40-Issue 5

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https://diglib.eg.org/handle/10.2312/2633079

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Globally Injective Geometry Optimization with Non-Injective Steps
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Overby, Matthew; Kaufman, Danny; Narain, Rahul; Digne, Julie and Crane, Keenan
We present a method to minimize distortion and compute globally injective mappings from non-injective initialization. Many approaches for distortion minimization subject to injectivity constraints require an injective initialization and feasible intermediate states. However, it is often the case that injective initializers are not readily available, and many distortion energies of interest have barrier terms that stall global progress. The alternating direction method of multipliers (ADMM) has recently gained traction in graphics due to its efficiency and generality. In this work we explore how to endow ADMM with global injectivity while retaining the ability to traverse non-injective iterates. We develop an iterated coupled-solver approach that evolves two solution states in tandem. Our primary solver rapidly drives down energy to a nearly injective state using a dynamic set of efficiently enforceable inversion and overlap constraints. Then, a secondary solver corrects the state, herding the solution closer to feasibility. The resulting method not only compares well to previous work, but can also resolve overlap with free boundaries.
Geodesic Distance Computation via Virtual Source Propagation
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Trettner, Philip; Bommes, David; Kobbelt, Leif; Digne, Julie and Crane, Keenan
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 theWild 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 6x speed-up on the CPU and 20x 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.
Surface Map Homology Inference
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Born, Janis; Schmidt, Patrick; Campen, Marcel; Kobbelt, Leif; Digne, Julie and Crane, Keenan
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.
On Landmark Distances in Polygons
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Gotsman, Craig; Hormann, Kai; Digne, Julie and Crane, Keenan
We study the landmark distance function between two points in a simply connected planar polygon. We show that if the polygon vertices are used as landmarks, then the resulting landmark distance function to any given point in the polygon has a maximum principle and also does not contain local minima. The latter implies that a path between any two points in the polygon may be generated by steepest descent on this distance without getting ''stuck'' at a local minimum. Furthermore, if landmarks are increasingly added along polygon edges, the steepest descent path converges to the minimal geodesic path. Therefore, the landmark distance can be used, on the one hand in robotic navigation for routing autonomous agents along close-to-shortest paths and on the other for efficiently computing approximate geodesic distances between any two domain points, a property which may be useful in an extension of our work to surfaces in 3D. In the discrete setting, the steepest descent strategy becomes a greedy routing algorithm along the edges of a triangulation of the interior of the polygon, and our experiments indicate that this discrete landmark routing always delivers (i.e., does not get stuck) on ''nice'' triangulations.
Learning Direction Fields for Quad Mesh Generation
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Dielen, Alexander; Lim, Isaak; Lyon, Max; Kobbelt, Leif; Digne, Julie and Crane, Keenan
State of the art quadrangulation methods are able to reliably and robustly convert triangle meshes into quad meshes. Most of these methods rely on a dense direction field that is used to align a parametrization from which a quad mesh can be extracted. In this context, the aforementioned direction field is of particular importance, as it plays a key role in determining the structure of the generated quad mesh. If there are no user-provided directions available, the direction field is usually interpolated from a subset of principal curvature directions. To this end, a number of heuristics that aim to identify significant surface regions have been proposed. Unfortunately, the resulting fields often fail to capture the structure found in meshes created by human experts. This is due to the fact that experienced designers can leverage their domain knowledge in order to optimize a mesh for a specific application. In the context of physics simulation, for example, a designer might prefer an alignment and local refinement that facilitates a more accurate numerical simulation. Similarly, a character artist may prefer an alignment that makes the resulting mesh easier to animate. Crucially, this higher level domain knowledge cannot be easily extracted from local curvature information alone. Motivated by this issue, we propose a data-driven approach to the computation of direction fields that allows us to mimic the structure found in existing meshes, which could originate from human experts or other sources. More specifically, we make use of a neural network that aggregates global and local shape information in order to compute a direction field that can be used to guide a parametrization-based quad meshing method. Our approach is a first step towards addressing this challenging problem with a fully automatic learning-based method. We show that compared to classical techniques our data-driven approach combined with a robust model-driven method, is able to produce results that more closely exhibit the ground truth structure of a synthetic dataset (i.e. a manually designed quad mesh template fitted to a variety of human body types in a set of different poses).
