Abstract The manufacturing industry faces an urgent need to transition from the linear “make‐take‐use‐dispose” production model towards more sustainable circular models that retain resources in the production chain. Motivated by this need, we introduce the new problem of approximating 3D surfaces by reusing panels from other surfaces. We present an algorithm that takes as input one or several existing shapes and relies on partial shape registration to identify a small set of simple panels that, once cut from the existing shapes and transformed rigidly, approximate a target shape within a user‐defined distance threshold. As a proof of concept, we demonstrate our algorithm in the context of rapid prototyping, where we harvest curved panels from plastic bottles and assemble them with custom connectors to fabricate medium‐size freeform structures.

Parametrization is a key element of many geometric modeling tasks. Seamless parametrization, in particular, is needed as a starting point for many algorithms for quadrangulation and conversion to high-order patches, as well as for the construction of seamless texture maps and displacement maps. Seamless parametrizations are difficult to compute robustly, in part because, in general, it is not known if one exists for a given mesh connectivity or for a particular configuration of singularities. Recently, Penner-coordinate-based methods that allow for connectivity changes have been shown to achieve a perfect success rate on a widely used dataset (Thingi10k). However, previously proposed Penner coordinate methods do not support sharp feature alignment or soft alignment with preferred directions on the surface, both of which are important for practical applications, especially those involving models with sharp features. In this paper, we extend Penner coordinates to surfaces with sharp features to which the parametrization needs to be aligned. Our algorithm extends the holonomy signature description of seamless parametrizations to surfaces with marked feature curves. We describe sufficient conditions for obtaining feasible solutions and describe a two-phase method to efficiently enforce feature constraints or minimize residual errors when solutions are unattainable. We demonstrate that the resulting algorithm works robustly on the Thingi10k dataset with automatic feature labeling, and the resulting seamless parametrizations can be optimized, quantized, and quadrangulated, completing the quad mesh generation pipeline.

Modern hexahedral mesh generation relies on integer-grid maps (IGM), which map the Cartesian grid of integer iso-surfaces to a structure-aligned and conforming hexahedral cell complex discretizing the target shape. The hexahedral mesh is formed by iso-surfaces of the map such that an extraction algorithm is needed to convert the implicit map representation into an explicit mesh. State-of-the-art algorithms have been designed with two goals in mind, i.e., (i) unconditional robustness and (ii) tolerance to map defects in the form of inverted or degenerate tetrahedra. Because of significant advancements in the generation of locally injective maps, the tolerance to map defects has become irrelevant. At the same time, there is a growing demand for efficiently handling significantly larger mesh complexities, unfortunately not well served by the state-of-the-art since the tolerance to map defects induces a high runtime cost. Consequently, we present HexHex, a novel (unconditionally robust) hexahedral mesh extraction algorithm for locally injective integer-grid maps designed for maximal performance and scalability. Key contributions include a novel and highly compact mesh data structure based on so-called propellers and a conservative rasterization technique, significantly reducing the number of required exact predicate tests. HexHex not only offers lower asymptotic runtime complexities from a theoretical perspective but also lower constants, enabling in practice a 30x speedup for medium-sized examples and a larger speedup for more complex inputs, specifically when the hex-to-tet ratio is large. We provide a C++ reference implementation, supporting multi-core parallelization and the extraction of curved (piecewise-linear) hexahedral mesh edges and faces, e.g., valuable for subsequent higher-order mesh generation.

Abstract Several state‐of‐the‐art algorithms for semi‐structured hexahedral meshing involve a so called quantization step to decide on the integer DoFs of the meshing problem, corresponding to the number of hexahedral elements to embed into certain regions of the domain. Existing reliable methods for quantization are based on solving a sequence of integer quadratic programs (IQP). Solving these in a timely and predictable manner with general‐purpose solvers is a challenge, even more so in the open‐source field. We present here an alternative robust and efficient quantization scheme that is instead based on solving a series of continuous linear programs (LP), for which solver availability and efficiency are not an issue. In our formulation, such LPs are used to determine where inflation or deflation of virtual hexahedral sheets are favorable. We compare our method to two implementations of the former IQP formulation (using a commercial and an open‐source MIP solver, respectively), finding that (a) the solutions found by our method are near‐optimal or optimal in most cases, (b) these solutions are found within a much more predictable time frame, and (c) the state of the art run time is outperformed, in the case of using the open‐source solver by orders of magnitude.

