Abstract While studying set function properties of Lebesgue measure, F. Barthe and M. Madiman proved that Lebesgue measure is fractionally superadditive on compact sets in $$\mathbb {R}^n$$ R n . In doing this they proved a fractional generalization of the Brunn–Minkowski–Lyusternik (BML) inequality in dimension $$n=1$$ n = 1 . In this paper we will prove the equality conditions for the fractional superadditive volume inequalites for any dimension. The non-trivial equality conditions are as follows. In the one-dimensional case we will show that for a fractional partition $$(\mathcal {G},\beta )$$ ( G , β ) and nonempty sets $$A_1,\dots ,A_m\subseteq \mathbb {R}$$ A 1 , ⋯ , A m ⊆ R , equality holds iff for each $$S\in \mathcal {G}$$ S ∈ G , the set $$\sum _{i\in S}A_i$$ ∑ i ∈ S A i is an interval. In the case of dimension $$n\ge 2$$ n ≥ 2 we will show that equality can hold if and only if the set $$\sum _{i=1}^{m}A_i$$ ∑ i = 1 m A i has measure 0.

Line work is a core element for the stylization of computer animations used by recent shows. However, existing stylization techniques are limited to edge treatments based on brush strokes or textures applied solely on top of curves. In this work, we propose new stylization effects by offering artists direct control over the inside and outside of surface contours. To this end, we introduce a method that creates ribbons, geometry strips of possibly varying width, that extrude from each side of the surface contour with temporally coherent orientations. Our contributions include the generation of spatially and temporally consistent normal orientations along visible contours and a trimming routine that converts arrangements of offset curves into ribbons free of intersections. We demonstrate the expressiveness and versatility of stylized ribbons by applying various effects on both character and shadow edges from animation sequences.

Linework on 3D animated characters is an important aspect of stylized looks for films. We present CurveCrafter, a system allowing animators to create new lines on 3D models and to edit the shape and opacity of silhouette curves. Our tools allow users to draw, redraw, erase, edit and retime user created curves. Silhouette curves can have their shape edited or reverted, and their opacity erased or revealed. Our algorithm for propagating edits over tracked silhouette curves ensures temporal consistency even as curves expand and merge. Five professional animators used our system to animate lines on three shots with different characters. Additionally, the effects lead from the short film Pete used our system to more easily recreate edits on a film shot. CurveCrafter was able to successfully enhance the resulting animations with additional linework.

Although Monte Carlo path tracing is a simple and effective algorithm to synthesize photo-realistic images, it is often very slow to converge to noise-free results when involving complex global illumination. One of the most successful variance-reduction techniques is path guiding, which can learn better distributions for importance sampling to reduce pixel noise. However, previous methods require a large number of path samples to achieve reliable path guiding. We present a novel neural path guiding approach that can reconstruct high-quality sampling distributions for path guiding from a sparse set of samples, using an offline trained neural network. We leverage photons traced from light sources as the primary input for sampling density reconstruction, which is effective for challenging scenes with strong global illumination. To fully make use of our deep neural network, we partition the scene space into an adaptive hierarchical grid, in which we apply our network to reconstruct high-quality sampling distributions for any local region in the scene. This allows for effective path guiding for arbitrary path bounce at any location in path tracing. We demonstrate that our photon-driven neural path guiding approach can generalize to diverse testing scenes, often achieving better rendering results than previous path guiding approaches and opening up interesting future directions.

The demand for creating rendered content continues to drastically grow. As it often is extremely computationally expensive and thus costly to render high-quality computer-generated images, there is a high incentive to reduce this computational burden. Recent advances in learning-based frame interpolation methods have shown exciting progress but still have not achieved the production-level quality which would be required to render fewer pixels and achieve savings in rendering times and costs. Therefore, in this paper we propose a method specifically targeted to achieve high-quality frame interpolation for rendered content. In this setting, we assume that we have full input for every n -th frame in addition to auxiliary feature buffers that are cheap to evaluate (e.g. depth, normals, albedo) for every frame. We propose solutions for leveraging such auxiliary features to obtain better motion estimates, more accurate occlusion handling, and to correctly reconstruct non-linear motion between keyframes. With this, our method is able to significantly push the state-of-the-art in frame interpolation for rendered content and we are able to obtain production-level quality results.

