We present a discretization of the dynamic optimal transport problem for which we can obtain the convergence rate for the value of the transport cost to its continuous value when the temporal and spatial stepsize vanish. This convergence result does not require any regularity assumption on the measures, though experiments suggest that the rate is not sharp. Via an analysis of the duality gap we also obtain the convergence rates for the gradient of the optimal potentials and the velocity field under mild regularity assumptions. To obtain such rates, we discretize the dual formulation of the dynamic optimal transport problem and use the mature literature related to the error due to discretizing the Hamilton--Jacobi equation.

@article{IshidaLavenant2024DualOT,
  author={Ishida, Sadashige
  and Lavenant, Hugo},
  title={Quantitative Convergence of a Discretization of Dynamic Optimal Transport Using the Dual Formulation},
  journal={Foundations of Computational Mathematics},
  year={2024},
  month={Nov},
  day={11},
  abstract={We present a discretization of the dynamic optimal transport problem for which we can obtain the convergence rate for the value of the transport cost to its continuous value when the temporal and spatial stepsize vanish. This convergence result does not require any regularity assumption on the measures, though experiments suggest that the rate is not sharp. Via an analysis of the duality gap we also obtain the convergence rates for the gradient of the optimal potentials and the velocity field under mild regularity assumptions. To obtain such rates, we discretize the dual formulation of the dynamic optimal transport problem and use the mature literature related to the error due to discretizing the Hamilton--Jacobi equation.},
  issn={1615-3383},
  doi={10.1007/s10208-024-09686-3},
  url={https://doi.org/10.1007/s10208-024-09686-3}
}

We present a computational approach for unfolding 3D shapes isometrically into the plane as a single patch without overlapping triangles. This is a hard, sometimes impossible, problem, which existing methods are forced to soften by allowing for map distortions or multiple patches. Instead, we propose a geometric relaxation of the problem: we modify the input shape until it admits an overlap-free unfolding. We achieve this by locally displacing vertices and collapsing edges, guided by the unfolding process. We validate our algorithm quantitatively and qualitatively on a large dataset of complex shapes and show its proficiency by fabricating real shapes from paper.

@article{https://doi.org/10.1111/cgf.15269,
  author = {Bhargava, M. and Schreck, C. and Freire, M. and Hugron, P. A. and Lefebvre, S. and Sellán, S. and Bickel, B.},
  title = {Mesh Simplification for Unfolding},
  journal = {Computer Graphics Forum},
  volume = {n/a},
  number = {n/a},
  pages = {e15269},
  keywords = {fabrication, single patch unfolding, mesh simplification},
  doi = {https://doi.org/10.1111/cgf.15269},
  url = {https://onlinelibrary.wiley.com/doi/abs/10.1111/cgf.15269},
  eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1111/cgf.15269},
}

We present a formula for the signed area of a spherical polygon via prequantization. In contrast to the traditional formula based on the Gauss–Bonnet theorem that requires measuring angles, the new formula mimics Green’s theorem and is applicable to a wider range of degenerate spherical curves and polygons.

@article{ChernIshida2024,
  author = {Chern, Albert and Ishida, Sadashige},
  title = {Area Formula for Spherical Polygons via Prequantization},
  journal = {SIAM Journal on Applied Algebra and Geometry},
  volume = {8},
  number = {3},
  pages = {782-796},
  year = {2024},
  doi = {10.1137/23M1565255},
  URL = {https://doi.org/10.1137/23M1565255},
  eprint = {https://doi.org/10.1137/23M1565255},
  abstract = {We present a formula for the signed area of a spherical polygon via prequantization. In contrast to the traditional formula based on the Gauss–Bonnet theorem that requires measuring angles, the new formula mimics Green’s theorem and is applicable to a wider range of degenerate spherical curves and polygons.}
}

The behavior of a rigid body primarily depends on its mass moments, which consist of the mass, center of mass, and moments of inertia. It is possible to manipulate these quantities without altering the geometric appearance of an object by introducing cavities in its interior. Algorithms that find cavities of suitable shapes and sizes have enabled the computational design of spinning tops, yo-yos, wheels, buoys, and statically balanced objects. Previous work is based, for example, on topology optimization on voxel grids, which introduces a large number of optimization variables and box constraints, or offset surface computation, which cannot guarantee that solutions to a feasible problem will always be found. In this work, we provide a mathematical analysis of constrained topology optimization problems that depend only on mass moments. This class of problems covers, among others, all applications mentioned above. Our main result is to show that no matter the outer shape of the rigid body to be optimized or the optimization objective and constraints considered, the optimal solution always features a quadric-shaped interface between material and cavities. This proves that optimal interfaces are always ellipsoids, hyperboloids, paraboloids, or one of a few degenerate cases, such as planes. This insight lets us replace a difficult topology optimization problem with a provably equivalent non-linear equation system in a small number (<10) of variables, which represent the coefficients of the quadric. This system can be solved in a few seconds for most examples, provides insights into the geometric structure of many specific applications, and lets us describe their solution properties. Finally, our method integrates seamlessly into modern fabrication workflows because our solutions are analytical surfaces that are native to the CAD domain.

@article{SpinItFasterHafner24,
  author = {Hafner, Christian and Ly, Mickaël and Wojtan, Chris},
  title = {Spin-It Faster: Quadrics Solve All Topology Optimization Problems That Depend Only On Mass Moments},
  year = {2024},
  issue_date = {October 2024},
  publisher = {Association for Computing Machinery},
  address = {New York, NY, USA},
  volume = {43},
  number = {4},
  url = {https://doi.org/10.1145/3658194},
  doi = {10.1145/3658194},
  journal = {ACM Trans. Graph.},
  month = {sep},
  articleno = {171},
  numpages = {13},
  keywords = {Topology Optimization, Mass Moments, Computational Geometry}
}

We introduce a multi-material non-manifold mesh-based surface tracking algorithm that converts self-intersections into topological changes. Our algorithm generalizes prior work on manifold surface tracking with topological changes: it preserves surface features like mesh-based methods, and it robustly handles topological changes like level set methods. Our method also offers improved efficiency and robustness over the state of the art. We demonstrate the effectiveness of the approach on a range of examples, including complex soap film simulations with thousands of interacting bubbles, and boolean unions of non-manifold meshes consisting of millions of triangles.

@article{MultimaterialMeshing24,
  author = {Heiss-Synak, Peter and Kalinov, Aleksei and Strugaru, Malina and Etemadi, Arian and Yang, Huidong and Wojtan, Chris},
  title = {Multi-Material Mesh-Based Surface Tracking with Implicit Topology Changes},
  year = {2024},
  issue_date = {October 2024},
  publisher = {Association for Computing Machinery},
  address = {New York, NY, USA},
  volume = {43},
  number = {4},
  journal = {ACM Trans. Graph.},
  month = {sep},
  articleno = {171},
  numpages = {14},
  keywords = {surface tracking, topology change, nonmanifold meshes, multi-material flows, solid modeling}
}

Current numerical algorithms for simulating friction fall in one of two camps: smooth solvers sacrifice the stable treatment of static friction in exchange for fast convergence, and non-smooth solvers accurately compute friction at convergence rates that are often prohibitive for large graphics applications. We introduce a novel bridge between these two ideas that computes static and dynamic friction stably and efficiently. Our key idea is to convert the highly constrained non-smooth problem into an unconstrained smooth problem using logarithmic barriers that converges to the exact solution as accuracy increases. We phrase the problem as an interior point primal-dual problem that can be solved efficiently with Newton iteration. We observe quadratic convergence despite the non-smooth nature of the original problem, and our method is well-suited for large systems of tightly packed objects with many contact points. We demonstrate the efficacy of our method with stable piles of grains and stacks of objects, complex granular flows, and robust interlocking assemblies of rigid bodies.

@article{PrimalDualChen24,
  author = {Chen, Yi-Lu and Ly, Mickaël and Wojtan, Chris},
  title = {Primal–Dual Non-Smooth Friction for Rigid Body Animation},
  year = {2024},
  issue_date = {October 2024},
  publisher = {Association for Computing Machinery},
  booktitle = {SIGGRAPH 2024 Conference Papers},
  address = {New York, NY, USA},
  month = {sep},
  numpages = {10},
  keywords = {physical simulation, frictional contact, rigid body mechanics, nonsmooth dynamics}
}

The Kirchhoff rod model describes the bending and twisting of slender elastic rods in three dimensions, and has been widely studied to enable the prediction of how a rod will deform, given its geometry and boundary conditions. In this work, we study a number of inverse problems with the goal of computing the geometry of a straight rod that will automatically deform to match a curved target shape after attaching its endpoints to a support structure. Our solution lets us finely control the static equilibrium state of a rod by varying the cross-sectional profiles along its length. We also show that the set of physically realizable equilibrium states admits a concise geometric description in terms of linear line complexes, which leads to very efficient computational design algorithms. Implemented in an interactive software tool, they allow us to convert three-dimensional hand-drawn spline curves to elastic rods, and give feedback about the feasibility and practicality of a design in real time. We demonstrate the efficacy of our method by designing and manufacturing several physical prototypes with applications to interior design and soft robotics.

@article{KirchhoffHafner23,
  author = {Hafner, Christian and Bickel, Bernd},
  title = {The Design Space of Kirchhoff Rods},
  year = {2023},
  issue_date = {October 2023},
  publisher = {Association for Computing Machinery},
  address = {New York, NY, USA},
  volume = {42},
  number = {5},
  issn = {0730-0301},
  url = {https://doi.org/10.1145/3606033},
  doi = {10.1145/3606033},
  journal = {ACM Trans. Graph.},
  month = {sep},
  articleno = {171},
  numpages = {20},
  keywords = {Kirchhoff Rods, Digital Fabrication, Active Bending, Computational Design}
}

We introduce a compact, intuitive procedural graph representation for cellular metamaterials, which are small-scale, tileable structures that can be architected to exhibit many useful material properties. Because the structures’ "architectures" vary widely—with elements such as beams, thin shells, and solid bulks—it is difficult to explore them using existing representations. Generic approaches like voxel grids are versatile, but it is cumbersome to represent and edit individual structures; architecture-specific approaches address these issues, but are incompatible with one another. By contrast, our procedural graph succinctly represents the construction process for any structure using a simple skeleton annotated with spatially varying thickness. To express the highly constrained triply periodic minimal surfaces (TPMS) in this manner, we present the first fully automated version of the conjugate surface construction method, which allows novices to create complex TPMS from intuitive input. We demonstrate our representation’s expressiveness, accuracy, and compactness by constructing a wide range of established structures and hundreds of novel structures with diverse architectures and material properties. We also conduct a user study to verify our representation’s ease-of-use and ability to expand engineers’ capacity for exploration.

@article{MakaturaMeta23,
  author = {Makatura, Liane and Wang, Bohan and Chen, Yi-Lu and Deng, Bolei and Wojtan, Chris and Bickel, Bernd and Matusik, Wojciech},
  title = {Procedural Metamaterials: A Unified Procedural Graph for Metamaterial Design},
  year = {2023},
  issue_date = {October 2023},
  publisher = {Association for Computing Machinery},
  address = {New York, NY, USA},
  volume = {42},
  number = {5},
  issn = {0730-0301},
  url = {https://doi.org/10.1145/3605389},
  doi = {10.1145/3605389},
  month = {jul},
  articleno = {168},
  numpages = {19},
  keywords = {microstructures, hybrid metamaterials, conjugate surface construction method, procedural representation, procedural graph, shellular, cellular metamaterials, Graph representation, truss structures, triply periodic minimal surfaces (TPMS)}
}

Tattoos are a highly popular medium, with both artistic and medical applications. Although the mechanical process of tattoo application has evolved historically, the results are reliant on the artisanal skill of the artist. This can be especially challenging for some skin tones, or in cases where artists lack experience. We provide the first systematic overview of tattooing as a computational fabrication technique. We built an automated tattooing rig and a recipe for the creation of silicone sheets mimicking realistic skin tones, which allowed us to create an accurate model predicting tattoo appearance. This enables several exciting applications including tattoo previewing, color retargeting, novel ink spectra optimization, color-accurate prosthetics, and more.