Geometry Processing 2021 CGF 40-5: Frontmatter
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Digne, Julie; Crane, Keenan; Digne, Julie and Crane, Keenan
Developable Approximation via Gauss Image Thinning
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Binninger, Alexandre; Verhoeven, Floor; Herholz, Philipp; Sorkine-Hornung, Olga; Digne, Julie and Crane, Keenan
Approximating 3D shapes with piecewise developable surfaces is an active research topic, driven by the benefits of developable geometry in fabrication. Piecewise developable surfaces are characterized by having a Gauss image that is a 1D object - a collection of curves on the Gauss sphere. We present a method for developable approximation that makes use of this classic definition from differential geometry. Our algorithm is an iterative process that alternates between thinning the Gauss image of the surface and deforming the surface itself to make its normals comply with the Gauss image. The simple, local-global structure of our algorithm makes it easy to implement and optimize. We validate our method on developable shapes with added noise and demonstrate its effectiveness on a variety of non-developable inputs. Compared to the state of the art, our method is more general, tessellation independent, and preserves the input mesh connectivity.
Blending of Hyperbolic Closed Curves
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Ikemakhen, Aziz; Ahanchaou, Taoufik; Digne, Julie and Crane, Keenan
In recent years, game developers are interested in developing games in the hyperbolic space. Shape blending is one of the fundamental techniques to produce animation and videos games. This paper presents two algorithms for blending between two closed curves in the hyperbolic plane in a manner that guarantees that the intermediate curves are closed. We deal with hyperbolic discrete curves on Poincaré disc which is a famous model of the hyperbolic plane. We use the linear interpolation approach of the geometric invariants of hyperbolic polygons namely hyperbolic side lengths, exterior angles and geodesic discrete curvature. We formulate the closing condition of a hyperbolic polygon in terms of its geodesic side lengths and exterior angles. This is to be able to generate closed intermediate curves. Finally, some experimental results are given to illustrate that the proposed methods generate aesthetic blending of closed hyperbolic curves.
A Data-Driven Approach to Functional Map Construction and Bases Pursuit
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Azencot, Omri; Lai, Rongjie; Digne, Julie and Crane, Keenan
We propose a method to simultaneously compute scalar basis functions with an associated functional map for a given pair of triangle meshes. Unlike previous techniques that put emphasis on smoothness with respect to the Laplace-Beltrami operator and thus favor low-frequency eigenfunctions, we aim for a basis that allows for better feature matching. This change of perspective introduces many degrees of freedom into the problem allowing to better exploit non-smooth descriptors. To effectively search in this high-dimensional space of solutions, we incorporate into our minimization state-of-the-art regularizers. We solve the resulting highly non-linear and non-convex problem using an iterative scheme via the Alternating Direction Method of Multipliers. At each step, our optimization involves simple to solve linear or Sylvester-type equations. In practice, our method performs well in terms of convergence, and we additionally show that it is similar to a provably convergent problem. We show the advantages of our approach by extensively testing it on multiple datasets in a few applications including shape matching, consistent quadrangulation and scalar function transfer.
Simpler Quad Layouts using Relaxed Singularities
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Lyon, Max; Campen, Marcel; Kobbelt, Leif; Digne, Julie and Crane, Keenan
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.
Stable and Efficient Differential Estimators on Oriented Point Clouds
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Lejemble, Thibault; Coeurjolly, David; Barthe, Loïc; Mellado, Nicolas; Digne, Julie and Crane, Keenan
Point clouds are now ubiquitous in computer graphics and computer vision. Differential properties of the point-sampled surface, such as principal curvatures, are important to estimate in order to locally characterize the scanned shape. To approximate the surface from unstructured points equipped with normal vectors, we rely on the Algebraic Point Set Surfaces (APSS) [GG07] for which we provide convergence and stability proofs for the mean curvature estimator. Using an integral invariant viewpoint, this first contribution links the algebraic sphere regression involved in the APSS algorithm to several surface derivatives of different orders. As a second contribution, we propose an analytic method to compute the shape operator and its principal curvatures from the fitted algebraic sphere. We compare our method to the state-of-the-art with several convergence and robustness tests performed on a synthetic sampled surface. Experiments show that our curvature estimations are more accurate and stable while being faster to compute compared to previous methods. Our differential estimators are easy to implement with little memory footprint and only require a unique range neighbors query per estimation. Its highly parallelizable nature makes it appropriate for processing large acquired data, as we show in several real-world experiments.
Gauss Stylization: Interactive Artistic Mesh Modeling based on Preferred Surface Normals
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Kohlbrenner, Maximilian; Finnendahl, Ugo; Djuren, Tobias; Alexa, Marc; Digne, Julie and Crane, Keenan
Extending the ARAP energy with a term that depends on the face normal, energy minimization becomes an effective stylization tool for shapes represented as meshes. Our approach generalizes the possibilities of Cubic Stylization: the set of preferred normals can be chosen arbitrarily from the Gauss sphere, including semi-discrete sets to model preference for cylinder- or cone-like shapes. The optimization is designed to retain, similar to ARAP, the constant linear system in the global optimization. This leads to convergence behavior that enables interactive control over the parameters of the optimization. We provide various examples demonstrating the simplicity and versatility of the approach.