Path guiding is a promising technique to reduce the variance of path tracing. Although existing online path guiding algorithms can eventually learn good sampling distributions given a large amount of time and samples, the speed of learning becomes a major bottleneck. In this paper, we accelerate the learning of sampling distributions by training a light-weight neural network offline to reconstruct from sparse samples. Uniquely, we design our neural network to directly operate convolutions on a sparse quadtree, which regresses a high-quality hierarchical sampling distribution. Our approach can reconstruct reasonably accurate sampling distributions faster, allowing for efficient path guiding and rendering. In contrast to the recent offline neural path guiding techniques that reconstruct low-resolution 2D images for sampling, our novel hierarchical framework enables more fine-grained directional sampling with less memory usage, effectively advancing the practicality and efficiency of neural path guiding. In addition, we take advantage of hybrid bidirectional samples including both path samples and photons, as we have found this more robust to different light transport scenarios compared to using only one type of sample as in previous work. Experiments on diverse testing scenes demonstrate that our approach often improves rendering results with better visual quality and lower errors. Our framework can also provide the proper balance of speed, memory cost, and robustness.

Abstract Although modern path tracers are successfully being applied to many rendering applications, there is considerable interest to push them towards ever‐decreasing sampling rates. As the sampling rate is substantially reduced, however, even Monte Carlo (MC) denoisers–which have been very successful at removing large amounts of noise–typically do not produce acceptable final results. As an orthogonal approach to this, we believe that good importance sampling of paths is critical for producing better‐converged, path‐traced images at low sample counts that can then, for example, be more effectively denoised. However, most recent importance‐sampling techniques for guiding path tracing (an area known as “path guiding”) involve expensive online (per‐scene) training and offer benefits only at high sample counts. In this paper, we propose an offline, scene‐independent deep‐learning approach that can importance sample first‐bounce light paths for general scenes without the need of the costly online training, and can start guiding path sampling with as little as 1 sample per pixel. Instead of learning to “overfit” to the sampling distribution of a specific scene like most previous work, our data‐driven approach is trained a priori on a set of training scenes on how to use a local neighborhood of samples with additional feature information to reconstruct the full incident radiance at a point in the scene, which enables first‐bounce importance sampling for new test scenes. Our solution is easy to integrate into existing rendering pipelines without the need for retraining, as we demonstrate by incorporating it into both the Blender/Cycles and Mitsuba path tracers. Finally, we show how our offline, deep importance sampler (ODIS) increases convergence at low sample counts and improves the results of an off‐the‐shelf denoiser relative to other state‐of‐the‐art sampling techniques.

Many applications in rendering rely on integrating functions over spherical polygons. We present a new numerical solution for computing the integral of spherical harmonics (SH) expansions clipped to polygonal domains. Our solution, based on zonal decompositions of spherical integrands and discrete contour integration, introduces an important numerical operating for SH expansions in rendering applications. Our method is simple, efficient, and scales linearly in the bandlimited integrand’s harmonic expansion. We apply our technique to problems in rendering, including surface and volume shading, hierarchical product importance sampling, and fast basis projection for interactive rendering. Moreover, we show how to handle general, nonpolynomial integrands in a Monte Carlo setting using control variates. Our technique computes the integral of bandlimited spherical functions with performance competitive to (or faster than) more general numerical integration methods for a broad class of problems, both in offline and interactive rendering contexts. Our implementation is simple, relying only on self-contained SH evaluation and discrete contour integration routines, and we release a full source CPU-only and shader-based implementations (<750 lines of commented code).

We present a modular convolutional architecture for denoising rendered images. We expand on the capabilities of kernel-predicting networks by combining them with a number of task-specific modules, and optimizing the assembly using an asymmetric loss. The source-aware encoder---the first module in the assembly---extracts low-level features and embeds them into a common feature space, enabling quick adaptation of a trained network to novel data. The spatial and temporal modules extract abstract, high-level features for kernel-based reconstruction, which is performed at three different spatial scales to reduce low-frequency artifacts. The complete network is trained using a class of asymmetric loss functions that are designed to preserve details and provide the user with a direct control over the variance-bias trade-off during inference. We also propose an error-predicting module for inferring reconstruction error maps that can be used to drive adaptive sampling. Finally, we present a theoretical analysis of convergence rates of kernel-predicting architectures, shedding light on why kernel prediction performs better than synthesizing the colors directly, complementing the empirical evidence presented in this and previous works. We demonstrate that our networks attain results that compare favorably to state-of-the-art methods in terms of detail preservation, low-frequency noise removal, and temporal stability on a variety of production and academic datasets.

