Surface Maps via Adaptive Triangulations

We present a new method to compute continuous and bijective maps (surface homeomorphisms) between two or more genus-0 triangle meshes. In contrast to previous approaches, we decouple the resolution at which a map is represented from the resolution of the input meshes. We discretize maps via common triangulations that approximate the input meshes while remaining in bijective correspondence to them. Both the geometry and the connectivity of these triangulations are optimized with respect to a single objective function that simultaneously controls mapping distortion, triangulation quality, and approximation error. A discrete-continuous optimization algorithm performs both energy-based remeshing as well as global second-order optimization of vertex positions, parametrized via the sphere. With this, we combine the disciplines of compatible remeshing and surface map optimization in a unified formulation and make a contribution in both fields. While existing compatible remeshing algorithms often operate on a fixed pre-computed surface map, we can now globally update this correspondence during remeshing. On the other hand, bijective surface-to-surface map optimization previously required computing costly overlay meshes that are inherently tied to the input mesh resolution. We achieve significant complexity reduction by instead assessing distortion between the approximating triangulations. This new map representation is inherently more robust than previous overlay-based approaches, is less intricate to implement, and naturally supports mapping between more than two surfaces. Moreover, it enables adaptive multi-resolution schemes that, e.g., first align corresponding surface regions at coarse resolutions before refining the map where needed. We demonstrate significant speedups and increased flexibility over state-of-the art mapping algorithms at similar map quality, and also provide a reference implementation of the method.     

A Shape Modulus for Fractal Geometry Generation

We present an efficient new method for computing Mandelbrot-like fractals (Julia sets) that approximate a user-defined shape. Our algorithm is orders of magnitude faster than previous methods, as it entirely sidesteps the need for a time-consuming numerical optimization. It is also more robust, succeeding on shapes where previous approaches failed. The key to our approach is a versor-modulus analysis of fractals that allows us to formulate a novel shape modulus function that directly controls the broad shape of a Julia set, while keeping fine-grained fractal details intact. Our formulation contains flexible artistic controls that allow users to seamlessly add fractal detail to desired spatial regions, while transitioning back to the original shape in others. No previous approach allows Mandelbrot-like details to be “painted” onto meshes.     

Optimizing Object Decomposition to Reduce Visual Artifacts in 3D Printing

We propose a method for the automatic segmentation of 3D objects into parts which can be individually 3D printed and then reassembled by preserving the visual quality of the final object. Our technique focuses on minimizing the surface affected by supports, decomposing the object into multiple parts whose printing orientation is automatically chosen. The segmentation reduces the visual impact on the fabricated model producing non-planar cuts that adapt to the object shape. This is performed by solving an optimization problem that balances the effects of supports and cuts, while trying to place both in occluded regions of the object surface. To assess the practical impact of the solution, we show a number of segmented, 3D printed and reassembled objects.     

HexBox: Interactive Box Modeling of Hexahedral Meshes

We introduce HexBox, an intuitive modeling method and interactive tool for creating and editing hexahedral meshes. Hexbox brings the major and widely validated surface modeling paradigm of surface box modeling into the world of hex meshing. The main idea is to allow the user to box-model a volumetric mesh by primarily modifying its surface through a set of topological and geometric operations. We support, in particular, local and global subdivision, various instantiations of extrusion, removal, and cloning of elements, the creation of non-conformal or conformal grids, as well as shape modifications through vertex positioning, including manual editing, automatic smoothing, or, eventually, projection on an externally-provided target surface. At the core of the efficient implementation of the method is the coherent maintenance, at all steps, of two parallel data structures: a hexahedral mesh representing the topology and geometry of the currently modeled shape, and a directed acyclic graph that connects operation nodes to the affected mesh hexahedra. Operations are realized by exploiting recent advancements in grid-based meshing, such as mixing of 3-refinement, 2-refinement, and face-refinement, and using templated topological bridges to enforce on-the-fly mesh conformity across pairs of adjacent elements. A direct manipulation user interface lets users control all operations. The effectiveness of our tool, released as open source to the community, is demonstrated by modeling several complex shapes hard to realize with competing tools and techniques.     

Cost Minimizing Local Anisotropic Quad Mesh Refinement

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.     

