Volume 40 (2021)

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https://diglib.eg.org/handle/10.2312/2633014

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Fast Updates for Least-Squares Rotational Alignment
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Zhang, Jiayi Eris; Jacobson, Alec; Alexa, Marc; Mitra, Niloy and Viola, Ivan
Across computer graphics, vision, robotics and simulation, many applications rely on determining the 3D rotation that aligns two objects or sets of points. The standard solution is to use singular value decomposition (SVD), where the optimal rotation is recovered as the product of the singular vectors. Faster computation of only the rotation is possible using suitable parameterizations of the rotations and iterative optimization. We propose such a method based on the Cayley transformations. The resulting optimization problem allows better local quadratic approximation compared to the Taylor approximation of the exponential map. This results in both faster convergence as well as more stable approximation compared to other iterative approaches. It also maps well to AVX vectorization. We compare our implementation with a wide range of alternatives on real and synthetic data. The results demonstrate up to two orders of magnitude of speedup compared to a straightforward SVD implementation and a 1.5-6 times speedup over popular optimized code.
Quad Layouts via Constrained T-Mesh Quantization
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Lyon, Max; Campen, Marcel; Kobbelt, Leif; Mitra, Niloy and Viola, Ivan
We present a robust and fast method for the creation of conforming quad layouts on surfaces. Our algorithm is based on the quantization of a T-mesh, i.e. an assignment of integer lengths to the sides of a non-conforming rectangular partition of the surface. This representation has the benefit of being able to encode an infinite number of layout connectivity options in a finite manner, which guarantees that a valid layout can always be found. We carefully construct the T-mesh from a given seamless parametrization such that the algorithm can provide guarantees on the results' quality. In particular, the user can specify a bound on the angular deviation of layout edges from prescribed directions. We solve an integer linear program (ILP) to find a coarse quad layout adhering to that maximal deviation. Our algorithm is guaranteed to yield a conforming quad layout free of T-junctions together with bounded angle distortion. Our results show that the presented method is fast, reliable, and achieves high quality layouts.
Stable and Efficient Differential Estimators on Oriented Point Clouds
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Lejemble, Thibault; Coeurjolly, David; Barthe, Loïc; Mellado, Nicolas; Digne, Julie and Crane, Keenan
Point clouds are now ubiquitous in computer graphics and computer vision. Differential properties of the point-sampled surface, such as principal curvatures, are important to estimate in order to locally characterize the scanned shape. To approximate the surface from unstructured points equipped with normal vectors, we rely on the Algebraic Point Set Surfaces (APSS) [GG07] for which we provide convergence and stability proofs for the mean curvature estimator. Using an integral invariant viewpoint, this first contribution links the algebraic sphere regression involved in the APSS algorithm to several surface derivatives of different orders. As a second contribution, we propose an analytic method to compute the shape operator and its principal curvatures from the fitted algebraic sphere. We compare our method to the state-of-the-art with several convergence and robustness tests performed on a synthetic sampled surface. Experiments show that our curvature estimations are more accurate and stable while being faster to compute compared to previous methods. Our differential estimators are easy to implement with little memory footprint and only require a unique range neighbors query per estimation. Its highly parallelizable nature makes it appropriate for processing large acquired data, as we show in several real-world experiments.
From Noon to Sunset: Interactive Rendering, Relighting, and Recolouring of Landscape Photographs by Modifying Solar Position
(© 2021 Eurographics ‐ The European Association for Computer Graphics and John Wiley & Sons Ltd, 2021) Türe, Murat; Çıklabakkal, Mustafa Ege; Erdem, Aykut; Erdem, Erkut; Satılmış, Pinar; Akyüz, Ahmet Oguz; Benes, Bedrich and Hauser, Helwig
Image editing is a commonly studied problem in computer graphics. Despite the presence of many advanced editing tools, there is no satisfactory solution to controllably update the position of the sun using a single image. This problem is made complicated by the presence of clouds, complex landscapes, and the atmospheric effects that must be accounted for. In this paper, we tackle this problem starting with only a single photograph. With the user clicking on the initial position of the sun, our algorithm performs several estimation and segmentation processes for finding the horizon, scene depth, clouds, and the sky line. After this initial process, the user can make both fine‐ and large‐scale changes on the position of the sun: it can be set beneath the mountains or moved behind the clouds practically turning a midday photograph into a sunset (or vice versa). We leverage a precomputed atmospheric scattering algorithm to make all of these changes not only realistic but also in real‐time. We demonstrate our results using both clear and cloudy skies, showing how to add, remove, and relight clouds, all the while allowing for advanced effects such as scattering, shadows, light shafts, and lens flares.
