We present an algorithm that produces high quality tetrahedral meshes conforming with input polyhedra. Our meshing algorithm is based on Ruppert's Delaunay refinement where convergence is guaranteed thanks to a novel chamfering approach that removes all acute angles from the input. On such a modified input Delaunay refinement produces a Delaunay tetrahedrization where all the faces have bounded angles. The input portions that were removed by the chamfering are re-inserted in this tetrahedrization to achieve exact conformance at the cost of a small number of bad-shaped tetrahedra near the formerly acute input angles. Numerical robustness is guaranteed along all the phases thanks to a clever use of modern indirect geometric predicates and the definition of a new type of implicit point to represent Steiner vertices on the input faces. In practice, our prototype implementation produces meshes having a quality comparable to the state-of-the-art tetgen software: while tetgen fails on 37% of the 3942 valid models in the Thingi10k dataset, our method succeeds on all of them.

Artist-created meshes in-the-wild often do not have a well defined interior. We observe that they typically consist of a mix of solid elements, faces that bound a volume, and shell elements that represent the medial surface of a thin shell. The lack of a well-defined interior prevents downstream applications, such as solid-modeling, simulation, and manufacturing. We present a method that takes as input a surface mesh and assigns to each face a label determining whether it belongs to a solid or shell. These labels reduce ambiguity by defining the interior for solid faces through thresholding the generalized winding number field, and for shell faces as the volume within an offset. We cast the labeling problem as an optimization that outputs a solid/shell label for each face, guided by a sparse set of user inputs. Once labeling is complete, we show how the shape can be volume meshed by passing the shell faces through an offset mesher and the solid faces to an off-the-shelf tetrahedral mesher, producing a final volumetric mesh by taking their union. Experiments on diverse meshes with defects and multiple solid and shell components demonstrate that our approach delivers the desired labels, enabling modeling and simulation on wild meshes in a way that respects the user intent.

High speed, high-resolution, and accurate 3D scanning would open doors to many new applications in graphics, robotics, science, and medicine by enabling the accurate scanning of deformable objects during interactions. Past attempts to use structured light, time-of-flight, and stereo in high-speed settings have usually required tradeoffs in resolution or inaccuracy. In this paper, we introduce a method that enables, for the first time, 3D scanning at 450 frames per second at 1 Megapixel, or 1,450 frames per second at 0.4 Megapixel in an environment with controlled lighting. The key idea is to use a per-pixel lookup table that maps colors to depths, which is built using a linear stage. Imperfections, such as lens-distortion and sensor defects are baked into the calibration. We describe our method and test it on a novel hardware prototype. We compare the system with both ground-truth geometry as well as commercially available dynamic sensors like the Microsoft Kinect and Intel Realsense. Our results show the system acquiring geometry of objects undergoing high-speed deformations and oscillations and demonstrate the ability to recover physical properties from the reconstructions.

A stretch sensor is a device that attaches to objects and measures the amount by which they deform. These sensors have shown great promise as an alternative to vision-based motion-capture systems, and for robotic sensing. Currently, they are generally limited to linear designs, and require a somewhat challenging calibration process. Our goal is to enable inverse design of such sensors, and to largely eliminate the calibration process. To this end, we introduce an accurate, differentiable simulator for capacitive stretch sensors, that treats both the elasto- and electro -static parts of the system. Differentiability allows optimizing the geometry of the sensor in order to improve its design for specific applications. We demonstrate the accuracy of our simulator and the effectiveness of our sensor optimization process for various use cases, such as human interfaces and robotics.

Barrier potentials gained popularity as a means for robust contact handling in physical modeling and for modeling self-avoiding shapes. The key to the success of these approaches is adherence to geometric constraints, i.e., avoiding intersections, which are the cause of most robustness problems in complex deformation simulation with contact. However, existing barrier-potential methods may lead to spurious forces and imperfect satisfaction of the geometric constraints. They may have strong resolution dependence, requiring careful adaptation of the potential parameters to the object discretizations. We present a systematic derivation of a continuum potential defined for smooth and piecewise smooth surfaces, starting from identifying a set of natural requirements for contact potentials, including the barrier property, locality, differentiable dependence on shape, and absence of forces in rest configurations. Our potential is formulated independently of surface discretization and addresses the shortcomings of existing potential-based methods while retaining their advantages. We present a discretization of our potential that is a drop-in replacement for the potential used in the incremental potential contact formulation [Li et al. 2020], and compare its behavior to other potential formulations, demonstrating that it has the expected behavior. The presented formulation connects existing barrier approaches, as all recent existing methods can be viewed as a variation of the presented potential, and lays a foundation for developing alternative (e.g., higher-order) versions.

We propose a conservative algorithm to test the geometrical validity of simplicial (triangles, tetrahedra), tensor product (quadrilaterals, hexahedra), and mixed (prisms) elements of arbitrary polynomial order as they deform linearly within a time interval. Our algorithm uses a combination of adaptive Bézier refinement and bisection search to determine if, when, and where the Jacobian determinant of an element’s polynomial geometric map becomes negative in the transition from one configuration to another. In elastodynamic simulation, our algorithm guarantees that the system remains physically valid during the entire trajectory, not only at discrete time steps. Unlike previous approaches, physical validity is preserved even when our method is implemented using floating point arithmetic. Hence, our algorithm is only slightly slower than existing non-conservative methods while providing guarantees and while being an easy drop-in replacement for current validity tests. To prove the practical effectiveness of our algorithm, we demonstrate its use in a high-order Incremental Potential Contact (IPC) elastodynamic simulator and experimentally show that it prevents invalid, simulation-breaking configurations that would otherwise occur using non-conservative methods.

Many problems in computer graphics can be formulated as finding the global minimum of a function subject to a set of non-linear constraints (Minimize), or finding all solutions of a system of non-linear constraints (Solve). We introduce MiSo, a domain-specific language and compiler for generating efficient C++ code for low-dimensional Minimize and Solve problems, that uses interval methods to guarantee conservative results while using floating point arithmetic. We demonstrate that MiSo-generated code shows competitive performance compared to hand-optimized codes for several computer graphics problems, including high-order collision detection with non-linear trajectories, surface-surface intersection, and geometrical validity checks for finite element simulation.

We introduce Topological Offsets , a novel approach to generate manifold and self-intersection-free offset surfaces that are topologically equivalent to an offset infinitesimally close to the surface. Our approach, by construction, creates a manifold, watertight, and self-intersection-free offset surface strictly enclosing the input, while doing a best effort to move it to a prescribed distance from the input. Differently from existing approaches, we embed the input in a background mesh and insert a topological offset around the input with purely combinatorial operations. The topological offset is then inflated/deflated to match the user-prescribed distance while enforcing that no intersections or non-manifold configurations are introduced. We evaluate the effectiveness and robustness of our approach on the Thingi10k dataset, and show that topological offsets are beneficial in multiple graphics applications, including (1) converting non-manifold surfaces to manifold ones, (2) creating layered offsets, and (3) reliably computing finite offsets.

Fossils are crucial for understanding our natural history and the digitalization of fossils has paved the way for paleontologists to share and study them in greater detail. Yet, many fossil-dense regions, in particular low- and middle-income countries, lack the resources to digitalize their vast collections. This project reports on a collaboration between paleontologists and computer scientists to design, build, and operate a device that can be deployed in the field for digitizing a collection of thousands of fossils. We introduce PaleoScan, a user-friendly, cost-effective, high-volume scanner designed to expedite the digitization of extensive fossil collections. PaleoScan is a self-contained 3D scanning system consisting of a light and compact mirrorless camera, a microcontroller, a ChArUco calibration board, and user-controlled LEDs. Software and data processing is cloud-based, where the user interacts with the system through a web application. We deployed PaleoScan in a museum in Brazil with a world-class fossil collection. Our early results reveal its potential to revolutionize the scanning process for fossils.

We present Image Sculpting, a new framework for editing 2D images by incorporating tools from 3D geometry and graphics. This approach differs markedly from existing methods, which are confined to 2D spaces and typically rely on textual instructions, leading to ambiguity and limited control. Image Sculpting converts 2D objects into 3D, enabling direct interaction with their 3D geometry. Post-editing, these objects are re-rendered into 2D, merging into the original image to produce high-fidelity results through a coarse-to-fine enhancement process. The framework supports precise, quantifiable, and physically-plausible editing options such as pose editing, rotation, translation, 3D composition, carving, and serial addition. It marks an initial step towards combining the creative freedom of generative models with the precision of graphics pipelines.

Abstract Two‐scale topology optimization, combined with the design of microstructure families with a broad range of effective material parameters, is widely used in many fabrication applications to achieve a target deformation behavior for a variety of objects. The main idea of this approach is to optimize the distribution of material properties in the object partitioned into relatively coarse cells, and then replace each cell with microstructure geometry that mimics these material properties. In this paper, we focus on adapting this approach to complex shapes in situations when preserving the shape's surface is essential. Our approach extends any regular (i.e. defined on a regular lattice grid) microstructure family to complex shapes, by enriching it with tiles adapted to the geometry of the cut‐cell. We propose a fully automated and robust pipeline based on this approach, and we show that the performance of the regular microstructure family is only minimally affected by our extension while allowing its use on 2D and 3D shapes of high complexity.

