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.

Abstract We propose a hierarchical graph for learning physics and a novel way to handle obstacles. The finest level of the graph consist of the particles itself. Coarser levels consist of the cells of sparse grids with successively doubling cell sizes covering the volume occupied by the particles. The hierarchical structure allows for the information to propagate at great distance in a single message passing iteration. The novel obstacle handling allows the simulation to be obstacle aware without the need for ghost particles. We train the network to predict effective acceleration produced by multiple sub‐steps of 3D multi‐material material point method (MPM) simulation consisting of water, sand and snow with complex obstacles. Our network produces lower error, trains up to 7.0X faster and inferences up to 11.3X faster than [SGGP*20]. It is also, on average, about 3.7X faster compared to Taichi Elements simulation running on the same hardware in our tests.

Abstract The shape matching method is a popular approach to simulate deformable objects in interactive applications due to its stability and simplicity. An important feature is that there is no need for a mesh since the method works on arbitrary local groups within a set of particles. A major drawback of shape matching is the fact that it is geometrically motivated and not derived from physical principles which makes calibration difficult. The fact that the method does not conserve volume can yield visual artifacts, e.g. when a tire is compressed but does not bulge. In this paper we present a new meshless simulation method that is related to shape matching but derived from continuous constitutive models. Volume conservation and stiffness can be specified with physical parameters. Further, if the elements of a tetrahedral mesh are used as groups, our method perfectly reproduces FEM based simulations.

Abstract We introduce a triangle mesh based convolutional neural network. The proposed network structure can be used for problems where input and/or output are defined on a manifold triangle mesh with or without boundary. We demonstrate its applications in cloth upsampling, adding back details to Principal Component Analysis (PCA) compressed cloth, regressing clothing deformation from character poses, and regressing hand skin deformation from bones' joint angles. The data used for training in this work are generated from high resolution extended position based dynamics (XPBD) physics simulations with small time steps and high iteration counts and from an offline FEM simulator, but it can come from other sources. The inference time of our prototype implementation, depending on the mesh resolution and the network size, can provide between 4 to 134 times faster than a GPU based simulator. The inference also only needs to be done for meshes currently visible by the camera.

Abstract 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.

Abstract We examine the relationship between primal, or force‐based, and dual, or constraint‐based formulations of dynamics. Variational frameworks such as Projective Dynamics have proved popular for deformable simulation, however they have not been adopted for contact‐rich scenarios such as rigid body simulation. We propose a new preconditioned frictional contact solver that is compatible with existing primal optimization methods, and competitive with complementarity‐based approaches. Our relaxed primal model generates improved contact force distributions when compared to dual methods, and has the advantage of being differentiable, making it well‐suited for trajectory optimization. We derive both primal and dual methods from a common variational point of view, and present a comprehensive numerical analysis of both methods with respect to conditioning. We demonstrate our method on scenarios including rigid body contact, deformable simulation, and robotic manipulation.

Abstract We present a rigid body simulation method that can resolve small temporal and spatial details by using a quasi explicit integration scheme that is unconditionally stable. Traditional rigid body simulators linearize constraints because they operate on the velocity level or solve the equations of motion implicitly thereby freezing the constraint directions for multiple iterations. Our method always works with the most recent constraint directions. This allows us to trace high speed motion of objects colliding against curved geometry, to reduce the number of constraints, to increase the robustness of the simulation, and to simplify the formulation of the solver. In this paper we provide all the details to implement a fully fledged rigid body solver that handles contacts, a variety of joint types and the interaction with soft objects.

We present a framework for the simulation of rigid and deformable bodies in the presence of contact and friction. Our method is based on a non-smooth Newton iteration that solves the underlying nonlinear complementarity problems (NCPs) directly. This approach allows us to support nonlinear dynamics models, including hyperelastic deformable bodies and articulated rigid mechanisms, coupled through a smooth isotropic friction model. The fixed-point nature of our method means it requires only the solution of a symmetric linear system as a building block. We propose a new complementarity preconditioner for NCP functions that improves convergence, and we develop an efficient GPU-based solver based on the conjugate residual (CR) method that is suitable for interactive simulations. We show how to improve robustness using a new geometric stiffness approximation and evaluate our method’s performance on a number of robotics simulation scenarios, including dexterous manipulation and training using reinforcement learning.

