Generation of bi-monotone patches from quadrilateral mesh for reverse engineering

Thanks to recent improvements, computational methods can now be used to convert triangular meshes into quadrilateral meshes so that the quadrilateral elements capture well the principal curvature directional fields of surfaces and intrinsically have surface parametric values. In this study, a quadrilateral mesh generated using the mixed integer quadrangulation technique of Bommes et al. is used for input. We first segment a quadrilateral mesh into four-sided patches. The feature curves inside these patches are then detected and are constrained to act as the patch boundaries. Finally, the patch configuration is improved to generate large patches. The proposed method produces bi-monotone patches, which are appropriate for use in reverse engineering to capture the surface details of an object. A shape control parameter that can be adjusted by the user during the patch generation process is also provided to support the creation of patches with good bi-monotone shapes. This study mainly targets shape models of mechanical parts consisting of major smooth surfaces with feature curves between them.     

Consistent quadrangulation for shape collections via feature line co-extraction

This paper presents a method that takes a collection of 3D surface shapes, and produces a consistent and individually feature preserving quadrangulation of each shape. By exploring the correspondence among shapes within a collection, we coherently extract a set of representative feature lines as the key characteristics for the given shapes. Then we compute a smooth cross-field interpolating sparsely distributed directional constraints induced from the feature lines and apply the mixed-integer quadrangulation to generate the quad meshes. We develop a greedy algorithm to extract aligned cut graphs across the shape collection so that the meshes can be aligned in a common parametric domain. Computational results demonstrate that our approach not only produces consistent quad meshes across the entire collection with significant geometry variation but also achieves a trade-off between global structural simplicity for the collection and local geometry fidelity for each shape.     

Computing Surface PolyCube‐Maps by Constrained Voxelization

We present a novel method to compute bijective PolyCube‐maps with low isometric distortion. Given a surface and its pre‐axis‐aligned shape that is not an exact PolyCube shape, the algorithm contains two steps: (i) construct a PolyCube shape to approximate the pre‐axis‐aligned shape; and (ii) generate a bijective, low isometric distortion mapping between the constructed PolyCube shape and the input surface. The PolyCube construction is formulated as a constrained optimization problem, where the objective is the number of corners in the constructed PolyCube, and the constraint is to bound the approximation error between the constructed PolyCube and the input pre‐axis‐aligned shape while ensuring topological validity. A novel erasing‐and‐filling solver is proposed to solve this challenging problem. Centeral to the algorithm for computing bijective PolyCube‐maps is a quad mesh optimization process that projects the constructed PolyCube onto the input surface with high‐quality quads. We demonstrate the efficacy of our algorithm on a data set containing 300 closed meshes. Compared to state‐of‐the‐art methods, our method achieves higher practical robustness and lower mapping distortion.     

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.     

Shape optimization of quad mesh elements

We study the problem of optimizing the face elements of a quad mesh surface, that is, re-sampling a given quad mesh to make it possess, as much as possible, face elements of some desired aspect ratio and size. Unlike previous quad mesh optimization/improvement methods based on local operations on a small group of elements, we adopt a global approach that does not introduce extra singularities and therefore preserves the original quad structure of the input mesh. Starting from a collection of quad patches extracted from an input quad mesh, two global operations, i.e. re-sampling and re-distribution, are performed to optimize the number and spacings of grid lines in each patch. Both operations are formulated as simple optimization problems with linear constraints.     

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.     

B‐Mesh: A Modeling System for Base Meshes of 3D Articulated Shapes

This paper presents a novel modeling system, called B‐Mesh, for generating base meshes of 3D articulated shapes. The user only needs to draw a one‐dimensional skeleton and to specify key balls at the skeletal nodes. The system then automatically generates a quad dominant initial mesh. Further subdivision and evolution are performed to refine the initial mesh and generate a quad mesh which has good edge flow along the skeleton directions. The user can also modify and manipulate the shape by editing the skeleton and the key balls and can easily compose new shapes by cutting and pasting existing models in our system. The mesh models generated in our system greatly benefit the sculpting operators for sculpting modeling and skeleton‐based animation.     