@inproceedings{Piovarci2023,
author = { Michal Piovar\v{c}i and Alexandre Chapiro] and Bernd Bickel},
title = {Skin-Screen: A Computational Fabrication Framework for Color Tattoos},
journal = {ACM Transactions on Graphics (Proc. SIGGRAPH)},
year = {2023},
volume = {42},
number = {4}
}

We propose a computational design approach for covering a surface with individually addressable RGB LEDs, effectively forming a low-resolution surface screen. To achieve a low-cost and scalable approach, we propose creating designs from flat PCB panels bent in-place along the surface of a 3D printed core. Working with standard rigid PCBs enables the use of established PCB manufacturing services, allowing the fabrication of designs with several hundred LEDs. Our approach optimizes the PCB geometry for folding, and then jointly optimizes the LED packing, circuit and routing, solving a challenging layout problem under strict manufacturing requirements. Unlike paper, PCBs cannot bend beyond a certain point without breaking. Therefore, we introduce parametric cut patterns acting as hinges, designed to allow bending while remaining compact. To tackle the joint optimization of placement, circuit and routing, we propose a specialized algorithm that splits the global problem into one sub-problem per triangle, which is then individually solved. Our technique generates PCB blueprints in a completely automated way. After being fabricated by a PCB manufacturing service, the boards are bent and glued by the user onto the 3D printed support. We demonstrate our technique on a range of physical models and virtual examples, creating intricate surface light patterns from hundreds of LEDs.

@article{freire:hal-04129354,
  TITLE = {PCBend: Light Up Your 3D Shapes With Foldable Circuit Boards},
  AUTHOR = {Freire, Marco and Bhargava, Manas and Schreck, Camille and Hugron, Pierre-Alexandre and Bickel, Bernd and Lefebvre, Sylvain},
  URL = {https://inria.hal.science/hal-04129354},
  JOURNAL = {ACM Transactions on Graphics},
  PUBLISHER = {Association for Computing Machinery},
  YEAR = {2023},
  DOI = {10.1145/3592411},
  KEYWORDS = {PCB design ; PCB bending ; automated placement and routing ; 3D electronics},
  PDF = {https://inria.hal.science/hal-04129354/file/main.pdf},
  HAL_ID = {hal-04129354},
  HAL_VERSION = {v1},
}

This paper introduces a novel method for simulating large bodies of water as a height field. At the start of each time step, we partition the waves into a bulk flow (which approximately satisfies the assumptions of the shallow water equations) and surface waves (which approximately satisfy the assumptions of Airy wave theory). We then solve the two wave regimes separately using appropriate state-of-the-art techniques, and re-combine the resulting wave velocities at the end of each step. This strategy leads to the first heightfield wave model capable of simulating complex interactions between both deep and shallow water effects, like the waves from a boat wake sloshing up onto a beach, or a dam break producing wave interference patterns and eddies. We also analyze the numerical dispersion created by our method and derive an exact correction factor for waves at a constant water depth, giving us a numerically perfect re-creation of theoretical water wave dispersion patterns.

@article{10.1145/3592098,
  author = {Jeschke, Stefan and Wojtan, Chris},
  title = {Generalizing Shallow Water Simulations with Dispersive Surface Waves},
  year = {2023},
  issue_date = {August 2023},
  publisher = {Association for Computing Machinery},
  address = {New York, NY, USA},
  volume = {42},
  number = {4},
  issn = {0730-0301},
  url = {https://doi.org/10.1145/3592098},
  doi = {10.1145/3592098},
  month = {jul},
  articleno = {83},
  numpages = {12},
  keywords = {real-time animation, natural phenomena, water animation}
}

We present a technique to optimize the reflectivity of a surface while preserving its overall shape. The naïve optimization of the mesh vertices using the gradients of reflectivity simulations results in undesirable distortion. In contrast, our robust formulation optimizes the surface normal as an independent variable that bridges the reflectivity term with differential rendering, and the regularization term with as-rigid-as-possible elastic energy. We further adaptively subdivide the input mesh to improve the convergence. Consequently, our method can minimize the retroreflectivity of a wide range of input shapes, resulting in sharply creased shapes ubiquitous among stealth aircraft and Sci-Fi vehicles. Furthermore, by changing the reward for the direction of the outgoing light directions, our method can be applied to other reflectivity design tasks, such as the optimization of architectural walls to concentrate light in a specific region. We have tested the proposed method using light-transport simulations and real-world 3D-printed objects.

@inproceedings{Tojo2023Stealth,
title = {Stealth Shaper: Reflectivity Optimization as Surface Stylization},
author = {Tojo, Kenji and Shamir, Ariel and Bickel, Bernd and Umetani, Nobuyuki},
booktitle = {ACM SIGGRAPH 2023 Conference Proceedings},
year = {2023},
series = {SIGGRAPH '23},
}

Color and gloss are fundamental aspects of surface appearance. State-of-the-art fabrication techniques can manipulate both properties of the printed 3D objects. However, in the context of appearance reproduction, perceptual aspects of color and gloss are usually handled separately, even though previous perceptual studies suggest their interaction. Our work is motivated by previous studies demonstrating a perceived color shift due to a change in the object's gloss, i.e., two samples with the same color but different surface gloss appear as they have different colors. In this paper, we conduct new experiments which support this observation and provide insights into the magnitude and direction of the perceived color change. We use the observations as guidance to design a new method that estimates and corrects the color shift enabling the fabrication of objects with the same perceived color but different surface gloss. We formulate the problem as an optimization procedure solved using differentiable rendering. We evaluate the effectiveness of our method in perceptual experiments with 3D objects fabricated using a multi-material 3D printer and demonstrate potential applications.

@inproceedings{Condor2023,
author = { Jorge Condor and Michal Piovar\v{c}i and Bernd Bickel and Piotr Didyk},
title = {Gloss-aware Color Correction for 3D Printing},
booktitle = {SIGGRAPH 2023 Conference Papers},
year = {2023},
} 

3D printing based on continuous deposition of materials, such as filament-based 3D printing, has seen widespread adoption thanks to its versatility in working with a wide range of materials. An important shortcoming of this type of technology is its limited multi-material capabilities. While there are simple hardware designs that enable multi-material printing in principle, the required software is heavily underdeveloped. A typical hardware design fuses together individual materials fed into a single chamber from multiple inlets before they are deposited. This design, however, introduces a time delay between the intended material mixture and its actual deposition. In this work, inspired by diverse path planning research in robotics, we show that this mechanical challenge can be addressed via improved printer control. We propose to formulate the search for optimal multi-material printing policies in a reinforcement learning setup. We put forward a simple numerical deposition model that takes into account the non-linear material mixing and delayed material deposition. To validate our system we focus on color fabrication, a problem known for its strict requirements for varying material mixtures at a high spatial frequency. We demonstrate that our learned control policy outperforms state-of-the-art hand-crafted algorithms.

@article{zhenyuan2023embroidery,
journal = {Computer Graphics Forum},
title = {Directionality-Aware Design of Embroidery Patterns},
author = {Zhenyuan, Liu and Piovar\v{c}i, Michal and Hafner, Christian and Charrondi\`{e}re, Rapha\"{e}l and Bickel, Bernd},
year = {2023},
volume = {42},
number = {2},
publisher = {The Eurographics Association and John Wiley & Sons Ltd.},
}

Embroidery is a long-standing and high-quality approach to making logos and images on textiles. Nowadays, it can also be performed via automated machines that weave threads with high spatial accuracy. A characteristic feature of the appearance of the threads is a high degree of anisotropy. The anisotropic behavior is caused by depositing thin but long strings of thread. As a result, the stitched patterns convey both color and direction. Artists leverage this anisotropic behavior to enhance pure color images with textures, illusions of motion, or depth cues. However, designing colorful embroidery patterns with prescribed directionality is a challenging task, one usually requiring an expert designer. In this work, we propose an interactive algorithm that generates machine-fabricable embroidery patterns from multi-chromatic images equipped with user-specified directionality fields. We cast the problem of finding a stitching pattern into vector theory. To find a suitable stitching pattern, we extract sources and sinks from the divergence field of the vector field extracted from the input and use them to trace streamlines. We further optimize the streamlines to guarantee a smooth and connected stitching pattern. The generated patterns approximate the color distribution constrained by the directionality field. To allow for further artistic control, the trade-off between color match and directionality match can be interactively explored via an intuitive slider. We showcase our approach by fabricating several embroidery paths.

@article{zhenyuan2023embroidery,
journal = {Computer Graphics Forum},
title = {Directionality-Aware Design of Embroidery Patterns},
author = {Zhenyuan, Liu and Piovar\v{c}i, Michal and Hafner, Christian and Charrondi\`{e}re, Rapha\"{e}l and Bickel, Bernd},
year = {2023},
volume = {42},
number = {2},
publisher = {The Eurographics Association and John Wiley & Sons Ltd.},
}

This paper presents a new representation of curve dynamics, with applications to vortex filaments in fluid dynamics. Instead of representing these filaments with explicit curve geometry and Lagrangian equations of motion, we represent curves implicitly with a new co-dimensional 2 level set description. Our implicit representation admits several redundant mathematical degrees of freedom in both the configuration and the dynamics of the curves, which can be tailored specifically to improve numerical robustness, in contrast to naive approaches for implicit curve dynamics that suffer from overwhelming numerical stability problems. Furthermore, we note how these hidden degrees of freedom perfectly map to a Clebsch representation in fluid dynamics. Motivated by these observations, we introduce untwisted level set functions and non-swirling dynamics which successfully regularize sources of numerical instability, particularly in the twisting modes around curve filaments. A consequence is a novel simulation method which produces stable dynamics for large numbers of interacting vortex filaments and effortlessly handles topological changes and re-connection events.

@article{
  iwc2022implicit_filaments,
  title = Hidden Degrees of Freedom in Implicit Vortex Filaments,
  author = {Sadashige Ishida and Chris Wojtan and Albert Chern}
  journal = {ACM Transactions on Graphics},
  year = 2022,
  volume = 41,
  number = 6,
  pages = {241:1--241:14},
  articleno = 241,
  url = {http://dx.doi.org/10.1145/3550454.3555459},
  doi = {10.1145/3550454.3555459},
  publisher = {ACM}
}

A good match of material appearance between real-world objects and their digital on-screen representations is critical for many applications such as fabrication, design, and e-commerce. However, faithful appearance reproduction is challenging, especially for complex phenomena, such as gloss. In most cases, the view-dependent nature of gloss and the range of luminance values required for reproducing glossy materials exceeds the current capabilities of display devices. As a result, appearance reproduction poses significant problems even with accurately rendered images. This paper studies the gap between the gloss perceived from real-world objects and their digital counterparts. Based on our psychophysical experiments on a wide range of 3D printed samples and their corresponding photographs, we derive insights on the influence of geometry, illumination, and the display's brightness and measure the change in gloss appearance due to the display limitations. Our evaluation experiments demonstrate that using the prediction to correct material parameters in a rendering system improves the match of gloss appearance between real objects and their visualization on a display device.

@inproceedings{Chen2022,
  author = {Chen, Bin and Piovar\v{c}i, Michal and Wang, Chao and Seidel, Hans-Peter and Didyk, Piotr and Myszkowski, Karol and Serrano, Ana},
  title = {Gloss Management for Consistent Reproduction of Real and Virtual Objects},
  year = {2022},
  isbn = {9781450394703},
  publisher = {Association for Computing Machinery},
  address = {New York, NY, USA},
  url = {https://doi.org/10.1145/3550469.3555406},
  doi = {10.1145/3550469.3555406},
  booktitle = {SIGGRAPH Asia 2022 Conference Papers},
  articleno = {35},
  numpages = {9},
  keywords = {glossiness reproduction, glossiness perception},
  location = {Daegu, Republic of Korea},
  series = {SA '22}
}

Portrait viewpoint and illumination editing is an important problem with several applications in VR/AR, movies, and photography. Comprehensive knowledge of geometry and illumination is critical for obtaining photorealistic results. Current methods are unable to explicitly model in 3D while handing both viewpoint and illumination editing from a single image. In this paper, we propose VoRF, a novel approach that can take evena single portrait image as input and relight human heads under novel illuminations that can be viewed from arbitrary viewpoints. VoRF represents a human head as a continuous volumetric field and learns a prior model of human heads using a coordinate-based MLP with individual latent spaces for identity and illumination. The prior model is learnt in an auto-decoder manner over a diverse class of head shapes and appearances, allowing VoRF to generalize to novel test identities from a single input image. Additionally, VoRF has a reflectance MLP that uses the intermediate features of the prior model for rendering One-Light-at-A-Time (OLAT) images under novel views. We synthesize novel illuminations by combining these OLAT images with target environment maps. Qualitative and quantitative evaluations demonstrate the effectiveness of VoRF for relighting and novel view synthesis, even when applied to unseen subjects under uncontrolled illuminations.

Moulding refers to a set of manufacturing techniques in which a mould, usually a cavity or a solid frame, is used to shape a liquid or pliable material into an object of the desired shape. The popularity of moulding comes from its effectiveness, scalability and versatility in terms of employed materials. Its relevance as a fabrication process is demonstrated by the extensive literature covering different aspects related to mould design, from material flow simulation to the automation of mould geometry design. In this state-of-the-art report, we provide an extensive review of the automatic methods for the design of moulds, focusing on contributions from a geometric perspective. We classify existing mould design methods based on their computational approach and the nature of their target moulding process. We summarize the relationships between computational approaches and moulding techniques, highlighting their strengths and limitations. Finally, we discuss potential future research directions.