Fast and Robust QEF Minimization using Probabilistic Quadrics

Error quadrics are a fundamental and powerful building block in many geometry processing algorithms. However, finding the minimizer of a given quadric is in many cases not robust and requires a singular value decomposition or some ad-hoc regularization. While classical error quadrics measure the squared deviation from a set of ground truth planes or polygons, we treat the input data as genuinely uncertain information and embed error quadrics in a probabilistic setting (“probabilistic quadrics”) where the optimal point minimizes the expected squared error. We derive closed form solutions for the popular plane and triangle quadrics subject to (spatially varying, anisotropic) Gaussian noise. Probabilistic quadrics can be minimized robustly by solving a simple linear system — 50× faster than SVD. We show that probabilistic quadrics have superior properties in tasks like decimation and isosurface extraction since they favor more uniform triangulations and are more tolerant to noise while still maintaining feature sensitivity. A broad spectrum of applications can directly benefit from our new quadrics as a drop-in replacement which we demonstrate with mesh smoothing via filtered quadrics and non-linear subdivision surfaces.     

FAKIR: An algorithm for revealing the anatomy and pose of statues from raw point sets

3D acquisition of archaeological artefacts has become an essential part of cultural heritage research for preservation or restoration purpose. Statues, in particular, have been at the center of many projects. In this paper, we introduce a way to improve the understanding of acquired statues representing real or imaginary creatures by registering a simple and pliable articulated model to the raw point set data. Our approach performs a Forward And bacKward Iterative Registration (FAKIR) which proceeds joint by joint, needing only a few iterations to converge. We are thus able to detect the pose and elementary anatomy of sculptures, with possibly non realistic body proportions. By adapting our simple skeleton, our method can work on animals and imaginary creatures.     

Displacement-Correlated XFEM for Simulating Brittle Fracture

We present a remeshing-free brittle fracture simulation method under the assumption of quasi-static linear elastic fracture mechanics (LEFM). To achieve this, we devise two algorithms. First, we develop an approximate volumetric simulation, based on the extended Finite Element Method (XFEM), to initialize and propagate Lagrangian crack-fronts. We model the geometry of fracture explicitly as a surface mesh, which allows us to generate high-resolution crack surfaces that are decoupled from the resolution of the deformation mesh. Our second contribution is a mesh cutting algorithm, which produces fragments of the input mesh using the fracture surface. We do this by directly operating on the half-edge data structures of two surface meshes, which enables us to cut general surface meshes including those of concave polyhedra and meshes with abutting concave polygons. Since we avoid triangulation for cutting, the connectivity of the resulting fragments is identical to the (uncut) input mesh except at edges introduced by the cut. We evaluate our simulation and cutting algorithms and show that they outperform state-of-the-art approaches both qualitatively and quantitatively.     

A survey of Optimal Transport for Computer Graphics and Computer Vision

Optimal transport is a long-standing theory that has been studied in depth from both theoretical and numerical point of views. Starting from the 50s this theory has also found a lot of applications in operational research. Over the last 30 years it has spread to computer vision and computer graphics and is now becoming hard to ignore. Still, its mathematical complexity can make it difficult to comprehend, and as such, computer vision and computer graphics researchers may find it hard to follow recent developments in their field related to optimal transport. This survey first briefly introduces the theory of optimal transport in layman's terms as well as most common numerical techniques to solve it. More importantly, it presents applications of these numerical techniques to solve various computer graphics and vision related problems. This involves applications ranging from image processing, geometry processing, rendering, fluid simulation, to computational optics, and many more. It is aimed at computer graphics researchers desiring to follow optimal transport research in their field as well as optimal transport researchers willing to find applications for their numerical algorithms.     

Facial Expression Synthesis using a Global-Local Multilinear Framework

We present a practical method to synthesize plausible 3D facial expressions for a particular target subject. The ability to synthesize an entire facial rig from a single neutral expression has a large range of applications both in computer graphics and computer vision, ranging from the efficient and cost-effective creation of CG characters to scalable data generation for machine learning purposes. Unlike previous methods based on multilinear models, the proposed approach is capable to extrapolate well outside the sample pool, which allows it to plausibly predict the identity of the target subject and create artifact free expression shapes while requiring only a small input dataset. We introduce global-local multilinear models that leverage the strengths of expression-specific and identity-specific local models combined with coarse motion estimations from a global model. Experimental results show that we achieve high-quality, plausible facial expression synthesis results for an individual that outperform existing methods both quantitatively and qualitatively.     