Mechanical shock is a common occurrence in various settings, there are two different scenarios for shock protection: catastrophic protection (e.g. car collisions and falls) and routine protection (e.g. shoe soles and mattresses). The former protects against one-time events, the latter against periodic shocks and loads. Common shock absorbers based on plasticity and fracturing materials are suitable for the former, while our focus is on the latter, where elastic structures are useful. Further, we optimize the effective elastic material properties which control the critical shock parameter, maximal stress, with energy dissipation by viscous forces assumed adequate. Improved elastic materials protecting against shock can be used in applications such as automotive suspension, furniture like sofas and mattresses, landing gear systems, etc. Materials offering optimal protection against shock have a highly non-linear elastic response: their reaction force needs to be as close as possible to constant with respect to deformation. In this paper, we use shape optimization and topology search to design 2D families of microstructures approximating the ideal behavior across a range of deformations, leading to superior shock protection. We present an algorithmic pipeline for the optimal design of such families combining differentiable nonlinear homogenization with self-contact and an optimization algorithm. We validate the effectiveness of our advanced 2D designs by extruding and fabricating them with 3D printing technologies and performing material and drop testing.

We introduce a general differentiable solver for time-dependent deformation problems with contact and friction. Our approach uses a finite element discretization with a high-order time integrator coupled with the recently proposed incremental potential contact method for handling contact and friction forces to solve ODE- and PDE-constrained optimization problems on scenes with complex geometry. It supports static and dynamic problems and differentiation with respect to all physical parameters involved in the physical problem description, which include shape, material parameters, friction parameters, and initial conditions. Our analytically derived adjoint formulation is efficient, with a small overhead (typically less than 10% for nonlinear problems) over the forward simulation, and shares many similarities with the forward problem, allowing the reuse of large parts of existing forward simulator code. We implement our approach on top of the open-source PolyFEM library and demonstrate the applicability of our solver to shape design, initial condition optimization, and material estimation on both simulated results and physical validations.

Inspired by surface mesh simplification methods, we present a technique for reducing the number of Bézier curves in a vector graphics while maintaining high fidelity. We propose a curve-to-curve distance metric to repeatedly conduct local segment removal operations. By construction, we identify all possible lossless removal operations ensuring the smallest possible zero-error representation of a given design. Subsequent lossy operations are computed via local Gauss-Newton optimization and processed in a priority queue. We tested our method on the OpenClipArts dataset of 20,000 real-world vector graphics images and show significant improvements over representative previous methods. The generality of our method allows us to show results for curves with varying thickness and for vector graphics animations.

We present a numerically robust algorithm for computing the constrained Delaunay tetrahedrization (CDT) of a piecewise-linear complex, which has a 100% success rate on the 4408 valid models in the Thingi10k dataset. We build on the underlying theory of the well-known tetgen software, but use a floating-point implementation based on indirect geometric predicates to implicitly represent Steiner points: this new approach dramatically simplifies the implementation, removing the need for ad-hoc tolerances in geometric operations. Our approach leads to a robust and parameter-free implementation, with an empirically manageable number of added Steiner points. Furthermore, our algorithm addresses a major gap in tetgen's theory which may lead to algorithmic failure on valid models, even when assuming perfect precision in the calculations. Our output tetrahedrization conforms with the input geometry without approximations. We can further round our output to floating-point coordinates for downstream applications, which almost always results in valid floating-point meshes unless the input triangulation is very close to being degenerate.

We propose In-Timestep Remeshing, a fully coupled, adaptive meshing algorithm for contacting elastodynamics where remeshing steps are tightly integrated, implicitly, within the timestep solve. Our algorithm refines and coarsens the domain automatically by measuring physical energy changes within each ongoing timestep solve. This provides consistent, degree-of-freedom-efficient, productive remeshing that, by construction, is physics-aware and so avoids the errors, over-refinements, artifacts, per-example hand-tuning, and instabilities commonly encountered when remeshing with timestepping methods. Our in-timestep computation then ensures that each simulation step's output is both a converged stable solution on the updated mesh and a temporally consistent trajectory with respect to the model and solution of the last timestep. At the same time, the output is guaranteed safe (intersection- and inversion-free) across all operations. We demonstrate applications across a wide range of extreme stress tests with challenging contacts, sharp geometries, extreme compressions, large timesteps, and wide material stiffness ranges - all scenarios well-appreciated to challenge existing remeshing methods.

Abstract We introduce the firstexactroot parity counter for continuous collision detection (CCD). That is, our algorithm computes the parity (even or odd) of the number of roots of the cubic polynomial arising from a CCD query. We note that the parity is unable to differentiate between zero (no collisions) and the rare case of two roots (collisions). Our method does not have numerical parameters to tune, has a performance comparable to efficient approximate algorithms, and is exact. We test our approach on a large collection of synthetic tests and real simulations, and we demonstrate that it can be easily integrated into existing simulators.

We introduce a code generator that converts unoptimized C++ code operating on sparse data into vectorized and parallel CPU or GPU kernels. Our approach unrolls the computation into a massive expression graph, performs redundant expression elimination, grouping, and then generates an architecture-specific kernel to solve the same problem, assuming that the sparsity pattern is fixed, which is a common scenario in many applications in computer graphics and scientific computing. We show that our approach scales to large problems and can achieve speedups of two orders of magnitude on CPUs and three orders of magnitude on GPUs, compared to a set of manually optimized CPU baselines. To demonstrate the practical applicability of our approach, we employ it to optimize popular algorithms with applications to physical simulation and interactive mesh deformation.

We introduce a novel approach to describe mesh generation, mesh adaptation, and geometric modeling algorithms relying on changing mesh connectivity using a high-level abstraction. The main motivation is to enable easy customization and development of these algorithms via a declarative specification consisting of a set of per-element invariants, operation scheduling, and attribute transfer for each editing operation. We demonstrate that widely used algorithms editing surfaces and volumes can be compactly expressed with our abstraction, and their implementation within our framework is simple, automatically parallelizable on shared-memory architectures, and with guaranteed satisfaction of the prescribed invariants. These algorithms are readable and easy to customize for specific use cases. We introduce a software library implementing this abstraction and providing automatic shared-memory parallelization.

We propose Deep Estimators of Features (DEFs), a learning-based framework for predicting sharp geometric features in sampled 3D shapes. Differently from existing data-driven methods, which reduce this problem to feature classification, we propose to regress a scalar field representing the distance from point samples to the closest feature line on local patches. Our approach is the first that scales to massive point clouds by fusing distance-to-feature estimates obtained on individual patches. We extensively evaluate our approach against related state-of-the-art methods on newly proposed synthetic and real-world 3D CAD model benchmarks. Our approach not only outperforms these (with improvements in Recall and False Positives Rates), but generalizes to real-world scans after training our model on synthetic data and fine-tuning it on a small dataset of scanned data. We demonstrate a downstream application, where we reconstruct an explicit representation of straight and curved sharp feature lines from range scan data. We make code, pre-trained models, and our training and evaluation datasets available at https://github.com/artonson/def.

The Finite Element Method (FEM) is widely used to solve discrete Partial Differential Equations (PDEs) in engineering and graphics applications. The popularity of FEM led to the development of a large family of variants, most of which require a tetrahedral or hexahedral mesh to construct the basis. While the theoretical properties of FEM basis (such as convergence rate, stability, etc.) are well understood under specific assumptions on the mesh quality, their practical performance, influenced both by the choice of the basis construction and quality of mesh generation, have not been systematically documented for large collections of automatically meshed 3D geometries. We introduce a set of benchmark problems involving most commonly solved elliptic PDEs, starting from simple cases with an analytical solution, moving to commonly used test problem setups, and using manufactured solutions for thousands of real-world, automatically meshed geometries. For all these cases, we use state-of-the-art meshing tools to create both tetrahedral and hexahedral meshes, and compare the performance of different element types for common elliptic PDEs. The goal of this benchmark is to enable comparison of complete FEM pipelines, from mesh generation to algebraic solver, and exploration of relative impact of different factors on the overall system performance. As a specific application of our geometry and benchmark dataset, we explore the question of relative advantages of unstructured (triangular/ tetrahedral) and structured (quadrilateral/hexahedral) discretizations. We observe that for Lagrange-type elements, while linear tetrahedral elements perform poorly, quadratic tetrahedral elements perform equally well or outperform hexahedral elements for our set of problems and currently available mesh generation algorithms. This observation suggests that for common problems in structural analysis, thermal analysis, and low Reynolds number flows, high-quality results can be obtained with unstructured tetrahedral meshes, which can be created robustly and automatically. We release the description of the benchmark problems, meshes, and reference implementation of our testing infrastructure to enable statistically significant comparisons between different FE methods, which we hope will be helpful in the development of new meshing and FEA techniques.

We describe a method for the generation of seamless surface parametrizations with guaranteed local injectivity and full control over holonomy. Previous methods guarantee only one of the two. Local injectivity is required to enable these parametrizations' use in applications such as surface quadrangulation and spline construction. Holonomy control is crucial to enable guidance or prescription of the parametrization's isocurves based on directional information, in particular from cross-fields or feature curves, and more generally to constrain the parametrization topologically. To this end we investigate the relation between cross-field topology and seamless parametrization topology. Leveraging previous results on locally injective parametrization and combining them with insights on this relation in terms of holonomy, we propose an algorithm that meets these requirements. A key component relies on the insight that arbitrary surface cut graphs, as required for global parametrization, can be homeomorphically modified to assume almost any set of turning numbers with respect to a given target cross-field.