This paper introduces a methodology for inverse-modeling of yarn-level mechanics of cloth, based on the mechanical response of fabrics in the real world. We compiled a database from physical tests of several different knitted fabrics used in the textile industry. These data span different types of complex knit patterns, yarn compositions, and fabric finishes, and the results demonstrate diverse physical properties like stiffness, nonlinearity, and anisotropy. We then develop a system for approximating these mechanical responses with yarn-level cloth simulation. To do so, we introduce an efficient pipeline for converting between fabric-level data and yarn-level simulation, including a novel swatch-level approximation for speeding up computation, and some small-but-necessary extensions to yarn-level models used in computer graphics.

@article{sperl2022eylsmpf,
    author    = {Sperl, Georg and Sánchez-Banderas,  Rosa M. and Li, Manwen and Wojtan, Chris and Otaduy, Miguel A.},
    title     = {Estimation of Yarn-Level Simulation Models for Production Fabrics},
    journal   = {ACM Transactions on Graphics (TOG)},
    number    = {4},
    volume    = {41},
    year      = {2022},
    publisher = {ACM}
}

Enabling additive manufacturing to employ a wide range of novel, functional materials can be a major boost to this technology. However, making such materials printable requires painstaking trial-and-error by an expert operator, as they typically tend to exhibit peculiar rheological or hysteresis properties. Even in the case of successfully finding the process parameters, there is no guarantee of print-to-print consistency due to material differences between batches. These challenges make closed-loop feedback an attractive option where the process parameters are adjusted on-the-fly. There are several challenges for designing an efficient controller: the deposition parameters are complex and highly coupled, artifacts occur after long time horizons, simulating the deposition is computationally costly, and learning on hardware is intractable. In this work, we demonstrate the feasibility of learning a closed-loop control policy for additive manufacturing using reinforcement learning. We show that approximate, but efficient, numerical simulation is sufficient as long as it allows learning the behavioral patterns of deposition that translate to real-world experiences. In combination with reinforcement learning, our model can be used to discover control policies that outperform baseline controllers. Furthermore, the recovered policies have a minimal sim-to-real gap. We showcase this by applying our control policy in-vivo on a single-layer printer using low and high viscosity materials.

@article{Piovarci2022,
  title = {Closed-Loop Control of Direct Ink Writing via Reinforcement Learning},
  author = {Michal Piovar\v{c}i and Michael Foshey and Jie Xu and Timothy Erps and Vahid Babaei and Piotr Didyk and Szymon Rusinkiewicz and Wojciech Matusik and Bernd Bickel},
  year = {2022},
  journal = {ACM Transactions on Graphics (Proc. SIGGRAPH)}, 
  volume = {41},
  number = {4}
}

Interlocking puzzles are intriguing geometric games where the puzzle pieces are held together based on their geometric arrangement, preventing the puzzle from falling apart. High-level-of-difficulty, or simply high-level, interlocking puzzles are a subclass of interlocking puzzles that require multiple moves to take out the first subassembly from the puzzle. Solving a high-level interlocking puzzle is a challenging task since one has to explore many different configurations of the puzzle pieces until reaching a configuration where the first subassembly can be taken out. Designing a high-level interlocking puzzle with a user-specified level of difficulty is even harder since the puzzle pieces have to be interlocking in all the configurations before the first subassembly is taken out. In this paper, we present a computational approach to design high-level interlocking puzzles. The core idea is to represent all possible configurations of an interlocking puzzle as well as transitions among these configurations using a rooted, undirected graph called a disassembly graph and leverage this graph to find a disassembly plan that requires a minimal number of moves to take out the first subassembly from the puzzle. At the design stage, our algorithm iteratively constructs the geometry of each puzzle piece to expand the disassembly graph incrementally, aiming to achieve a user-specified level of difficulty. We show that our approach allows efficient generation of high-level interlocking puzzles of various shape complexities, including new solutions not attainable by state-of-the-art approaches.

@article{Chen-2022-HighLevelPuzzle,
  title = {Computational Design of High-level Interlocking Puzzles},
  author = {Rulin Chen and Ziqi Wang and Peng Song and Bernd Bickel},
  year = {2022},
  journal = {ACM Transactions on Graphics (Proc. SIGGRAPH)},
  volume = {41},
  number = {4}
}

We study structural rigidity for assemblies with mechanical joints. Existing methods identify whether an assembly is structurally rigid by assuming parts are perfectly rigid. Yet, an assembly identified as rigid may not be that “rigid” in practice, and existing methods cannot quantify how rigid an assembly is. We address this limitation by developing a new measure, worst-case rigidity, to quantify the rigidity of an assembly as the largest possible deformation that the assembly undergoes for arbitrary external loads of fixed magnitude. Computing worst-case rigidity is non-trivial due to non-rigid parts and different joint types. We encode parts and their connections into a stiffness matrix, in which parts are modeled as deformable objects and joints as soft constraints. Based on this, we formulate worst-case rigidity analysis as an optimization that seeks the worst-case deformation of an assembly for arbitrary external loads. Furthermore, we present methods to optimize the geometry and topology of various assemblies to enhance their rigidity, as guided by our rigidity measure. We validate our analysis and optimization on various assembly structures with fabrication.

@article{liu2022rigidity,
  title = {Worst-Case Rigidity Analysis and Optimization for Assemblies with Mechanical Joints},
  author = {Zhenyuan Liu and Jingyu Hu and Hao Xu and Peng Song and Ran Zhang and Bernd Bickel and Chi-Wing Fu},
  year = {2022},
  journal = {Computer Graphics Forum}, 
  volume = {41},
  number = {2},
  doi = {10.1111/cgf.14490}
}

This paper proposes a method for simulating liquids in large bodies of water by coupling together a water surface wave simulator with a 3D Navier-Stokes simulator. The surface wave simulation uses the equivalent sources method (ESM) to efficiently animate large bodies of water with precisely controllable wave propagation behavior. The 3D liquid simulator animates complex non-linear fluid behaviors like splashes and breaking waves using off-the-shelf simulators using FLIP or the level set method with semi-Lagrangian advection. We combine the two approaches by using the 3D solver to animate localized non-linear behaviors, and the 2D wave solver to animate larger regions with linear surface physics. We use the surface motion from the 3D solver as boundary conditions for 2D surface wave simulator, and we use the velocity and surface heights from the 2D surface wave simulator as boundary conditions for the 3D fluid simulation. We also introduce a novel technique for removing visual artifacts caused by numerical errors in 3D fluid solvers: we use experimental data to estimate the artificial dispersion caused by the 3D solver and we then carefully tune the wave speeds of the 2D solver to match it, effectively eliminating any differences in wave behavior across the boundary. To the best of our knowledge, this is the first time such a empirically driven error compensation approach has been used to remove coupling errors from a physics simulator. Our coupled simulation approach leverages the strengths of each simulation technique, animating large environments with seamless transitions between 2D and 3D physics.

@article {10.1111:cgf.14478,
journal   = "Computer Graphics Forum",
title     = "Coupling 3D Liquid Simulation with 2D Wave Propagation for Large Scale Water Surface Animation Using the Equivalent Sources Method",
author    = "Schreck, Camille and Wojtan, Chris",
year      = "2022",
publisher = "The Eurographics Association and John Wiley & Sons Ltd.",
ISSN      = "1467-8659",
DOI       = "10.1111/cgf.14478"
}

We introduce a novel technique to automatically decompose an input object's volume into a set of parts that can be represented by two opposite height fields. Such decomposition enables the manufacturing of individual parts using two-piece reusable rigid molds. Our decomposition strategy relies on a new energy formulation that utilizes a pre-computed signal on the mesh volume representing the accessibility for a predefined set of extraction directions. Thanks to this novel formulation, our method allows for efficient optimization of a fabrication-aware partitioning of volumes in a completely automatic way. We demonstrate the efficacy of our approach by generating valid volume partitionings for a wide range of complex objects and physically reproducing several of them.

@article{alderighi2021volume,
  author    = {Alderighi, Thomas and Malomo, Luigi and Bickel, Bernd and Cignoni, Paolo and Pietroni, Nico},
  title     = {Volume decomposition for two-piece rigid casting},
  journal   = {ACM Transactions on Graphics (TOG)},
  number    = {6},
  volume    = {40},
  year      = {2021},
  publisher = {ACM}
}

Tactile feedback of an object’s surface enables us to discern its material properties and affordances. This understanding is used in digital fabrication processes by creating objects with high-resolution surface variations to influence a user’s tactile perception. As the design of such surface haptics commonly relies on knowledge from real-life experiences, it is unclear how to adapt this information for digital design methods. In this work, we investigate replicating the haptics of real materials. Using an existing process for capturing an object’s microgeometry, we digitize and reproduce the stable surface information of a set of 15 fabric samples. In a psychophysical experiment, we evaluate the tactile qualities of our set of original samples and their replicas. From our results, we see that direct reproduction of surface variations is able to influence different psychophysical dimensions of the tactile perception of surface textures. While the fabrication process did not preserve all properties, our approach underlines that replication of surface microgeometries benefits fabrication methods in terms of haptic perception by covering a large range of tactile variations. Moreover, by changing the surface structure of a single fabricated material, its material perception can be influenced. We conclude by proposing strategies for capturing and reproducing digitized textures to better resemble the perceived haptics of the originals.

@article{degraen2021capturing,
  author    = {Degraen, Donald and Piovar{\v{c}}i, Michal and Bickel, Bernd and Kr{\"u}ger, Antonio},
  title     = {Capturing Tactile Properties of Real Surfaces for Haptic Reproduction},
  booktitle = {The 34th Annual ACM Symposium on User Interface Software and Technology},
  pages     = {954--971},
  year      = {2021}
}

Material appearance hinges on material reflectance properties but also surface geometry and illumination. The unlimited number of potential combinations between these factors makes understanding and predicting material appearance a very challenging task. In this work, we collect a large-scale dataset of perceptual ratings of appearance attributes with more than 215,680 responses for 42,120 distinct combinations of material, shape, and illumination. The goal of this dataset is twofold. First, we analyze for the first time the effects of illumination and geometry in material perception across such a large collection of varied appearances. We connect our findings to those of the literature, discussing how previous knowledge generalizes across very diverse materials, shapes, and illuminations. Second, we use the collected dataset to train a deep learning architecture for predicting perceptual attributes that correlate with human judgments. We demonstrate the consistent and robust behavior of our predictor in various challenging scenarios, which, for the first time, enables estimating perceived material attributes from general 2D images. Since the predictor relies on the final appearance in an image, it can compare appearance properties across different geometries and illumination conditions. Finally, we demonstrate several applications that use our predictor, including appearance reproduction using 3D printing, BRDF editing by integrating our predictor in a differentiable renderer, illumination design, or material recommendations for scene design.

@article{Serrano2021,
  author = { Ana Serrano and Bin Chen and Chao Wang and Michal Piovar\v{c}i and Hans-Peter Seidel and Piotr Didyk and Karol Myszkowski},
  title = {The effect of shape and illumination on material perception: model and applications},
  journal = {ACM Trans. on Graph.},
  volume = {40},
  number = {4},
  year = {2021},
} 

Elastic bending of initially flat slender elements allows the realization and economic fabrication of intriguing curved shapes. In this work, we derive an intuitive but rigorous geometric characterization of the design space of plane elastic rods with variable stiffness. It enables designers to determine which shapes are physically viable with active bending by visual inspection alone. Building on these insights, we propose a method for efficiently designing the geometry of a flat elastic rod that realizes a target equilibrium curve, which only requires solving a linear program. We implement this method in an interactive computational design tool that gives feedback about the feasibility of a design, and computes the geometry of the structural elements necessary to realize it within an instant. The tool also offers an iterative optimization routine that improves the fabricability of a model while modifying it as little as possible. In addition, we use our geometric characterization to derive an algorithm for analyzing and recovering the stability of elastic curves that would otherwise snap out of their unstable equilibrium shapes by buckling. We show the efficacy of our approach by designing and manufacturing several physical models that are assembled from flat elements.