Optimal Axes for Data Value Estimation in Star Coordinates and Radial Axes Plots
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Rubio-Sánchez, Manuel; Lehmann, Dirk J.; Sanchez, Alberto; Rojo-Álvarez, Jose Luis; Borgo, Rita and Marai, G. Elisabeta and Landesberger, Tatiana von
Radial axes plots are projection methods that represent high-dimensional data samples as points on a two-dimensional plane. These techniques define mappings through a set of axis vectors, each associated with a data variable, which users can manipulate interactively to create different plots and analyze data from multiple points of view. However, updating the direction and length of an axis vector is far from trivial. Users must consider the data analysis task, domain knowledge, the directions in which values should increase, the relative importance of each variable, or the correlations between variables, among other factors. Another issue is the difficulty to approximate high-dimensional data values in the two-dimensional visualizations, which can hamper searching for data with particular characteristics, analyzing the most common data values in clusters, inspecting outliers, etc. In this paper we present and analyze several optimization approaches for enhancing radial axes plots regarding their ability to represent high-dimensional data values. The techniques can be used not only to approximate data values with greater accuracy, but also to guide users when updating axis vectors or extending visualizations with new variables, since they can reveal poor choices of axis vectors. The optimal axes can also be included in nonlinear plots. In particular, we show how they can be used within RadViz to assess the quality of a variable ordering. The in-depth analysis carried out is useful for visualization designers developing radial axes techniques, or planning to incorporate axes into other visualization methods.
Towards a Neural Graphics Pipeline for Controllable Image Generation
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Chen, Xuelin; Cohen-Or, Daniel; Chen, Baoquan; Mitra, Niloy J.; Mitra, Niloy and Viola, Ivan
In this paper, we leverage advances in neural networks towards forming a neural rendering for controllable image generation, and thereby bypassing the need for detailed modeling in conventional graphics pipeline. To this end, we present Neural Graphics Pipeline (NGP), a hybrid generative model that brings together neural and traditional image formation models. NGP decomposes the image into a set of interpretable appearance feature maps, uncovering direct control handles for controllable image generation. To form an image, NGP generates coarse 3D models that are fed into neural rendering modules to produce view-specific interpretable 2D maps, which are then composited into the final output image using a traditional image formation model. Our approach offers control over image generation by providing direct handles controlling illumination and camera parameters, in addition to control over shape and appearance variations. The key challenge is to learn these controls through unsupervised training that links generated coarse 3D models with unpaired real images via neural and traditional (e.g., Blinn- Phong) rendering functions, without establishing an explicit correspondence between them. We demonstrate the effectiveness of our approach on controllable image generation of single-object scenes. We evaluate our hybrid modeling framework, compare with neural-only generation methods (namely, DCGAN, LSGAN, WGAN-GP, VON, and SRNs), report improvement in FID scores against real images, and demonstrate that NGP supports direct controls common in traditional forward rendering. Code is available at http://geometry.cs.ucl.ac.uk/projects/2021/ngp.
IMAT: The Iterative Medial Axis Transform
(© 2021 Eurographics ‐ The European Association for Computer Graphics and John Wiley & Sons Ltd, 2021) Lee, Yonghyeon; Baek, Jonghyuk; Kim, Young Min; Park, Frank Chongwoo; Benes, Bedrich and Hauser, Helwig
We present the iterative medial axis transform (IMAT), an iterative descent method that constructs a medial axis transform (MAT) for a sparse, noisy, oriented point cloud sampled from an object's boundary. We first establish the equivalence between the traditional definition of the MAT of an object, i.e., the set of centres and corresponding radii of all balls maximally inscribed inside the object, with an alternative characterization matching the boundary enclosing the union of the balls with the object boundary. Based on this boundary equivalence characterization, a new MAT algorithm is proposed, in which an error function that reflects the difference between the two boundaries is minimized while restricting the number of balls to within some a priori specified upper limit. An iterative descent method with guaranteed local convergence is developed for the minimization that is also amenable to parallelization. Both quantitative and qualitative analyses of diverse 2D and 3D objects demonstrate the noise robustness, shape fidelity, and representation efficiency of the resulting MAT.