The Cut&Glue mesh generation algorithm

This paper describes a technique for generating quadrilateral finite element meshes on convex, four-sided patches, given an arbitrary number of elements along each side of the patch. The technique first generates a subdivision with the correct topological structure and smoothes the subdivision to obtain elements of acceptance shape for finite element analysis. The correct mesh topology is obtained from a regular subdivision by cutting rectangular corners of appropriate size and interconnecting the sides introduced by the cuts.The method can also be applied on three-dimensional patches producing meshes of brick elements with gradations in all directions.     

Automatic unstructured all-hexahedral mesh generation from B-Reps for non-manifold CAD assemblies

This paper describes an automatic and robust approach to convert non-manifold CAD assemblies into unstructured all-hexahedral meshes conformal to the given B-Reps (boundary-representations) and with sharp feature preservation. In previous works, we developed an octree-based isocontouring method to construct unstructured hexahedral meshes for arbitrary non-manifold and manifold domains. However, sharp feature preservation still remains a challenge, especially for non-manifold CAD assemblies. In this paper, boundary features such as NURBS (non-uniform rational B-Splines) curves and surface patches are first extracted from the given B-Reps. Features shared by multiple components are identified and distinguished. To preserve these non-manifold features, one given surface patch may need to be split into several small ones. An octree-based algorithm is then carried out to create an unstructured all-hexahedral base mesh, detecting and preserving all the sharp features via a curve and surface parametrization. Two sets of local refinement templates are provided for adaptive mesh generation, along with a novel 2-refinement implementation. Vertices in the base mesh are categorized into four groups based on the given non-manifold topology, and each group is relocated using various methods with all sharp features preserved. After this stage, a novel two-step pillowing technique is developed for such complicated non-manifold domains to eliminate triangle-shaped quadrilateral elements along the curves and “doublets”, handling non-manifold and manifold features in different ways. Finally, a combination of smoothing and optimization is used to further improve the mesh quality. Our algorithm is automatic and robust for non-manifold and manifold domains. We have applied our algorithm to several complicated CAD assemblies.     

Tracing Field‐Coherent Quad Layouts

Given a cross field over a triangulated surface we present a practical and robust method to compute a field aligned coarse quad layout over the surface. The method works directly on a triangle mesh without requiring any parametrization and it is based on a new technique for tracing field‐coherent geodesic paths directly on a triangle mesh, and on a new relaxed formulation of a binary LP problem, which allows us to extract both conforming quad layouts and coarser layouts containing t‐junctions. Our method is easy to implement, very robust, and, being directly based on the input cross field, it is able to generate better aligned layouts, even with complicated fields containing many singularities. We show results on a number of datasets and comparisons with state‐of‐the‐art methods.     

Isogeometric shape optimization of photonic crystals via Coons patches

In this paper, we present an approach that extends isogeometric shape optimization from optimization of rectangular-like NURBS patches to the optimization of topologically complex geometries. We have successfully applied this approach in designing photonic crystals where complex geometries have been optimized to maximize the band gaps.Salient features of this approach include the following: (1) multi-patch Coons representation of design geometry. The design geometry is represented as a collection of Coons patches where the four boundaries of each patch are represented as NURBS curves. The use of multiple patches is motivated by the need for representing topologically complex geometries. The Coons patches are used as a design representation so that designers do not need to specify interior control points and they provide a mechanism to compute analytical sensitivities for internal nodes in shape optimization, (2) exact boundary conversion to the analysis geometry with guaranteed mesh injectivity. The analysis geometry is a collection of NURBS patches that are converted from the multi-patch Coons representation with geometric exactness in patch boundaries. The internal NURBS control points are embedded in the parametric domain of the Coons patches with a built-in mesh rectifier to ensure the injectivity of the resulting B-spline geometry, i.e. every point in the physical domain is mapped to one point in the parametric domain, (3) analytical sensitivities. Sensitivities of objective functions and constraints with respect to design variables are derived through nodal sensitivities. The nodal sensitivities for the boundary control points are directly determined by the design parameters and those for internal nodes are obtained via the corresponding Coons patches.     