@article{hafner2021tdsopec,
  author    = {Hafner, Christian and Bickel, Bernd},
  title     = {The Design Space of Plane Elastic Curves},
  journal   = {ACM Transactions on Graphics (TOG)},
  number    = {4},
  volume    = {40},
  year      = {2021},
  publisher = {ACM}
}

Triangle mesh-based simulations are able to produce satisfying animations of knitted and woven cloth; however, they lack the rich geometric detail of yarn-level simulations. Naive texturing approaches do not consider yarn-level physics, while full yarn-level simulations may become prohibitively expensive for large garments. We propose a method to animate yarn-level cloth geometry on top of an underlying deforming mesh in a mechanics-aware fashion. Using triangle strains to interpolate precomputed yarn geometry, we are able to reproduce effects such as knit loops tightening under stretching. In combination with precomputed mesh animation or real-time mesh simulation, our method is able to animate yarn-level cloth in real-time at large scales.

@article{sperl2021madypg,
  author    = {Sperl, Georg and Narain, Rahul and Wojtan, Chris},
  title     = {Mechanics-Aware Deformation of Yarn Pattern Geometry},
  journal   = {ACM Transactions on Graphics (TOG)},
  number    = {4},
  volume    = {40},
  year      = {2021},
  publisher = {ACM}
}

This paper presents a method for designing planar multistable compliant structures. Given a sequence of desired stable states and the corresponding poses of the structure, we identify the topology and geometric realization of a mechanism—consisting of bars and joints—that is able to physically reproduce the desired multistable behavior. In order to solve this problem efficiently, we build on insights from minimally rigid graph theory to identify simple but effective topologies for the mechanism. We then optimize its geometric parameters, such as joint positions and bar lengths, to obtain correct transitions between the given poses. Simultaneously, we ensure adequate stability of each pose based on an effective approximate error metric related to the elastic energy Hessian of the bars in the mechanism. As demonstrated by our results, we obtain functional multistable mechanisms of manageable complexity that can be fabricated using 3D printing. Further, we evaluated the effectiveness of our method on a large number of examples in the simulation and fabricated several physical prototypes.

@article{zhang2021multistable,
  author    = {Zhang, Ran and Auzinger, Thomas and Bickel, Bernd},
  title     = {Computational Design of Planar Multistable Compliant Structures},
  journal   = {ACM Transactions on Graphics (TOG)},
  volume    = {40},
  number    = {5},
  pages     = {1--16},
  year      = {2021},
  publisher = {ACM}
}

Photorealistic editing of head portraits is a challenging task as humans are very sensitive to inconsistencies in faces. We present an approach for high-quality intuitive editing of the camera viewpoint and scene illumination (parameterised with an environment map) in a portrait image. This requires our method to capture and control the full reflectance field of the person in the image. Most editing approaches rely on supervised learning using training data captured with setups such as light and camera stages. Such datasets are expensive to acquire, not readily available and do not capture all the rich variations of in-the-wild portrait images. In addition, most supervised approaches only focus on relighting, and do not allow camera viewpoint editing. Thus, they only capture and control a subset of the reflectance field. Recently, portrait editing has been demonstrated by operating in the generative model space of StyleGAN. While such approaches do not require direct supervision, there is a significant loss of quality when compared to the supervised approaches. In this paper, we present a method which learns from limited supervised training data. The training images only include people in a fixed neutral expression with eyes closed, without much hair or background variations. Each person is captured under 150 one-light-at-a-time conditions and under 8 camera poses. Instead of training directly in the image space, we design a supervised problem which learns transformations in the latent space of StyleGAN. This combines the best of supervised learning and generative adversarial modeling. We show that the StyleGAN prior allows for generalisation to different expressions, hairstyles and backgrounds. This produces high-quality photorealistic results for in-the-wild images and significantly outperforms existing methods. Our approach can edit the illumination and pose simultaneously, and runs at interactive rates.

@article{mallikarjun2021photoapp,
  title={PhotoApp: Photorealistic appearance editing of head portraits},
  author={Mallikarjun, BR and Tewari, Ayush and Dib, Abdallah and Weyrich, Tim and Bickel, Bernd and Seidel, Hans Peter and Pfister, Hanspeter and Matusik, Wojciech and Chevallier, Louis and Elgharib, Mohamed A and others},
  journal={ACM Transactions on Graphics},
  volume={40},
  number={4},
  year={2021}
}

The reflectance field of a face describes the reflectance properties responsible for complex lighting effects including diffuse, specular, inter-reflection and self shadowing. Most existing methods for estimating the face reflectance from a monocular image assume faces to be diffuse with very few approaches adding a specular component. This still leaves out important perceptual aspects of reflectance as higher-order global illumination effects and self-shadowing are not modeled. We present a new neural representation for face reflectance where we can estimate all components of the reflectance responsible for the final appearance from a single monocular image. Instead of modeling each component of the reflectance separately using parametric models, our neural representation allows us to generate a basis set of faces in a geometric deformation-invariant space, parameterized by the input light direction, viewpoint and face geometry. We learn to reconstruct this reflectance field of a face just from a monocular image, which can be used to render the face from any viewpoint in any light condition. Our method is trained on a light-stage training dataset, which captures 300 people illuminated with 150 light conditions from 8 viewpoints. We show that our method outperforms existing monocular reflectance reconstruction methods, in terms of photorealism due to better capturing of physical premitives, such as sub-surface scattering, specularities, self-shadows and other higher-order effects.

@inproceedings{tewari2021monocular,
  title={Monocular Reconstruction of Neural Face Reflectance Fields},
  author={Tewari, Ayush and Oh, Tae-Hyun and Weyrich, Tim and Bickel, Bernd and Seidel, Hans-Peter and Pfister, Hanspeter and Matusik, Wojciech and Elgharib, Mohamed and Theobalt, Christian and others},
  booktitle={Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition},
  pages={4791--4800},
  year={2021}
}

The understanding of material appearance perception is a complex problem due to interactions between material reflectance, surface geometry, and illumination. Recently, Serrano et al. collected the largest dataset to date with subjective ratings of material appearance attributes, including glossiness, metallicness, sharpness and contrast of reflections. In this work, we make use of their dataset to investigate for the first time the impact of the interactions between illumination, geometry, and eight different material categories in perceived appearance attributes. After an initial analysis, we select for further analysis the four material categories that cover the largest range for all perceptual attributes: fabric, plastic, ceramic, and metal. Using a cumulative link mixed model (CLMM) for robust regression, we discover interactions between these material categories and four representative illuminations and object geometries. We believe that our findings contribute to expanding the knowledge on material appearance perception and can be useful for many applications, such as scene design, where any particular material in a given shape can be aligned with dominant classes of illumination, so that a desired strength of appearance attributes can be achieved.

@article{Chen2021,
  author = { Bin Chen and Chao Wang and Michal Piovar\v{c}i and Hans-Peter Seidel and Piotr Didyk and Karol Myszkowski and Ana Serrano},
  title = {The effect of geometry and illumination on appearance perception of different material categories},
  journal = {The Visual Computer},
  volume = {37},
  year = {2021},
}

With the wider availability of full-color 3D printers, color-accurate 3D-print preparation has received increased attention. A key challenge lies in the inherent translucency of commonly used print materials that blurs out details of the color texture. Previous work tries to compensate for these scattering effects through strategic assignment of colored primary materials to printer voxels. To date, the highest-quality approach uses iterative optimization that relies on computationally expensive Monte Carlo light transport simulation to predict the surface appearance from subsurface scattering within a given print material distribution; that optimization, however, takes in the order of days on a single machine. In our work, we dramatically speed up the process by replacing the light transport simulation with a data-driven approach. Leveraging a deep neural network to predict the scattering within a highly heterogeneous medium, our method performs around two orders of magnitude faster than Monte Carlo rendering while yielding optimization results of similar quality level. The network is based on an established method from atmospheric cloud rendering, adapted to our domain and extended by a physically motivated weight sharing scheme that substantially reduces the network size. We analyze its performance in an end-to-end print preparation pipeline and compare quality and runtime to alternative approaches, and demonstrate its generalization to unseen geometry and material values. This for the first time enables full heterogenous material optimization for 3D-print preparation within time frames in the order of the actual printing time.

@article{rittig2021neural,
  title={Neural Acceleration of Scattering-Aware Color 3D Printing},
  author={Rittig, Tobias and Sumin, Denis and Babaei, Vahid and Didyk, Piotr and Voloboy, Alexey and Wilkie, Alexander and Bickel, Bernd and Myszkowski, Karol and Weyrich, Tim and K{\v{r}}iv{\'a}nek, Jaroslav},
  booktitle={Computer Graphics Forum},
  volume={40},
  number={2},
  pages={205--219},
  year={2021},
  organization={Wiley Online Library}
}

Volumetric light transport is a pervasive physical phenomenon, and therefore its accurate simulation is important for a broad array of disciplines. While suitable mathematical models for computing the transport are now available, obtaining the necessary material parameters needed to drive such simulations is a challenging task: direct measurements of these parameters from material samples are seldom possible. Building on the inverse scattering paradigm, we present a novel measurement approach which indirectly infers the transport parameters from extrinsic observations of multiple-scattered radiance. The novelty of the proposed approach lies in replacing structured illumination with a structured reflector bonded to the sample, and a robust fitting procedure that largely compensates for potential systematic errors in the calibration of the setup. We show the feasibility of our approach by validating simulations of complex 3D compositions of the measured materials against physical prints, using photo-polymer resins. As presented in this paper, our technique yields colorspace data suitable for accurate appearance reproduction in the area of 3D printing. Beyond that, and without fundamental changes to the basic measurement methodology, it could equally well be used to obtain spectral measurements that are useful for other application areas.

@article{Elek:21,
  author = {Oskar Elek and Ran Zhang and Denis Sumin and Karol Myszkowski and Bernd Bickel and Alexander Wilkie and Jaroslav K\v{r}iv\'{a}nek and Tim Weyrich},
  journal = {Opt. Express},
  keywords = {Illumination design; Inverse scattering; Medical imaging; Optical properties; Scattering media; Structured illumination microscopy},
  number = {5},
  pages = {7568--7588},
  publisher = {OSA},
  title = {Robust and practical measurement of volume transport parameters in solid photo-polymer materials for 3D printing},
  volume = {29},
  month = {Mar},
  year = {2021},
  url = {http://www.osapublishing.org/oe/abstract.cfm?URI=oe-29-5-7568},
  doi = {10.1364/OE.406095}
}

Cold bent glass is a promising and cost-efficient method for realizing doubly curved glass façades. They are produced by attaching planar glass sheets to curved frames and must keep the occurring stress within safe limits. However, it is very challenging to navigate the design space of cold bent glass panels because of the fragility of the material, which impedes the form finding for practically feasible and aesthetically pleasing cold bent glass façades. We propose an interactive, data-driven approach for designing cold bent glass façades that can be seamlessly integrated into a typical architectural design pipeline. Our method allows non-expert users to interactively edit a parametric surface while providing real-time feedback on the deformed shape and maximum stress of cold bent glass panels. The designs are automatically refined to minimize several fairness criteria, while maximal stresses are kept within glass limits. We achieve interactive frame rates by using a differentiable Mixture Density Network trained from more than a million simulations. Given a curved boundary, our regression model is capable of handling multistable configurations and accurately predicting the equilibrium shape of the panel and its corresponding maximal stress. We show that the predictions are highly accurate and validate our results with a physical realization of a cold bent glass surface.

@article{Gavriil2020,
author = {Gavriil, Konstantinos and Guseinov, Ruslan and P{\'e}rez, Jes{\'u}s and Pellis, Davide and Henderson, Paul and Rist, Florian and Pottmann, Helmut and Bickel, Bernd},
title = {Computational Design of Cold Bent Glass Fa{\c c}ades},
journal = {ACM Transactions on Graphics (SIGGRAPH Asia 2020)},
year = {2020},
month = {12},
volume = {39},
number = {6}
articleno = {208},
numpages = {16}
}

The “procedural” approach to animating ocean waves is the dominant algorithm for animating larger bodies of water in interactive applications as well as in off-line productions — it provides high visual quality with a low computational demand. In this paper, we widen the applicability of procedural water wave animation with an extension that guarantees the satisfaction of boundary conditions imposed by terrain while still approximating physical wave behavior. In combination with a particle system that models wave breaking, foam, and spray, this allows us to naturally model waves interacting with beaches and rocks. Our system is able to animate waves at large scales at interactive frame rates on a commodity PC.