Interpolated corrected curvature measures for polygonal surfaces

A consistent and yet practically accurate definition of curvature onto polyhedral meshes remains an open problem. We propose a new framework to define curvature measures, based on the Corrected Normal Current, which generalizes the normal cycle: it uncouples the positional information of the polyhedral mesh from its geometric normal vector field, and the user can freely choose the corrected normal vector field at vertices for curvature computations. A smooth surface is then built in the Grassmannian ℝ³ × 𝕊² by simply interpolating the given normal vector field. Curvature measures are then computed using the usual Lipschitz–Killing forms, and we provide closed-form formulas per triangle. We prove a stability result with respect to perturbations of positions and normals. Our approach provides a natural scale-space for all curvature estimations, where the scale is given by the radius of the measuring ball. We show on experiments how this method outperforms state-of-the-art methods on clean and noisy data, and even achieves pointwise convergence on difficult polyhedral meshes like digital surfaces. The framework is also well suited to curvature computations using normal map information.     

HalfedgeCNN for Native and Flexible Deep Learning on Triangle Meshes

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.     

Face Editing Using Part-Based Optimization of the Latent Space

We propose an approach for interactive 3D face editing based on deep generative models. Most of the current face modeling methods rely on linear methods and cannot express complex and non-linear deformations. In contrast to 3D morphable face models based on Principal Component Analysis (PCA), we introduce a novel architecture based on variational autoencoders. Our architecture has multiple encoders (one for each part of the face, such as the nose and mouth) which feed a single decoder. As a result, each sub-vector of the latent vector represents one part. We train our model with a novel loss function that further disentangles the space based on different parts of the face. The output of the network is a whole 3D face. Hence, unlike part-based PCA methods, our model learns to merge the parts intrinsically and does not require an additional merging process. To achieve interactive face modeling, we optimize for the latent variables given vertex positional constraints provided by a user. To avoid unwanted global changes elsewhere on the face, we only optimize the subset of the latent vector that corresponds to the part of the face being modified. Our editing optimization converges in less than a second. Our results show that the proposed approach supports a broader range of editing constraints and generates more realistic 3D faces.     

Persistence Analysis of Multi-scale Planar Structure Graph in Point Clouds

Modern acquisition techniques generate detailed point clouds that sample complex geometries. For instance, we are able to produce millimeter-scale acquisition of whole buildings. Processing and exploring geometrical information within such point clouds requires scalability, robustness to acquisition defects and the ability to model shapes at different scales. In this work, we propose a new representation that enriches point clouds with a multi-scale planar structure graph. We define the graph nodes as regions computed with planar segmentations at increasing scales and the graph edges connect regions that are similar across scales. Connected components of the graph define the planar structures present in the point cloud within a scale interval. For instance, with this information, any point is associated to one or several planar structures existing at different scales. We then use topological data analysis to filter the graph and provide the most prominent planar structures. Our representation naturally encodes a large range of information. We show how to efficiently extract geometrical details (e.g. tiles of a roof), arrangements of simple shapes (e.g. steps and mean ramp of a staircase), and large-scale planar proxies (e.g. walls of a building) and present several interactive tools to visualize, select and reconstruct planar primitives directly from raw point clouds. The effectiveness of our approach is demonstrated by an extensive evaluation on a variety of input data, as well as by comparing against state-of-the-art techniques and by showing applications to polygonal mesh reconstruction.     

Interactive Modeling of Cellular Structures on Surfaces with Application to Additive Manufacturing

The rich and evocative patterns of natural tessellations endow them with an unmistakable artistic appeal and structural properties which are echoed across design, production, and manufacturing. Unfortunately, interactive control of such patterns-as modeled by Voronoi diagrams, is limited to the simple two dimensional case and does not extend well tofreeform surfaces. We present an approach for direct modeling and editing of such cellular structures on surface meshes. The overall modeling experience is driven by a set of editing primitives which are efficiently implemented on graphics hardware. We feature a novel application for 3D printing on modern support-free additive manufacturing platforms. Our method decomposes the input surface into a cellular skeletal structure which hosts a set of overlay shells. In this way, material saving can be channeled to the shells while structural stability is channeled to the skeleton. To accommodate the available printer build volume, the cellular structure can be further split into moderately sized parts. Together with shells, they can be conveniently packed to save on production time. The assembly of the printed parts is streamlined by a part numbering scheme which respects the geometric layout of the input model.     