Abstract Grid preserving maps of triangulated surfaces were introduced for quad meshing because the 2D unit grid in such maps corresponds to a sub‐division of the surface into quad‐shaped charts. These maps can be obtained by solving a mixed integer optimization problem: Real variables define the geometry of the charts and integer variables define the combinatorial structure of the decomposition. To make this optimization problem tractable, a common strategy is to ignore integer constraints at first, then to enforce them in a so‐called quantization step. Actual quantization algorithms exploit the geometric interpretation of integer variables to solve an equivalent problem: They consider that the final quad mesh is a sub‐division of a T‐mesh embedded in the surface, and optimize the number of sub‐divisions for each edge of this T‐mesh. We propose to operate on a decimated version of the original surface instead of the T‐mesh. It is easier to implement and to adapt to constraints such as free boundaries, complex feature curves network etc .

We present a novel algorithm to map ball-topology tetrahedral meshes onto star-shaped domains with guarantees regarding bijectivity. Our algorithm is based on the recently introduced idea of Shrink-and-Expand, where images of interior vertices are initially clustered at one point (Shrink-), before being sequentially moved to non-degenerate positions yielding a bijective map (-and-Expand). In this context, we introduce the concept of the cluster mesh , i.e. the unexpanded interior mesh consisting of geometrically degenerate simplices. Using local, per-vertex connectivity information solely from the cluster mesh, we show that a viable expansion sequence guaranteed to produce a bijective map can always be found as long as the mesh is shellable. In addition to robustness guarantees for this ubiquitous class of inputs, other practically relevant benefits include improved parsimony and reduced algorithmic complexity. While inheriting some of the worst-case high run time requirements of the state of the art, significant acceleration for the average case is experimentally demonstrated.

Subdividing non-conforming T-mesh layouts into conforming quadrangular meshes is a core component of state-of-the-art (re-)meshing methods. Typically, the required constrained assignment of integer lengths to T-Mesh edges is left to generic branch-and-cut solvers, greedy heuristics, or a combination of the two. This either does not scale well with input complexity or delivers suboptimal result quality. We introduce the Minimum-Deviation-Flow Problem in bi-directed networks (Bi-MDF) and demonstrate its use in modeling and efficiently solving a variety of T-Mesh quantization problems. We develop a fast approximate solver as well as an iterative refinement algorithm based on matching in graphs that solves Bi-MDF exactly. Compared to the state-of-the-art QuadWild [Pietroni et al. 2021] implementation on the authors' 300 dataset, our exact solver finishes after only 0.49% (total 17.06s) of their runtime (3491s) and achieves 11% lower energy while an approximation is computed after 0.09% (3.19s) of their runtime at the cost of 24% increased energy. A novel half-arc-based T-Mesh quantization formulation extends the feasible solution space to include previously unattainable quad meshes. The Bi-MDF problem is more general than our application in layout quantization, potentially enabling similar speedups for other optimization problems that fit into the scheme, such as quad mesh refinement.

The main robustness issue of state-of-the-art frame field based hexahedral mesh generation algorithms originates from non-meshable topological configurations, which do not admit the construction of an integer-grid map but frequently occur in smooth frame fields. In this article, we investigate the topology of frame fields and derive conditions on their meshability, which are the basis for a novel algorithm to automatically turn a given non-meshable frame field into a similar but locally meshable one. Despite local meshability is only a necessary but not sufficient condition for the stronger requirement of meshability, our algorithm increases the 2% success rate of generating valid integer-grid maps with state-of-the-art methods to 58%, when compared on the challenging HexMe dataset [Beaufort et al. 2022]. The source code of our implementation and the data of our experiments are available at https://lib.algohex.eu.