Frame Field Operators
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Palmer, David; Stein, Oded; Solomon, Justin; Digne, Julie and Crane, Keenan
Differential operators are widely used in geometry processing for problem domains like spectral shape analysis, data interpolation, parametrization and mapping, and meshing. In addition to the ubiquitous cotangent Laplacian, anisotropic second-order operators, as well as higher-order operators such as the Bilaplacian, have been discretized for specialized applications. In this paper, we study a class of operators that generalizes the fourth-order Bilaplacian to support anisotropic behavior. The anisotropy is parametrized by a symmetric frame field, first studied in connection with quadrilateral and hexahedral meshing, which allows for fine-grained control of local directions of variation. We discretize these operators using a mixed finite element scheme, verify convergence of the discretization, study the behavior of the operator under pullback, and present potential applications.
Scalable Surface Reconstruction with Delaunay-Graph Neural Networks
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Sulzer, Raphael; Landrieu, Loic; Marlet, Renaud; Vallet, Bruno; Digne, Julie and Crane, Keenan
We introduce a novel learning-based, visibility-aware, surface reconstruction method for large-scale, defect-laden point clouds. Our approach can cope with the scale and variety of point cloud defects encountered in real-life Multi-View Stereo (MVS) acquisitions. Our method relies on a 3D Delaunay tetrahedralization whose cells are classified as inside or outside the surface by a graph neural network and an energy model solvable with a graph cut. Our model, making use of both local geometric attributes and line-of-sight visibility information, is able to learn a visibility model from a small amount of synthetic training data and generalizes to real-life acquisitions. Combining the efficiency of deep learning methods and the scalability of energybased models, our approach outperforms both learning and non learning-based reconstruction algorithms on two publicly available reconstruction benchmarks.
Practical Computation of the Cut Locus on Discrete Surfaces
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Mancinelli, Claudio; Livesu, Marco; Puppo, Enrico; Digne, Julie and Crane, Keenan
We present a novel method to compute the cut locus of a distance function encoded on a polygonal mesh. Our method exploits theoretical findings about the cut locus and - with a combination of analytic, geometric and topological tools - it is able to compute a topologically correct and geometrically accurate approximation of it. Our result can be either restricted to the mesh edges, or aligned with the real cut locus. Both outputs may be useful for practical applications. We also provide a convenient tool to optionally prune the weak branches of the cut locus, simplifying its structure. Our approach supersedes prior art, in that it is easier to use and also orders of magnitude faster. In fact, it depends on just one parameter, and it flawlessly operates on meshes with high genus and very high element count at interactive rates. We experiment with different datasets and methods for geodesic distance estimation. We also present applications to local and global surface parameterization.
A Benchmark Dataset for Repetitive Pattern Recognition on Textured 3D Surfaces
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Lengauer, Stefan; Sipiran, Ivan; Preiner, Reinhold; Schreck, Tobias; Bustos, Benjamin; Digne, Julie and Crane, Keenan
In digital archaeology, a large research area is concerned with the computer-aided analysis of 3D captured ancient pottery objects. A key aspect thereby is the analysis of motifs and patterns that were painted on these objects' surfaces. In particular, the automatic identification and segmentation of repetitive patterns is an important task serving different applications such as documentation, analysis and retrieval. Such patterns typically contain distinctive geometric features and often appear in repetitive ornaments or friezes, thus exhibiting a significant amount of symmetry and structure. At the same time, they can occur at varying sizes, orientations and irregular placements, posing a particular challenge for the detection of similarities. A key prerequisite to develop and evaluate new detection approaches for such repetitive patterns is the availability of an expressive dataset of 3D models, defining ground truth sets of similar patterns occurring on their surfaces. Unfortunately, such a dataset has not been available so far for this particular problem. We present an annotated dataset of 82 different 3D models of painted ancient Peruvian vessels, exhibiting different levels of repetitiveness in their surface patterns. To serve the evaluation of detection techniques of similar patterns, our dataset was labeled by archaeologists who identified clearly definable pattern classes. Those given, we manually annotated their respective occurrences on the mesh surfaces. Along with the data, we introduce an evaluation benchmark that can rank different recognition techniques for repetitive patterns based on the mean average precision of correctly segmented 3D mesh faces. An evaluation of different incremental sampling-based detection approaches, as well as a domain specific technique, demonstrates the applicability of our benchmark. With this benchmark we especially want to address the geometry processing community, and expect it will induce novel approaches for pattern analysis based on geometric reasoning like 2D shape and symmetry analysis. This can enable novel research approaches in the Digital Humanities and related fields, based on digitized 3D Cultural Heritage artifacts. Alongside the source code for our evaluation scripts we provide our annotation tools for the public to extend the benchmark and further increase its variety.