Mesh blending

A new method for smoothly connecting different patches on triangle meshes with arbitrary connectivity, called mesh blending, is presented. A major feature of mesh blending is to move vertices of the blending region to a virtual blending surface by choosing an appropriate parameterization of those vertices. Once blending is completed, the parameterization optimization is performed to perfect the final meshes. Combining mesh blending with multiresolution techniques, an effective blending technique for meshes is obtained. Our method has several advantages: (1) the user can intuitively control the blending result using different blending radii, (2) the shape of cross-section curves can be adjusted to flexibly design complex models, and (3) the resulting mesh has the same connectivity as the original mesh. In this paper, some examples about smoothing, sharpening, and mesh editing show the efficiency of the method.     

A Projective Framework for Polyhedral Mesh Modelling

We present a novel framework for polyhedral mesh editing with face‐based projective maps that preserves planarity by definition. Such meshes are essential in the field of architectural design and rationalization. By using homogeneous coordinates to describe vertices, we can parametrize the entire shape space of planar‐preserving deformations with bilinear equations. The generality of this space allows for polyhedral geometric processing methods to be conducted with ease. We demonstrate its usefulness in planar‐quadrilateral mesh subdivision, a resulting multi‐resolution editing algorithm, and novel shape‐space exploration with prescribed transformations. Furthermore, we show that our shape space is a discretization of a continuous space of conjugate‐preserving projective transformation fields on surfaces. Our shape space directly addresses planar‐quad meshes, on which we put a focus, and we further show that our framework naturally extends to meshes with faces of more than four vertices as well.     

Shape aware normal interpolation for curved surface shading from polyhedral approximation

Independent interpolation of local surface patches and local normal patches is an efficient way for fast rendering of smooth curved surfaces from rough polyhedral meshes. However, the independently interpolating normals may deviate greatly from the analytical normals of local interpolating surfaces, and the normal deviation may cause severe rendering defects when the surface is shaded using the interpolating normals. In this paper we propose two novel normal interpolation schemes along with interpolation of cubic Bézier triangles for rendering curved surfaces from rough triangular meshes. Firstly, the interpolating normal is computed by a Gregory normal patch to each Bézier triangle by a new definition of quadratic normal functions along cubic space curves. Secondly, the interpolating normal is obtained by blending side-vertex normal functions along side-vertex parametric curves of the interpolating Bézier surface. The normal patches by these two methods can not only interpolate given normals at vertices or boundaries of a triangle but also match the shape of the local interpolating surface very well. As a result, more realistic shading results are obtained by either of the two new normal interpolation schemes than by the traditional quadratic normal interpolation method for rendering rough triangular meshes.     

A framework for quad/triangle subdivision surface fitting: Application to mechanical objects

In this paper we present a new framework for subdivision surface approximation of three‐dimensional models represented by polygonal meshes. Our approach, particularly suited for mechanical or Computer Aided Design (CAD) parts, produces a mixed quadrangle‐triangle control mesh, optimized in terms of face and vertex numbers while remaining independent of the connectivity of the input mesh. Our algorithm begins with a decomposition of the object into surface patches. The main idea is to approximate the region boundaries first and then the interior data. Thus, for each patch, a first step approximates the boundaries with subdivision curves (associated with control polygons) and creates an initial subdivision surface by linking the boundary control points with respect to the lines of curvature of the target surface. Then, a second step optimizes the initial subdivision surface by iteratively moving control points and enriching regions according to the error distribution. The final control mesh defining the whole model is then created assembling every local subdivision control meshes. This control polyhedron is much more compact than the original mesh and visually represents the same shape after several subdivision steps, hence it is particularly suitable for compression and visualization tasks. Experiments conducted on several mechanical models have proven the coherency and the efficiency of our algorithm, compared with existing methods.     

