Volume 34 (2015)

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A Review of Eye Gaze in Virtual Agents, Social Robotics and HCI: Behaviour Generation, User Interaction and Perception
(Copyright © 2015 The Eurographics Association and John Wiley & Sons Ltd., 2015) Ruhland, K.; Peters, C. E.; Andrist, S.; Badler, J. B.; Badler, N. I.; Gleicher, M.; Mutlu, B.; McDonnell, R.; Deussen, Oliver and Zhang, Hao (Richard)
A person's emotions and state of mind are apparent in their face and eyes. As a Latin proverb states: ‘The face is the portrait of the mind; the eyes, its informers’. This presents a significant challenge for Computer Graphics researchers who generate artificial entities that aim to replicate the movement and appearance of the human eye, which is so important in human–human interactions. This review article provides an overview of the efforts made on tackling this demanding task. As with many topics in computer graphics, a cross‐disciplinary approach is required to fully understand the workings of the eye in the transmission of information to the user. We begin with a discussion of the movement of the eyeballs, eyelids and the head from a physiological perspective and how these movements can be modelled, rendered and animated in computer graphics applications. Furthermore, we present recent research from psychology and sociology that seeks to understand higher level behaviours, such as attention and eye gaze, during the expression of emotion or during conversation. We discuss how these findings are synthesized in computer graphics and can be utilized in the domains of Human–Robot Interaction and Human–Computer Interaction for allowing humans to interact with virtual agents and other artificial entities. We conclude with a summary of guidelines for animating the eye and head from the perspective of a character animator.A person's emotions and state of mind are apparent in their face and eyes. As a Latin proverb states: ‘The face is the portrait of the mind; the eyes, its informers’. This presents a significant challenge for Computer Graphics researchers who generate artificial entities that aim to replicate the movement and appearance of the human eye, which is so important in human–human interactions. This review article provides an overview of the efforts made on tackling this demanding task. As with many topics in computer graphics, a cross‐disciplinary approach is required to fully understand the workings of the eye in the transmission of information to the user.
CPH: A Compact Representation for Hierarchical Meshes Generated by Primal Refinement
(Copyright © 2015 The Eurographics Association and John Wiley & Sons Ltd., 2015) Untereiner, L.; Kraemer, P.; Cazier, D.; Bechmann, D.; Deussen, Oliver and Zhang, Hao (Richard)
We present CPH (): a compact representation of the hierarchical connectivity of surface and volume manifold meshes generated through primal subdivision refinements. CPH is consistently defined in several dimensions and supports multiple kinds of tessellations and refinements, whether regular or adaptive. The basic idea is to store only the finest mesh, encoded in a classical monoresolution structure that is enriched with a minimal set of labels. These labels allow traversal of any intermediate level of the mesh concurrently without having to extract it in an additional structure. Our structure allows attributes to be stored on the cells not only on the finest level, but also on any intermediate level. We study the trade‐off between the memory cost of this compact representation and the time complexity of mesh traversals at any resolution level.We present CPH (): a compact representation of the hierarchical connectivity of surface and volume manifold meshes generated through primal subdivision refinements. CPH is consistently defined in several dimensions and supports multiple kinds of tessellations and refinements, whether regular or adaptive. The basic idea is to store only the finest mesh, encoded in a classical monoresolution structure that is enriched with a minimal set of labels. These labels allow traversal of any intermediate level of the mesh concurrently without having to extract it in an additional structure. Our structure allows attributes to be stored on the cells not only on the finest level, but also on any intermediate level.
Statics Aware Grid Shells
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Pietroni, Nico; Tonelli, Davide; Puppo, Enrico; Froli, Maurizio; Scopigno, Roberto; Cignoni, Paolo; Olga Sorkine-Hornung and Michael Wimmer
We introduce a framework for the generation of polygonal grid-shell architectural structures, whose topology is designed in order to excel in static performances. We start from the analysis of stress on the input surface and we use the resulting tensor field to induce an anisotropic non-Euclidean metric over it. This metric is derived by studying the relation between the stress tensor over a continuous shell and the optimal shape of polygons in a corresponding grid-shell. Polygonal meshes with uniform density and isotropic cells under this metric exhibit variable density and anisotropy in Euclidean space, thus achieving a better distribution of the strain energy over their elements. Meshes are further optimized taking into account symmetry and regularity of cells to improve aesthetics. We experiment with quad meshes and hex-dominant meshes, demonstrating that our grid-shells achieve better static performances than state-of-the-art grid-shells.