Fabrication of curved shells plays an important role in modern design, industry, and science. Among their remarkable properties are, for example, aesthetics of organic shapes, ability to evenly distribute loads, or efficient flow separation. They find applications across vast length scales ranging from sky-scraper architecture to microscopic devices. But, at the same time, the design of curved shells and their manufacturing process pose a variety of challenges. In this thesis, they are addressed from several perspectives. In particular, this thesis presents approaches based on the transformation of initially flat sheets into the target curved surfaces. This involves problems of interactive design of shells with nontrivial mechanical constraints, inverse design of complex structural materials, and data-driven modeling of delicate and time-dependent physical properties. At the same time, two ewly-developed self-morphing mechanisms targeting flat-to-curved transformation are presented. In architecture, doubly curved surfaces can be realized as cold bent glass panelizations. Originally flat glass panels are bent into frames and remain stressed. This is a cost-efficient fabrication approach compared to hot bending, when glass panels are shaped plastically. However such constructions are prone to breaking during bending, and it is highly nontrivial to navigate the design space, keeping the panels fabricable and aesthetically pleasing at the same time. We introduce an interactive design system for cold bent glass façades, while previously even offline optimization for such scenarios has not been sufficiently developed. Our method is based on a deep learning approach providing quick and high precision estimation of glass panel shape and stress while handling the shape multimodality. Fabrication of smaller objects of scales below 1 m, can also greatly benefit from shaping originally flat sheets. In this respect, we designed new self-morphing shell mechanisms transforming from an initial flat state to a doubly curved state with high precision and detail. Our so-called CurveUps demonstrate the encodement of the geometric information into the shell. Furthermore, we explored the frontiers of programmable materials and showed how temporal information can additionally be encoded into a flat shell. This allows prescribing deformation sequences for doubly curved surfaces and, thus, facilitates self-collision avoidance enabling complex shapes and functionalities otherwise impossible. Both of these methods include inverse design tools keeping the user in the design loop.

@phdthesis{GuseinovCDoCTS2020,
author = {Guseinov, Ruslan},
title = {Computational design of curved thin shells: from glass façades to programmable matter},
school = {IST Austria},
year = {2020},
month = {9}
}

We propose a method to enhance the visual detail of a water surface simulation. Our method works as a post-processing step which takes a simulation as input and increases its apparent resolution by simulating many detailed Lagrangian water waves on top of it. We extend linear water wave theory to work in non-planar domains which deform over time, and we discretize the theory using Lagrangian wave packets attached to spline curves. The method is numerically stable and trivially parallelizable, and it produces high frequency ripples with dispersive wave-like behaviors customized to the underlying fluid simulation.

@article{skrivan2020wc,
  author    = {Skřivan, Tomáš and Söderström, Andreas and Johansson, John and Sprenger, Christoph and Museth, Ken and Wojtan, Chris},
  title     = {Wave Curves: Simulating Lagrangian water waves on dynamically deforming surfaces},
  journal   = {ACM Transactions on Graphics (TOG)},
  number    = {4},
  volume    = {39},
  year      = {2020},
  publisher = {ACM}
}

We present a method for animating yarn-level cloth effects using a thin-shell solver. We accomplish this through numerical homogenization: we first use a large number of yarn-level simulations to build a model of the potential energy density of the cloth, and then use this energy density function to compute forces in a thin shell simulator. We model several yarn-based materials, including both woven and knitted fabrics. Our model faithfully reproduces expected effects like the stiffness of woven fabrics, and the highly deformable nature and anisotropy of knitted fabrics. Our approach does not require any real-world experiments nor measurements; because the method is based entirely on simulations, it can generate entirely new material models quickly, without the need for testing apparatuses or human intervention. We provide data-driven models of several woven and knitted fabrics, which can be used for efficient simulation with an off-the-shelf cloth solver.

@article{sperl2020hylc,
  author    = {Sperl, Georg and Narain, Rahul and Wojtan, Chris},
  title     = {Homogenized Yarn-Level Cloth},
  journal   = {ACM Transactions on Graphics (TOG)},
  number    = {4},
  volume    = {39},
  year      = {2020},
  publisher = {ACM}
}

Previous research on animations of soap bubbles, films, and foams largely focuses on the motion and geometric shape of the bubble surface. These works neglect the evolution of the bubble's thickness, which is normally responsible for visual phenomena like surface vortices, Newton's interference patterns, capillary waves, and deformation-dependent rupturing of films in a foam. In this paper, we model these natural phenomena by introducing the film thickness as a reduced degree of freedom in the Navier-Stokes equations and deriving their equations of motion. We discretize the equations on a non-manifold triangle mesh surface and couple it to an existing bubble solver. In doing so, we also introduce an incompressible fluid solver for 2.5D films and a novel advection algorithm for convecting fields across non-manifold surface junctions. Our simulations enhance state-of-the-art bubble solvers with additional effects caused by convection, rippling, draining, and evaporation of the thin film.

@article{isnhw2020soapfilm_with_thickness,
  author    = {Sadashige Ishida and Peter Synak and Fumiya Narita and Toshiya Hachisuka and Chris Wojtan}
  title     = {A Model for Soap Film Dynamics with Evolving Thickness},
  journal   = {ACM Trans. on Graphics},
  number    = {4},
  volume    = {39},
  year      = {2020},
  pages     = {31:1--31:11},
  articleno = 31,
  url       = {http://dx.doi.org/10.1145/3386569.3392405},
  doi       = {10.1145/3386569.3392405},
  publisher = {ACM}
}

This paper introduces a simple method for simulating highly anisotropic elastoplastic material behaviors like the dissolution of fibrous phenomena (splintering wood, shredding bales of hay) and materials composed of large numbers of irregularly-shaped bodies (piles of twigs, pencils, or cards). We introduce a simple transformation of the anisotropic problem into an equivalent isotropic one, and we solve this new ``fictitious'' isotropic problem using an existing simulator based on the material point method. Our approach results in minimal changes to existing simulators, and it allows us to re-use popular isotropic plasticity models like the Drucker-Prager yield criterion instead of inventing new anisotropic plasticity models for every phenomenon we wish to simulate.

@article{SW_ampm20,
author       = "Schreck, Camille and Wojtan, Chris"
title        = "A Practical Method for Animating Anisotropic Elastoplastic Materials",
journal      = "Computer Graphics Forum - Eurographics 2020",
number       = "2",
volume       = "39",
year         = "2020",
}

Advances in shape-morphing materials, such as hydrogels, shape-memory polymers and light-responsive polymers have enabled prescribing self-directed deformations of initially flat geometries. However, most proposed solutions evolve towards a target geometry without considering time-dependent actuation paths. To achieve more complex geometries and avoid self-collisions, it is critical to encode a spatial and temporal shape evolution within the initially flat shell. Recent realizations of time-dependent morphing are limited to the actuation of few, discrete hinges and cannot form doubly curved surfaces. Here, we demonstrate a method for encoding temporal shape evolution in architected shells that assume complex shapes and doubly curved geometries. The shells are non-periodic tessellations of pre-stressed contractile unit cells that soften in water at rates prescribed locally by mesostructure geometry. The ensuing midplane contraction is coupled to the formation of encoded curvatures. We propose an inverse design tool based on a data-driven model for unit cells’ temporal responses.

@article{Guseinov2020,
author={Guseinov, Ruslan and McMahan, Connor and P{\'e}rez, Jes{\'u}s and Daraio, Chiara and Bickel, Bernd},
title={Programming temporal morphing of self-actuated shells},
journal={Nature Communications},
year={2020},
volume={11},
number={1},
pages={237},
issn={2041-1723},
doi={10.1038/s41467-019-14015-2},
url={https://doi.org/10.1038/s41467-019-14015-2}
}

We present a computational design system that assists users to model, optimize, and fabricate quad-robots with soft skins.Our system addresses the challenging task of predicting their physical behavior by fully integrating the multibody dynamics of the mechanical skeleton and the elastic behavior of the soft skin. The developed motion control strategy uses an alternating optimization scheme to avoid expensive full space time-optimization, interleaving space-time optimization for the skeleton and frame-by-frame optimization for the full dynamics. The output are motor torques to drive the robot to achieve a user prescribed motion trajectory.We also provide a collection of convenient engineering tools and empirical manufacturing guidance to support the fabrication of the designed quad-robot. We validate the feasibility of designs generated with our system through physics simulations and with a physically-fabricated prototype.

@article{alderighi2021volume,
  author    = {Feng, Xudong and Liu, Jiafeng and Wang, Huamin and Yang, Yin and Bao, Hujun and Bickel, Bernd and Xu, Weiwei},
  title     = {Computational Design of Skinned Quad-Robots},
  journal   = {IEEE Transactions on Visualization and Computer Graphics},
  number    = {6},
  volume    = {27},
  year      = {2019},
  publisher = {IEEE}
}

We propose a novel generic shape optimization method for CAD models based on the eXtended Finite Element Method (XFEM). Our method works directly on the intersection between the model and a regular simulation grid, without the need to mesh or remesh, thus removing a bottleneck of classical shape optimization strategies. This is made possible by a novel hierarchical integration scheme that accurately integrates finite element quantities with sub-element precision. For optimization, we efficiently compute analytical shape derivatives of the entire framework, from model intersection to in- tegration rule generation and XFEM simulation. Moreover, we describe a differentiable projection of shape parameters onto a constraint manifold spanned by user-specified shape preservation, consistency, and manufactura- bility constraints. We demonstrate the utility of our approach by optimizing mass distribution, strength-to-weight ratio, and inverse elastic shape design objectives directly on parameterized 3D CAD models.

@article{Hafner:2019,
author = {Hafner, Christian and Schumacher, Christian and Knoop, Espen and Auzinger, Thomas and Bickel, Bernd and B\"{a}cher, Moritz},
title = {X-CAD: Optimizing CAD Models with Extended Finite Elements},
journal = {ACM Trans. Graph.},
issue_date = {November 2019},
volume = {38},
number = {6},
month = nov,
year = {2019},
issn = {0730-0301},
pages = {157:1--157:15},
articleno = {157},
numpages = {15},
url = {http://doi.acm.org/10.1145/3355089.3356576},
doi = {10.1145/3355089.3356576},
acmid = {3356576},
publisher = {ACM},
address = {New York, NY, USA},
keywords = {CAD processing, XFEM, shape optimization, simulation},
} 

Bending-active structures are able to efficiently produce complex curved shapes starting from flat panels. The desired deformation of the panels derives from the proper selection of their elastic properties. Optimized panels, called FlexMaps, are designed such that, once they are bent and assembled, the resulting static equilibrium configuration matches a desired input 3D shape. The FlexMaps elastic properties are controlled by locally varying spiraling geometric mesostructures, which are optimized in size and shape to match the global curvature (i.e., bending requests) of the target shape. The design pipeline starts from a quad mesh representing the input 3D shape, which determines the edge size and the total amount of spirals: every quad will embed one spiral. Then, an optimization algorithm tunes the geometry of the spirals by using a simplified pre-computed rod model. This rod model is derived from a non-linear regression algorithm which approximates the non-linear behavior of solid FEM spiral models subject to hundreds of load combinations. This innovative pipeline has been applied to the project of a lightweight plywood pavilion named FlexMaps Pavilion, which is a single-layer piecewise twisted arc that fits a bounding box of 3.90x3.96x3.25 meters.

@InProceedings\{LMPPPBC19,
author       = "Laccone, Francesco and Malomo, Luigi and P\'erez, Jes\'us and Pietroni, Nico and Ponchio, Federico and Bickel, Bernd and Cignoni, Paolo",
title        = "FlexMaps Pavilion: a twisted arc made of mesostructured flat flexible panels",
booktitle    = "FORM and FORCE, IASS Symposium 2019, Structural Membranes 2019",
pages        = "498-504",
month        = "oct",
year         = "2019",
editor       = "Carlos L\'azaro, Kai-Uwe Bletzinger, Eugenio O\~{n}ate",
publisher    = "International Centre for Numerical Methods in Engineering (CIMNE)",
url          = "http://vcg.isti.cnr.it/Publications/2019/LMPPPBC19"
}

Commercially available full-color 3D printing allows for detailed control of material deposition in a volume, but an exact reproduction of a target surface appearance is hampered by the strong subsurface scattering that causes nontrivial volumetric cross-talk at the print surface. Previous work showed how an iterative optimization scheme based on accumulating absorptive materials at the surface can be used to find a volumetric distribution of print materials that closely approximates a given target appearance. In this work, we first revisit the assumption that pushing the absorptive materials to the surface results in minimal volumetric cross-talk. We design a full-fledged optimization on a small domain for this task and confirm this previously reported heuristic. Then, we extend the above approach that is critically limited to color reproduction on planar surfaces, to arbitrary 3D shapes. Our method enables high-fidelity color texture reproduction on 3D prints by effectively compensating for internal light scattering within arbitrarily shaped objects. In addition, we propose a content-aware gamut mapping that significantly improves color reproduction for the pathological case of thin geometric features. Using a wide range of sample objects with complex textures and geometries, we demonstrate color reproduction whose fidelity is superior to state-of-the-art drivers for color 3D printers.