Volumetric mapping is a ubiquitous and difficult problem in Geometry Processing and has been the subject of research in numerous and various directions. While several methods show encouraging results, the field still lacks a general approach with guarantees regarding map bijectivity. Through this work, we aim at opening the door to a new family of methods by providing a novel framework based on the concept of progressive expansion. Starting from an initial map of a tetrahedral mesh whose image may contain degeneracies but no inversions, we incrementally adjust vertex images to expand degenerate elements. By restricting movement to so-called expansion cones , it is done in such a way that the number of degenerate elements decreases in a strictly monotonic manner, without ever introducing any inversion. Adaptive local refinement of the mesh is performed to facilitate this process. We describe a prototype algorithm in the realm of this framework for the computation of maps from ball-topology tetrahedral meshes to convex or star-shaped domains. This algorithm is evaluated and compared to state-of-the-art methods, demonstrating its benefits in terms of bijectivity. We also discuss the associated cost in terms of sometimes significant mesh refinement to obtain the necessary degrees of freedom required for establishing a valid mapping. Our conclusions include that while this algorithm is only of limited immediate practical utility due to efficiency concerns, the general framework has the potential to inspire a range of novel methods improving on the efficiency aspect.

In this article, we provide a detailed survey of techniques for hexahedral mesh generation. We cover the whole spectrum of alternative approaches to mesh generation, as well as post-processing algorithms for connectivity editing and mesh optimization. For each technique, we highlight capabilities and limitations, also pointing out the associated unsolved challenges. Recent relaxed approaches, aiming to generate not pure-hex but hex-dominant meshes, are also discussed. The required background, pertaining to geometrical as well as combinatorial aspects, is introduced along the way.

We present a robust technique to build a topologically optimal all-hexahedral layer on the boundary of a model with arbitrarily complex ridges and corners. The generated boundary layer mesh strictly respects the geometry of the input surface mesh, and it is optimal in the sense that the hexahedral valences of the boundary edges are as close as possible to their ideal values (local dihedral angle divided by 90°). Starting from a valid watertight surface mesh (all-quad in practice), we build a global optimization integer programming problem to minimize the mismatch between the hexahedral valences of the boundary edges and their ideal values. The formulation of the integer programming problem relies on the duality between boundary hexahedral configurations and triangulations of the disk, which we reframe in terms of integer constraints. The global problem is solved efficiently by performing combinatorial branch-and-bound searches on a series of sub-problems defined in the vicinity of complicated ridges/corners, where the local mesh topology is necessarily irregular because of the inherent constraints in hexahedral meshes. From the integer solution, we build the topology of the all-hexahedral layer, and the mesh geometry is computed by untangling/smoothing. Our approach is fully automated, topologically robust, and fast.

HexMe consists of 189 tetrahedral meshes with tagged features and a workflow to generate them. The primary purpose of HexMe meshes is to enable consistent and practically meaningful evaluation of hexahedral meshing algorithms and related techniques, specifically regarding the correct meshing of specified feature points, curves, and surfaces. The tetrahedral meshes have been generated with Gmsh, starting from 63 computer-aided design (CAD) models from various databases. To highlight and label the diverse and challenging aspects of hexahedral mesh generation, the CAD models are classified into three categories: simple, nasty, and industrial. For each CAD model, we provide three kinds of tetrahedral meshes (uniform, curvature-adapted, and box-embedded). The mesh generation pipeline is defined with the help of Snakemake, a modern workflow management system, which allows us to specify a fully automated, extensible, and sustainable workflow. It is possible to download the whole dataset or select individual meshes by browsing the online catalog. The HexMe dataset is built with evolution in mind and prepared for future developments. A public GitHub repository hosts the HexMe workflow, where external contributions and future releases are possible and encouraged. We demonstrate the value of HexMe by exploring the robustness limitations of state-of-the-art frame-field-based hexahedral meshing algorithm. Only for 19 of 189 tagged tetrahedral inputs all feature entities are meshed correctly, while the average success rates are 70.9% / 48.5% / 34.6% for feature points/curves/surfaces.

