Volume Quantization with Flexible Singularities for Hexahedral Meshing - Brückler, Hendrik; Campen, Marcel (Computer Graphics Forum)
Abstract We present a novel algorithm for quantization and subsequent hexahedral mesh generation from seamless volumetric maps. Quantization is the process of choosing integers that represent the numbers of hexahedral elements to be placed in each region of the volume, and transforming the seamless map into an integer‐grid map matching that choice, inducing a hexahedral mesh. Previous work computes such quantizations under the restriction of a fixed predetermined singularity graph. Our novel approach allows for implicit modification and, in particular, simplification of the map's singularity structure wherever that benefits the chosen objective, such as matching target hexahedron sizes as closely as possible. It comes with two novel ingredients: A feature‐focused distortion measure guiding the quantization, and constraints ensuring map injectivity and structure preservation of geometric and topological features, both without relying on a fixed singularity structure. We demonstrate the benefit of the added flexibility offered by this approach: it allows for the generation of hexahedral meshes that more accurately match a desired resolution globally, as well as of meshes exhibiting a simpler block structure.

DiskScissors: Cutting Arbitrary‐Topology Solids for Bijective Mapping - Hinderink, Steffen; Campen, Marcel (Computer Graphics Forum)
Abstract An algorithm for cutting solid objects in a topology‐controlled manner is presented. Concretely, given a loop on the object boundary, a disk‐topology cut surface bounded by the loop is constructed in the interior. In contrast to various previous approaches, both disk topology and conformance to the prescribed loop are ensured by construction, while supporting not only contractible but also incontractible loops on the boundaries of manifold objects of higher genus and arbitrary non‐trivial topology. We describe an implementation of this algorithm in the discrete setting, with triangle mesh cut surfaces embedded in tetrahedral mesh objects. Making use of this novel cutting algorithm, we describe a method for the reliable construction of bijective volumetric maps between solid objects, demonstrating the algorithm's utility. This mapping method overcomes restrictions of the state of the art to topological balls, extending coverage to objects of arbitrary genus, specifically so‐called 1 ‐handlebodies.

Integer‐Sheet‐Pump Quantization for Hexahedral Meshing - Brückler, Hendrik; Bommes, David; Campen, Marcel (Comput. Graph. Forum)
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.

HalfedgeCNN for Native and Flexible Deep Learning on Triangle Meshes - Ludwig, Ingmar; Tyson, Daniel; Campen, Marcel (Computer Graphics Forum)
Abstract We describe HalfedgeCNN, a collection of modules to build neural networks that operate on triangle meshes. Taking inspiration from the (edge‐based) MeshCNN, convolution, pooling, and unpooling layers are consistently defined on the basis of halfedges of the mesh, pairs of oppositely oriented virtual instances of each edge. This provides benefits over alternative definitions on the basis of vertices, edges, or faces. Additional interface layers enable support for feature data associated with such mesh entities in input and output as well. Due to being defined natively on mesh entities and their neighborhoods, lossy resampling or interpolation techniques (to enable the application of operators adopted from image domains) do not need to be employed. The operators have various degrees of freedom that can be exploited to adapt to application‐specific needs.

TinyAD: Automatic Differentiation in Geometry Processing Made Simple - Schmidt, Patrick; Born, Janis; Bommes, David; Campen, Marcel; Kobbelt, Leif (Computer Graphics Forum)
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.

The 3D Motorcycle Complex for Structured Volume Decomposition - Brückler, Hendrik; Gupta, Ojaswi; Mandad, Manish; Campen, Marcel (Computer Graphics Forum)
Abstract The so‐called motorcycle graph has been employed in recent years for various purposes in the context of structured and aligned block decomposition of 2D shapes and 2‐manifold surfaces. Applications are in the fields of surface parametrization, spline space construction, semi‐structured quad mesh generation, or geometry data compression. We describe a generalization of this motorcycle graph concept to the three‐dimensional volumetric setting. Through careful extensions aware of topological intricacies of this higher‐dimensional setting, we are able to guarantee important block decomposition properties also in this case. We describe algorithms for the construction of this 3D motorcycle complex on the basis of either hexahedral meshes or seamless volumetric parametrizations. Its utility is illustrated on examples in hexahedral mesh generation and volumetric T‐spline construction.