LoBSTr: Real-time Lower-body Pose Prediction from Sparse Upper-body Tracking Signals
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Yang, Dongseok; Kim, Doyeon; Lee, Sung-Hee; Mitra, Niloy and Viola, Ivan
With the popularization of games and VR/AR devices, there is a growing need for capturing human motion with a sparse set of tracking data. In this paper, we introduce a deep neural network (DNN) based method for real-time prediction of the lowerbody pose only from the tracking signals of the upper-body joints. Specifically, our Gated Recurrent Unit (GRU)-based recurrent architecture predicts the lower-body pose and feet contact states from a past sequence of tracking signals of the head, hands, and pelvis. A major feature of our method is that the input signal is represented by the velocity of tracking signals. We show that the velocity representation better models the correlation between the upper-body and lower-body motions and increases the robustness against the diverse scales and proportions of the user body than position-orientation representations. In addition, to remove foot-skating and floating artifacts, our network predicts feet contact state, which is used to post-process the lower-body pose with inverse kinematics to preserve the contact. Our network is lightweight so as to run in real-time applications. We show the effectiveness of our method through several quantitative evaluations against other architectures and input representations with respect to wild tracking data obtained from commercial VR devices.
Deep Compositional Denoising for High-quality Monte Carlo Rendering
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Zhang, Xianyao; Manzi, Marco; Vogels, Thijs; Dahlberg, Henrik; Gross, Markus; Papas, Marios; Bousseau, Adrien and McGuire, Morgan
We propose a deep-learning method for automatically decomposing noisy Monte Carlo renderings into components that kernelpredicting denoisers can denoise more effectively. In our model, a neural decomposition module learns to predict noisy components and corresponding feature maps, which are consecutively reconstructed by a denoising module. The components are predicted based on statistics aggregated at the pixel level by the renderer. Denoising these components individually allows the use of per-component kernels that adapt to each component's noisy signal characteristics. Experimentally, we show that the proposed decomposition module consistently improves the denoising quality of current state-of-the-art kernel-predicting denoisers on large-scale academic and production datasets.
Wavelet‐based Heat Kernel Derivatives: Towards Informative Localized Shape Analysis
(© 2021 Eurographics ‐ The European Association for Computer Graphics and John Wiley & Sons Ltd, 2021) Kirgo, Maxime; Melzi, Simone; Patanè, Giuseppe; Rodolà, Emanuele; Ovsjanikov, Maks; Benes, Bedrich and Hauser, Helwig
In this paper, we propose a new construction for the Mexican hat wavelets on shapes with applications to partial shape matching. Our approach takes its main inspiration from the well‐established methodology of diffusion wavelets. This novel construction allows us to rapidly compute a multi‐scale family of Mexican hat wavelet functions, by approximating the derivative of the heat kernel. We demonstrate that this leads to a family of functions that inherit many attractive properties of the heat kernel (e.g. local support, ability to recover isometries from a single point, efficient computation). Due to its natural ability to encode high‐frequency details on a shape, the proposed method reconstructs and transfers ‐functions more accurately than the Laplace‐Beltrami eigenfunction basis and other related bases. Finally, we apply our method to the challenging problems of partial and large‐scale shape matching. An extensive comparison to the state‐of‐the‐art shows that it is comparable in performance, while both simpler and much faster than competing approaches.