Abstract Non‐linear optimization is essential to many areas of geometry processing research. However, when experimenting with different problem formulations or when prototyping new algorithms, a major practical obstacle is the need to figure out derivatives of objective functions, especially when second‐order derivatives are required. Deriving and manually implementing gradients and Hessians is both time‐consuming and error‐prone. Automatic differentiation techniques address this problem, but can introduce a diverse set of obstacles themselves, e.g. limiting the set of supported language features, imposing restrictions on a program's control flow, incurring a significant run time overhead, or making it hard to exploit sparsity patterns common in geometry processing. We show that for many geometric problems, in particular on meshes, the simplest form of forward‐mode automatic differentiation is not only the most flexible, but also actually the most efficient choice. We introduce TinyAD: a lightweight C++ library that automatically computes gradients and Hessians, in particular of sparse problems, by differentiating small (tiny) sub‐problems. Its simplicity enables easy integration; no restrictions on, e.g., looping and branching are imposed. TinyAD provides the basic ingredients to quickly implement first and second order Newton‐style solvers, allowing for flexible adjustment of both problem formulations and solver details. By showcasing compact implementations of methods from parametrization, deformation, and direction field design, we demonstrate how TinyAD lowers the barrier to exploring non‐linear optimization techniques. This enables not only fast prototyping of new research ideas, but also improves replicability of existing algorithms in geometry processing. TinyAD is available to the community as an open source library.

Developments in the field of parametrization-based quad mesh generation on surfaces have been impactful over the past decade. In this context, an important advance has been the replacement of error-prone rounding in the generation of integer-grid maps, by robust quantization methods. In parallel, parametrization-based hex mesh generation for volumes has been advanced. In this volumetric context, however, the state-of-the-art still relies on fragile rounding, not rarely producing defective meshes, especially when targeting a coarse mesh resolution. We present a method to robustly quantize volume parametrizations, i.e., to determine guaranteed valid choices of integers for 3D integer-grid maps. Inspired by the 2D case, we base our construction on a non-conforming cell decomposition of the volume, a 3D analogue of a T-mesh. In particular, we leverage the motorcycle complex, a recent generalization of the motorcycle graph, for this purpose. Integer values are expressed in a differential manner on the edges of this complex, enabling the efficient formulation of the conditions required to strictly prevent forcing the map into degeneration. Applying our method in the context of hexahedral meshing, we demonstrate that hexahedral meshes can be generated with significantly improved flexibility.

Abstract We present a highly practical, efficient, and versatile approach for computing approximate geodesic distances. The method is designed to operate on triangle meshes and a set of point sources on the surface. We also show extensions for all kinds of geometric input including inconsistent triangle soups and point clouds, as well as other source types, such as lines. The algorithm is based on the propagation of virtual sources and hence easy to implement. We extensively evaluate our method on about 10000 meshes taken from the Thingi10k and the Tet Meshing in the Wild data sets. Our approach clearly outperforms previous approximate methods in terms of runtime efficiency and accuracy. Through careful implementation and cache optimization, we achieve runtimes comparable to other elementary mesh operations (e.g. smoothing, curvature estimation) such that geodesic distances become a “first‐class citizen” in the toolbox of geometric operations. Our method can be parallelized and we observe up to 6× speed‐up on the CPU and 20× on the GPU. We present a number of mesh processing tasks easily implemented on the basis of fast geodesic distances. The source code of our method is provided as a C++ library under the MIT license.

Abstract Quad meshes as a surface representation have many conceptual advantages over triangle meshes. Their edges can naturally be aligned to principal curvatures of the underlying surface and they have the flexibility to create strongly anisotropic cells without causing excessively small inner angles. While in recent years a lot of progress has been made towards generating high quality uniform quad meshes for arbitrary shapes, their adaptive and anisotropic refinement remains difficult since a single edge split might propagate across the entire surface in order to maintain consistency. In this paper we present a novel refinement technique which finds the optimal trade‐off between number of resulting elements and inserted singularities according to a user prescribed weighting. Our algorithm takes as input a quad mesh with those edges tagged that are prescribed to be refined. It then formulates a binary optimization problem that minimizes the number of additional edges which need to be split in order to maintain consistency. Valence 3 and 5 singularities have to be introduced in the transition region between refined and unrefined regions of the mesh. The optimization hence computes the optimal trade‐off and places singularities strategically in order to minimize the number of consistency splits — or avoids singularities where this causes only a small number of additional splits. When applying the refinement scheme iteratively, we extend our binary optimization formulation such that previous splits can be undone if this prevents degenerate cells with small inner angles that otherwise might occur in anisotropic regions or in the vicinity of singularities. We demonstrate on a number of challenging examples that the algorithm performs well in practice.