Modeling Polyhedral Meshes with Affine Maps

We offer a framework for editing and modeling of planar meshes, focusing on planar quad, and hexagonal‐dominant meshes, which are held in high demand in the field of architectural design. Our framework manipulates these meshes by affine maps that are assigned per‐face, and which naturally ensure the planarity of these faces throughout the process, resulting in a linear subspace of compatible planar deformations for any given mesh. Our modeling metaphors include classical handle‐based editing, mesh interpolation, and shape‐space exploration, all of which allow for an intuitive way to produce new polyhedral and near‐polyhedral meshes by editing.     

Perfect Matching Quad Layouts for Manifold Meshes

This paper introduces a new approach to automatically generate pure quadrilateral patch layouts on manifold meshes. The algorithm is based on a careful construction of a singularity graph of a given input frame field or a given periodic global parameterization. A pure quadrilateral patch layout is then derived as a constrained minimum weight perfect matching of that graph. The resulting layout is optimal relative to a balance between coarseness and geometric feature alignment. We formulate the problem of finding pure quadrilateral patch layouts as a global optimization problem related to a well‐known concept in graph theory. The main advantage of the new method is its simplicity and its computation speed. Patch layouts generated by the present algorithm are high quality and are very competitive compared to current state of the art.     

High quality refinable G‐splines for locally quad‐dominant meshes with T‐gons

Polyhedral modeling and re‐meshing algorithms use T‐junctions to add or remove feature lines in a quadrilateral mesh. In many ways this is akin to adaptive knot insertion in a tensor‐product spline, but differs in that the designer or meshing algorithm does not necessarily protect the consistent combinatorial structure that is required to interpret the resulting quad‐dominant mesh as the control net of a hierarchical spline – and so associate a smooth surface with the mesh as in the popular tensor‐product spline paradigm. While G‐splines for multi‐sided holes or generalized subdivision can, in principle, convert quad‐dominant meshes with T‐junctions into smooth surfaces, they do not preserve the two preferred directions and so cause visible shape artifacts. Only recently have n‐gons with T‐junctions (T‐gons) in unstructured quad‐dominant meshes been recognized as a distinct challenge for generalized splines. This paper makes precise the notion of locally quad‐dominant mesh as quad‐meshes including τ‐nets, i.e. T‐gons surrounded by quads; and presents the first high‐quality G‐spline construction that can use τ‐nets as control nets for spline surfaces suitable, e.g., for automobile outer surfaces. Remarkably, T‐gons can be neighbors, separated by only one quad, both of T‐gons and of points where many quads meet. A τ‐net surface cap consists of 16 polynomial pieces of degree (3,5) and is refinable in a way that is consistent with the surrounding surface. An alternative, everywhere bi‐3 cap is not formally smooth, but achieves the same high‐quality highlight line distribution.     

Local Painting and Deformation of Meshes on the GPU

We present a novel method to adaptively apply modifications to scene data stored in GPU memory. Such modifications may include interactive painting and sculpting operations in an authoring tool, or deformations resulting from collisions between scene objects detected by a physics engine. We only allocate GPU memory for the faces affected by these modifications to store fine‐scale colour or displacement values. This requires dynamic GPU memory management in order to assign and adaptively apply edits to individual faces at runtime. We present such a memory management technique based on a scan‐operation that is efficiently parallelizable. Since our approach runs entirely on the GPU, we avoid costly CPU–GPU memory transfer and eliminate typical bandwidth limitations. This minimizes runtime overhead to under a millisecond and makes our method ideally suited to many real‐time applications such as video games and interactive authoring tools. In addition, our algorithm significantly reduces storage requirements and allows for much higher resolution content compared to traditional global texturing approaches. Our technique can be applied to various mesh representations, including Catmull–Clark subdivision surfaces, as well as standard triangle and quad meshes. In this paper, we demonstrate several scenarios for these mesh types where our algorithm enables adaptive mesh refinement, local surface deformations and interactive on‐mesh painting and sculpting.