3D Fabrication of 2D Mechanisms
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Hergel, Jean; Lefebvre, Sylvain; Olga Sorkine-Hornung and Michael Wimmer
The success of physics sandbox applications and physics-based puzzle games is a strong indication that casual users and hobbyists enjoy designing mechanisms, for educational or entertainment purposes. In these applications, a variety of mechanisms are designed by assembling two-dimensional shapes, creating gears, cranks, cams, and racks. The experience is made enjoyable by the fact that the user does not need to worry about the intricate geometric details that would be necessary to produce a real mechanism. In this paper, we propose to start from such casual designs of mechanisms and turn them into a 3D model that can be printed onto widely available, inexpensive filament based 3D printers. Our intent is to empower the users of such tools with the ability to physically realize their mechanisms and see them operate in the real world. To achieve this goal we tackle several challenges. The input 2D mechanism allows for some parts to overlap during simulation. These overlapping parts have to be resolved into non-intersecting 3D parts in the real mechanism. We introduce a novel scheme based on the idea of including moving parts into one another whenever possible. This reduces bending stresses on axles compared to previous methods. Our approach supports sliding parts and arbitrarily shaped mechanical parts in the 2D input. The exact 3D shape of the parts is inferred from the 2D input and the simulation of the mechanism, using boolean operations between shapes. The input mechanism is often simply attached to the background. We automatically synthesize a chassis by formulating a topology optimization problem, taking into account the stresses exerted by the mechanism on the chassis through time.
A One-dimensional Homologically Persistent Skeleton of an Unstructured Point Cloud in any Metric Space
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Kurlin, Vitaliy; Mirela Ben-Chen and Ligang Liu
Real data are often given as a noisy unstructured point cloud, which is hard to visualize. The important problem is to represent topological structures hidden in a cloud by using skeletons with cycles. All past skeletonization methods require extra parameters such as a scale or a noise bound. We define a homologically persistent skeleton, which depends only on a cloud of points and contains optimal subgraphs representing 1-dimensional cycles in the cloud across all scales. The full skeleton is a universal structure encoding topological persistence of cycles directly on the cloud. Hence a 1-dimensional shape of a cloud can be now easily predicted by visualizing our skeleton instead of guessing a scale for the original unstructured cloud. We derive more subgraphs to reconstruct provably close approximations to an unknown graph given only by a noisy sample in any metric space. For a cloud of n points in the plane, the full skeleton and all its important subgraphs can be computed in time O(n log n).
Register Efficient Dynamic Memory Allocator for GPUs
(Copyright © 2015 The Eurographics Association and John Wiley & Sons Ltd., 2015) Vinkler, M.; Havran, V.; Deussen, Oliver and Zhang, Hao (Richard)
We compare five existing dynamic memory allocators optimized for GPUs and show their strengths and weaknesses. In the measurements, we use three generic evaluation tests proposed in the past and we add one with a real workload, where dynamic memory allocation is used in building the ‐d tree data structure. Following the performance analysis we propose a new dynamic memory allocator and its variants that address the limitations of the existing dynamic memory allocators. The new dynamic memory allocator uses few resources and is targeted towards large and variably sized memory allocations on massively parallel hardware architectures.We compare five existing dynamic memory allocators optimized for GPUs and show their strengths and weaknesses. In the measurements, we use three generic evaluation tests proposed in the past and we add one with a real workload, where dynamic memory allocation is used in building the ‐d tree data structure. Following the performance analysis we propose a new dynamic memory allocator and its variants that address the limitations of the existing dynamic memory allocators. The new dynamic memory allocator uses few resources and is targeted towards large and variably sized memory allocations on massively parallel hardware architectures.