Abstract A major challenge in line drawing vectorization is segmenting the input bitmap into separate curves. This segmentation is especially problematic for rough sketches, where curves are depicted using multiple overdrawn strokes. Inspired by feature‐aligned mesh quadrangulation methods in geometry processing, we propose to extract vector curve networks by parametrizing the image with local drawing‐aligned integer grids. The regular structure of the grid facilitates the extraction of clean line junctions; due to the grid's discrete nature, nearby strokes are implicitly grouped together. We demonstrate that our method successfully vectorizes both clean and rough line drawings, whereas previous methods focused on only one of those drawing types.

Field-guided parameterization methods have proven effective for quad meshing of surfaces; these methods compute smooth cross fields to guide the meshing process and then integrate the fields to construct a discrete mesh. A key challenge in extending these methods to three dimensions, however, is representation of field values. Whereas cross fields can be represented by tangent vector fields that form a linear space, the 3D analog—an octahedral frame field—takes values in a nonlinear manifold. In this work, we describe the space of octahedral frames in the language of differential and algebraic geometry. With this understanding, we develop geometry-aware tools for optimization of octahedral fields, namely geodesic stepping and exact projection via semidefinite relaxation. Our algebraic approach not only provides an elegant and mathematically sound description of the space of octahedral frames but also suggests a generalization to frames whose three axes scale independently, better capturing the singular behavior we expect to see in volumetric frame fields. These new odeco frames , so called as they are represented by orthogonally decomposable tensors, also admit a semidefinite program–based projection operator. Our description of the spaces of octahedral and odeco frames suggests computing frame fields via manifold-based optimization algorithms; we show that these algorithms efficiently produce high-quality fields while maintaining stability and smoothness.

We present a method for designing smooth cross fields on surfaces that automatically align to sharp features of an underlying geometry. Our approach introduces a novel class of energies based on a representation of cross fields in the spherical harmonic basis. We provide theoretical analysis of these energies in the smooth setting, showing that they penalize deviations from surface creases while otherwise promoting intrinsically smooth fields. We demonstrate the applicability of our method to quad meshing and include an extensive benchmark comparing our fields to other automatic approaches for generating feature-aligned cross fields on triangle meshes.

Abstract 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.

The generation of quad meshes based on surface parametrization techniques has proven to be a versatile approach. These techniques quantize an initial seamless parametrization so as to obtain an integer grid map implying a pure quad mesh. State-of-the-art methods following this approach have to assume that the surface to be meshed either has no boundary, or has a boundary which the resulting mesh is supposed to be aligned to. In a variety of applications this is not desirable and non-boundary-aligned meshes or grid-parametrizations are preferred. We thus present a technique to robustly generate integer grid maps which are either boundary-aligned, non-boundary-aligned, or partially boundary-aligned, just as required by different applications. We thereby generalize previous work to this broader setting. This enables the reliable generation of trimmed quad meshes with partial elements along the boundary, preferable in various scenarios, from tiled texturing over design and modeling to fabrication and architecture, due to fewer constraints and hence higher overall mesh quality and other benefits in terms of aesthetics and flexibility.