Generating locally optimal UAV-trajectories is challenging due to the non-convex constraints of collision avoidance and actuation limits. We present the first local, optimization-based UAV-trajectory generator that simultane-ously guarantees validity and asymptotic optimality for known environments. Validity: Given a feasible initial guess, our algo-rithm guarantees the satisfaction of all constraints throughout the process of optimization. Asymptotic Optimality: We use an asymptotic exact piecewise approximation of the trajectory with an automatically adjustable resolution of its discretization. The trajectory converges under refinement to the first-order stationary point of the exact non-convex programming problem. Our method has additional practical advantages including joint optimality in terms of trajectory and time-allocation, and robustness to challenging environments as demonstrated in our experiments.

We describe an efficient algorithm to compute a discrete metric with prescribed Gaussian curvature at all interior vertices and prescribed geodesic curvature along the boundary of a mesh. The metric is (discretely) conformally equivalent to the input metric. Its construction is based on theory developed in [Gu et al. 2018b] and [Springborn 2020], relying on results on hyperbolic ideal Delaunay triangulations. Generality is achieved by considering the surface's intrinsic triangulation as a degree of freedom, and particular attention is paid to the proper treatment of surface boundaries. While via a double cover approach the case with boundary can be reduced to the case without boundary quite naturally, the implied symmetry of the setting causes additional challenges related to stable Delaunay-critical configurations that we address explicitly. We furthermore explore the numerical limits of the approach and derive continuous maps from the discrete metrics.

We introduce the first implicit time-stepping algorithm for rigid body dynamics, with contact and friction, that guarantees intersection-free configurations at every time step. Our algorithm explicitly models the curved trajectories traced by rigid bodies in both collision detection and response. For collision detection, we propose a conservative narrow phase collision detection algorithm for curved trajectories, which reduces the problem to a sequence of linear CCD queries with minimal separation. For time integration and contact response, we extend the recently proposed incremental potential contact framework to reduced coordinates and rigid body dynamics. We introduce a benchmark for rigid body simulation and show that our approach, while less efficient than alternatives, can robustly handle a wide array of complex scenes, which cannot be simulated with competing methods, without requiring per-scene parameter tuning.

We introduce a robust and automatic algorithm to convert linear triangle meshes with feature annotated into coarse tetrahedral meshes with curved elements. Our construction guarantees that the high-order meshes are free of element inversion or self-intersection. A user-specified maximal geometrical error from the input mesh controls the faithfulness of the curved approximation. The boundary of the output mesh is in bijective correspondence to the input, enabling attribute transfer between them, such as boundary conditions for simulations, making our curved mesh an ideal replacement or complement for the original input geometry. The availability of a bijective shell around the input surface is employed to ensure robust curving, prevent self-intersections, and compute a bijective map between the linear input and curved output surface. As necessary building blocks of our algorithm, we extend the bijective shell formulation to support features and propose a robust approach for boundary-preserving linear tetrahedral meshing. We demonstrate the robustness and effectiveness of our algorithm by generating high-order meshes for a large collection of complex 3D models.

Establishing a consistent normal orientation for point clouds is a notoriously difficult problem in geometry processing, requiring attention to both local and global shape characteristics. The normal direction of a point is a function of the local surface neighborhood; yet, point clouds do not disclose the full underlying surface structure. Even assuming known geodesic proximity, calculating a consistent normal orientation requires the global context. In this work, we introduce a novel approach for establishing a globally consistent normal orientation for point clouds. Our solution separates the local and global components into two different sub-problems. In the local phase, we train a neural network to learn a coherent normal direction per patch ( i.e. , consistently oriented normals within a single patch). In the global phase, we propagate the orientation across all coherent patches using a dipole propagation. Our dipole propagation decides to orient each patch using the electric field defined by all previously orientated patches. This gives rise to a global propagation that is stable, as well as being robust to nearby surfaces, holes, sharp features and noise.

We introduce a large-scale benchmark for continuous collision detection (CCD) algorithms, composed of queries manually constructed to highlight challenging degenerate cases and automatically generated using existing simulators to cover common cases. We use the benchmark to evaluate the accuracy, correctness, and efficiency of state-of-the-art continuous collision detection algorithms, both with and without minimal separation. We discover that, despite the widespread use of CCD algorithms, existing algorithms are (1) correct but impractically slow; (2) efficient but incorrect, introducing false negatives that will lead to interpenetration; or (3) correct but over conservative, reporting a large number of false positives that might lead to inaccuracies when integrated in a simulator. By combining the seminal interval root finding algorithm introduced by Snyder in 1992 with modern predicate design techniques, we propose a simple and efficient CCD algorithm. This algorithm is competitive with state-of-the-art methods in terms of runtime while conservatively reporting the time of impact and allowing explicit tradeoff between runtime efficiency and number of false positives reported.

The ScanAllFish project is a large-scale effort to scan all the world's 33,100 known species of fishes. It has already generated thousands of volumetric CT scans of fish species which are available on open access platforms such as the Open Science Framework. To achieve a scanning rate required for a project of this magnitude, many specimens are grouped together into a single tube and scanned all at once. The resulting data contain many fish which are often bent and twisted to fit into the scanner. Our system, Unwind, is a novel interactive visualization and processing tool which extracts, unbends, and untwists volumetric images of fish with minimal user interaction. Our approach enables scientists to interactively unwarp these volumes to remove the undesired torque and bending using a piecewise-linear skeleton extracted by averaging isosurfaces of a harmonic function connecting the head and tail of each fish. The result is a volumetric dataset of a individual, straight fish in a canonical pose defined by the marine biologist expert user. We have developed Unwind in collaboration with a team of marine biologists: Our system has been deployed in their labs, and is presently being used for dataset construction, biomechanical analysis, and the generation of figures for scientific publication.

We introduce an optimal control method for electromagnetic haptic guidance systems. Our real-time approach assists users in pen-based tasks such as drawing, sketching or designing. The key to our control method is that it guides users, yet does not take away agency. Existing approaches force the stylus to a continuously advancing setpoint on a target trajectory, leading to undesirable behavior such as loss of haptic guidance or unintended snapping. Our control approach, in contrast, gently pulls users towards the target trajectory, allowing them to always easily override the system to adapt their input spontaneously and draw at their own speed. To achieve this flexible guidance, our optimization iteratively predicts the motion of an input device such as a pen, and adjusts the position and strength of an underlying dynamic electromagnetic actuator accordingly. To enable real-time computation, we additionally introduce a novel and fast approximate model of an electromagnet. We demonstrate the applicability of our approach by implementing it on a prototypical hardware platform based on an electromagnet moving on a bi-axial linear stage, as well as a set of applications. Experimental results show that our approach is more accurate and preferred by users compared to open-loop and time-dependent closed-loop approaches.

Abstract Sparse matrix computations are among the most important computational patterns, commonly used in geometry processing, physical simulation, graph algorithms, and other situations where sparse data arises. In many cases, the structure of a sparse matrix is known a priori, but the values may change or depend on inputs to the algorithm. We propose a new methodology for compile‐time specialization of algorithms relying on mixing sparse and dense linear algebra operations, using an extension to the widely‐used open source Eigen package. In contrast to library approaches optimizing individual building blocks of a computation (such as sparse matrix product), we generate reusable sparsity‐specific implementations for a given algorithm, utilizing vector intrinsics and reducing unnecessary scanning through matrix structures. We demonstrate the effectiveness of our technique on a benchmark of artificial expressions to quantitatively evaluate the benefit of our approach over the state‐of‐the‐art library Intel MKL. To further demonstrate the practical applicability of our technique we show that our technique can improve performance, with minimal code changes, for mesh smoothing, mesh parametrization, volumetric deformation, optical flow, and computation of the Laplace operator.

We propose a new tetrahedral meshing method, fTetWild, to convert triangle soups into high-quality tetrahedral meshes. Our method builds on the TetWild algorithm, replacing the rational triangle insertion with a new incremental approach to construct and optimize the output mesh, interleaving triangle insertion and mesh optimization. Our approach makes it possible to maintain a valid floating-point tetrahedral mesh at all algorithmic stages, eliminating the need for costly constructions with rational numbers used by TetWild, while maintaining full robustness and similar output quality. This allows us to improve on TetWild in two ways. First, our algorithm is significantly faster, with running time comparable to less robust Delaunay-based tetrahedralization algorithms. Second, our algorithm is guaranteed to produce a valid tetrahedral mesh with floating-point vertex coordinates, while TetWild produces a valid mesh with rational coordinates which is not guaranteed to be valid after floating-point conversion. As a trade-off, our algorithm no longer guarantees that all input triangles are present in the output mesh, but in practice, as confirmed by our tests on the Thingi10k dataset, the algorithm always succeeds in inserting all input triangles.

We introduce an algorithm to convert a self-intersection free, orientable, and manifold triangle mesh T into a generalized prismatic shell equipped with a bijective projection operator to map T to a class of discrete surfaces contained within the shell whose normals satisfy a simple local condition. Properties can be robustly and efficiently transferred between these surfaces using the prismatic layer as a common parametrization domain. The combination of the prismatic shell construction and corresponding projection operator is a robust building block readily usable in many downstream applications, including the solution of PDEs, displacement maps synthesis, Boolean operations, tetrahedral meshing, geometric textures, and nested cages.