Unified Shape and SVBRDF Recovery using Differentiable Monte Carlo Rendering
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Luan, Fujun; Zhao, Shuang; Bala, Kavita; Dong, Zhao; Bousseau, Adrien and McGuire, Morgan
Reconstructing the shape and appearance of real-world objects using measured 2D images has been a long-standing inverse rendering problem. In this paper, we introduce a new analysis-by-synthesis technique capable of producing high-quality reconstructions through robust coarse-to-fine optimization and physics-based differentiable rendering. Unlike most previous methods that handle geometry and reflectance largely separately, our method unifies the optimization of both by leveraging image gradients with respect to both object reflectance and geometry. To obtain physically accurate gradient estimates, we develop a new GPU-based Monte Carlo differentiable renderer leveraging recent advances in differentiable rendering theory to offer unbiased gradients while enjoying better performance than existing tools like PyTorch3D [RRN*20] and redner [LADL18]. To further improve robustness, we utilize several shape and material priors as well as a coarse-to-fine optimization strategy to reconstruct geometry. Using both synthetic and real input images, we demonstrate that our technique can produce reconstructions with higher quality than previous methods.
Color Nameability Predicts Inference Accuracy in Spatial Visualizations
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Reda, Khairi; Salvi, Amey A.; Gray, Jack; Papka, Michael E.; Borgo, Rita and Marai, G. Elisabeta and Landesberger, Tatiana von
Color encoding is foundational to visualizing quantitative data. Guidelines for colormap design have traditionally emphasized perceptual principles, such as order and uniformity. However, colors also evoke cognitive and linguistic associations whose role in data interpretation remains underexplored. We study how two linguistic factors, name salience and name variation, affect people's ability to draw inferences from spatial visualizations. In two experiments, we found that participants are better at interpreting visualizations when viewing colors with more salient names (e.g., prototypical 'blue', 'yellow', and 'red' over 'teal', 'beige', and 'maroon'). The effect was robust across four visualization types, but was more pronounced in continuous (e.g., smooth geographical maps) than in similar discrete representations (e.g., choropleths). Participants' accuracy also improved as the number of nameable colors increased, although the latter had a less robust effect. Our findings suggest that color nameability is an important design consideration for quantitative colormaps, and may even outweigh traditional perceptual metrics. In particular, we found that the linguistic associations of color are a better predictor of performance than the perceptual properties of those colors. We discuss the implications and outline research opportunities. The data and materials for this study are available at https://osf.io/asb7n
TourVis: Narrative Visualization of Multi-Stage Bicycle Races
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Díaz, Jose; Fort, Marta; Vázquez, Pere-Pau; Borgo, Rita and Marai, G. Elisabeta and Landesberger, Tatiana von
There are many multiple-stage racing competitions in various sports such as swimming, running, or cycling. The wide availability of affordable tracking devices facilitates monitoring the position along with the race of all participants, even for non-professional contests. Getting real-time information of contenders is useful but also unleashes the possibility of creating more complex visualization systems that ease the understanding of the behavior of all participants during a simple stage or throughout the whole competition. In this paper we focus on bicycle races, which are highly popular, especially in Europe, being the Tour de France its greatest exponent. Current visualizations from TV broadcasting or real-time tracking websites are useful to understand the current stage status, up to a certain extent. Unfortunately, still no current system exists that visualizes a whole multi-stage contest in such a way that users can interactively explore the relevant events of a single stage (e.g. breakaways, groups, virtual leadership: : :), as well as the full competition. In this paper, we present an interactive system that is useful both for aficionados and professionals to visually analyze the development of multi-stage cycling competitions.
Write Like You: Synthesizing Your Cursive Online Chinese Handwriting via Metric-based Meta Learning
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Tang, Shusen; Lian, Zhouhui; Mitra, Niloy and Viola, Ivan
In this paper, we propose a novel Sequence-to-Sequence model based on metric-based meta learning for the arbitrary style transfer of online Chinese handwritings. Unlike most existing methods that treat Chinese handwritings as images and are unable to reflect the human writing process, the proposed model directly handles sequential online Chinese handwritings. Generally, our model consists of three sub-models: a content encoder, a style encoder and a decoder, which are all Recurrent Neural Networks. In order to adaptively obtain the style information, we introduce an attention-based adaptive style block which has been experimentally proven to bring considerable improvement to our model. In addition, to disentangle the latent style information from characters written by any writers effectively, we adopt metric-based meta learning and pre-train the style encoder using a carefully-designed discriminative loss function. Then, our entire model is trained in an end-to-end manner and the decoder adaptively receives the style information from the style encoder and the content information from the content encoder to synthesize the target output. Finally, by feeding the trained model with a content character and several characters written by a given user, our model can write that Chinese character in the user's handwriting style by drawing strokes one by one like humans. That is to say, as long as you write several Chinese character samples, our model can imitate your handwriting style when writing. In addition, after fine-tuning the model with a few samples, it can generate more realistic handwritings that are difficult to be distinguished from the real ones. Both qualitative and quantitative experiments demonstrate the effectiveness and superiority of our method.