Despite high practical demand, algorithmic hexahedral meshing with guarantees on robustness and quality remains unsolved. A promising direction follows the idea of integer-grid maps, which pull back the Cartesian hexahedral grid formed by integer isoplanes from a parametric domain to a surface-conforming hexahedral mesh of the input object. Since directly optimizing for a high-quality integer-grid map is mathematically challenging, the construction is usually split into two steps: (1) generation of a surface-aligned octahedral field and (2) generation of an integer-grid map that best aligns to the octahedral field. The main robustness issue stems from the fact that smooth octahedral fields frequently exhibit singularity graphs that are not appropriate for hexahedral meshing and induce heavily degenerate integer-grid maps. The first contribution of this work is an enumeration of all local configurations that exist in hex meshes with bounded edge valence, and a generalization of the Hopf-Poincaré formula to octahedral fields, leading to necessary local and global conditions for the hex-meshability of an octahedral field in terms of its singularity graph. The second contribution is a novel algorithm to generate octahedral fields with prescribed hex-meshable singularity graphs, which requires the solution of a large nonlinear mixed-integer algebraic system. This algorithm is an important step toward robust automatic hexahedral meshing since it enables the generation of a hex-meshable octahedral field.

The computation of smooth fields of orthogonal directions within a volume is a critical step in hexahedral mesh generation, used to guide placement of edges and singularities. While this problem shares high-level structure with surface-based frame field problems, critical aspects are lost when extending to volumes, while new structure from the flat Euclidean metric emerges. Taking these considerations into account, this article presents an algorithm for computing such “octahedral” fields. Unlike existing approaches, our formulation achieves infinite resolution in the interior of the volume via the boundary element method (BEM), continuously assigning frames to points in the interior from only a triangle mesh discretization of the boundary. The end result is an orthogonal direction field that can be sampled anywhere inside the mesh, with smooth variation and singular structure in the interior, even with a coarse boundary. We illustrate our computed frames on a number of challenging test geometries. Since the octahedral frame field problem is relatively new, we also contribute a thorough discussion of theoretical and practical challenges unique to this problem.

Surface remeshing is a key component in many geometry processing applications. The typical goal consists in finding a mesh that is (1) geometrically faithful to the original geometry, (2) as coarse as possible to obtain a low-complexity representation and (3) free of bad elements that would hamper the desired application (e.g., the minimum interior angle is above an application-dependent threshold). Our algorithm is designed to address all three optimization goals simultaneously by targeting prescribed bounds on approximation error , minimal interior angle and maximum mesh complexity (number of vertices). The approximation error bound is a hard constraint, while the other two criteria are modeled as optimization goals to guarantee feasibility. Our optimization framework applies carefully prioritized local operators in order to greedily search for the coarsest mesh with minimal interior angle above and approximation error bounded by . Fast runtime is enabled by a local approximation error estimation, while implicit feature preservation is obtained by specifically designed vertex relocation operators. Experiments show that for reasonable angle bounds ( ) our approach delivers high-quality meshes with implicitly preserved features (no tagging required) and better balances between geometric fidelity, mesh complexity and element quality than the state-of-the-art.

Abstract Direction fields and vector fields play an increasingly important role in computer graphics and geometry processing. The synthesis of directional fields on surfaces, or other spatial domains, is a fundamental step in numerous applications, such as mesh generation, deformation, texture mapping, and many more. The wide range of applications resulted in definitions for many types of directional fields: from vector and tensor fields, over line and cross fields, to frame and vector‐set fields. Depending on the application at hand, researchers have used various notions of objectives and constraints to synthesize such fields. These notions are defined in terms of fairness, feature alignment, symmetry, or field topology, to mention just a few. To facilitate these objectives, various representations, discretizations, and optimization strategies have been developed. These choices come with varying strengths and weaknesses. This report provides a systematic overview of directional field synthesis for graphics applications, the challenges it poses, and the methods developed in recent years to address these challenges.

