalar displacement. Given a high-frequency triangle mesh, a coarse PN (point-normal) quad mesh is computed as the base surface. The parameters used to evaluate the base surface are precomputed such that scalar displacement from the evaluated points reaches the features of the original surface. The parameters are stored in the displacement map together with the displacement scalars. The run-time algorithm uses the hardware tessellation capability of GPU and reconstructs the high-frequency surface. Using the proposed method, surface features are accurately preserved, surface deformation is well supported, LOD control becomes quite flexible, and the base surface can be extremely simplified.

Sketch-based garment design with quad meshes

Garment creation continues to be the most tedious part of the virtual clothing process. In this paper, we present an easy to use sketch-based cloth modeling approach. Contours can be easily sketched on a mannequin to generate quad meshes to represent pieces of cloth already fitted and draped. Typically, the clothing process depends greatly on the meshing scheme that has to infer its geometry from the input boundary. Our quad meshing scheme is based on discrete Coons patches but with arbitrary boundary input. We also apply the permanence principle to our topological solution to allow more control over the influence of the input boundary polyline on the interior output polygonal mesh. This facilitates the creation of folds that are strongest in curvature at the boundary and which diminish towards the interior. The generated garments can then be easily animated in a simulation system based on finite elements, using a rediscretization of the generated mesh and a reconstructed metric of the cloth surface.

Interactive Quadrangulation with Reeb Atlases and Connectivity Textures

Creating high-quality quad meshes from triangulated surfaces is a highly nontrivial task that necessitates consideration of various application specific metrics of quality. In our work, we follow the premise that automatic reconstruction techniques may not generate outputs meeting all the subjective quality expectations of the user. Instead, we put the user at the center of the process by providing a flexible, interactive approach to quadrangulation design. By combining scalar field topology and combinatorial connectivity techniques, we present a new framework, following a coarse to fine design philosophy, which allows for explicit control of the subjective quality criteria on the output quad mesh, at interactive rates. Our quadrangulation framework uses the new notion of Reeb atlas editing, to define with a small amount of interactions a coarse quadrangulation of the model, capturing the main features of the shape, with user prescribed extraordinary vertices and alignment. Fine grain tuning is easily achieved with the notion of connectivity texturing, which allows for additional extraordinary vertices specification and explicit feature alignment, to capture the high-frequency geometries. Experiments demonstrate the interactivity and flexibility of our approach, as well as its ability to generate quad meshes of arbitrary resolution with high-quality statistics, while meeting the user’s own subjective requirements.

Design of self-supporting surfaces

Self-supporting masonry is one of the most ancient and elegant techniques for building curved shapes. Because of the very geometric nature of their failure, analyzing and modeling such strutures is more a geometry processing problem than one of classical continuum mechanics. This paper uses the thrust network method of analysis and presents an iterative nonlinear optimization algorithm for efficiently approximating freeform shapes by self-supporting ones. The rich geometry of thrust networks leads us to close connections between diverse topics in discrete differential geometry, such as a finite-element discretization of the Airy stress potential, perfect graph Laplacians, and computing admissible loads via curvatures of polyhedral surfaces. This geometric viewpoint allows us, in particular, to remesh self-supporting shapes by self-supporting quad meshes with planar faces, and leads to another application of the theory: steel/glass constructions with low moments in nodes.

Design of self-supporting surfaces

Self-supporting masonry is one of the most ancient and elegant techniques for building curved shapes. Because of the very geometric nature of their failure, analyzing and modeling such strutures is more a geometry processing problem than one of classical continuum mechanics. This paper uses the thrust network method of analysis and presents an iterative nonlinear optimization algorithm for efficiently approximating freeform shapes by self-supporting ones. The rich geometry of thrust networks leads us to close connections between diverse topics in discrete differential geometry, such as a finite-element discretization of the Airy stress potential, perfect graph Laplacians, and computing admissible loads via curvatures of polyhedral surfaces. This geometric viewpoint allows us, in particular, to remesh self-supporting shapes by self-supporting quad meshes with planar faces, and leads to another application of the theory: steel/glass constructions with low moments in nodes.

Dual loops meshing

We present a theoretical framework and practical method for the automatic construction of simple, all-quadrilateral patch layouts on manifold surfaces. The resulting layouts are coarse, surface-embedded cell complexes well adapted to the geometric structure, hence they are ideally suited as domains and base complexes for surface parameterization, spline fitting, or subdivision surfaces and can be used to generate quad meshes with a high-level patch structure that are advantageous in many application scenarios. Our approach is based on the careful construction of the layout graph's combinatorial dual. In contrast to the primal this dual perspective provides direct control over the globally interdependent structural constraints inherent to quad layouts. The dual layout is built from curvature-guided, crossing loops on the surface. A novel method to construct these efficiently in a geometry- and structure-aware manner constitutes the core of our approach.

Dual loops meshing

We present a theoretical framework and practical method for the automatic construction of simple, all-quadrilateral patch layouts on manifold surfaces. The resulting layouts are coarse, surface-embedded cell complexes well adapted to the geometric structure, hence they are ideally suited as domains and base complexes for surface parameterization, spline fitting, or subdivision surfaces and can be used to generate quad meshes with a high-level patch structure that are advantageous in many application scenarios. Our approach is based on the careful construction of the layout graph's combinatorial dual. In contrast to the primal this dual perspective provides direct control over the globally interdependent structural constraints inherent to quad layouts. The dual layout is built from curvature-guided, crossing loops on the surface. A novel method to construct these efficiently in a geometry- and structure-aware manner constitutes the core of our approach.