Regression-based algorithms have shown to be good at denoising Monte Carlo (MC) renderings by leveraging its inexpensive by-products (e.g., feature buffers). However, when using higher-order models to handle complex cases, these techniques often overfit to noise in the input. For this reason, supervised learning methods have been proposed that train on a large collection of reference examples, but they use explicit filters that limit their denoising ability. To address these problems, we propose a novel, supervised learning approach that allows the filtering kernel to be more complex and general by leveraging a deep convolutional neural network (CNN) architecture. In one embodiment of our framework, the CNN directly predicts the final denoised pixel value as a highly non-linear combination of the input features. In a second approach, we introduce a novel, kernel-prediction network which uses the CNN to estimate the local weighting kernels used to compute each denoised pixel from its neighbors. We train and evaluate our networks on production data and observe improvements over state-of-the-art MC denoisers, showing that our methods generalize well to a variety of scenes. We conclude by analyzing various components of our architecture and identify areas of further research in deep learning for MC denoising.

This paper introduces a new computational method to solve differential equations on subdivision surfaces. Our approach adapts the numerical framework of Discrete Exterior Calculus (DEC) from the polygonal to the subdivision setting by exploiting the refin-ability of subdivision basis functions. The resulting Subdivision Exterior Calculus (SEC) provides significant improvements in accuracy compared to existing polygonal techniques, while offering exact finite-dimensional analogs of continuum structural identities such as Stokes' theorem and Helmholtz-Hodge decomposition. We demonstrate the versatility and efficiency of SEC on common geometry processing tasks including parameterization, geodesic distance computation, and vector field design.

Feature-adaptive subdivision (FAS) is one of the state-of-the art real-time rendering methods for subdivision surfaces on modern GPUs. It enables efficient and accurate rendering of subdivision surfaces in many interactive applications, such as video games or authoring tools. In this paper, we present dynamic feature-adaptive subdivision (DFAS), which improves upon FAS by enabling an independent subdivision depth for every irregularity. Our subdivision kernels fill a dynamic patch buffer on-the-fly with the appropriate number of patches corresponding to the chosen level-of-detail scheme. By reducing the number of generated and processed patches, DFAS significantly improves upon the performance of static FAS.

Subspace deformable body simulations can be very fast, but can behave unrealistically when behaviors outside the prescribed subspace such as novel external collisions, are encountered. We address this limitation by presenting a fast, flexible new method that allows full space computation to be activated in the neighborhood of novel events while the rest of the body still computes in a subspace. We achieve this using a method we call subspace condensation , a variant on the classic static condensation precomputation. However, instead of a precomputation, we use the speed of subspace methods to perform the condensation at every frame. This approach allows the full space regions to be specified arbitrarily at runtime, and forms a natural two-way coupling with the subspace regions. While condensation is usually only applicable to linear materials, the speed of our technique enables its application to non-linear materials as well. We show the effectiveness of our approach by applying it to a variety of articulated character scenarios.

We present a new approach to clothing simulation using low-dimensional linear subspaces with temporally adaptive bases. Our method exploits full-space simulation training data in order to construct a pool of low-dimensional bases distributed across pose space. For this purpose, we interpret the simulation data as offsets from a kinematic deformation model that captures the global shape of clothing due to body pose. During subspace simulation, we select low-dimensional sets of basis vectors according to the current pose of the character and the state of its clothing. Thanks to this adaptive basis selection scheme, our method is able to reproduce diverse and detailed folding patterns with only a few basis vectors. Our experiments demonstrate the feasibility of subspace clothing simulation and indicate its potential in terms of quality and computational efficiency.

Abstract Environment‐mapped rendering of Lambertian isotropic surfaces is common, and a popular technique is to use a quadratic spherical harmonic expansion. This compact irradiance map representation is widely adopted in interactive applications like video games. However, many materials are anisotropic, and shading is determined by the local tangent direction, rather than the surface normal. Even for visualization and illustration, it is increasingly common to define a tangent vector field, and use anisotropic shading. In this paper, we extend spherical harmonic irradiance maps to anisotropic surfaces, replacing Lambertian reflectance with the diffuse term of the popular Kajiya‐Kay model. We show that there is a direct analogy, with the surface normal replaced by the tangent. Our main contribution is an analytic formula for the diffuse Kajiya‐Kay BRDF in terms of spherical harmonics; this derivation is more complicated than for the standard diffuse lobe. We show that the terms decay even more rapidly than for Lambertian reflectance, going as l–3, where l is the spherical harmonic order, and with only 6 terms (l = 0 and l = 2) capturing 99.8% of the energy. Existing code for irradiance environment maps can be trivially adapted for real‐time rendering with tangent irradiance maps. We also demonstrate an application to offline rendering of the diffuse component of fibers, using our formula as a control variate for Monte Carlo sampling.