State-of-the-art hex meshing algorithms consist of three steps: Frame-field design, parametrization generation, and mesh extraction. However, while the first two steps are usually discussed in detail, the last step is often not well studied. In this paper, we fully concentrate on reliable mesh extraction. Parametrization methods employ computationally expensive countermeasures to avoid mapping input tetrahedra to degenerate or flipped tetrahedra in the parameter domain because such a parametrization does not define a proper hexahedral mesh. Nevertheless, there is no known technique that can guarantee the complete absence of such artifacts. We tackle this problem from the other side by developing a mesh extraction algorithm which is extremely robust against typical imperfections in the parametrization. First, a sanitization process cleans up numerical inconsistencies of the parameter values caused by limited precision solvers and floating-point number representation. On the sanitized parametrization, we extract vertices and so-called darts based on intersections of the integer grid with the parametric image of the tetrahedral mesh. The darts are reliably interconnected by tracing within the parametrization and thus define the topology of the hexahedral mesh. In a postprocessing step, we let certain pairs of darts cancel each other, counteracting the effect of flipped regions of the parametrization. With this strategy, our algorithm is able to robustly extract hexahedral meshes from imperfect parametrizations which previously would have been considered defective. The algorithm will be published as an open source library [Lyon et al. 2016].

Global surface parametrization often requires the use of cuts or charts due to non-trivial topology. In recent years a focus has been on so-calledseamlessparametrizations, where the transition functions across the cuts are rigid transformations with a rotation about some multiple of 90°. Of particular interest, e.g. for quadrilateral meshing, paneling, or texturing, are those instances where in addition the translational part of these transitions is integral (or more generally: quantized). We show that finding not even the optimal, but just an arbitrary valid quantization (one that does not imply parametric degeneracies), is a complex combinatorial problem. We present a novel method that allows us to solve it, i.e. to find valid as well as good quality quantizations. It is based on an original approach to quickly construct solutions to linear Diophantine equation systems, exploiting the specific geometric nature of the parametrization problem. We thereby largely outperform the state-of-the-art, sometimes by several orders of magnitude.

Designers frequently draw curvature lines to convey bending of smooth surfaces in concept sketches. We present a method to extrapolate curvature lines in a rough concept sketch, recovering the intended 3D curvature field and surface normal at each pixel of the sketch. This 3D information allows to enrich the sketch with 3D-looking shading and texturing. We first introduce the concept of regularized curvature lines that model the lines designers draw over curved surfaces, encompassing curvature lines and their extension as geodesics over flat or umbilical regions. We build on this concept to define the orthogonal cross field that assigns two regularized curvature lines to each point of a 3D surface. Our algorithm first estimates the projection of this cross field in the drawing, which is nonorthogonal due to foreshortening. We formulate this estimation as a scattered interpolation of the strokes drawn in the sketch, which makes our method robust to sketchy lines that are typical for design sketches. Our interpolation relies on a novel smoothness energy that we derive from our definition of regularized curvature lines. Optimizing this energy subject to the stroke constraints produces a dense nonorthogonal 2D cross field which we then lift to 3D by imposing orthogonality. Thus, one central concept of our approach is the generalization of existing cross field algorithms to the nonorthogonal case. We demonstrate our algorithm on a variety of concept sketches with various levels of sketchiness. We also compare our approach with existing work that takes clean vector drawings as input.

The most effective and popular tools for obtaining feature aligned quad meshes from triangular input meshes are based on cross field guided parametrization. These methods are incarnations of a conceptual three-step pipeline: (1) cross field computation, (2) field-guided surface parametrization, (3) quad mesh extraction. While in most meshing scenarios the user prescribes a desired target quad size or edge length, this information is typically taken into account from step 2 onwards only, but not in the cross field computation step. This turns into a problem in the presence of small scale geometric or topological features or noise in the input mesh: closely placed singularities are induced in the cross field, which are not properly reproducible by vertices in a quad mesh with the prescribed edge length, causing severe distortions or even failure of the meshing algorithm. We reformulate the construction of cross fields as well as field-guided parametrizations in a scale-aware manner which effectively suppresses densely spaced features and noise of geometric as well as topological kind. Dominant large-scale features are adequately preserved in the output by relying on the unaltered input mesh as the computational domain.

Abstract Triangle meshes have been nearly ubiquitous in computer graphics, and a large body of data structures and geometry processing algorithms based on them has been developed in the literature. At the same time, quadrilateral meshes, especially semi‐regular ones, have advantages for many applications, and significant progress was made in quadrilateral mesh generation and processing during the last several years. In this survey we discuss the advantages and problems of techniques operating on quadrilateral meshes, including surface analysis and mesh quality, simplification, adaptive refinement, alignment with features, parametrisation and remeshing.