Emotion Analysis and Classification: Understanding the Performers' Emotions Using the LMA Entities
(Copyright © 2015 The Eurographics Association and John Wiley & Sons Ltd., 2015) Aristidou, Andreas; Charalambous, Panayiotis; Chrysanthou, Yiorgos; Deussen, Oliver and Zhang, Hao (Richard)
The increasing availability of large motion databases, in addition to advancements in motion synthesis, has made motion indexing and classification essential for better motion composition. However, in order to achieve good connectivity in motion graphs, it is important to understand human behaviour; human movement though is complex and difficult to completely describe. In this paper, we investigate the similarities between various emotional states with regards to the arousal and valence of the Russell's circumplex model. We use a variety of features that encode, in addition to the raw geometry, stylistic characteristics of motion based on Laban Movement Analysis (LMA). Motion capture data from acted dance performances were used for training and classification purposes. The experimental results show that the proposed features can partially extract the LMA components, providing a representative space for indexing and classification of dance movements with regards to the emotion. This work contributes to the understanding of human behaviour and actions, providing insights on how people express emotional states using their body, while the proposed features can be used as complement to the standard motion similarity, synthesis and classification methods.The increasing availability of large motion databases, in addition to advancements in motion synthesis, has made motion indexing and classification essential for better motion composition. However, in order to achieve good connectivity in motion graphs, it is important to understand human behaviour; human movement though is complex and difficult to completely describe. In this paper, we investigate the similarities between various emotional states with regards to the arousal and valence of the Russell's circumplex model. We use a variety of features that encode, in addition to the raw geometry, stylistic characteristics of motion based on Laban Movement Analysis (LMA).
Local Painting and Deformation of Meshes on the GPU
(Copyright © 2015 The Eurographics Association and John Wiley & Sons Ltd., 2015) Schäfer, H.; Keinert, B.; Nießner, M.; Stamminger, M.; Deussen, Oliver and Zhang, Hao (Richard)
We present a novel method to adaptively apply modifications to scene data stored in GPU memory. Such modifications may include interactive painting and sculpting operations in an authoring tool, or deformations resulting from collisions between scene objects detected by a physics engine. We only allocate GPU memory for the faces affected by these modifications to store fine‐scale colour or displacement values. This requires dynamic GPU memory management in order to assign and adaptively apply edits to individual faces at runtime. We present such a memory management technique based on a scan‐operation that is efficiently parallelizable. Since our approach runs entirely on the GPU, we avoid costly CPU–GPU memory transfer and eliminate typical bandwidth limitations. This minimizes runtime overhead to under a millisecond and makes our method ideally suited to many real‐time applications such as video games and interactive authoring tools. In addition, our algorithm significantly reduces storage requirements and allows for much higher resolution content compared to traditional global texturing approaches. Our technique can be applied to various mesh representations, including Catmull–Clark subdivision surfaces, as well as standard triangle and quad meshes. In this paper, we demonstrate several scenarios for these mesh types where our algorithm enables adaptive mesh refinement, local surface deformations and interactive on‐mesh painting and sculpting.We present a novel method to adaptively apply modifications to scene data stored in GPU memory. Such modifications may include interactive painting and sculpting operations in an authoring tool, or deformations resulting from collisions between scene objects detected by a physics engine. We only allocate GPU memory for the faces affected by these modifications to store fine‐scale color or displacement values.
Collective Crowd Formation Transform with Mutual Information–Based Runtime Feedback
(Copyright © 2015 The Eurographics Association and John Wiley & Sons Ltd., 2015) Xu, Mingliang; Wu, Yunpeng; Ye, Yangdong; Farkas, Illes; Jiang, Hao; Deng, Zhigang; Deussen, Oliver and Zhang, Hao (Richard)
This paper introduces a new crowd formation transform approach to achieve visually pleasing group formation transition and control. Its core idea is to transform crowd formation shapes with a least effort pair assignment using the Kuhn–Munkres algorithm, discover clusters of agent subgroups using affinity propagation and Delaunay triangulation algorithms and apply subgroup‐based social force model (SFM) to the agent subgroups to achieve alignment, cohesion and collision avoidance. Meanwhile, mutual information of the dynamic crowd is used to guide agents' movement at runtime. This approach combines both macroscopic (involving least effort position assignment and clustering) and microscopic (involving SFM) controls of the crowd transformation to maximally maintain subgroups' local stability and dynamic collective behaviour, while minimizing the overall effort (i.e. travelling distance) of the agents during the transformation. Through simulation experiments and comparisons, we demonstrate that this approach is efficient and effective to generate visually pleasing and smooth transformations and outperform several existing crowd simulation approaches including reciprocal velocity avoidances, optimal reciprocal collision avoidance and OpenSteer.This paper introduces a new crowd formation transform approach to achieve visually pleasing group formation transition and control. Its core idea is to transform crowd formation shapes with a least‐effort pair assignment using the Kuhn–Munkres algorithm, discover clusters of agent subgroups using affinity propagation and Delaunay triangulation algorithms, and apply subgroup‐based SFM (social force model) to the agent subgroups to achieve alignment, cohesion and collision avoidance.