Contacts weave through every aspect of our physical world, from daily household chores to acts of nature. Modeling and predictive computation of these phenomena for solid mechanics is important to every discipline concerned with the motion of mechanical systems, including engineering and animation. Nevertheless, efficiently time-stepping accurate and consistent simulations of real-world contacting elastica remains an outstanding computational challenge. To model the complex interaction of deforming solids in contact we propose Incremental Potential Contact (IPC) - a new model and algorithm for variationally solving implicitly time-stepped nonlinear elastodynamics. IPC maintains an intersection- and inversion-free trajectory regardless of material parameters, time step sizes, impact velocities, severity of deformation, or boundary conditions enforced. Constructed with a custom nonlinear solver, IPC enables efficient resolution of time-stepping problems with separate, user-exposed accuracy tolerances that allow independent specification of the physical accuracy of the dynamics and the geometric accuracy of surface-to-surface conformation. This enables users to decouple, as needed per application, desired accuracies for a simulation's dynamics and geometry. The resulting time stepper solves contact problems that are intersection-free (and thus robust), inversion-free, efficient (at speeds comparable to or faster than available methods that lack both convergence and feasibility), and accurate (solved to user-specified accuracies). To our knowledge this is the first implicit time-stepping method, across both the engineering and graphics literature that can consistently enforce these guarantees as we vary simulation parameters. In an extensive comparison of available simulation methods, research libraries and commercial codes we confirm that available engineering and computer graphics methods, while each succeeding admirably in custom-tuned regimes, often fail with instabilities, egregious constraint violations and/or inaccurate and implausible solutions, as we vary input materials, contact numbers and time step. We also exercise IPC across a wide range of existing and new benchmark tests and demonstrate its accurate solution over a broad sweep of reasonable time-step sizes and beyond (up to h = 2s) across challenging large-deformation, large-contact stress-test scenarios with meshes composed of up to 2.3M tetrahedra and processing up to 498K contacts per time step. For applications requiring high-accuracy we demonstrate tight convergence on all measures. While, for applications requiring lower accuracies, e.g. animation, we confirm IPC can ensure feasibility and plausibility even when specified tolerances are lowered for efficiency.

New fabrication technologies have significantly decreased the cost of fabrication of shapes with highly complex geometric structure. One important application of complex fine-scale geometric structures is to create variable effective elastic material properties in shapes manufactured from a single material. Modification of material properties has a variety of uses, from aerospace applications to soft robotics and prosthetic devices. Due to its scalability and effectiveness, an increasingly common approach to creating spatially varying materials is to partition a shape into cells and use a parametric family of small-scale geometric structures with known effective properties to fill the cells. We propose a new approach to solving this problem for extruded, planar microstructures. Differently from existing methods for two-scale optimization based on regular grids with square periodic cells, which cannot conform to an arbitrary boundary, we introduce cell decompositions consisting of (nearly) rhombic cells. These meshes have far greater flexibility than those with square cells in terms of approximating arbitrary shapes, and, at the same time, have a number of properties simplifying small-scale structure construction. Our main contributions include a new family of 2D cell geometry structures, explicitly parameterized by their effective Young's moduli E , Poisson's ratios v , and rhombic angle α with the geometry parameters expressed directly as smooth spline functions of E, v , and α. This family leads to smooth transitions between the tiles and can handle a broad range of rhombic cell shapes. We introduce a complete material design pipeline based on this microstructure family, composed of an algorithm to generate rhombic tessellation from quadrilateral meshes and an algorithm to synthesize the microstructure geometry. We fabricated a number of models and experimentally demonstrated how our method, in combination with material optimization, can be used to achieve the desired deformation behavior.

We introduce a new technique to check containment of a triangle within an envelope built around a given triangle mesh. While existing methods conservatively check containment within a Euclidean envelope, our approach makes use of a non-Euclidean envelope where containment can be checked both exactly and efficiently. Exactness is crucial to address major robustness issues in existing geometry processing algorithms, which we demonstrate by integrating our technique in two surface triangle remeshing algorithms and a volumetric tetrahedral meshing algorithm. We provide a quantitative comparison of our method and alternative algorithms, showing that our solution, in addition to being exact, is also more efficient. Indeed, while containment within large envelopes can be checked in a comparable time, we show that our algorithm outperforms alternative methods when the envelope becomes thin.

3-axis milling enables cheap and precise fabrication of target objects from precut slabs of materials such as wood or stone. However, the space of directly millable shapes is limited since a 3-axis mill can only carve a height-field (HF) surface during each milling and their size is bounded by the slab dimensions, one of which, the height , is typically significantly smaller than the other two for many typical materials. Extending 3-axis milling of precut slabs to general arbitrarily-sized shapes requires decomposing them into bounded-height 3-axis millable parts, or slices , which can be individually milled and then assembled to form the target object. We present DHFSlicer , a novel decomposition method that satisfies the above constraints and significantly reduces both milling time and material waste compared to alternative approaches. We satisfy the fabrication constraints by partitioning target objects into double height-field (DHF) slices, which can be fabricated using two milling passes: the HF surface accessible from one side is milled first, the slice is then flipped using appropriate fixtures, and then the second, remaining, HF surface is milled. DHFSlicer uses an efficient coarse-to-fine decomposition process: It first partitions the inputs into maximally coarse blocks that satisfy a local DHF criterion with respect to per-block milling axes, and then cuts each block into well-sized DHF slices. It minimizes milling time and material waste by keeping the slice count small, and maximizing slice height. We validate our method by embedding it within an end-to-end DHF milling pipeline and fabricating objects from slabs of foam, wood, and MDF; demonstrate that using the obtained slices reduces milling time and material waste by 42% on average compared to existing automatic alternatives; and highlight the benefits of DHFSlicer via extensive ablation studies.

We present an efficient, trivially parallelizable algorithm to compute offset surfaces of shapes discretized using a dexel data structure. Our algorithm is based on a two-stage sweeping procedure that is simple to implement and efficient, entirely avoiding volumetric distance field computations typical of existing methods. Our construction is based on properties of half-space power diagrams, where each seed is only visible by a half space, which were never used before for the computation of surface offsets. The primary application of our method is interactive modeling for digital fabrication. Our technique enables a user to interactively process high-resolution models. It is also useful in a plethora of other geometry processing tasks requiring fast, approximate offsets, such as topology optimization, collision detection, and skeleton extraction. We present experimental timings, comparisons with previous approaches, and provide a reference implementation in the supplemental material.

We introduce ABC-Dataset, a collection of one million Computer-Aided Design (CAD) models for research of geometric deep learning methods and applications. Each model is a collection of explicitly parametrized curves and surfaces, providing ground truth for differential quantities, patch segmentation, geometric feature detection, and shape reconstruction. Sampling the parametric descriptions of surfaces and curves allows generating data in different formats and resolutions, enabling fair comparisons for a wide range of geometric learning algorithms. As a use case for our dataset, we perform a large-scale benchmark for estimation of surface normals, comparing existing data driven methods and evaluating their performance against both the ground truth and traditional normal estimation methods.

The reconstruction of a discrete surface from a point cloud is a fundamental geometry processing problem that has been studied for decades, with many methods developed. We propose the use of a deep neural network as a geometric prior for surface reconstruction. Specifically, we overfit a neural network representing a local chart parameterization to part of an input point cloud using the Wasserstein distance as a measure of approximation. By jointly fitting many such networks to overlapping parts of the point cloud, while enforcing a consistency condition, we compute a manifold atlas. By sampling this atlas, we can produce a dense reconstruction of the surface approximating the input cloud. The entire procedure does not require any training data or explicit regularization, yet, we show that it is able to perform remarkably well: not introducing typical overfitting artifacts, and approximating sharp features closely at the same time. We experimentally show that this geometric prior produces good results for both man-made objects containing sharp features and smoother organic objects, as well as noisy inputs. We compare our method with a number of well-known reconstruction methods on a standard surface reconstruction benchmark.

We present a theoretical and empirical study of the gradient dynamics of overparameterized shallow ReLU networks with one-dimensional input, solving least-squares interpolation. We show that the gradient dynamics of such networks are determined by the gradient flow in a non-redundant parameterization of the network function. We examine the principal qualitative features of this gradient flow. In particular, we determine conditions for two learning regimes: \emph{kernel} and \emph{adaptive}, which depend both on the relative magnitude of initialization of weights in different layers and the asymptotic behavior of initialization coefficients in the limit of large network widths. We show that learning in the kernel regime yields smooth interpolants, minimizing curvature, and reduces to \emph{cubic splines} for uniform initializations. Learning in the adaptive regime favors instead \emph{linear splines}, where knots cluster adaptively at the sample points.

Abstract We propose an octree‐based algorithm to tessellate the interior of a closed surface with hexahedral cells. The generated hexahedral mesh (1) explicitly preserves sharp features of the original input, (2) has a maximal, user‐controlled distance deviation from the input surface, (3) is composed of elements with only positive scaled jacobians (measured by the eight corners of a hex [SEK*07]), and (4) does not have self‐intersections. We attempt to achieve these goals by proposing a novel pipeline to create an initial pure hexahedral mesh from an octree structure, taking advantage of recent developments in the generation of locally injective 3D parametrizations to warp the octree boundary to conform to the input surface. Sharp features in the input are bijectively mapped to poly‐lines in the output and preserved by the deformation, which takes advantage of a scaffold mesh to prevent local and global intersections. The robustness of our technique is experimentally validated by batch processing a large collection of organic and CAD models, without any manual cleanup or parameter tuning. All results including mesh data and statistics in the paper are provided in the additional material. The open‐source implementation will be made available online to foster further research in this direction.