Fabrication-Aware Reverse Engineering for Carpentry
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Noeckel, James; Zhao, Haisen; Curless, Brian; Schulz, Adriana; Digne, Julie and Crane, Keenan
We propose a novel method to generate fabrication blueprints from images of carpentered items. While 3D reconstruction from images is a well-studied problem, typical approaches produce representations that are ill-suited for computer-aided design and fabrication applications. Our key insight is that fabrication processes define and constrain the design space for carpentered objects, and can be leveraged to develop novel reconstruction methods. Our method makes use of domain-specific constraints to recover not just valid geometry, but a semantically valid assembly of parts, using a combination of image-based and geometric optimization techniques. We demonstrate our method on a variety of wooden objects and furniture, and show that we can automatically obtain designs that are both easy to edit and accurate recreations of the ground truth. We further illustrate how our method can be used to fabricate a physical replica of the captured object as well as a customized version, which can be produced by directly editing the reconstructed model in CAD software.
A Robust Multi-View System for High-Fidelity Human Body Shape Reconstruction
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Zhang, Qitong; Wang, Lei; Ge, Linlin; Luo, Shan; Zhu, Taihao; Jiang, Feng; Ding, Jimmy; Feng, Jieqing; Digne, Julie and Crane, Keenan
This paper proposes a passive multi-view system for human body shape reconstruction, namely RHF-Human, to overcome several challenges including accurate calibration and stereo matching in self-occluded and low-texture skin regions. The reconstruction process includes four steps: capture, multi-view camera calibration, dense reconstruction, and meshing. The capture system, which consists of 90 digital single-lens reflex cameras, is single-shot to avoid nonrigid deformation of the human body. Two technical contributions are made: (1) a two-step robust multi-view calibration approach that improves calibration accuracy and saves calibration time for each new human body acquired and (2) an accurate PatchMatch multi-view stereo method for dense reconstruction to perform correct matching in self-occluded and low-texture skin regions and to reduce the noise caused by body hair. Experiments on models of various genders, poses, and skin with different amounts of body hair show the robustness of the proposed system. A high-fidelity human body shape dataset with 227 models is constructed, and the average accuracy is within 1.5 mm. The system provides a new scheme for the accurate reconstruction of nonrigid human models based on passive vision and has good potential in fashion design and health care.
Normal-Driven Spherical Shape Analogies
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Liu, Hsueh-Ti Derek; Jacobson, Alec; Digne, Julie and Crane, Keenan
This paper introduces a new method to stylize 3D geometry. The key observation is that the surface normal is an effective instrument to capture different geometric styles. Centered around this observation, we cast stylization as a shape analogy problem, where the analogy relationship is defined on the surface normal. This formulation can deform a 3D shape into different styles within a single framework. One can plug-and-play different target styles by providing an exemplar shape or an energy-based style description (e.g., developable surfaces). Our surface stylization methodology enables Normal Captures as a geometric counterpart to material captures (MatCaps) used in rendering, and the prototypical concept of Spherical Shape Analogies as a geometric counterpart to image analogies in image processing.
SimJEB: Simulated Jet Engine Bracket Dataset
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Whalen, Eamon; Beyene, Azariah; Mueller, Caitlin; Digne, Julie and Crane, Keenan
This paper introduces the Simulated Jet Engine Bracket Dataset (SimJEB) [WBM21]: a new, public collection of crowdsourced mechanical brackets and accompanying structural simulations. SimJEB is applicable to a wide range of geometry processing tasks; the complexity of the shapes in SimJEB offer a challenge to automated geometry cleaning and meshing, while categorical labels and structural simulations facilitate classification and regression (i.e. engineering surrogate modeling). In contrast to existing shape collections, SimJEB's models are all designed for the same engineering function and thus have consistent structural loads and support conditions. On the other hand, SimJEB models are more complex, diverse, and realistic than the synthetically generated datasets commonly used in parametric surrogate model evaluation. The designs in SimJEB were derived from submissions to the GrabCAD Jet Engine Bracket Challenge: an open engineering design competition with over 700 hand-designed CAD entries from 320 designers representing 56 countries. Each model has been cleaned, categorized, meshed, and simulated with finite element analysis according to the original competition specifications. The result is a collection of 381 diverse, high-quality and application-focused designs for advancing geometric deep learning, engineering surrogate modeling, automated cleaning and related geometry processing tasks.

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