We introduce a deep reinforcement learning method that learns to control articulated humanoid bodies to imitate given target motions closely when simulated in a physics simulator. The target motion, which may not have been seen by the agent and can be noisy, is supplied at runtime. Our method can recover balance from moderate external disturbances and keep imitating the target motion. When subjected to large disturbances that cause the humanoid to fall down, our method can control the character to get up and recover to track the motion. Our method is trained to imitate the mocap clips from the CMU motion capture database and a number of other publicly available databases. We use a state-of-the-art deep reinforcement learning algorithm to learn to dynamically control the gain of PD controllers, whose target angles are derived from the mocap clip and to apply corrective torques with the goal of imitating the provided motion clip as closely as possible. Both the simulation and the learning algorithms are parallelized and run on the GPU. We demonstrate that the proposed method can control the character to imitate a wide variety of motions such as running, walking, dancing, jumping, kicking, punching, standing up, and so on.

Abstract Robustly and efficiently simulating cables and ropes that are part of a larger system such as cable driven machines, cable cars or tendons in a human or robot is a challenging task. To be able to adapt to the environment, cables are typically modeled as a large number of small segments that are connected via joints. The two main difficulties with this approach are to satisfy the inextensibility constraint and to handle the typically large mass ratio between the small segments and the larger objects they connect. In this paper we present a new approach which solves these problems in a simple and effective way. Our method is based on the idea to simulate the effect of the cables instead of the cables themselves. To this end we propose a new special type of distance constraint we call cable joint that changes both its attachment points and its rest length dynamically. A cable connecting a series of objects is then modeled as a sequence of cable joints which reduces the complexity of the simulation from the order of the number of segments to just the number of connected objects. This makes simulations both faster and more robust as we will demonstrate on a variety of examples.

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 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.

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.

Abstract 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.

Abstract 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, pointwise 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.

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.

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.

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 first 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 floorplan editing, and it 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.

In this paper, we introduce a novel scene representation for the visualization of large-scale point clouds accompanied by a set of high-resolution photographs. Many real-world applications deal with very densely sampled point-cloud data, which are augmented with photographs that often reveal lighting variations and inaccuracies in registration. Consequently, the high-quality representation of the captured data, i.e., both point clouds and photographs together, is a challenging and time-consuming task. We propose a two-phase approach, in which the first (preprocessing) phase generates multiple overlapping surface patches and handles the problem of seamless texture generation locally for each patch. The second phase stitches these patches at render-time to produce a high-quality visualization of the data. As a result of the proposed localization of the global texturing problem, our algorithm is more than an order of magnitude faster than equivalent mesh-based texturing techniques. Furthermore, since our preprocessing phase requires only a minor fraction of the whole data set at once, we provide maximum flexibility when dealing with growing data sets.

Abstract This paper presents a parallel, implementation‐friendly analytic visibility method for triangular meshes. Together with an analytic filter convolution, it allows for a fully analytic solution to anti‐aliased 3D mesh rendering on parallel hardware. Building on recent works in computational geometry, we present a new edge‐triangle intersection algorithm and a novel method to complete the boundaries of all visible triangle regions after a hidden line elimination step. All stages of the method are embarrassingly parallel and easily implementable on parallel hardware. A GPU implementation is discussed and performance characteristics of the method are shown and compared to traditional sampling‐based rendering methods.

Abstract This paper presents an analytic formulation for anti‐aliased sampling of 2D polygons and 3D polyhedra. Our framework allows the exact evaluation of the convolution integral with a linear function defined on the polytopes. The filter is a spherically symmetric polynomial of any order, supporting approximations to refined variants such as the Mitchell‐Netravali filter family. This enables high‐quality rasterization of triangles and tetrahedra with linearly interpolated vertex values to regular and non‐regular grids. A closed form solution of the convolution is presented and an efficient implementation on the GPU using DirectX and CUDA C is described.

Abstract Diffusion curves are a powerful vector graphic representation that stores an image as a set of 2D Bezier curves with colors defined on either side. These colors are diffused over the image plane, resulting in smooth color regions as well as sharp boundaries. In this paper, we introduce a new automatic diffusion curve coloring algorithm. We start by defining a geometric heuristic for the maximum density of color control points along the image curves. Following this, we present a new algorithm to set the colors of these points so that the resulting diffused image is as close as possible to a source image in a least squares sense. We compare our coloring solution to the existing one which fails for textured regions, small features, and inaccurately placed curves. The second contribution of the paper is to extend the diffusion curve representation to include texture details based on Gabor noise. Like the curves themselves, the defined texture is resolution independent, and represented compactly. We define methods to automatically make an initial guess for the noise texure, and we provide intuitive manual controls to edit the parameters of the Gabor noise. Finally, we show that the diffusion curve representation itself extends to storing any number of attributes in an image, and we demonstrate this functionality with image stippling an hatching applications.