Most additive manufacturing processes fabricate objects by stacking planar layers of solidified material. As a result, produced parts exhibit a so-called staircase effect, which results from sampling slanted surfaces with parallel planes. Using thinner slices reduces this effect, but it always remains visible where layers almost align with the input surfaces. In this research we exploit the ability of some additive manufacturing processes to deposit material slightly out of plane to dramatically reduce these artifacts. We focus in particular on the widespread Fused Filament Fabrication (FFF) technology, since most printers in this category can deposit along slightly curved paths, under deposition slope and thickness constraints. Our algorithm curves the layers, making them either follow the natural slope of the input surface or on the contrary, make them intersect the surfaces at a steeper angle thereby improving the sampling quality. Rather than directly computing curved layers, our algorithm optimizes for a deformation of the model which is then sliced with a standard planar approach. We demonstrate that this approach enables us to encode all fabrication constraints, including the guarantee of generating collision-free toolpaths, in a convex optimization that can be solved using a QP solver. We produce a variety of models and compare print quality between curved deposition and planar slicing.

We propose a hardware and software pipeline to fabricate flexible wearable sensors and use them to capture deformations without line-of-sight. Our first contribution is a low-cost fabrication pipeline to embed multiple aligned conductive layers with complex geometries into silicone compounds. Overlapping conductive areas from separate layers form local capacitors that measure dense area changes. Contrary to existing fabrication methods, the proposed technique only requires hardware that is readily available in modern fablabs. While area measurements alone are not enough to reconstruct the full 3D deformation of a surface, they become sufficient when paired with a data-driven prior. A novel semi-automatic tracking algorithm, based on an elastic surface geometry deformation, allows us to capture ground-truth data with an optical mocap system, even under heavy occlusions or partially unobservable markers. The resulting dataset is used to train a regressor based on deep neural networks, directly mapping the area readings to global positions of surface vertices. We demonstrate the flexibility and accuracy of the proposed hardware and software in a series of controlled experiments and design a prototype of wearable wrist, elbow, and biceps sensors, which do not require line-of-sight and can be worn below regular clothing.

We propose a stretch-sensing soft glove to interactively capture hand poses with high accuracy and without requiring an external optical setup. We demonstrate how our device can be fabricated and calibrated at low cost, using simple tools available in most fabrication labs. To reconstruct the pose from the capacitive sensors embedded in the glove, we propose a deep network architecture that exploits the spatial layout of the sensor itself. The network is trained only once, using an inexpensive off-the-shelf hand pose reconstruction system to gather the training data. The per-user calibration is then performed on-the-fly using only the glove. The glove's capabilities are demonstrated in a series of ablative experiments, exploring different models and calibration methods. Comparing against commercial data gloves, we achieve a 35% improvement in reconstruction accuracy.

We propose a robust 2D meshing algorithm, TriWild , to generate curved triangles reproducing smooth feature curves, leading to coarse meshes designed to match the simulation requirements necessary by applications and avoiding the geometrical errors introduced by linear meshes. The robustness and effectiveness of our technique are demonstrated by batch processing an SVG collection of 20k images, and by comparing our results against state of the art linear and curvilinear meshing algorithms. We demonstrate for our algorithm the practical utility of computing diffusion curves, fluid simulations, elastic deformations, and shape inflation on complex 2D geometries.

We introduce an integrated meshing and finite-element method pipeline enabling solution of partial differential equations in the volume enclosed by a boundary representation. We construct a hybrid hexahedral-dominant mesh, which contains a small number of star-shaped polyhedra, and build a set of high-order bases on its elements, combining triquadratic B-splines, triquadratic hexahedra, and harmonic elements. We demonstrate that our approach converges cubically under refinement, while requiring around 50% of the degrees of freedom than a similarly dense hexahedral mesh composed of triquadratic hexahedra. We validate our approach solving Poisson’s equation on a large collection of models, which are automatically processed by our algorithm, only requiring the user to provide boundary conditions on their surface.

Tutte embedding is one of the most common building blocks in geometry processing algorithms due to its simplicity and provable guarantees. Although provably correct in infinite precision arithmetic, it fails in challenging cases when implemented using floating point arithmetic, largely due to the induced exponential area changes. We propose Progressive Embedding , with similar theoretical guarantees to Tutte embedding, but more resilient to the rounding error of floating point arithmetic. Inspired by progressive meshes, we collapse edges on an invalid embedding to a valid, simplified mesh, then insert points back while maintaining validity. We demonstrate the robustness of our method by computing embeddings for a large collection of disk topology meshes. By combining our robust embedding with a variant of the matchmaker algorithm, we propose a general algorithm for the problem of mapping multiply connected domains with arbitrary hard constraints to the plane, with applications in texture mapping and remeshing.

We study data-driven representations for three-dimensional triangle meshes, which are one of the prevalent objects used to represent 3D geometry. Recent works have developed models that exploit the intrinsic geometry of manifolds and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants, which learn from the local metric tensor via the Laplacian operator. Despite offering excellent sample complexity and built-in invariances, intrinsic geometry alone is invariant to isometric deformations, making it unsuitable for many applications. To overcome this limitation, we propose several upgrades to GNNs to leverage extrinsic differential geometry properties of three-dimensional surfaces, increasing its modeling power. In particular, we propose to exploit the Dirac operator, whose spectrum detects principal curvature directions - this is in stark contrast with the classical Laplace operator, which directly measures mean curvature. We coin the resulting models Surface Networks (SN). We prove that these models define shape representations that are stable to deformation and to discretization, and we demonstrate the efficiency and versatility of SNs on two challenging tasks: temporal prediction of mesh deformations under non-linear dynamics and generative models using a variational autoencoder framework with encoders/decoders given by SNs.

In this video we present soft self-sensing capacitive arrays and demonstrate their use in capturing dense surface deformations without requiring line of sight. The capacitive arrays are made of two electrode patterns embedded into a single silicone compound. The overlaps of the electrode strip patterns form local capacitors. As the sensor is stretched the local capacitance measurements change. We introduce a fabrication technique that allows to produce such sensors while only requiring hardware readily available in modern fablabs. The resulting sensors are able to densely capture area changes as they deform. Since they do not directly measure bend, a prior is required to fully reconstruct the underlying motion. We propose a deep neural network regressing the sensor geometry from the local area measurements. A motion capture system is used for training data acquisition. At runtime vertex positions are predicted and used as positional constraints to deform a mesh using a state-of-the-art elastic surface energy. The flexibility and accuracy of the introduced sensors is demonstrated in a series of controlled experiments and by fabricating a prototype sensor and applying it to capture deforming skeletal and non-skeletal objects.

We introduce the first fully automatic pipeline to convert arbitrary 3D shapes into knit models. Our pipeline is based on a global parametrization remeshing pipeline to produce an isotropic quad-dominant mesh aligned with a 2-RoSy field. The knitting directions over the surface are determined using a set of custom topological operations and a two-step global optimization that minimizes the number of irregularities. The resulting mesh is converted into a valid stitch mesh that represents the knit model. The yarn curves are generated from the stitch mesh and the final yarn geometry is computed using a yarn-level relaxation process. Thus, we produce topologically valid models that can be used with a yarn-level simulation. We validate our algorithm by automatically generating knit models from complex 3D shapes and processing over a hundred models with various shapes without any user input or parameter tuning. We also demonstrate applications of our approach for custom knit model generation using fabrication via 3D printing.

We propose a novel tetrahedral meshing technique that is unconditionally robust, requires no user interaction, and can directly convert a triangle soup into an analysis-ready volumetric mesh. The approach is based on several core principles: (1) initial mesh construction based on a fully robust, yet efficient, filtered exact computation (2) explicit (automatic or user-defined) tolerancing of the mesh relative to the surface input (3) iterative mesh improvement with guarantees, at every step, of the output validity. The quality of the resulting mesh is a direct function of the target mesh size and allowed tolerance: increasing allowed deviation from the initial mesh and decreasing the target edge length both lead to higher mesh quality. Our approach enables "black-box" analysis, i.e. it allows to automatically solve partial differential equations on geometrical models available in the wild, offering a robustness and reliability comparable to, e.g., image processing algorithms, opening the door to automatic, large scale processing of real-world geometric data.

We propose a novel algorithm for decomposing general three-dimensional geometries into a small set of overlap-free height-field blocks , volumes enclosed by a flat base and a height-field surface defined with respect to this base. This decomposition is useful for fabrication methodologies such as 3-axis CNC milling, where a single milling pass can only carve a single height-field surface defined with respect to the machine tray but can also benefit other fabrication settings. Computing our desired decomposition requires solving a highly constrained discrete optimization problem, variants of which are known to be NP-hard. We effectively compute a high-quality decomposition by using a two-step process that leverages the unique characteristics of our setup. Specifically, we notice that if the height-field directions are constrained to the major axes, then we can always produce a valid decomposition starting from a suitable surface segmentation. Our method first produces a compact set of large, possibly overlapping, height-field blocks that jointly cover the model surface by recasting this discrete constrained optimization problem as an unconstrained optimization of a continuous function, which allows for an efficient solution. We then cast the computation of an overlap-free, final decomposition as an ordering problem on a graph and solve it via a combination of cycle elimination and topological sorting. The combined algorithm produces a compact set of height-field blocks that jointly describe the input model within a user given tolerance. We demonstrate our method on a range of inputs and showcase a number of real life models manufactured using our technique.

We introduce a practical pipeline to create UV T-layouts for real-world quad dominant semi-regular meshes. Our algorithm creates large rectangular patches by relaxing the notion of motorcycle graphs and making it insensitive to local irregularities in the mesh structure such as non-quad elements, redundant irregular vertices, T-junctions, and others. Each surface patch, which can contain multiple singularities and/or polygonal elements, is mapped to an axis-aligned rectangle, leading to a simple and efficient UV layout, which is ideal for texture mapping (allowing for mipmapping and artifact-free bilinear interpolation). We demonstrate that our algorithm is an ideal solution for both recent semi-regular, quad-dominant meshing methods, and for the low-poly meshes typically used in games and movies.