We present a novel method for high-performance GPU-based rendering of Catmull-Clark subdivision surfaces. Unlike previous methods, our algorithm computes the true limit surface up to machine precision, and is capable of rendering surfaces that conform to the full RenderMan specification for Catmull-Clark surfaces. Specifically, our algorithm can accommodate base meshes consisting of arbitrary valence vertices and faces, and the surface can contain any number and arrangement of semisharp creases and hierarchically defined detail. We also present a variant of the algorithm which guarantees watertight positions and normals, meaning that even displaced surfaces can be rendered in a crack-free manner. Finally, we describe a view-dependent level-of-detail scheme which adapts to both the depth of subdivision and the patch tessellation density. Though considerably more general, the performance of our algorithm is comparable to the best approximating method, and is considerably faster than Stam's exact method.

Soft shadows from area lights are one of the most crucial effects in high-quality and production rendering, but Monte-Carlo sampling of visibility is often the main source of noise in rendered images. Indeed, it is common to use deterministic uniform sampling for the smoother shading effects in direct lighting, so that all of the Monte Carlo noise arises from visibility sampling alone. In this article, we analyze theoretically and empirically, using both statistical and Fourier methods, the effectiveness of different nonadaptive Monte Carlo sampling patterns for rendering soft shadows. We start with a single image scanline and a linear light source, and gradually consider more complex visibility functions at a pixel. We show analytically that the lowest expected variance is in fact achieved by uniform sampling (albeit at the cost of visual banding artifacts). Surprisingly, we show that for two or more discontinuities in the visibility function, a comparable error to uniform sampling is obtained by “uniform jitter” sampling, where a constant jitter is applied to all samples in a uniform pattern (as opposed to jittering each stratum as in standard stratified sampling). The variance can be reduced by up to a factor of two, compared to stratified or quasi-Monte Carlo techniques, without the banding in uniform sampling. We augment our statistical analysis with a novel 2D Fourier analysis across the pixel-light space. This allows us to characterize the banding frequencies in uniform sampling, and gives insights into the behavior of uniform jitter and stratified sampling. We next extend these results to planar area light sources. We show that the best sampling method can vary, depending on the type of light source (circular, Gaussian, or square/rectangular). The correlation of adjacent “light scanlines” in square light sources can reduce the effectiveness of uniform jitter sampling, while the smoother shape of circular and Gaussian-modulated sources preserves its benefits—these findings are also exposed through our frequency analysis. In practical terms, the theory in this article provides guidelines for selecting visibility sampling strategies, which can reduce the number of shadow samples by 20--40%, with simple modifications to existing rendering code.

In this paper we consider the problem of creating and controlling volume deformations used to articulate characters for use in high-end applications such as computer generated feature films. We introduce a method we call harmonic coordinates that significantly improves upon existing volume deformation techniques. Our deformations are controlled using a topologically flexible structure, called a cage, that consists of a closed three dimensional mesh. The cage can optionally be augmented with additional interior vertices, edges, and faces to more precisely control the interior behavior of the deformation. We show that harmonic coordinates are generalized barycentric coordinates that can be extended to any dimension. Moreover, they are the first system of generalized barycentric coordinates that are non-negative even in strongly concave situations, and their magnitude falls off with distance as measured within the cage.

Many applications in Computer Graphics contain computationally expensive calculations. These calculations are often performed at many points to produce a full solution, even though the subspace of reasonable solutions may be of a relatively low dimension. The calculation of facial articulation and rendering of scenes with global illumination are two example applications that require these sort of computations. In this paper, we present Key Point Subspace Acceleration and Soft Caching, a technique for accelerating these types of computations. Key Point Subspace Acceleration (KPSA) is a statistical acceleration scheme that uses examples to compute a statistical subspace and a set of characteristic key points. The full calculation is then computed only at these key points and these points are used to provide a subspace based estimate of the entire calculation. The soft caching process is an extension to the KPSA technique where the key points are also used to provide a confidence estimate for the KPSA result. In cases with high anticipated error the calculation will then "fail through" to a full evaluation of all points (a cache miss), while frames with low error can use the accelerated statistical evaluation (a cache hit).