@article{sumin19geometry-aware,
author = {Sumin, Denis and Rittig, Tobias and Babaei, Vahid and Nindel, Thomas and Wilkie, Alexander and Didyk, Piotr and Bickel, Bernd and K\v{r}iv\'anek, Jaroslav and Myszkowski, Karol and Weyrich, Tim},
title = {Geometry-Aware Scattering Compensation for {3D} Printing},
journal = {ACM Transactions on Graphics (Proc. SIGGRAPH)},
year = 2019,
month = jul,
volume = 38,
numpages = 14,
keywords = {computational fabrication, appearance reproduction, appearance enhancement, sub-surface light transport, volu- metric optimization, gradient rendering},
}

This paper investigates the use of fundamental solutions for animating detailed linear water surface waves. We first propose an analytical solution for efficiently animating circular ripples in closed form. We then show how to adapt the method of fundamental solutions (MFS) to create ambient waves interacting with complex obstacles. Subsequently, we present a novel wavelet-based discretization which outperforms the state of the art MFS approach for simulating time-varying water surface waves with moving obstacles. Our results feature high-resolution spatial details, interactions with complex boundaries, and large open ocean domains. Our method compares favorably with previous work as well as known analytical solutions. We also present comparisons between our method and real world examples.

@article{SHW_fsww19,
author       = "Schreck, Camille and Hafner, Christian and Wojtan, Chris"
title        = "Fundamental Solutions for Water Wave Animation",
journal      = "ACM Trans. on Graphics - Siggraph 2019",
number       = "4",
volume       = "38",
pages        = "14",
month        = "July",
year         = "2019",
note         = "https://doi.org/10.1145/3306346.3323002"
}

We propose a novel technique for the automatic design of molds to cast highly complex shapes. The technique generates composite, two-piece molds. Each mold piece is made up of a hard plastic shell and a flexible silicone part. Thanks to the thin, soft, and smartly shaped silicone part, which is kept in place by a hard plastic shell, we can cast objects of unprecedented complexity. An innovative algorithm based on a volumetric analysis defines the layout of the internal cuts in the silicone mold part. Our approach can robustly handle thin protruding features and intertwined topologies that have caused previous methods to fail. We compare our results with state of the art techniques, and we demonstrate the casting of shapes with extremely complex geometry.

@article{Alderighi:2019,
author = {Alderighi, Thomas and Malomo, Luigi and Giorgi, Daniela and Bickel, Bernd and Cignoni, Paolo and Pietroni, Nico},
title = {Volume-aware Design of Composite Molds},
journal = {ACM Trans. Graph.},
issue_date = {July 2019},
volume = {38},
number = {4},
month = jul,
year = {2019},
issn = {0730-0301},
pages = {110:1--110:12},
articleno = {110},
numpages = {12},
url = {http://doi.acm.org/10.1145/3306346.3322981},
doi = {10.1145/3306346.3322981},
acmid = {3322981},
publisher = {ACM},
address = {New York, NY, USA},
keywords = {casting, fabrication, mold design},
}

We propose FlexMaps, a novel framework for fabricating smooth shapes out of flat, flexible panels with tailored mechanical properties. We start by mapping the 3D surface onto a 2D domain as in traditional UV mapping to design a set of deformable flat panels called FlexMaps. For these panels, we design and obtain specific mechanical properties such that, once they are assembled, the static equilibrium configuration matches the desired 3D shape. FlexMaps can be fabricated from an almost rigid material, such as wood or plastic, and are made flexible in a controlled way by using computationally designed spiraling microstructures.

@article{MPIPMCB18,
author       = "Malomo, Luigi and Per\'ez, Jes\'us and Iarussi, Emmanuel and Pietroni, Nico and Miguel, Eder and Cignoni, Paolo and Bickel, Bernd",
title        = "FlexMaps: Computational Design of Flat Flexible Shells for Shaping 3D Objects",
journal      = "ACM Trans. on Graphics - Siggraph Asia 2018",
number       = "6",
volume       = "37",
pages        = "14",
month        = "dec",
year         = "2018",
note         = "https://doi.org/10.1145/3272127.3275076"
}

We introduce dynamically warping grids for adaptive liquid simulation. Our primary contributions are a strategy for dynamically deforming regular grids over the course of a simulation and a method for efficiently utilizing these deforming grids for liquid simulation. Prior work has shown that unstructured grids are very effective for adaptive fluid simulations. However, unstructured grids often lead to complicated implementations and a poor cache hit rate due to inconsistent memory access. Regular grids, on the other hand, provide a fast, fixed memory access pattern and straightforward implementation. Our method combines the advantages of both: we leverage the simplicity of regular grids while still achieving practical and controllable spatial adaptivity. We demonstrate that our method enables adaptive simulations that are fast, flexible, and robust to null-space issues. At the same time, our method is simple to implement and takes advantage of existing highly-tuned algorithms.

@article{ibayashi2018simulating,
  title={Simulating liquids on dynamically warping grids},
  author={Ibayashi, Hikaru and Wojtan, Chris and Thuerey, Nils and Igarashi, Takeo and Ando, Ryoichi},
  journal={IEEE transactions on visualization and computer graphics},
  volume={26},
  number={6},
  pages={2288--2302},
  year={2018},
  publisher={IEEE}
  doi={10.1109/TVCG.2018.2883628}
}

Molding is a popular mass production method, in which the initial expenses for the mold are offset by the low per-unit production cost. However, the physical fabrication constraints of the molding technique commonly restrict the shape of moldable objects. For a complex shape, a decomposition of the object into moldable parts is a common strategy to address these constraints, with plastic model kits being a popular and illustrative example. However, conducting such a decomposition requires considerable expertise, and it depends on the technical aspects of the fabrication technique, as well as aesthetic considerations. We present an interactive technique to create such decompositions for two-piece molding, in which each part of the object is cast between two rigid mold pieces. Given the surface description of an object, we decompose its thin-shell equivalent into moldable parts by first performing a coarse decomposition and then utilizing an active contour model for the boundaries between individual parts. Formulated as an optimization problem, the movement of the contours is guided by an energy reflecting fabrication constraints to ensure the moldability of each part. Simultaneously the user is provided with editing capabilities to enforce aesthetic guidelines. Our interactive interface provides control of the contour positions by allowing, for example, the alignment of part boundaries with object features. Our technique enables a novel workflow, as it empowers novice users to explore the design space, and it generates fabrication-ready two-piece molds that can be used either for casting or industrial injection molding of free-form objects.

@article{Nakashima:2018:10.1145/3197517.3201341,
author = {Nakashima, Kazutaka and Auzinger, Thomas and Iarussi, Emmanuel and Zhang, Ran and Igarashi, Takeo and Bickel, Bernd},
title = {CoreCavity: Interactive Shell Decomposition for Fabrication with Two-Piece Rigid Molds},
journal = {ACM Transactions on Graphics (SIGGRAPH 2018)},
year = {2018},
volume = {37},
number = {4},
pages = {135:1--135:13},
articleno = {135},
numpages = {16},
url = {https://dx.doi.org/10.1145/3197517.3201341},
doi = {10.1145/3197517.3201341},
acmid = {3201341},
publisher = {ACM},
address = {New York, NY, USA},
keywords = {molding, fabrication, height field, decomposition}
}

Additive manufacturing has recently seen drastic improvements in resolution, making it now possible to fabricate features at scales of hundreds or even dozens of nanometers, which previously required very expensive lithographic methods. As a result, additive manufacturing now seems poised for optical applications, including those relevant to computer graphics, such as material design, as well as display and imaging applications. In this work, we explore the use of additive manufacturing for generating structural colors, where the structures are designed using a fabrication-aware optimization process. This requires a combination of full-wave simulation, a feasible parameterization of the design space, and a tailored optimization procedure. Many of these components should be re-usable for the design of other optical structures at this scale. We show initial results of material samples fabricated based on our designs. While these suffer from the prototype character of state-of-the-art fabrication hardware, we believe they clearly demonstrate the potential of additive nanofabrication for structural colors and other graphics applications.

@article{Auzinger:2018:10.1145/3197517.3201376,
author = {Auzinger, Thomas and Heidrich, Wolfgang and Bickel, Bernd},
title = {Computational Design of Nanostructural Color for Additive Manufacturing},
journal = {ACM Transactions on Graphics (SIGGRAPH 2018)},
year = {2018},
volume = {37},
number = {4},
pages = {159:1--159:16},
articleno = {159},
numpages = {16},
url = {http://doi.acm.org/10.1145/3197517.3201376},
doi = {10.1145/3197517.3201376},
acmid = {3201376},
publisher = {ACM},
address = {New York, NY, USA},
keywords = {structural colorization, appearance, multiphoton lithography, direct laser writing, computational fabrication, computational design, shape optimization, FDTD, diffraction, Nanoscribe}
}

We propose a new method for fabricating digital objects through reusable silicone molds. Molds are generated by casting liquid silicone into custom 3D printed containers called metamolds. Metamolds automatically define the cuts that are needed to extract the cast object from the silicone mold. The shape of metamolds is designed through a novel segmentation technique, which takes into account both geometric and topological constraints involved in the process of mold casting. Our technique is simple, does not require changing the shape or topology of the input objects, and only requires offthe-shelf materials and technologies. We successfully tested our method on a set of challenging examples with complex shapes and rich geometric detail.

@article{Alderighi:2018,
 author = {Alderighi, Thomas and Malomo, Luigi and Giorgi, Daniela and Pietroni, Nico and Bickel, Bernd and Cignoni, Paolo},
 title = {Metamolds: Computational Design of Silicone Molds},
 journal = {ACM Trans. Graph.},
 issue_date = {August 2018},
 volume = {37},
 number = {4},
 month = jul,
 year = {2018},
 issn = {0730-0301},
 pages = {136:1--136:13},
 articleno = {136},
 numpages = {13},
 url = {http://doi.acm.org/10.1145/3197517.3201381},
 doi = {10.1145/3197517.3201381},
 acmid = {3201381},
 publisher = {ACM},
 address = {New York, NY, USA},
 keywords = {casting, fabrication, molding},
}

We present a data-driven technique to instantly predict how fluid flows around various three-dimensional objects. Such simulation is useful for computational fabrication and engineering, but is usually computationally expensive since it requires solving the Navier-Stokes equation for many time steps. To accelerate the process, we propose a machine learning framework which predicts aerodynamic forces and velocity and pressure fields given a threedimensional shape input. Handling detailed free-form three-dimensional shapes in a data-driven framework is challenging because machine learning approaches usually require a consistent parametrization of input and output. We present a novel PolyCube maps-based parametrization that can be computed for three-dimensional shapes at interactive rates. This allows us to efficiently learn the nonlinear response of the flow using a Gaussian process regression. We demonstrate the effectiveness of our approach for the interactive design and optimization of a car body.

@article{Umetani:2018,
author = {Umetani, Nobuyuki and Bickel, Bernd},
title = {Learning Three-dimensional Flow for Interactive Aerodynamic Design},
journal = {ACM Transactions on Graphics (SIGGRAPH 2018)},
year = {2018},
volume = {37},
number = {4},
articleno = {89},
numpages = {10},
url = {https://doi.org/10.1145/3197517.3201325},
doi = {10.1145/3197517.3201325},
publisher = {ACM},
address = {New York, NY, USA}
}

The current state of the art in real-time two-dimensional water wave simulation requires developers to choose between efficient Fourier-based methods, which lack interactions with moving obstacles, and finite-difference or finite element methods, which handle environmental interactions but are significantly more expensive. This paper attempts to bridge this long-standing gap between complexity and performance, by proposing a new wave simulation method that can faithfully simulate wave interactions with moving obstacles in real time while simultaneously preserving minute details and accommodating very large simulation domains. Previous methods for simulating 2D water waves directly compute the change in height of the water surface, a strategy which imposes limitations based on the CFL condition (fast moving waves require small time steps) and Nyquist's limit (small wave details require closely-spaced simulation variables). This paper proposes a novel wavelet transformation that discretizes the liquid motion in terms of amplitude-like functions that vary over {\em space, frequency, and direction}, effectively generalizing Fourier-based methods to handle local interactions. Because these new variables change much more slowly over space than the original water height function, our change of variables drastically reduces the limitations of the CFL condition and Nyquist limit, allowing us to simulate highly detailed water waves at very large visual resolutions. Our discretization is amenable to fast summation and easy to parallelize. We also present basic extensions like pre-computed wave paths and two-way solid fluid coupling. Finally, we argue that our discretization provides a convenient set of variables for artistic manipulation, which we illustrate with a novel wave-painting interface.