For a given PDE problem, three main factors affect the accuracy of FEM solutions: basis order, mesh resolution, and mesh element quality. The first two factors are easy to control, while controlling element shape quality is a challenge, with fundamental limitations on what can be achieved. We propose to use p -refinement (increasing element degree) to decouple the approximation error of the finite element method from the domain mesh quality for elliptic PDEs. Our technique produces an accurate solution even on meshes with badly shaped elements, with a slightly higher running time due to the higher cost of high-order elements. We demonstrate that it is able to automatically adapt the basis to badly shaped elements, ensuring an error consistent with high-quality meshing, without any per-mesh parameter tuning. Our construction reduces to traditional fixed-degree FEM methods on high-quality meshes with identical performance. Our construction decreases the burden on meshing algorithms, reducing the need for often expensive mesh optimization and automatically compensates for badly shaped elements, which are present due to boundary constraints or limitations of current meshing methods. By tackling mesh generation and finite element simulation jointly, we obtain a pipeline that is both more efficient and more robust than combinations of existing state of the art meshing and FEM algorithms.

Modern geometry processsing algorithms depend on an ever-growing toolbox of fundamental sub-routines and data structures. Prototyping from scratch requires much time building basic tools rather than focusing on the novel research idea. Many existing code libraries have unsatisfactory APIs and the time spent implementing sub-routines is often replaced with time spent learning complex, templated object hierarchies or memory layouts.

We propose a robust and efficient field-aligned volumetric meshing algorithm that produces hex-dominant meshes, i.e. meshes that are predominantly composed of hexahedral elements while containing a small number of irregular polyhedra. The latter are placed according to the singularities of two optimized guiding fields, which allow our method to generate meshes with an exceptionally high amount of isotropy. The field design phase of our method relies on a compact quaternionic representation of volumetric octa-fields and a corresponding optimization that explicitly models the discrete matchings between neighboring elements. This optimization naturally supports alignment constraints and scales to very large datasets. We also propose a novel extraction technique that uses field-guided mesh simplification to convert the optimized fields into a hexdominant output mesh. Each simplification operation maintains topological validity as an invariant, ensuring manifold output. These steps easily generalize to other dimensions or representations, and we show how they can be an asset in existing 2D surface meshing techniques. Our method can automatically and robustly convert any tetrahedral mesh into an isotropic hex-dominant mesh and (with minor modifications) can also convert any triangle mesh into a corresponding isotropic quad-dominant mesh, preserving its genus, number of holes, and manifoldness. We demonstrate the benefits of our algorithm on a large collection of shapes provided in the supplemental material along with all generated results.

We introduce a robust and automatic algorithm to simplify the structure and reduce the singularities of a hexahedral mesh. Our algorithm interleaves simplification operations to collapse sheets and chords of the base complex of the input mesh with a geometric optimization, which improves the elements quality. All our operations are guaranteed not to introduce elements with negative Jacobians, ensuring that our algorithm always produces valid hex-meshes, and not to increase the Hausdorff distance from the original shape more than a user-defined threshold, ensuring a faithful approximation of the input geometry. Our algorithm can improve meshes produced with any existing hexahedral meshing algorithm --- we demonstrate its effectiveness by processing a dataset of 194 hex-meshes created with octree-based, polycube-based, and field-aligned methods.

Bijective maps are commonly used in many computer graphics and scientific computing applications, including texture, displacement, and bump mapping. However, their computation is numerically challenging due to the global nature of the problem, which makes standard smooth optimization techniques prohibitively expensive. We propose to use a scaffold structure to reduce this challenging and global problem to a local injectivity condition. This construction allows us to benefit from the recent advancements in locally injective maps optimization to efficiently compute large scale bijective maps (both in 2D and 3D), sidestepping the need to explicitly detect and avoid collisions. Our algorithm is guaranteed to robustly compute a globally bijective map, both in 2D and 3D. To demonstrate the practical applicability, we use it to compute globally bijective single patch parametrizations, to pack multiple charts into a single UV domain, to remove self-intersections from existing models, and to deform 3D objects while preventing self-intersections. Our approach is simple to implement, efficient (two orders of magnitude faster than competing methods), and robust, as we demonstrate in a stress test on a parametrization dataset with over a hundred meshes.

T-junctions occur where surface strips start or terminate. This paper develops a new way to create smooth piecewise polynomial free-form spline surfaces from quad-meshes that include T-junctions. All mesh nodes are interpreted as control points of GT-splines, that is, geometrically smoothly joined piecewise polynomials. GT-splines are akin to and compatible with B-splines and cover simple T-junctions by two polynomial pieces of degree bi-4 and more complex ones by four such patches. They complement multi-sided surface constructions in generating free-form surfaces with adaptive layout. Since GT-splines do not require a global coordination of knot intervals, GT-constructions are easy to deploy and can provide smooth surfaces with T-junctions where T-splines cannot have a smooth parameterization. GT-constructions display a uniform highlight linedistribution on input meshes where alternatives, such as Catmull-Clark subdivision, exhibit oscillations.

We propose a novel framework for the computational design of tensegrity structures, which are constructions made of struts and cables, held rigid by continuous tension between the elements. Tensegrities are known to be difficult to design---existing design methods are often restricted to using symmetric or templated configurations, limiting the design space to simple constructions. We introduce an algorithm to automatically create free-form stable tensegrity designs that satisfy both fabrication and geometric constraints, and faithfully approximate input geometric shapes. Our approach sidesteps the usual force-based approach in favor of a geometric optimization on the positions of the elements. Equipped with this formulation, we provide a design framework to explore the highly constrained space of tensegrity structures. We validate our method with simulations and real-world constructions.

We propose a UV mapping algorithm that jointly optimizes for cuts and distortion, sidestepping heuristics for placing the cuts. The energy we minimize is a state-of-the-art geometric distortion measure, generalized to take seams into account. Our algorithm is designed to support an interactive workflow: it optimizes UV maps on the fly, while the user can interactively move vertices, cut mesh parts, join seams, separate overlapping regions, and control the placement of the parameterization patches in the UV space. Our UV maps are of high quality in terms of both geometric distortion and cut placement, and compare favorably to those designed with traditional modeling tools. The UV maps can be created in a fraction of the time as existing methods, since our algorithm drastically alleviates the trial-and-error, iterative procedures that plague traditional UV mapping approaches.

We present a scalable approach for the optimization of flip-preventing energies in the general context of simplicial mappings and specifically for mesh parameterization. Our iterative minimization is based on the observation that many distortion energies can be optimized indirectly by minimizing a family of simpler proxy energies. Minimization of these proxies is a natural extension of the local/global minimization of the ARAP energy. Our algorithm is simple to implement and scales to datasets with millions of faces. We demonstrate our approach for the computation of maps that minimize a conformal or isometric distortion energy, both in two and three dimensions. In addition to mesh parameterization, we show that our algorithm can be applied to mesh deformation and mesh quality improvement.

Today's 3D scanning pipelines can be classified into two overarching categories: offline, high accuracy methods that rely on global optimization to reconstruct complex scenes with hundreds of millions of samples, and online methods that produce real-time but low-quality output, usually from structure-from-motion or depth sensors. The method proposed in this paper is the first to combine the benefits of both approaches, supporting online reconstruction of scenes with hundreds of millions of samples from high-resolution sensing modalities such as structured light or laser scanners. The key property of our algorithm is that it sidesteps the signed-distance computation of classical reconstruction techniques in favor of direct filtering, parametrization, and mesh and texture extraction. All of these steps can be realized using only weak notions of spatial neighborhoods, which allows for an implementation that scales approximately linearly with the size of each dataset that is integrated into a partial reconstruction. Combined, these algorithmic differences enable a drastically more efficient output-driven interactive scanning and reconstruction workflow, where the user is able to see the final quality field-aligned textured mesh during the entirety of the scanning procedure. Holes or parts with registration problems are displayed in real-time to the user and can be easily resolved by adding further localized scans, or by adjusting the input point cloud using our interactive editing tools with immediate visual feedback on the output mesh. We demonstrate the effectiveness of our algorithm in conjunction with a state-of-the-art structured light scanner and optical tracking system and test it on a large variety of challenging models.

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.

We propose a novel approach to digital character animation, combining the benefits of tangible input devices and sophisticated rig animation algorithms. A symbiotic software and hardware approach facilitates the animation process for novice and expert users alike. We overcome limitations inherent to all previous tangible devices by allowing users to directly control complex rigs using only a small set (5-10) of physical controls. This avoids oversimplification of the pose space and excessively bulky device configurations. Our algorithm derives a small device configuration from complex character rigs, often containing hundreds of degrees of freedom, and a set of sparse sample poses. Importantly, only the most influential degrees of freedom are controlled directly, yet detailed motion is preserved based on a pose interpolation technique. We designed a modular collection of joints and splitters, which can be assembled to represent a wide variety of skeletons. Each joint piece combines a universal joint and two twisting elements, allowing to accurately sense its configuration. The mechanical design provides a smooth inverse kinematics-like user experience and is not prone to gimbal locking. We integrate our method with the professional 3D software Autodesk Maya ® and discuss a variety of results created with characters available online. Comparative user experiments show significant improvements over the closest state-of-the-art in terms of accuracy and time in a keyframe posing task.