Rational Bézier Guarding - Khanteimouri, Payam; Mandad, Manish; Campen, Marcel (Computer Graphics Forum)
Abstract We present a reliable method to generate planar meshes of nonlinear rational triangular elements. The elements are guaranteed to be valid, i.e. defined by injective rational functions. The mesh is guaranteed to conform exactly, without geometric error, to arbitrary rational domain boundary and feature curves. The method generalizes the recent Bézier Guarding technique, which is applicable only to polynomial curves and elements. This generalization enables the accurate handling of practically important cases involving, for instance, circular or elliptic arcs and NURBS curves, which cannot be matched by polynomial elements. Furthermore, although many practical scenarios are concerned with rational functions of quadratic and cubic degree only, our method is fully general and supports arbitrary degree. We demonstrate the method on a variety of test cases.

Surface Map Homology Inference - Born, Janis; Schmidt, Patrick; Campen, Marcel; Kobbelt, Leif (Computer Graphics Forum)
Abstract A homeomorphism between two surfaces not only defines a (continuous and bijective) geometric correspondence of points but also (by implication) an identification of topological features, i.e. handles and tunnels, and how the map twists around them. However, in practice, surface maps are often encoded via sparse correspondences or fuzzy representations that merely approximate a homeomorphism and are therefore inherently ambiguous about map topology. In this work, we show a way to infer topological information from an imperfect input map between two shapes. In particular, we compute a homology map, a linear map that transports homology classes of cycles from one surface to the other, subject to a global consistency constraint. Our inference robustly handles imperfect (e.g., partial, sparse, fuzzy, noisy, outlier‐ridden, non‐injective) input maps and is guaranteed to produce homology maps that are compatible with true homeomorphisms between the input shapes. Homology maps inferred by our method can be directly used to transfer homological information between shapes, or serve as foundation for the construction of a proper homeomorphism guided by the input map, e.g., via compatible surface decomposition.

Quad Layouts via Constrained T‐Mesh Quantization - Lyon, Max; Campen, Marcel; Kobbelt, Leif (Computer Graphics Forum)
Abstract We present a robust and fast method for the creation of conforming quad layouts on surfaces. Our algorithm is based on the quantization of a T‐mesh, i.e. an assignment of integer lengths to the sides of a non‐conforming rectangular partition of the surface. This representation has the benefit of being able to encode an infinite number of layout connectivity options in a finite manner, which guarantees that a valid layout can always be found. We carefully construct the T‐mesh from a given seamless parametrization such that the algorithm can provide guarantees on the results' quality. In particular, the user can specify a bound on the angular deviation of layout edges from prescribed directions. We solve an integer linear program (ILP) to find a coarse quad layout adhering to that maximal deviation. Our algorithm is guaranteed to yield a conforming quad layout free of T‐junctions together with bounded angle distortion. Our results show that the presented method is fast, reliable, and achieves high quality layouts.

Simpler Quad Layouts using Relaxed Singularities - Lyon, Max; Campen, Marcel; Kobbelt, Leif (Computer Graphics Forum)
Abstract A common approach to automatic quad layout generation on surfaces is to, in a first stage, decide on the positioning of irregular layout vertices, followed by finding sensible layout edges connecting these vertices and partitioning the surface into quadrilateral patches in a second stage. While this two‐step approach reduces the problem's complexity, this separation also limits the result quality. In the worst case, the set of layout vertices fixed in the first stage without consideration of the second may not even permit a valid quad layout. We propose an algorithm for the creation of quad layouts in which the initial layout vertices can be adjusted in the second stage. Whenever beneficial for layout quality or even validity, these vertices may be moved within a prescribed radius or even be removed. Our algorithm is based on a robust quantization strategy, turning a continuous T‐mesh structure into a discrete layout. We show the effectiveness of our algorithm on a variety of inputs.