In the context of robotic manipulation and grasping, the shift from a view that is static (force closure of a single posture) and contact-deprived (only contact for force closure is allowed, everything else is obstacle) towards a view that is dynamic and contact-rich (soft manipulation) has led to an increased interest in soft hands. These hands can easily exploit environmental constraints and object surfaces without risk, and safely interact with humans, but present also some challenges. Designing them is difficult, as well as predicting, modelling, and “programming” their interactions with the objects and the environment. This paper tackles the problem of simulating them in a fast and effective way, leveraging on novel and existing simulation technologies. We present a triple-layered simulation framework where dynamic properties such as stiffness are determined from slow but accurate FEM simulation data once, and then condensed into a lumped parameter model that can be used to fast simulate soft fingers and soft hands. We apply our approach to the simulation of soft pneumatic fingers.

@inproceedings{pozziefficient,
title = {Efficient FEM-Based Simulation of Soft Robots Modeled as Kinematic Chains},
booktitle = {IEEE International Conference on Robotics and Automation 2018},
author = {Pozzi, Maria and Miguel, Eder and Deimel, Raphael and Malvezzi, Monica and Bickel, Bernd and Brock, Oliver and Prattichizzo, Domenico},
year = {2018}
}

Color texture reproduction in 3D printing commonly ignores volumetric light transport (cross-talk) between surface points on a 3D print. Such light diffusion leads to significant blur of details and color bleeding, and is particularly severe for highly translucent resin-based print materials. Given their widely varying scattering properties, this cross-talk between surface points strongly depends on the internal structure of the volume surrounding each surface point. Existing scattering-aware methods use simplified models for light diffusion, and often accept the visual blur as an immutable property of the print medium. In contrast, our work counteracts heterogeneous scattering to obtain the impression of a crisp albedo texture on top of the 3D print, by optimizing for a fully volumetric material distribution that preserves the target appearance. Our method employs an efficient numerical optimizer on top of a general Monte-Carlo simulation of heterogeneous scattering, supported by a practical calibration procedure to obtain scattering parameters from a given set of printer materials. Despite the inherent translucency of the medium, we reproduce detailed surface textures on 3D prints. We evaluate our system using a commercial, five-tone 3D print process and compare against the printer's native color texturing mode, demonstrating that our method preserves high-frequency features well without having to compromise on color gamut.

@article{ElekSumin2017SGA,
author = {Elek, Oskar and Sumin, Denis and Zhang, Ran and Weyrich, Tim and Myszkowski, Karol and Bickel, Bernd and Wilkie, Alexander and K\v{r}iv\'{a}nek, Jaroslav},
title = {Scattering-aware Texture Reproduction for 3{D} Printing},
journal = {ACM Transactions on Graphics (Proceedings of SIGGRAPH Asia)},
volume = {36},
number = {6},
year = {2017},
pages = {241:1--241:15}
}

This paper presents a method for simulating water surface waves as a displacement field on a 2D domain. Our method relies on Lagrangian particles that carry packets of water wave energy; each packet carries information about an entire group of wave trains, as opposed to only a single wave crest. Our approach is unconditionally stable and can simulate high resolution geometric details. This approach also presents a straightforward interface for artistic control, because it is essentially a particle system with intuitive parameters like wavelength and amplitude. Our implementation parallelizes well and runs in real time for moderately challenging scenarios.

@article{Jeschke2017,
author = {Stefan Jeschke and Chris Wojtan},
title = {Water Wave Packets},
journal = {ACM Transactions on Graphics (SIGGRAPH 2017)},
year = {2017},
volume = {36},
number = {4}
}

We present an interactive design system to create functional mechanical objects. Our computational approach allows novice users to retarget an existing mechanical template to a user-specified input shape. Our proposed representation for a mechanical template encodes a parameterized mechanism, mechanical constraints that ensure a physically valid configuration, spatial relationships of mechanical parts to the user-provided shape, and functional constraints that specify an intended functionality. We provide an intuitive interface and optimization-in-the-loop approach for finding a valid configuration of the mechanism and the shape to ensure that higher-level functional goals are met. Our algorithm interactively optimizes the mechanism while the user manipulates the placement of mechanical components and the shape. Our system allows users to efficiently explore various design choices and to synthesize customized mechanical objects that can be fabricated with rapid prototyping technologies. We demonstrate the efficacy of our approach by retargeting various mechanical templates to different shapes and fabricating the resulting functional mechanical objects.

@article{Zhang2017,
author = {Zhang, Ran and Auzinger, Thomas and Ceylan, Duygu and Li, Wilmot and Bickel, Bernd},
title = {Functionality-aware Retargeting of Mechanisms to 3D Shapes},
journal = {ACM Transactions on Graphics (SIGGRAPH 2017)},
year = {2017},
volume = {36},
number = {4}
}

We present a computational approach for designing CurveUps, curvy shells that form from an initially flat state. They consist of small rigid tiles that are tightly held together by two pre-stretched elastic sheets attached to them. Our method allows the realization of smooth, doubly curved surfaces that can be fabricated as a flat piece. Once released, the restoring forces of the pre-stretched sheets support the object to take shape in 3D. CurveUps are structurally stable in their target configuration. The design process starts with a target surface. Our method generates a tile layout in 2D and optimizes the distribution, shape, and attachment areas of the tiles to obtain a configuration that is fabricable and in which the curved up state closely matches the target. Our approach is based on an efficient approximate model and a local optimization strategy for an otherwise intractable nonlinear optimization problem. We demonstrate the effectiveness of our approach for a wide range of shapes, all realized as physical prototypes.

@article{Guseinov2017,
author = {Guseinov, Ruslan and Miguel, Eder and Bickel, Bernd},
title = {CurveUps: Shaping Objects from Flat Plates with Tension-Actuated Curvature},
year = {2017},
volume = {36},
number = {4}
journal = {ACM Trans. Graph.},
month = {7},
articleno = {64},
numpages = {12},
doi = {10.1145/3072959.3073709},
}

We present an interactive system for computational design, optimization, and fabrication of multicopters. Our computational approach allows non-experts to design, explore, and evaluate a wide range of different multicopters. We provide users with an intuitive interface for assembling a multicopter from a collection of components (e.g., propellers, motors, and carbon fiber rods). Our algorithm interactively optimizes shape and controller parameters of the current design to ensure its proper operation. In addition, we allow incorporating a variety of other metrics (such as payload, battery usage, size, and cost) into the design process and exploring tradeoffs between them. We show the efficacy of our method and system by designing, optimizing, fabricating, and operating multicopters with complex geometries and propeller configurations. We also demonstrate the ability of our optimization algorithm to improve the multicopter performance under different metrics.

We present FlexMolds, a novel computational approach to automatically design flexible, reusable molds that, once 3D printed, allow us to physically fabricate, by means of liquid casting, multiple copies of complex shapes with rich surface details and complex topology. The approach to design such flexible molds is based on a greedy bottom-up search of possible cuts over an object, evaluating for each possible cut the feasibility of the resulting mold. We use a dynamic simulation approach to evaluate candidate molds, providing a heuristic to generate forces that are able to open, detach, and remove a complex mold from the object it surrounds. We have tested the approach with a number of objects with nontrivial shapes and topologies.

Computer graphics is an extremely exciting field for two reasons. On the one hand, there is a healthy injection of pragmatism coming from the visual effects industry that want robust algorithms that work so they can produce results at an increasingly frantic pace. On the other hand, they must always try to push the envelope and achieve the impossible to wow their audiences in the next blockbuster, which means that the industry has not succumb to conservatism, and there is *plenty* of room to try out new and *crazy* ideas if there is a chance that it will pan into something useful. Water simulation has been in visual effects for decades, however it still remains extremely challenging because of its high computational cost and difficult art-directability. The work in this thesis tries to address some of these difficulties. Specifically, we make the following three novel contributions to the state-of-the-art in water simulation for visual effects.

@phdthesis{TCaAWSW2016,
author = {Morten Bojsen-Hansen},
title = {Tracking, Correcting and Absorbing Water Surface Waves},
school = {IST Austria},
year = {2016},
month = {9}
}

When aiming to seamlessly integrate a fluid simulation into a larger scenario (like an open ocean), careful attention must be paid to boundary conditions. In particular, one must implement special "non-reflecting" boundary conditions, which dissipate out-going waves as they exit the simulation. Unfortunately, the state of the art in non-reflecting boundary conditions (perfectly-matched layers, or PMLs) only permits trivially simple inflow/outflow conditions, so there is no reliable way to integrate a fluid simulation into a more complicated environment like a stormy ocean or a turbulent river. This paper introduces the first method for combining non-reflecting boundary conditions based on PMLs with inflow/outflow boundary conditions that vary arbitrarily throughout space and time. Our algorithm is a generalization of state-of-the-art mean-flow boundary conditions in the computational fluid dynamics literature, and it allows for seamless integration of a fluid simulation into much more complicated environments. Our method also opens the door for previously-unseen post-process effects like retroactively changing the location of solid obstacles, and locally increasing the visual detail of a pre-existing simulation.

@article{GNRBfFRS2016,
author = {Morten Bojsen-Hansen and Chris Wojtan},
title = {Generalized Non-Reflecting Boundaries for Fluid Re-Simulation},
journal = {ACM Transactions on Graphics (SIGGRAPH 2016)},
year = {2016},
volume = {35},
number = {4},
}

We present a computational method for designing wire sculptures consisting of interlocking wires. Our method allows the computation of aesthetically pleasing structures that are structurally stable, efficiently fabricatable with a 2D wire bending machine, and assemblable without the need of additional connectors. Starting from a set of planar contours provided by the user, our method automatically tests for the feasibility of a design, determines a discrete ordering of wires at intersection points, and optimizes for the rest shape of the individual wires to maximize structural stability under frictional contact. In addition to their application to art, wire sculptures present an extremely efficient and fast alternative for low-fidelity rapid prototyping because manufacturing time and required material linearly scales with the physical size of objects. We demonstrate the effectiveness of our approach on a varied set of examples, all of which we fabricated.

@article{CDoSPRS2016,
author = {Eder Miguel and Mathias Lepoutre and Bernd Bickel},
title = {Computational Design of Stable Planar-Rod Structures},
journal = {ACM Transactions on Graphics (SIGGRAPH 2016)},
year = {2016},
volume = {35},
number = {4}
}

This paper generalizes the well-known Diffusion Curves Images (DCI), which are composed of a set of Bezier curves with colors specified on either side. These colors are diffused as Laplace functions over the image domain, which results in smooth color gradients interrupted by the Bezier curves. Our new formulation allows for more color control away from the boundary, providing a similar expressive power as recent Bilaplace image models without introducing associated issues and computational costs. The new model is based on a special Laplace function blending and a new edge blur formulation. We demonstrate that given some user-defined boundary curves over an input raster image, fitting colors and edge blur from the image to the new model and subsequent editing and animation is equally convenient as with DCIs. Numerous examples and comparisons to DCIs are presented.

@article{GDCI2016,
author = {Stefan Jeschke},
title = {Generalized Diffusion Curves: An Improved Vector Representation for Smooth-Shaded Images},
journal = {Computer Graphics Forum},
year = {2016},
volume = {35},
number = {2},
pages = {1--9}
}

We present a novel optimization-based algorithm for the design and fabrication of customized, deformable input devices, capable of continuously sensing their deformation. We propose to embed piezoresistive sensing elements into flexible 3D printed objects. These sensing elements are then utilized to recover rich and natural user interactions at runtime. Designing such objects manually is a challenging and hard problem for all but the simplest geometries and deformations. Our method simultaneously optimizes the internal routing of the sensing elements and computes a mapping from low-level sensor readings to user-specified outputs in order to minimize reconstruction error. We demonstrate the power and flexibility of the approach by designing and fabricating a set of flexible input devices. Our results indicate that the optimization-based design greatly outperforms manual routings in terms of reconstruction accuracy and thus interaction fidelity.

Creating mechanical automata that can walk in stable and pleasing manners is a challenging task that requires both skill and expertise. We propose to use computational design to offset the technical difficulties of this process. A simple drag-and-drop interface allows casual users to create personalized walking toys from a library of pre-defined template mechanisms. Provided with this input, our method leverages physical simulation and evolutionary optimization to refine the mechanical designs such that the resulting toys are able to walk. The optimization process is guided by an intuitive set of objectives that measure the quality of the walking motions. We demonstrate our approach on a set of simulated mechanical toys with different numbers of legs and various distinct gaits. Two fabricated prototypes showcase the feasibility of our designs.