We propose a method to fabricate textured 3D models using thermoforming. Differently from industrial techniques, which target mass production of a specific shape, we propose a combined hardware and software solution to manufacture customized, unique objects. Our method simulates the forming process and converts the texture of a given digital 3D model into a pre-distorted image that we transfer onto a plastic sheet. During thermoforming, the sheet deforms to create a faithful physical replica of the digital model. Our hardware setup uses off-the-shelf components and can be calibrated with an automatic algorithm that extracts the simulation parameters from a single calibration object produced by the same process.

Abstract In the digital world, assigning arbitrary colors to an object is a simple operation thanks to texture mapping. However, in the real world, the same basic function of applying colors onto an object is far from trivial. One can specify colors during the fabrication process using a color 3D printer, but this does not apply to already existing objects. Paint and decals can be used during post‐fabrication, but they are challenging to apply on complex shapes. In this paper, we develop a method to enable texture mapping of physical objects, that is, we allow one to map an arbitrary color image onto a three‐dimensional object. Our approach builds upon hydrographics, a technique to transfer pigments printed on a sheet of polymer onto curved surfaces. We first describe a setup that makes the traditional water transfer printing process more accurate and consistent across prints. We then simulate the transfer process using a specialized parameterization to estimate the mapping between the planar color map and the object surface. We demonstrate that our approach enables the application of detailed color maps onto complex shapes such as 3D models of faces and anatomical casts.

We present a novel approach to remesh a surface into an isotropic triangular or quad-dominant mesh using a unified local smoothing operator that optimizes both the edge orientations and vertex positions in the output mesh. Our algorithm produces meshes with high isotropy while naturally aligning and snapping edges to sharp features. The method is simple to implement and parallelize, and it can process a variety of input surface representations, such as point clouds, range scans and triangle meshes. Our full pipeline executes instantly (less than a second) on meshes with hundreds of thousands of faces, enabling new types of interactive workflows. Since our algorithm avoids any global optimization, and its key steps scale linearly with input size, we are able to process extremely large meshes and point clouds, with sizes exceeding several hundred million elements. To demonstrate the robustness and effectiveness of our method, we apply it to hundreds of models of varying complexity and provide our cross-platform reference implementation in the supplemental material.

We present a framework for designing curl-free tangent vector fields on discrete surfaces. Such vector fields are gradients of locally-defined scalar functions, and this property is beneficial for creating surface parameterizations, since the gradients of the parameterization coordinate functions are then exactly aligned with the designed fields. We introduce a novel definition for discrete curl between unordered sets of vectors (PolyVectors), and devise a curl-eliminating continuous optimization that is independent of the matchings between them. Our algorithm naturally places the singularities required to satisfy the user-provided alignment constraints, and our fields are the gradients of an inversion-free parameterization by design.

We propose an interactive quadrangulation method based on a large collection of patterns that are learned from models manually designed by artists. The patterns are distilled into compact quadrangulation rules and stored in a database. At run-time, the user draws strokes to define patches and desired edge flows, and the system queries the database to extract fitting patterns to tessellate the sketches' interiors. The quadrangulation patterns are general and can be applied to tessellate large regions while controlling the positions of the singularities and the edge flow. We demonstrate the effectiveness of our algorithm through a series of live retopology sessions and an informal user study with three professional artists.

Abstract We introduce N‐PolyVector fields, a generalization of N‐RoSy fields for which the vectors are neither necessarily orthogonal nor rotationally symmetric. We formally define a novel representation for N‐PolyVectors as the root sets of complex polynomials and analyze their topological and geometric properties. A smooth N‐PolyVector field can be efficiently generated by solving a sparse linear system without integer variables. We exploit the flexibility of N‐PolyVector fields to design conjugate vector fields, offering an intuitive tool to generate planar quadrilateral meshes.

Abstract We present a method to automatically segment indoor scenes by detecting repeated objects. Our algorithm scales to datasets with 198 million points and does not require any training data. We propose a trivially parallelizable preprocessing step, which compresses a point cloud into a collection of nearly‐planar patches related by geometric transformations. This representation enables us to robustly filter out noise and greatly reduces the computational cost and memory requirements of our method, enabling execution at interactive rates. We propose a patch similarity measure based on shape descriptors and spatial configurations of neighboring patches. The patches are clustered in a Euclidean embedding space based on the similarity matrix to yield the segmentation of the input point cloud. The generated segmentation can be used to compress the raw point cloud, create an object database, and increase the clarity of the point cloud visualization.

Abstract We present an interactive method to restore severely damaged historical parchments. When damaged by heat in a fire, such manuscripts undergo a complex deformation and contain various geometric distortions such as wrinkling, buckling, and shrinking, rendering them nearly illegible. They cannot be physically flattened due to the risk of further damage. We propose a virtual restoration framework to estimate the non‐rigid deformation the parchment underwent and to revert it, making reading the text significantly easier whilst maintaining the veracity of the textual content. We estimate the deformation by combining automatically extracted constraints with user‐provided hints informed by domain knowledge. We demonstrate that our method successfully flattens and straightens the text on a variety of pages scanned from a 17th century document which fell victim to fire damage.

Abstract We propose an algorithm to quadrangulate an N‐sided patch (2 ≤ N ≤ 6) with prescribed numbers of edge subdivisions at its boundary. Our algorithm is guaranteed to succeed for arbitrary valid input, which is proved using a canonical simplification of the input and a small set of topological patterns that are sufficient for supporting all possible cases. Our algorithm produces solutions with minimal number of irregular vertices by default, but it also allows the user to choose other feasible solutions by solving a set of small integer linear programs. We demonstrate the effectiveness of our algorithm by integrating it into a sketch‐based quad remeshing system. A reference C++ implementation of our algorithm is provided as a supplementary material.

Abstract In the context of real‐time, GPU‐based rendering of animated skinned meshes, we propose a new algorithm to compute surface normals with minimal overhead both in terms of the memory footprint and the required per‐vertex operations. By accounting for the variation of the skinning weights over the surface, we achieve a higher visual quality compared to the standard approximation ubiquitously used in video‐game engines and other real‐time applications. Our method supports Linear Blend Skinning and Dual Quaternion Skinning. We demonstrate the advantages of our technique on a variety of datasets and provide a complete open‐source implementation, including GLSL shaders.

Articulation of 3D characters requires control over many degrees of freedom: a difficult task with standard 2D interfaces. We present a tangible input device composed of interchangeable, hot-pluggable parts. Embedded sensors measure the device's pose at rates suitable for real-time editing and animation. Splitter parts allow branching to accommodate any skeletal tree. During assembly, the device recognizes topological changes as individual parts or pre-assembled subtrees are plugged and unplugged. A novel semi-automatic registration approach helps the user quickly map the device's degrees of freedom to a virtual skeleton inside the character. User studies report favorable comparisons to mouse and keyboard interfaces for the tasks of target acquisition and pose replication. Our device provides input for character rigging and automatic weight computation, direct skeletal deformation, interaction with physical simulations, and handle-based variational geometric modeling.

Self-supporting structures are prominent in historical and contemporary architecture due to advantageous structural properties and efficient use of material. Computer graphics research has recently contributed new design tools that allow creating and interactively exploring self-supporting freeform designs. However, the physical construction of such freeform structures remains challenging, even on small scales. Current construction processes require extensive formwork during assembly, which quickly leads to prohibitively high construction costs for realizations on a building scale. This greatly limits the practical impact of the existing freeform design tools. We propose to replace the commonly used dense formwork with a sparse set of temporary chains. Our method enables gradual construction of the masonry model in stable sections and drastically reduces the material requirements and construction costs. We analyze the input using a variational method to find stable sections, and devise a computationally tractable divide-and-conquer strategy for the combinatorial problem of finding an optimal construction sequence. We validate our method on 3D printed models, demonstrate an application to the restoration of historical models, and create designs of recreational, collaborative self-supporting puzzles.

We consider the problem of reproducing the look and the details of a 3D object on a surface that is confined to a given volume. Classic examples of such "appearance-mimicking" surfaces are bas-reliefs: decorations and artwork depicting recognizable 3D scenes using only a thin volumetric space. The design of bas-reliefs has fascinated humankind for millennia and it is extensively used on coins, medals, pottery and other art forms. We propose a unified framework to create surfaces that depict certain shapes from prescribed viewpoints, as a generalization of bas-reliefs. Given target shapes, viewpoints and space restrictions, our method finds a globally optimal surface that delivers the desired appearance when observed from the designated viewpoints, while guaranteeing exact, per-vertex depth bounds. We use 3D printing to validate our approach and demonstrate our results in a variety of applications, ranging from standard bas-reliefs to optical illusions and carving of complex geometries.

We introduce frame fields, which are a non-orthogonal and non-unit-length generalization of cross fields. Frame fields represent smoothly varying linear transformations on tangent spaces of a surface. We propose an algorithm to create discrete, dense frame fields that satisfy a sparse set of constraints. By computing a surface deformation that warps a frame field into a cross field, we generalize existing quadrangulation algorithms to generate anisotropic and non-uniform quad meshes whose elements shapes match the frame field. With this, our framework enables users to control not only the alignment but also the density and anisotropy of the elements' distribution, resulting in high-quality adaptive quad meshing.