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.

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.

Abstract In mechanical engineering and architecture, structural elements with low material consumption and high load‐bearing capabilities are essential for light‐weight and even self‐supporting constructions. This paper deals with so called point‐folding elements – non‐planar, pyramidal panels, usually formed from thin metal sheets, which exploit the increased structural capabilities emerging from folds or creases. Given a triangulated free‐form surface, a corresponding point‐folding structure is a collection of pyramidal elements basing on the triangles. User‐specified or material‐induced geometric constraints often imply that each individual folding element has a different shape, leading to immense fabrication costs. We present a rationalization method for such structures which respects the prescribed aesthetic and production constraints and finds a minimal set of molds for the production process, leading to drastically reduced costs. For each base triangle we compute and parametrize the range of feasible folding elements that satisfy the given constraints within the allowed tolerances. Then we pose the rationalization task as a geometric intersection problem, which we solve so as to maximize the re‐use of mold dies. Major challenges arise from the high precision requirements and the non‐trivial parametrization of the search space. We evaluate our method on a number of practical examples where we achieve rationalization gains of more than 90%.

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.

Abstract We introduce a fully automatic algorithm which optimizes the high‐level structure of a given quadrilateral mesh to achieve a coarser quadrangular base complex. Such a topological optimization is highly desirable, since state‐of‐the‐art quadrangulation techniques lead to meshes which have an appropriate singularity distribution and an anisotropic element alignment, but usually they are still far away from the high‐level structure which is typical for carefully designed meshes manually created by specialists and used e.g. in animation or simulation. In this paper we show that the quality of the high‐level structure is negatively affected by helical configurations within the quadrilateral mesh. Consequently we present an algorithm which detects helices and is able to remove most of them by applying a novel grid preserving simplification operator (GP‐operator) which is guaranteed to maintain an all‐quadrilateral mesh. Additionally it preserves the given singularity distribution and in particular does not introduce new singularities. For each helix we construct a directed graph in which cycles through the start vertex encode operations to remove the corresponding helix. Therefore a simple graph search algorithm can be performed iteratively to remove as many helices as possible and thus improve the high‐level structure in a greedy fashion. We demonstrate the usefulness of our automatic structure optimization technique by showing several examples with varying complexity.

We present a novel method for quadrangulating a given triangle mesh. After constructing an as smooth as possible symmetric cross field satisfying a sparse set of directional constraints (to capture the geometric structure of the surface), the mesh is cut open in order to enable a low distortion unfolding. Then a seamless globally smooth parametrization is computed whose iso-parameter lines follow the cross field directions. In contrast to previous methods, sparsely distributed directional constraints are sufficient to automatically determine the appropriate number, type and position of singularities in the quadrangulation. Both steps of the algorithm (cross field and parametrization) can be formulated as a mixed-integer problem which we solve very efficiently by an adaptive greedy solver. We show several complex examples where high quality quad meshes are generated in a fully automatic manner.

In an increasing number of applications triangle meshes represent a flexible and efficient alternative to traditional NURBS-based surface representations. Especially in engineering applications it is crucial to guarantee that a prescribed approximation tolerance to a given reference geometry is respected for any combination of geometric algorithms that are applied when processing a triangle mesh. We propose a simple and generic method for computing the distance of a given polygonal mesh to the reference surface, based on a linear approximation of its signed distance field. Exploiting the hardware acceleration of modern GPUs allows us to perform up to 3M triangle checks per second, enabling real-time distance evaluations even for complex geometries. An additional feature of our approach is the accurate high-quality distance visualization of dynamically changing meshes at a rate of 15M triangles per second. Due to its generality, the presented approach can be used to enhance any mesh processing method by global error control, guaranteeing the resulting mesh to stay within a prescribed error tolerance. The application examples that we present include mesh decimation, mesh smoothing and freeform mesh deformation.