In recent years we have seen numerous improvements on 3D scanning and tracking of human faces, greatly advancing the creation of digital doubles for film and video games. However, despite the high-resolution quality of the reconstruction approaches available, current methods are unable to capture one of the most important regions of the face – the eye region. In this work we present the first method for detailed spatio-temporal reconstruction of eyelids. Tracking and reconstructing eyelids is extremely challenging, as this region exhibits very complex and unique skin deformation where skin is folded under while opening the eye. Furthermore, eyelids are often only partially visible and obstructed due to selfocclusion and eyelashes. Our approach is to combine a geometric deformation model with image data, leveraging multi-view stereo, optical flow, contour tracking and wrinkle detection from local skin appearance. Our deformation model serves as a prior that enables reconstruction of eyelids even under strong self-occlusions caused by rolling and folding skin as the eye opens and closes. The output is a person-specific, time-varying eyelid reconstruction with anatomically plausible deformations. Our high-resolution detailed eyelids couple naturally with current facial performance capture approaches. As a result, our method can largely increase the fidelity of facial capture and the creation of digital doubles.

This paper introduces OmniAD, a novel data-driven pipeline to model and acquire the aerodynamics of three-dimensional rigid objects. Traditionally, aerodynamics are examined through elaborate wind tunnel experiments or expensive fluid dynamics computations, and are only measured for a small number of discrete wind directions. OmniAD allows the evaluation of aerodynamic forces, such as drag and lift, for any incoming wind direction using a novel representation based on spherical harmonics. Our data-driven technique acquires the aerodynamic properties of an object simply by capturing its falling motion using a single camera. Once model parameters are estimated, OmniAD enables realistic real-time simulation of rigid bodies, such as the tumbling and gliding of leaves, without simulating the surrounding air. In addition, we propose an intuitive user interface based on OmniAD to interactively design three-dimensional kites that actually fly. Various non-traditional kites were designed to demonstrate the physical validity of our model.

We propose a method for fabricating deformable objects with spatially varying elasticity using 3D printing. Using a single, relatively stiff printer material, our method designs an assembly of small-scale microstructures that have the effect of a softer material at the object scale, with properties depending on the microstructure used in each part of the object. We build on work in the area of metamaterials, using numerical optimization to design tiled microstructures with desired properties, but with the key difference that our method designs families of related structures that can be interpolated to smoothly vary the material properties over a wide range. To create an object with spatially varying elastic properties, we tile the object's interior with microstructures drawn from these families, generating a different microstructure for each cell using an efficient algorithm to select compatible structures for neighboring cells. We show results computed for both 2D and 3D objects, validating several 2D and 3D printed structures using standard material tests as well as demonstrating various example applications.

We present a method to learn and propagate shape placements in 2D polygonal scenes from a few examples provided by a user. The placement of a shape is modeled as an oriented bounding box. Simple geometric relationships between this bounding box and nearby scene polygons define a feature set for the placement. The feature sets of all example placements are then used to learn a probabilistic model over all possible placements and scenes. With this model we can generate a new set of placements with similar geometric relationships in any given scene. We introduce extensions that enable propagation and generation of shapes in 3D scenes, as well as the application of a learned modeling session to large scenes without additional user interaction. These concepts allow us to generate complex scenes with thousands of objects with relatively little user interaction.

@article{guerrero-2015-lsp,
title =      "Learning Shape Placements by Example",
author =     "Paul Guerrero and Stefan Jeschke and Michael Wimmer and Peter Wonka",
year =       "2015",
pages =      "1--13",
month =      aug,
event =      "ACM SIGGRAPH 2015",
journal =    "ACM Transactions on Graphics",
location =   "Los Angeles, CA",
keywords =   "complex model generation, modeling by example",
}

We present a computational tool for fabrication-oriented design of flexible rod meshes. Given a deformable surface and a set of deformed poses as input, our method automatically computes a printable rod mesh that, once manufactured, closely matches the input poses under the same boundary conditions. The core of our method is formed by an optimization scheme that adjusts the cross-sectional profiles of the rods and their rest centerline in order to best approximate the target deformations. This approach allows us to locally control the bending and stretching resistance of the surface with a single material, yielding high design flexibility and low fabrication cost.

We present an efficient wavefront tracking algorithm for animating bodies of water that interact with their environment. Our contributions include: a novel wavefront tracking technique that enables dispersion, refraction, reflection, and diffraction in the same simulation; a unique multivalued function interpolation method that enables our simulations to elegantly sidestep the Nyquist limit; a dispersion approximation for efficiently amplifying the number of simulated waves by several orders of magnitude; and additional extensions that allow for time-dependent effects and interactive artistic editing of the resulting animation. Our contributions combine to give us multitudes more wave details than similar algorithms, while maintaining high frame rates and allowing close camera zooms.

@article{WWAvWPI2015,
author = {Stefan Jeschke and Chris Wojtan},
title = {Water Wave Animation via Wavefront Parameter Interpolation},
journal = {ACM Transactions on Graphics},
year = {2015},
volume = {34},
number = {3},
pages = {1--14}
}

Facial appearance capture is now firmly established within academic research and used extensively across various application domains, perhaps most prominently in the entertainment industry through the design of virtual characters in video games and films. While significant progress has occurred over the last two decades, no single survey currently exists that discusses the similarities, differences, and practical considerations of the available appearance capture techniques as applied to human faces. A central difficulty of facial appearance capture is the way light interacts with skin—which has a complex multi-layered structure—and the interactions that occur below the skin surface can, by definition, only be observed indirectly. In this report, we distinguish between two broad strategies for dealing with this complexity. “Image-based methods” try to exhaustively capture the exact face appearance under different lighting and viewing conditions, and then render the face through weighted image combinations. “Parametric methods” instead fit the captured reflectance data to some parametric appearance model used during rendering, allowing for a more lightweight and flexible representation but at the cost of potentially increased rendering complexity or inexact reproduction. The goal of this report is to provide an overview that can guide practitioners and researchers in assessing the tradeoffs between current approaches and identifying directions for future advances in facial appearance capture.

In this paper, we present a method for non-rigid, partial shape matching in vector graphics. Given a user-specified query region in a 2D shape, similar regions are found, even if they are non-linearly distorted. Furthermore, a non-linear mapping is established between the query regions and these matches, which allows the automatic transfer of editing operations such as texturing. This is achieved by a two-step approach. First, point-wise correspondences between the query region and the whole shape are established. The transformation parameters of these correspondences are registered in an appropriate transformation space. For transformations between similar regions, these parameters form surfaces in transformation space, which are extracted in the second step of our method. The extracted regions may be related to the query region by a non-rigid transform, enabling non-rigid shape matching.

@article{Guerrero-2014-TPS,
author = {Paul Guerrero and Thomas Auzinger and Michael Wimmer and Stefan Jeschke},
title = {Partial Shape Matching using Transformation Parameter Similarity},
journal = {Computer Graphics Forum},
year = {2014},
volume = {33},
number = {8},
pages = {1--14}
issn =  {1467-8659}
}

Spinning tops and yo-yos have long fascinated cultures around the world with their unexpected, graceful motions that seemingly elude gravity. We present an algorithm to generate designs for spinning objects by optimizing rotational dynamics properties. As input, the user provides a solid 3D model and a desired axis of rotation. Our approach then modifies the mass distribution such that the principal directions of the moment of inertia align with the target rotation frame. We augment the model by creating voids inside its volume, with interior fill represented by an adaptive multi-resolution voxelization. The discrete voxel fill values are optimized using a continuous, nonlinear formulation. Further, we optimize for rotational stability by maximizing the dominant principal moment. We extend our technique to incorporate deformation and multiple materials for cases where internal voids alone are insufficient. Our method is well-suited for a variety of 3D printed models, ranging from characters to abstract shapes. We demonstrate tops and yo-yos that spin surprisingly stably despite their asymmetric appearance.

We present a generalization of the bilateral filter that can be applied to feature-preserving smoothing of signals on images, meshes, and other domains within a single unified framework. Our discretization is competitive with state-of-the-art smoothing techniques in terms of both accuracy and speed, is easy to implement, and has parameters that are straightforward to understand. Unlike previous bilateral filters developed for meshes and other irregular domains, our construction reduces exactly to the image bilateral on rectangular domains and comes with a rigorous foundation in both the smooth and discrete settings. These guarantees allow us to construct unconditionally convergent mean-shift schemes that handle a variety of extremely noisy signals. We also apply our framework to geometric edge-preserving effects like feature enhancement and show how it is related to local histogram techniques.

We propose a method for propagating edit operations in 2D vector graphics, based on geometric relationship functions. These functions quantify the geometric relationship of a point to a polygon, such as the distance to the boundary or the direction to the closest corner vertex. The level sets of the relationship functions describe points with the same relationship to a polygon. For a given query point we ?rst determine a set of relationships to local features, construct all level sets for these relationships and accumulate them. The maxima of the resulting distribution are points with similar geometric relationships. We show extensions to handle mirror symmetries, and discuss the use of relationship functions as local coordinate systems. Our method can be applied for example to interactive ?oor-plan editing, and is especially useful for large layouts, where individual edits would be cumbersome. We demonstrate populating 2D layouts with tens to hundreds of objects by propagating relatively few edit operations.

@article{Guerrero-2014-GRF,
author = {Paul Guerrero and Stefan Jeschke and Michael Wimmer and Peter Wonka},
title = {Edit Propagation using Geometric Relationship Functions},
journal = {ACM Transactions on Graphics},
year = {2014},
volume = {33},
number = {2},
pages = {15:1--15:15}
}

Our work concerns the combination of an Eulerian liquid simulation with a high-resolution surface tracker (e.g. the level set method or a Lagrangian triangle mesh). The naive application of a high-resolution surface tracker to a low-resolution velocity field can produce many visually disturbing physical and topological artifacts that limit their use in practice. We address these problems by defining an error function which compares the current state of the surface tracker to the set of physically valid surface states. By reducing this error with a gradient descent technique, we introduce a novel physics-based surface fairing method. Similarly, by treating this error function as a potential energy, we derive a new surface correction force that mimics the vortex sheet equations. We demonstrate our results with both level set and mesh-based surface trackers.

@article{LSTwEC2013,
author = {Morten Bojsen-Hansen and Chris Wojtan},
title = {Liquid Surface Tracking with Error Compensation},
journal = {ACM Transactions on Graphics (SIGGRAPH 2013)},
year = {2013},
volume = {32},
number = {4},
pages = {79:1--79:10}
}

We present an interactive design system that allows non-expert users to create animated mechanical characters. Given an articulated character as input, the user iteratively creates an animation by sketching motion curves indicating how different parts of the character should move. For each motion curve, our framework creates an optimized mechanism that reproduces it as closely as possible. The resulting mechanisms are attached to the character and then connected to each other using gear trains, which are created in a semi-automated fashion. The mechanical assemblies generated with our system can be driven with a single input driver, such as a hand-operated crank or an electric motor, and they can be fabricated using rapid prototyping devices. We demonstrate the versatility of our approach by designing a wide range of mechanical characters, several of which we manufactured using 3D printing. While our pipeline is designed for characters driven by planar mechanisms, significant parts of it extend directly to non-planar mechanisms, allowing us to create characters with compelling 3D motions.

We present a method for recovering a temporally coherent, deforming triangle mesh with arbitrarily changing topology from an incoherent sequence of static closed surfaces. We solve this problem using the surface geometry alone, without any prior information like surface templates or velocity fields. Our system combines a proven strategy for triangle mesh improvement, a robust multi-resolution non-rigid registration routine, and a reliable technique for changing surface mesh topology. We also introduce a novel topological constraint enforcement algorithm to ensure that the output and input always have similar topology. We apply our technique to a series of diverse input data from video reconstructions, physics simulations, and artistic morphs. The structured output of our algorithm allows us to efficiently track information like colors and displacement maps, recover velocity information, and solve PDEs on the mesh as a post process.

@article{TSwET2012,
author = {Morten Bojsen-Hansen and Hao Li and Chris Wojtan},
title = {Tracking Surfaces with Evolving Topology},
journal = {ACM Transactions on Graphics (SIGGRAPH 2012)},
year = {2012},
volume = {31},
number = {4},
pages = {53:1--53:10}
}

We have measured 3D face geometry, skin reflectance, and subsurface scattering using custom-built devices for 149 subjects of varying age, gender, and race. We developed a novel skin reflectance model whose parameters can be estimated from measurements. The model decomposes the large amount of measured skin data into a spatially-varying analytic BRDF, a diffuse albedo map, and diffuse subsurface scattering. Our model is intuitive, physically plausible, and – since we do not use the original measured data – easy to edit as well. High-quality renderings come close to reproducing real photographs. The analysis of the model parameters for our sample population reveals variations according to subject age, gender, skin type, and external factors (e.g., sweat, cold, or makeup). Using our statistics, a user can edit the overall appearance of a face (e.g., changing skin type and age) or change small-scale features using texture synthesis (e.g., adding moles and freckles). We are making the collected statistics publicly available to the research community for applications in face synthesis and analysis.