Abstract Geometric meshes that model animated characters must be designed while taking into account the deformations that the shape will undergo during animation. We analyze an input sequence of meshes with point‐to‐point correspondence, and we automatically produce a quadrangular mesh that fits well the input animation. We first analyze the local deformation that the surface undergoes at each point, and we initialize a cross field that remains as aligned as possible to the principal directions of deformation throughout the sequence. We then smooth this cross field based on an energy that uses a weighted combination of the initial field and the local amount of stretch. Finally, we compute a field‐aligned quadrangulation with an off‐the‐shelf method. Our technique is fast and very simple to implement, and it significantly improves the quality of the output quad mesh and its suitability for character animation, compared to creating the quad mesh based on a single pose. We present experimental results and comparisons with a state‐of‐the‐art quadrangulation method, on both sequences from 3D scanning and synthetic sequences obtained by a rough animation of a triangulated model.

Abstract Decades of research have culminated in a robust geometry processing pipeline for surfaces. Most steps in this pipeline, like deformation, smoothing, subdivision and decimation, may create self‐intersections. Volumetric processing of solid shapes then becomes difficult, because obtaining a correct volumetric discretization is impossible: existing tet‐meshing methods require watertight input. We propose an algorithm that produces a tetrahedral mesh that overlaps itself consistently with the self‐intersections in the input surface. This enables volumetric processing on self‐intersecting models. We leverage conformalized mean‐curvature flow, which removes self‐intersections, and define an intrinsically similar reverse flow, which prevents them. We tetrahedralize the resulting surface and map the mesh inside the original surface. We demonstrate the effectiveness of our method with applications to automatic skinning weight computation, physically based simulation and geodesic distance computation.

Abstract Mappings and deformations are ubiquitous in geometry processing, shape modeling, and animation. Numerous deformation energies have been proposed to tackle problems like mesh parameterization and volumetric deformations. We present an algorithm that modifies any deformation energy to guarantee a locally injective mapping, i.e., without inverted elements. Our formulation can be used to compute continuous planar or volumetric piecewise‐linear maps and it uses a barrier term to prevent inverted elements. Differently from previous methods, we carefully design both the barrier term and the associated numerical techniques to be able to provide immediate feedback to the user, enabling interactive manipulation of inversion‐free mappings. Stress tests show that our method robustly handles extreme deformations where previous techniques converge very slowly or even fail. We demonstrate that enforcing local injectivity increases fidelity of the results in applications such as shape deformation and parameterization.

We consider the problem of generalizing affine combinations in Euclidean spaces to triangle meshes: computing weighted averages of points on surfaces. We address both the forward problem , namely computing an average of given anchor points on the mesh with given weights, and the inverse problem , which is computing the weights given anchor points and a target point. Solving the forward problem on a mesh enables applications such as splines on surfaces, Laplacian smoothing and remeshing. Combining the forward and inverse problems allows us to define a correspondence mapping between two different meshes based on provided corresponding point pairs, enabling texture transfer, compatible remeshing, morphing and more. Our algorithm solves a single instance of a forward or an inverse problem in a few microseconds. We demonstrate that anchor points in the above applications can be added/removed and moved around on the meshes at interactive framerates, giving the user an immediate result as feedback.

We present a complete design pipeline that allows non-expert users to design and analyze masonry structures without any structural knowledge. We optimize the force layouts both geometrically and topologically, finding a self-supported structure that is as close as possible to a given target surface. The generated structures are tessellated into hexagonal blocks with a pattern that prevents sliding failure. The models can be used in physically plausible virtual environments or 3D printed and assembled without reinforcements.

Coarse quad meshes are the preferred representation for animating characters in movies and video games. In these scenarios, artists want explicit control over the edge flows and the singularities of the quad mesh. Despite the significant advances in recent years, existing automatic quad remeshing algorithms are not yet able to achieve the quality of manually created remeshings. We present an interactive system for manual quad remeshing that provides the user with a high degree of control while avoiding the tediousness involved in existing manual tools. With our sketch-based interface the user constructs a quad mesh by defining patches consisting of individual quads. The desired edge flow is intuitively specified by the sketched patch boundaries, and the mesh topology can be adjusted by varying the number of edge subdivisions at patch boundaries. Our system automatically inserts singularities inside patches if necessary, while providing the user with direct control of their topological and geometrical locations. We developed a set of novel user interfaces that assist the user in constructing a curve network representing such patch boundaries. The effectiveness of our system is demonstrated through a user evaluation with professional artists. Our system is also useful for editing automatically generated quad meshes.

Abstract We propose the space of axis‐aligned deformations as the meaningful space for content‐aware image retargeting. Such deformations exclude local rotations, avoiding harmful visual distortions, and they are parameterized in 1D. We show that standard warping energies for image retargeting can be minimized in the space of axis‐aligned deformations while guaranteeing that bijectivity constraints are satisfied, leading to high‐quality, smooth and robust retargeting results. Thanks to the 1D parameterization, our method only requires solving a small quadratic program, which can be done within a few milliseconds on the CPU with no precomputation overhead. We demonstrate how the image size and the saliency map can be changed in real time with our approach, and present results on various input images, including the RetargetMebenchmark. We compare our results with six other algorithms in a user study to demonstrate that the space of axis‐aligned deformations is suitable for the problem at hand.

Direction fields, line fields and cross fields are used in a variety of computer graphics applications ranging from non-photorealistic rendering to remeshing. In many cases, it is desirable that fields adhere to symmetry, which is predominant in natural as well as man-made shapes. We present an algorithm for designing smooth N-symmetry fields on surfaces respecting generalized symmetries of the shape, while maintaining alignment with local features. Our formulation for constructing symmetry fields is based on global symmetries, which are given as input to the algorithm, with no isometry assumptions. We explore in detail the properties of generalized symmetries (reflections in particular), and we also develop an algorithm for the robust computation of such symmetry maps, based on a small number of correspondences, for surfaces of genus zero.

Abstract We present a method for producing quad‐dominant subdivided meshes, which supports both adaptive refinement and adaptive coarsening. A hierarchical structure is stored implicitly in a standard half‐edge data structure, while allowing us to efficiently navigate through the different level of subdivision. Subdivided meshes contain a majority of quad elements and a moderate amount of triangles and pentagons in the regions of transition across different levels of detail. Topological LOD editing is controlled with local conforming operators, which support both mesh refinement and mesh coarsening. We show two possible applications of this method: we define an adaptive subdivision surface scheme that is topologically and geometrically consistent with the Catmull–Clark subdivision; and we present a remeshing method that produces semi‐regular adaptive meshes.

While often a requirement for geometric models, there has been little research in resolving the interaction of deforming surfaces during real-time modeling sessions. To address this important topic, we introduce an interference algorithm specifically designed for the domain of geometric modeling. This algorithm is general, easily working within existing modeling paradigms to maintain their important properties. Our algorithm is fast, and is able to maintain interactive rates on complex deforming meshes of over 75K faces, while robustly removing intersections. Lastly, our method is controllable, allowing fine-tuning to meet the specific needs of the user. This includes support for minimum separation between surfaces and control over the relative rigidity of interacting objects.

We present a method for the global parametrization of meshes that preserves alignment to a cross field in input while obtaining a parametric domain made of few coarse axis-aligned rectangular patches, which form an abstract base complex without T-junctions. The method is based on the topological simplification of the cross field in input, followed by global smoothing.

We present an automatic method to produce a Catmull-Clark subdivision surface that fits a given input mesh. Its control mesh is coarse and adaptive, and it is obtained by simplifying an initial mesh at high resolution. Simplification occurs progressively via local operators and addresses both quality of surface and faithfulness to the input shape throughout the whole process. The method is robust and performs well on rather complex shapes. Displacement mapping or normal mapping can be applied to approximate the input shape arbitrarily well.

Abstract In this paper we present an innovative approach to incremental quad mesh simplification, i.e. the task of producing a low complexity quad mesh starting from a high complexity one. The process is based on a novel set of strictly local operations which preserve quad structure. We show how good tessellation quality (e.g. in terms of vertex valencies) can be achieved by pursuing uniform length and canonical proportions of edges and diagonals. The decimation process is interleaved with smoothing in tangent space. The latter strongly contributes to identify a suitable sequence of local modification operations. The method is naturally extended to manage preservation of feature lines (e.g. creases) and varying (e.g. adaptive) tessellation densities. We also present an original Triangle‐to‐Quad conversion algorithm that behaves well in terms of geometrical complexity and tessellation quality, which we use to obtain the initial quad mesh from a given triangle mesh.

We introduce the RGB Subdivision: an adaptive subdivision scheme for triangle meshes, which is based on the iterative application of local refinement and coarsening operators, and generates the same limit surface of the Loop subdivision, independently on the order of application of local operators. Our scheme supports dynamic selective refinement, as in Continuous Level Of Detail models, and it generates conforming meshes at all intermediate steps. The RGB subdivision is encoded in a standard topological data structure, extended with few attributes, which can be used directly for further processing. We present an interactive tool that permits to start from a base mesh and use RGB subdivision to dynamically adjust its level of detail.

News articles contain a wealth of implicit geographic content that if exposed to readers improves understanding of today's news. However, most articles are not explicitly geotagged with their geographic content, and few news aggregation systems expose this content to users. A new system named NewsStand is presented that collects, analyzes, and displays news stories in a map interface, thus leveraging on their implicit geographic content. NewsStand monitors RSS feeds from thousands of online news sources and retrieves articles within minutes of publication. It then extracts geographic content from articles using a custom-built geotagger, and groups articles into story clusters using a fast online clustering algorithm. By panning and zooming in NewsStand's map interface, users can retrieve stories based on both topical significance and geographic region, and see substantially different stories depending on position and zoom level.