Combinatorial Construction of Seamless Parameter Domains - Zhou, Jiaran; Tu, Changhe; Zorin, Denis; Campen, Marcel (Computer Graphics Forum)
Abstract The problem of seamless parametrization of surfaces is of interest in the context of structured quadrilateral mesh generation and spline‐based surface approximation. It has been tackled by a variety of approaches, commonly relying on continuous numerical optimization to ultimately obtain suitable parameter domains. We present a general combinatorial seamless parameter domain construction, free from the potential numerical issues inherent to continuous optimization techniques in practice. The domains are constructed as abstract polygonal complexes which can be embedded in a discrete planar grid space, as unions of unit squares. We ensure that the domain structure matches any prescribed parametrization singularities (cones) and satisfies seamlessness conditions. Surfaces of arbitrary genus are supported. Once a domain suitable for a given surface is constructed, a seamless and locally injective parametrization over this domain can be obtained using existing planar disk mapping techniques, making recourse to Tutte's classical embedding theorem.

Visibility‐Aware Progressive Farthest Point Sampling on the GPU - Brandt, Sascha; Jähn, Claudius; Fischer, Matthias; Meyer auf der Heide, Friedhelm (Computer Graphics Forum)

Exact Constraint Satisfaction for Truly Seamless Parametrization - Mandad, Manish; Campen, Marcel (Computer Graphics Forum)
Abstract In the field of global surface parametrization a recent focus has been on so‐called seamless parametrization. This term refers to parametrization approaches which, while using an atlas of charts to enable the handling of surfaces of arbitrary topology, relate the parametrization across the cuts between charts via transition functions from special classes of transformations. This effectively makes the cuts invisible to applications which are invariant to these specific transformations in some sense. In actual implementations of these parametrization approaches, however, these restrictions are obeyed only approximately; errors stem from the tolerances of numerical solvers employed and, ultimately, from the limited accuracy of floating point arithmetic. In practice, robustness issues arise from these flaws in the seamlessness of a parametrization, no matter how small. We present a robust global algorithm that turns a given approximately seamless parametrization into an exactly seamless one ‐ that still is representable by standard floating point numbers. It supports common practically relevant additional constraints regarding boundary and feature curve alignment or isocurve connectivity, and ensures that these are likewise fulfilled exactly. This allows subsequent algorithms to operate robustly on the resulting truly seamless parametrization. We believe that the core of our method will furthermore be of benefit in a broader range of applications involving linearly constrained numerical optimization.

Partitioning Surfaces Into Quadrilateral Patches: A Survey - Campen, Marcel (Computer Graphics Forum)
Abstract The efficient and practical representation and processing of geometrically or topologically complex shapes often demands a partitioning into simpler patches. Possibilities range from unstructured arrangements of arbitrarily shaped patches on the one end, to highly structured conforming networks of all‐quadrilateral patches on the other end of the spectrum. Due to its regularity, this latter extreme of conforming partitions with quadrilateral patches, called quad layouts, is most beneficial in many application scenarios, for instance enabling the use of tensor‐product representations based on splines or Bézier patches, grid‐based multi‐resolution techniques and discrete pixel‐based map representations. However, this type of partition is also most complicated to create due to the strict inherent structural restrictions. Traditionally often performed manually in a tedious and demanding process, research in computer graphics and geometry processing has led to a number of computer‐assisted, semi‐automatic, as well as fully automatic approaches to address this problem more efficiently. This survey provides a detailed discussion of this range of methods, treats their strengths and weaknesses and outlines open problems in this field of research.

Scale‐Invariant Directional Alignment of Surface Parametrizations - Campen, Marcel; Ibing, Moritz; Ebke, Hans‐Christian; Zorin, Denis; Kobbelt, Leif (Computer Graphics Forum)
Abstract Various applications of global surface parametrization benefit from the alignment of parametrization isolines with principal curvature directions. This is particularly true for recent parametrization‐based meshing approaches, where this directly translates into a shape‐aware edge flow, better approximation quality, and reduced meshing artifacts. Existing methods to influence a parametrization based on principal curvature directions suffer from scale‐dependence, which implies the necessity of parameter variation, or try to capture complex directional shape features using simple 1D curves. Especially for non‐sharp features, such as chamfers, fillets, blends, and even more for organic variants thereof, these abstractions can be unfit. We present a novel approach which respects and exploits the 2D nature of such directional feature regions, detects them based on coherence and homogeneity properties, and controls the parametrization process accordingly. This approach enables us to provide an intuitive, scale‐invariant control parameter to the user. It also allows us to consider non‐local aspects like the topology of a feature, enabling further improvements. We demonstrate that, compared to previous approaches, global parametrizations of higher quality can be generated without user intervention.