Visualization of Tensor Fields in Mechanics
(© 2021 Eurographics ‐ The European Association for Computer Graphics and John Wiley & Sons Ltd, 2021) Hergl, Chiara; Blecha, Christian; Kretzschmar, Vanessa; Raith, Felix; Günther, Fabian; Stommel, Markus; Jankowai, Jochen; Hotz, Ingrid; Nagel, Thomas; Scheuermann, Gerik; Benes, Bedrich and Hauser, Helwig
Tensors are used to describe complex physical processes in many applications. Examples include the distribution of stresses in technical materials, acting forces during seismic events, or remodeling of biological tissues. While tensors encode such complex information mathematically precisely, the semantic interpretation of a tensor is challenging. Visualization can be beneficial here and is frequently used by domain experts. Typical strategies include the use of glyphs, color plots, lines, and isosurfaces. However, data complexity is nowadays accompanied by the sheer amount of data produced by large‐scale simulations and adds another level of obstruction between user and data. Given the limitations of traditional methods, and the extra cognitive effort of simple methods, more advanced tensor field visualization approaches have been the focus of this work. This survey aims to provide an overview of recent research results with a strong application‐oriented focus, targeting applications based on continuum mechanics, namely the fields of structural, bio‐, and geomechanics. As such, the survey is complementing and extending previously published surveys. Its utility is twofold: (i) It serves as basis for the visualization community to get an overview of recent visualization techniques. (ii) It emphasizes and explains the necessity for further research for visualizations in this context.
Implicit Modeling of Patient-Specific Aortic Dissections with Elliptic Fourier Descriptors
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Mistelbauer, Gabriel; Rössl, Christian; Bäumler, Kathrin; Preim, Bernhard; Fleischmann, Dominik; Borgo, Rita and Marai, G. Elisabeta and Landesberger, Tatiana von
Aortic dissection is a life-threatening vascular disease characterized by abrupt formation of a new flow channel (false lumen) within the aortic wall. Survivors of the acute phase remain at high risk for late complications, such as aneurysm formation, rupture, and death. Morphologic features of aortic dissection determine not only treatment strategies in the acute phase (surgical vs. endovascular vs. medical), but also modulate the hemodynamics in the false lumen, ultimately responsible for late complications. Accurate description of the true and false lumen, any communications across the dissection membrane separating the two lumina, and blood supply from each lumen to aortic branch vessels is critical for risk prediction. Patient-specific surface representations are also a prerequisite for hemodynamic simulations, but currently require time-consuming manual segmentation of CT data. We present an aortic dissection cross-sectional model that captures the varying aortic anatomy, allowing for reliable measurements and creation of high-quality surface representations. In contrast to the traditional spline-based cross-sectional model, we employ elliptic Fourier descriptors, which allows users to control the accuracy of the cross-sectional contour of a flow channel. We demonstrate (i) how our approach can solve the requirements for generating surface and wall representations of the flow channels, (ii) how any number of communications between flow channels can be specified in a consistent manner, and (iii) how well branches connected to the respective flow channels are handled. Finally, we discuss how our approach is a step forward to an automated generation of surface models for aortic dissections from raw 3D imaging segmentation masks.
Adversarial Single-Image SVBRDF Estimation with Hybrid Training
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Zhou, Xilong; Kalantari, Nima Khademi; Mitra, Niloy and Viola, Ivan
In this paper, we propose a deep learning approach for estimating the spatially-varying BRDFs (SVBRDF) from a single image. Most existing deep learning techniques use pixel-wise loss functions which limits the flexibility of the networks in handling this highly unconstrained problem. Moreover, since obtaining ground truth SVBRDF parameters is difficult, most methods typically train their networks on synthetic images and, therefore, do not effectively generalize to real examples. To avoid these limitations, we propose an adversarial framework to handle this application. Specifically, we estimate the material properties using an encoder-decoder convolutional neural network (CNN) and train it through a series of discriminators that distinguish the output of the network from ground truth. To address the gap in data distribution of synthetic and real images, we train our network on both synthetic and real examples. Specifically, we propose a strategy to train our network on pairs of real images of the same object with different lighting. We demonstrate that our approach is able to handle a variety of cases better than the state-of-the-art methods.