Parallel Transformation of Bounding Volume Hierarchies into Oriented Bounding Box Trees

Oriented bounding box (OBB) hierarchies can be used instead of hierarchies based on axis-aligned bounding boxes (AABB), providing tighter fitting to the underlying geometric structures and resulting in improved interference tests, such as ray-geometry intersections. In this paper, we present a method for the fast, parallel transformation of an existing bounding volume hierarchy (BVH), based on AABBs, into a hierarchy based on oriented bounding boxes. To this end, we parallelise a high-quality OBB extraction algorithm from the literature to operate as a standalone OBB estimator and further extend it to efficiently build an OBB hierarchy in a bottom up manner. This agglomerative approach allows for fast parallel execution and the formation of arbitrary, high-quality OBBs in bounding volume hierarchies. The method is fully implemented on the GPU and extensively evaluated with ray intersections.     

Two-stage Resampling for Bidirectional Path Tracing with Multiple Light Sub-paths

Recent advances in bidirectional path tracing (BPT) reveal that the use of multiple light sub-paths and the resampling of a small number of these can improve the efficiency of BPT. By increasing the number of pre-sampled light sub-paths, the possibility of generating light paths that provide large contributions can be better explored and this can alleviate the correlation of light paths due to the reuse of pre-sampled light sub-paths by all eye sub-paths. The increased number of pre-sampled light subpaths, however, also incurs a high computational cost. In this paper, we propose a two-stage resampling method for BPT to efficiently handle a large number of pre-sampled light sub-paths. We also derive a weighting function that can treat the changes in path probability due to the two-stage resampling. Our method can handle a two orders of magnitude larger number of presampled light sub-paths than previous methods in equal-time rendering, resulting in stable and better noise reduction than state-of-the-art methods.     

Exact Constraint Satisfaction for Truly Seamless Parametrization

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.     

Polygon Laplacian Made Simple

The discrete Laplace-Beltrami operator for surface meshes is a fundamental building block for many (if not most) geometry processing algorithms. While Laplacians on triangle meshes have been researched intensively, yielding the cotangent discretization as the de-facto standard, the case of general polygon meshes has received much less attention. We present a discretization of the Laplace operator which is consistent with its expression as the composition of divergence and gradient operators, and is applicable to general polygon meshes, including meshes with non-convex, and even non-planar, faces. By virtually inserting a carefully placed point we implicitly refine each polygon into a triangle fan, but then hide the refinement within the matrix assembly. The resulting operator generalizes the cotangent Laplacian, inherits its advantages, and is empirically shown to be on par or even better than the recent polygon Laplacian of Alexa and Wardetzky [AW11] — while being simpler to compute.     

Interactive Sculpting of Digital Faces Using an Anatomical Modeling Paradigm

Digitally sculpting 3D human faces is a very challenging task. It typically requires either 1) highly-skilled artists using complex software packages for high quality results, or 2) highly-constrained simple interfaces for consumer-level avatar creation, such as in game engines. We propose a novel interactive method for the creation of digital faces that is simple and intuitive to use, even for novice users, while consistently producing plausible 3D face geometry, and allowing editing freedom beyond traditional video game avatar creation. At the core of our system lies a specialized anatomical local face model (ALM), which is constructed from a dataset of several hundred 3D face scans. User edits are propagated to constraints for an optimization of our data-driven ALM model, ensuring the resulting face remains plausible even for simple edits like clicking and dragging surface points. We show how several natural interaction methods can be implemented in our framework, including direct control of the surface, indirect control of semantic features like age, ethnicity, gender, and BMI, as well as indirect control through manipulating the underlying bony structures. The result is a simple new method for creating digital human faces, for artists and novice users alike. Our method is attractive for low-budget VFX and animation productions, and our anatomical modeling paradigm can complement traditional game engine avatar design packages.