Developing a theory of differential geometry of smooth surfaces was one of the great triumphs of 19th century mathematics. Countless breakthroughs in science, from Einstein's theory of relativity to quantum mechanics to acoustic, rest on this foundation. A key assumption of the theory, though, is that the surface be smooth: what do we do when we're working with a computer, which can only handle discrete, "chunky" data like triangle meshes, or noisy clouds of points acquired using laser scanners or the Microsoft Kinect? One of the most exciting programs of the last few decades in computer graphics is the development of DDG,

Discrete differential geometry allows us to "port" key ideas and tools from smooth differential geometry to the discrete world. In addition to better understanding how to physically simulate everyday objects, many other applications such as surface reconstruction, editing/fairing/smoothing noisy data, shape matching and registration, etc. are built on ideas from discrete differential geometry.

In addition to pushing DDG further, I'm interested in using it to solve

Click the project titles or thumbnails to go to the project page/paper.

*discrete differential geometry*: the key idea is to take smooth differential geometry, and to build a*parallel theory*for discrete geometry that nevertheless preserves the most important relationships and structures of the smooth theory.Discrete differential geometry allows us to "port" key ideas and tools from smooth differential geometry to the discrete world. In addition to better understanding how to physically simulate everyday objects, many other applications such as surface reconstruction, editing/fairing/smoothing noisy data, shape matching and registration, etc. are built on ideas from discrete differential geometry.

In addition to pushing DDG further, I'm interested in using it to solve

*inverse*or*design problems*. In a physical simulation, you have some objects in a starting state, and ask, "how will these objects behave in the future?" In an inverse problem, you do the reverse: you have some desired behavior in mind, and ask "how can I modify the initial state of the objects so that they will do what I want"? How can you take a rough architectural design and modify it into a stable building? Modify growth of tissue so that it grows into any desired shape? 3D-print objects that not only look like, but also work, the way you expect?Click the project titles or thumbnails to go to the project page/paper.

On the Incompressibility of Cylindrical Origami PatternsFriedrich Bös, Max Wardetzky, Etienne Vouga, and Omer Gottesman Journal of Mechanical Design, 2016 The art and science of folding intricate three-dimensional structures out of paper has occupied artists, designers, engineers, and mathematicians for decades, culminating in the design of deployable structures and mechanical metamaterials. Here we investigate the axial compressibility of origami cylinders, i.e., cylindrical structures folded from rectangular sheets of paper. We prove, using geometric arguments, that a general fold pattern only allows for a finite number of isometric cylindrical embeddings. Therefore, compressibility of such structures requires either stretching the material or deforming the folds. Our result considerably restricts the space of constructions that must be searched when designing new types of origami-based rigid-foldable deployable structures and metamaterials. |

Programming Curvature Using Origami TessellationsLevi H. Dudte, Etienne Vouga, Tomohiro Tachi, and L. Mahadevan Nature Materials, 2016 Origami describes rules for creating folded structures from patterns on a flat sheet, but does not prescribe how patterns can be designed to fit target shapes. Here, starting from the simplest periodic origami pattern that yields one-degree-of-freedom collapsible structures—we show that scale-independent elementary geometric constructions and constrained optimization algorithms can be used to determine spatially modulated patterns that yield approximations to given surfaces of constant or varying curvature. Paper models confirm the feasibility of our calculations. We also assess the difficulty of realizing these geometric structures by quantifying the energetic barrier that separates the metastable flat and folded states. Moreover, we characterize the trade-off between the accuracy to which the pattern conforms to the target surface, and the effort associated with creating finer folds. Our approach enables the tailoring of origami patterns to drape complex surfaces independent of absolute scale, as well as the quantification of the energetic and material cost of doing so. |

Capturing Dynamic Textured Surfaces of Moving TargetsRuizhe Wang, Lingyu Wei, Etienne Vouga, Qixing Huang, Duygu Ceylan, Gerard Medioni, and Hao LiECCV (spotlight), 2016We present an end-to-end system for reconstructing complete watertight and textured models of moving subjects such as clothed humans and animals, using only three or four handheld sensors. The heart of our framework is a new pairwise registration algorithm that minimizes, using a particle swarm strategy, an alignment error metric based on mutual visibility and occlusion. We show that this algorithm reliably registers partial scans with as little as 15% overlap without requiring any initial correspondences, and outperforms alternative global registration algorithms. This registration algorithm allows us to reconstruct moving subjects from free-viewpoint video produced by consumer-grade sensors, without extensive sensor calibration, constrained capture volume, expensive arrays of cameras, or templates of the subject geometry. |

Dense Human Body Correspondences Using Convolutional NetworksLingyu Wei, Qixing Huang, Duygu Ceylan, Etienne Vouga, and Hao Li CVPR (oral), 2016We propose a deep learning approach for finding dense correspondences between 3D scans of people. Our method requires only partial geometric information in the form of two depth maps or partial reconstructed surfaces, works for humans in arbitrary poses and wearing any clothing, does not require the two people to be scanned from similar viewpoints, and runs in real time. We use a deep convolutional neural network to train a feature descriptor on depth map pixels, but crucially, rather than training the network to solve the shape correspondence problem directly, we train it to solve a body region classification problem, modified to increase the smoothness of the learned descriptors near region boundaries. This approach ensures that nearby points on the human body are nearby in feature space, and vice versa, rendering the feature descriptor suitable for computing dense correspondences between the scans. We validate our method on real and synthetic data for both clothed and unclothed humans, and show that our correspondences are more robust than is possible with state-of-the-art unsupervised methods, and more accurate than those found using methods that require full watertight 3D geometry. |