We present a method designed to address some limitations of typical route map displays of driving directions. The main goal of our system is to generate a printable version of a route map that shows the overview and detail views of the route within a single, consistent visual frame. Our proposed visualization provides a more intuitive spatial context than a simple list of turns. We present a novel multifocus technique to achieve this goal, where the foci are defined by points of interest (POI) along the route. A detail lens that encapsulates the POI at a finer geospatial scale is created for each focus. The lenses are laid out on the map to avoid occlusion with the route and each other, and to optimally utilize the free space around the route. We define a set of layout metrics to evaluate the quality of a lens layout for a given route map visualization. We compare standard lens layout methods to our proposed method and demonstrate the effectiveness of our method in generating aesthetically pleasing layouts. Finally, we perform a user study to evaluate the effectiveness of our layout choices.

Abstract This paper proposes to use relief‐mapped conical frusta (cones cut by planes) to skin skeletal objects. Based on this representation, current programmable graphics hardware can perform the rendering with only minimal communication between the CPU and GPU. A consistent definition of conical frusta including texture parametrization and a continuous surface normal is provided. Rendering is performed by analytical ray casting of the relief‐mapped frusta directly on the GPU. We demonstrate both static and animated objects rendered using our technique and compare to polygonal renderings of similar quality.

Abstract In this paper we present dart throwing algorithms to generate maximal Poisson disk point sets directly on 3D surfaces. We optimize dart throwing by efficiently excluding areas of the domain that are already covered by existing darts. In the case of triangle meshes, our algorithm shows dramatic speed improvement over comparable sampling methods. The simplicity of our basic algorithm naturally extends to the sampling of other surface types, including spheres, NURBS, subdivision surfaces, and implicits. We further extend the method to handle variable density points, and the placement of arbitrary ellipsoids without overlap. Finally, we demonstrate how to adapt our algorithm to work with geodesic instead of Euclidean distance. Applications for our method include fur modeling, the placement of mosaic tiles and polygon remeshing.

Abstract In this paper we address the question of how to quickly model glyph‐based Geographic Information System visualizations. Our solution is based on using shape grammars to set up the different aspects of a visualization, including the geometric content of the visualization, methods for resolving layout conflicts and interaction methods. Our approach significantly increases modelling efficiency over similarly flexible systems currently in use.

We present a new Laplacian solver for minimal surfaces---surfaces having a mean curvature of zero everywhere except at some fixed (Dirichlet) boundary conditions. Our solution has two main contributions: First, we provide a robust rasterization technique to transform continuous boundary values (diffusion curves) to a discrete domain. Second, we define a variable stencil size diffusion solver that solves the minimal surface problem. We prove that the solver converges to the right solution, and demonstrate that it is at least as fast as commonly proposed multigrid solvers, but much simpler to implement. It also works for arbitrary image resolutions, as well as 8 bit data. We show examples of robust diffusion curve rendering where our curve rasterization and diffusion solver eliminate the strobing artifacts present in previous methods. We also show results for real-time seamless cloning and stitching of large image panoramas.

Diffusion curve images (DCI) provide a powerful tool for efficient 2D image generation, storage and manipulation. A DCI consist of curves with colors defined on either side. By diffusing these colors over the image, the final result includes sharp boundaries along the curves with smoothly shaded regions between them. This paper extends the application of diffusion curves to render high quality surface details on 3D objects. The first extension is a view dependent warping technique that dynamically reallocates texture space so that object parts that appear large on screen get more texture for increased detail. The second extension is a dynamic feature embedding technique that retains crisp, anti-aliased curve details even in extreme closeups. The third extension is the application of dynamic feature embedding to displacement mapping and geometry images. Our results show high quality renderings of diffusion curve textures, displacements, and geometry images, all rendered interactively.

Abstract We present a method for rendering approximate soft shadows and diffuse indirect illumination in dynamic scenes. The proposed method approximates the original scene geometry with a set of tightly fitting spheres. In previous work, such spheres have been used to dynamically evaluate the visibility function to render soft shadows. In this paper, each sphere also acts as a low‐frequency secondary light source, thereby providing diffuse one‐bounce indirect illumination. The method is completely dynamic and proceeds in two passes: In a first pass, the light intensity distribution on each sphere is updated based on sample points on the corresponding object surface and converted into the spherical harmonics basis. In a second pass, this radiance information and the visibility are accumulated to shade final image pixels. The sphere approximation allows us to compute visibility and diffuse reflections of an object at interactive frame rates of over 20 fps for moderately complex scenes.

Impostors are image-based primitives commonly used to replace complex geometry in order to reduce the rendering time needed for displaying complex scenes. However, a big problem is the huge amount of memory required for impostors. This paper presents an algorithm that automatically places impostors into a scene so that a desired frame rate and image quality is always met, while at the same time not requiring enormous amounts of impostor memory. The low memory requirements are provided by a new placement method and through the simultaneous use of other acceleration techniques like visibility culling and geometric levels of detail.