Curve Complexity Heuristic KD-trees for Neighborhood-based Exploration of 3D Curves
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Lu, Yucheng; Cheng, Luyu; Isenberg, Tobias; Fu, Chi-Wing; Chen, Guoning; Liu, Hui; Deussen, Oliver; Wang, Yunhai; Mitra, Niloy and Viola, Ivan
We introduce the curve complexity heuristic (CCH), a KD-tree construction strategy for 3D curves, which enables interactive exploration of neighborhoods in dense and large line datasets. It can be applied to searches of k-nearest curves (KNC) as well as radius-nearest curves (RNC). The CCH KD-tree construction consists of two steps: (i) 3D curve decomposition that takes into account curve complexity and (ii) KD-tree construction, which involves a novel splitting and early termination strategy. The obtained KD-tree allows us to improve the speed of existing neighborhood search approaches by at least an order of magnitude (i. e., 28× for KNC and 12× for RNC with 98% accuracy) by considering local curve complexity. We validate this performance with a quantitative evaluation of the quality of search results and computation time. Also, we demonstrate the usefulness of our approach for supporting various applications such as interactive line queries, line opacity optimization, and line abstraction.
Texture Browser: Feature-based Texture Exploration
(The Eurographics Association and John Wiley & Sons Ltd., 2021) Luo, Xuejiao; Scandolo, Leonardo; Eisemann, Elmar; Borgo, Rita and Marai, G. Elisabeta and Landesberger, Tatiana von
Texture is a key characteristic in the definition of the physical appearance of an object and a crucial element in the creation process of 3D artists. However, retrieving a texture that matches an intended look from an image collection is difficult. Contrary to most photo collections, for which object recognition has proven quite useful, syntactic descriptions of texture characteristics is not straightforward, and even creating appropriate metadata is a very difficult task. In this paper, we propose a system to help explore large unlabeled collections of texture images. The key insight is that spatially grouping textures sharing similar features can simplify navigation. Our system uses a pre-trained convolutional neural network to extract high-level semantic image features, which are then mapped to a 2-dimensional location using an adaptation of t-SNE, a dimensionality-reduction technique. We describe an interface to visualize and explore the resulting distribution and provide a series of enhanced navigation tools, our prioritized t-SNE, scalable clustering, and multi-resolution embedding, to further facilitate exploration and retrieval tasks. Finally, we also present the results of a user evaluation that demonstrates the effectiveness of our solution.
Turbulent Details Simulation for SPH Fluids via Vorticity Refinement
(© 2021 Eurographics ‐ The European Association for Computer Graphics and John Wiley & Sons Ltd, 2021) Liu, Sinuo; Wang, Xiaokun; Ban, Xiaojuan; Xu, Yanrui; Zhou, Jing; Kosinka, Jiří; Telea, Alexandru C.; Benes, Bedrich and Hauser, Helwig
A major issue in smoothed particle hydrodynamics (SPH) approaches is the numerical dissipation during the projection process, especially under coarse discretizations. High‐frequency details, such as turbulence and vortices, are smoothed out, leading to unrealistic results. To address this issue, we introduce a vorticity refinement (VR) solver for SPH fluids with negligible computational overhead. In this method, the numerical dissipation of the vorticity field is recovered by the difference between the theoretical and the actual vorticity, so as to enhance turbulence details. Instead of solving the Biot‐Savart integrals, a stream function, which is easier and more efficient to solve, is used to relate the vorticity field to the velocity field. We obtain turbulence effects of different intensity levels by changing an adjustable parameter. Since the vorticity field is enhanced according to the curl field, our method can not only amplify existing vortices, but also capture additional turbulence. Our VR solver is straightforward to implement and can be easily integrated into existing SPH methods.

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