Global illumination provides important visual cues to an animation, however its computational expense limits its use in practice. In this paper, we present an easy to implement technique for accelerating the computation of indirect illumination for an animated sequence using stochastic ray tracing. We begin by computing a quick but noisy solution using a small number of sample rays at each sample location. The variation of these noisy solutions over time is then used to create a smooth basis. Finally, the noisy solutions are projected onto the smooth basis to produce the final solution. The resulting animation has greatly reduced spatial and temporal noise, and a computational cost roughly equivalent to the noisy, low sample computation.

Parameterization of discrete surfaces is a fundamental and widely-used operation in graphics, required, for instance, for texture mapping or remeshing. As 3D data becomes more and more detailed, there is an increased need for fast and robust techniques to automatically compute least-distorted parameterizations of large meshes. In this paper, we present new theoretical and practical results on the parameterization of triangulated surface patches. Given a few desirable properties such as rotation and translation invariance, we show that the only admissible parameterizations form a two-dimensional set and each parameterization in this set can be computed using a simple, sparse, linear system. Since these parameterizations minimize the distortion of different intrinsic measures of the original mesh, we call them Intrinsic Parameterizations. In addition to this partial theoretical analysis, we propose robust, efficient and tunable tools to obtain least-distorted parameterizations automatically. In particular, we give details on a novel, fast technique to provide an optimal mapping without fixing the boundary positions, thus providing a unique Natural Intrinsic Parameterization. Other techniques based on this parameterization family, designed to ease the rapid design of parameterizations, are also proposed.

We present a novel technique, both flexible and efficient, for interactive remeshing of irregular geometry. First, the original (arbitrary genus) mesh is substituted by a series of 2D maps in parameter space. Using these maps, our algorithm is then able to take advantage of established signal processing and halftoning tools that offer real-time interaction and intricate control. The user can easily combine these maps to create a control map --- a map which controls the sampling density over the surface patch. This map is then sampled at interactive rates allowing the user to easily design a tailored resampling. Once this sampling is complete, a Delaunay triangulation and fast optimization are performed to perfect the final mesh.As a result, our remeshing technique is extremely versatile and general, being able to produce arbitrarily complex meshes with a variety of properties including: uniformity, regularity, semi-regularity, curvature sensitive resampling, and feature preservation. We provide a high level of control over the sampling distribution allowing the user to interactively custom design the mesh based on their requirements thereby increasing their productivity in creating a wide variety of meshes.

Abstract Modeling and animation of cloth have experienced important developments in recent years. As a consequence, complex textile models can be used to realistically drape objects or human characters in a fairly efficient way. However, real‐time realistic simulation remains a major challenge, even if applications are numerous, from rapid prototyping to e‐commerce. In this paper, we present a stable, real‐time algorithm for animating cloth‐like materials. Using a hybrid explicit/implicit algorithm, we perform fast and stable time integration of a physically based model with rapid collision detection and response, as well as wind or liquid drag effects to enhance realism. We demonstrate our approach through a series of examples in virtual reality environments, proving that real‐time animation of cloth, even on low‐end computers, is now achievable. Copyright © 2001 John Wiley & Sons, Ltd.

In this paper, we develop methods to rapidly remove rough features from irregularly triangulated data intended to portray a smooth surface.The main task is to remove undesirable noise and uneven edges while retaining desirable geometric features.The problem arises mainly when creating high-fidelity computer graphics objects using imperfectly-measured data from the real world.Our approach contains three novel features: an implicit integration method to achieve efficiency, stability, and large time-steps; a scale-dependent Laplacian operator to improve the diffusion process; and finally, a robust curvature flow operator that achieves a smoothing of the shape itself, distinct from any parameterization.Additional features of the algorithm include automatic exact volume preservation, and hard and soft constraints on the positions of the points in the mesh.We compare our method to previous operators and related algorithms, and prove that our curvature and Laplacian operators have several mathematically-desirable qualities that improve the appearance of the resulting surface.In consequence, the user can easily select the appropriate operator according to the desired type of fairing.Finally, we provide a series of examples to graphically and numerically demonstrate the quality of our results.

We present a new interaction metaphor for object-centric tasks in the form of a prototype VR system, ALCOVE. Through analytic calculations, we quantitatively demonstrate the benefits of restructuring the interaction volume offered by current systems. Our metrics show that many applications' interaction volume increases by 1.5 to 2.6 times when using the ALCOVE system. We also offer an informal user task analysis and evaluations of previous VR systems that qualitatively support this improved interaction volume as well as demonstrate the need for a shift from room and desk-sized systems to desktop units. We present some testbed applications which benefit from this object-centric design and discuss some of the advantages and shortcomings of our system.