Directional Field Synthesis, Design, and Processing - Vaxman, Amir; Campen, Marcel; Diamanti, Olga; Panozzo, Daniele; Bommes, David; Hildebrandt, Klaus; Ben‐Chen, Mirela (Computer Graphics Forum)
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.

Quad Layout Embedding via Aligned Parameterization - Campen, Marcel; Kobbelt, Leif (Computer Graphics Forum)
Abstract Quad layouting, i.e. the partitioning of a surface into a coarse network of quadrilateral patches, is a fundamental step in application scenarios ranging from animation and simulation to reverse engineering and meshing. This process involves determining the layout's combinatorial structure as well as its geometric embedding in the surface. We present a novel quad layout algorithm that focuses on the embedding optimization, thereby complementing recent methods focusing on the structure optimization aspect. It takes as input a description of the target layout structure and computes a complete embedding in form of a parameterization globally optimized for isometry and, in particular, principal direction alignment. Besides being suited for fully automatic workflows, our method can also incorporate user constraints and support the tedious but common procedure of manual layouting .

Spherical Visibility Sampling - Eikel, Benjamin; Jähn, Claudius; Fischer, Matthias; Meyer auf der Heide, Friedhelm (Computer Graphics Forum)
Many 3D scenes (e.g. generated from CAD data) are composed of a multitude of objects that are nested in each other. A showroom, for instance, may contain multiple cars and every car has a gearbox with many gearwheels located inside. Because the objects occlude each other, only few are visible from outside. We present a new technique, Spherical Visibility Sampling (SVS), for real-time 3D rendering of such -- possibly highly complex -- scenes. SVS exploits the occlusion and annotates hierarchically structured objects with directional visibility information in a preprocessing step. For different directions, the directional visibility encodes which objects of a scene's region are visible from the outside of the regions' enclosing bounding sphere. Since there is no need to store a separate view space subdivision as in most techniques based on preprocessed visibility, a small memory footprint is achieved. Using the directional visibility information for an interactive walkthrough, the potentially visible objects can be retrieved very efficiently without the need for further visibility tests. Our evaluation shows that using SVS allows to preprocess complex 3D scenes fast and to visualize them in real time (e.g. a Power Plant model and five animated Boeing 777 models with billions of triangles). Because SVS does not require hardware support for occlusion culling during rendering, it is even applicable for rendering large scenes on mobile devices.

Practical Anisotropic Geodesy - Campen, Marcel; Heistermann, Martin; Kobbelt, Leif (Computer Graphics Forum)
Abstract The computation of intrinsic, geodesic distances and geodesic paths on surfaces is a fundamental low‐level building block in countless Computer Graphics and Geometry Processing applications. This demand led to the development of numerous algorithms – some for the exact, others for the approximative computation, some focussing on speed, others providing strict guarantees. Most of these methods are designed for computing distances according to the standard Riemannian metric induced by the surface's embedding in Euclidean space. Generalization to other, especially anisotropic, metrics – which more recently gained interest in several application areas – is not rarely hampered by fundamental problems. We explore and discuss possibilities for the generalization and extension of well‐known methods to the anisotropic case, evaluate their relative performance in terms of accuracy and speed, and propose a novel algorithm, the Short‐Term Vector Dijkstra . This algorithm is strikingly simple to implement and proves to provide practical accuracy at a higher speed than generalized previous methods.

Rationalization of Triangle‐Based Point‐Folding Structures - Zimmer, Henrik; Campen, Marcel; Bommes, David; Kobbelt, Leif (Computer Graphics Forum)
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%.