Reconciling Elastic and Equilibrium Methods for Static AnalysisHijung V. Shin, Christopher F. Porst, Etienne Vouga, John Ochsendorf, and Frédo Durand ACM Transactions on Graphics, 2016 We examine two widely used classes of methods for static analysis of masonry buildings: linear elasticity analysis using finite elements and equilibrium methods. It is often claimed in the literature that finite element analysis is less accurate than equilibrium analysis when it comes to masonry analysis; we examine and qualify this claimed inaccuracy, provide a systematic explanation for the discrepancy observed between their results, and present a unified formulation of the two approaches to stability analysis. We prove that both approaches can be viewed as equivalent, dual methods for getting the same answer to the same problem. We validate our observations with simulations and physical tilt experiments of structures. |

Nested CagesLeonardo Sacht, Etienne Vouga, and Alec Jacobson SIGGRAPH Asia ( ACM Transactions on Graphics ), 2015 Many tasks in geometry processing and physical simulation benefit from multiresolution hierarchies. One important characteristic across a variety of applications is that coarser layers strictly encage finer layers, nesting one another. Existing techniques such as surface mesh decimation, voxelization, or contouring distance level sets do not provide sufficient control over the quality of the output surfaces while maintaining strict nesting. We propose a solution that enables use of application-specific decimation and quality metrics. The method constructs each next-coarsest level of the hierarchy, using a sequence of decimation, flow, and contact-aware optimization steps. From coarse to fine, each layer then fully encages the next while retaining a snug fit. The method is applicable to a wide variety of shapes of complex geometry and topology. We demonstrate the effectiveness of our nested cages not only for multigrid solvers, but also for conservative collision detection, domain discretization for elastic simulation, and cage-based geometric modeling. |

Urban Pattern: Layout Design by Hierarchical Domain SplittingY. L. Yang, J. Wang, Etienne Vouga, and Peter Wonka SIGGRAPH Asia ( ACM Transactions on Graphics ), 2013 We present a framework for generating street networks and parcel layouts. Our goal is the generation of high-quality layouts that can be used for urban planning and virtual environments. We propose a solution based on hierarchical domain splitting using two splitting types: streamline-based splitting, which splits a region along one or multiple streamlines of a cross field, and template-based splitting, which warps pre-designed templates to a region and uses the interior geometry of the template as the splitting lines. We combine these two splitting approaches into a hierarchical framework, providing automatic and interactive tools to explore the design space. |

3D Self-PortraitsHao Li, Etienne Vouga, Anton Gudym, Jonathan T. Barron, Linjie Luo, Gleb Gusev SIGGRAPH Asia ( ACM Transactions on Graphics ), 2013 We develop an automatic pipeline that allows ordinary users to capture complete and fully textured 3D models of themselves in minutes, using only a single Kinect sensor, in the uncontrolled lighting environment of their own home. Our method requires neither a turntable nor a second operator, and is robust to the small deformations and changes of pose that inevitably arise during scanning. After the users rotate themselves with the same pose for a few scans from different views, our system stitches together the captured scans using multi-view non-rigid registration, and produces watertight final models. To ensure consistent texturing, we recover the underlying albedo from each scanned texture and generate seamless global textures using Poisson blending. Despite the minimal requirements we place on the hardware and users, our method is suitable for full body capture of challenging scenes that cannot be handled well using previous methods, such as those involving loose clothing, complex poses, and props. |

Design of Self-supporting SurfacesEtienne Vouga, Mathias Höbinger, Johannes Wallner, and Helmut Pottmann SIGGRAPH ( ACM Transactions on Graphics ), 2012 Self-supporting masonry is one of the most ancient and elegant techniques for building curved shapes. Because of the very geometric nature of their failure, analyzing and modeling such strutures is more a geometry processing problem than one of classical continuum mechanics. This paper uses the thrust network method of analysis and presents an iterative nonlinear optimization algorithm for efficiently approximating freeform shapes by self-supporting ones. The rich geometry of thrust networks leads us to close connections between diverse topics in discrete differential geometry, such as a finite-element discretization of the Airy stress potential, perfect graph Laplacians, and computing admissible loads via curvatures of polyhedral surfaces. This geometric viewpoint allows us, in particular, to remesh self-supporting shapes by self-supporting quad meshes with planar faces, and leads to another application of the theory: steel/glass constructions with low moments in nodes. |

Flexible Developable SurfacesJustin Solomon, Etienne Vouga, Max Wardetzky, and Eitan Grinspun Eurographics Symposium on Geometry Processing, 2012 We introduce a discrete paradigm for developable surface modeling. Unlike previous attempts at interactive developable surface modeling, our system is able to enforce exact developability at every step, ensuring that users do not inadvertently suggest configurations that leave the manifold of admissible folds of a flat two-dimensional sheet. With methods for navigation of this highly nonlinear constraint space in place, we show how to formulate a discrete mean curvature bending energy measuring how far a given discrete developable surface is from being flat. This energy enables relaxation of user-generated configurations and suggests a straightforward subdivision scheme that produces admissible smoothed versions of bent regions of our discrete developable surfaces. |