Probabilistic Connections for Bidirectional Path Tracing
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Popov, Stefan; Ramamoorthi, Ravi; Durand, Fredo; Drettakis, George; Jaakko Lehtinen and Derek Nowrouzezahrai
Bidirectional path tracing (BDPT) with Multiple Importance Sampling is one of the most versatile unbiased rendering algorithms today. BDPT repeatedly generates sub-paths from the eye and the lights, which are connected for each pixel and then discarded. Unfortunately, many such bidirectional connections turn out to have low contribution to the solution. Our key observation is that we can importance sample connections to an eye sub-path by considering multiple light sub-paths at once and creating connections probabilistically. We do this by storing light paths, and estimating probability mass functions of the discrete set of possible connections to all light paths. This has two key advantages: we efficiently create connections with low variance by Monte Carlo sampling, and we reuse light paths across different eye paths. We also introduce a caching scheme by deriving an approximation to sub-path contribution which avoids high-dimensional path distance computations. Our approach builds on caching methods developed in the different context of VPLs. Our Probabilistic Connections for Bidirectional Path Tracing approach raises a major challenge, since reuse results in high variance due to correlation between paths. We analyze the problem of path correlation and derive a conservative upper bound of the variance, with computationally tractable sample weights. We present results of our method which shows significant improvement over previous unbiased global illumination methods, and evaluate our algorithmic choices.
Frontmatter: Pacific Graphics 2015
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Stam, Jos; Mitra, Niloy J.; Xu, Kun; -
Visualizing the Evolution of Communities in Dynamic Graphs
(Copyright © 2015 The Eurographics Association and John Wiley & Sons Ltd., 2015) Vehlow, C.; Beck, F.; Auwärter, P.; Weiskopf, D.; Deussen, Oliver and Zhang, Hao (Richard)
The community structure of graphs is an important feature that gives insight into the high‐level organization of objects within the graph. In real‐world systems, the graph topology is oftentimes not static but changes over time and hence, also the community structure changes. Previous timeline‐based approaches either visualize the dynamic graph or the dynamic community structure. In contrast, our approach combines both in a single image and therefore allows users to investigate the community structure together with the underlying dynamic graph. Our optimized ordering of vertices and selection of colours in combination with interactive highlighting techniques increases the traceability of communities along the time axis. Users can identify visual signatures, estimate the reliability of the derived community structure and investigate whether community evolution interacts with changes in the graph topology. The utility of our approach is demonstrated in two application examples.The community structure of graphs is an important feature that gives insight into the high‐level organization of objects within the graph. In real‐world systems, the graph topology is oftentimes not static but changes over time and hence, also the community structure changes. Previous timeline‐based approaches either visualize the dynamic graph or the dynamic community structure. In contrast, our approach combines both in a single image and therefore allows users to investigate the community structure together with the underlying dynamic graph. Our optimized ordering of vertices and selection of colours in combination with interactive highlighting techniques increases the traceability of communities along the time axis. Users can identify visual signatures, estimate the reliability of the derived community structure and investigate whether community evolution interacts with changes in the graph topology. The utility of our approach is demonstrated in two application examples.
Visual Analytics for Exploring Local Impact of Air Traffic
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Buchmüller, Juri; Janetzko, Halldor; Andrienko, Gennady; Andrienko, Natalia; Fuchs, Georg; Keim, Daniel A.; H. Carr, K.-L. Ma, and G. Santucci
The environmental and noise impact of airports often causes extensive political discussion which in some cases even lead to transnational tensions. Analyzing local approach and departure patterns around an airport is difficult since it depends on a variety of complex variables like weather, local and general regulations and many more. Yet, understanding these movements and the expected amount of flights during arrival and departure is of great interest to both casual and expert users, as planes have a higher impact on the areas beneath during these phases. We present a Visual Analytics framework that enables users to develop an understanding of local flight behavior through visual exploration of historical data and interactive manipulation of prediction models with direct feedback, as well as a classification quality visualization using a random noise metaphor. We showcase our approach using real world data from the Zurich International Airport region, where aircraft noise has led to an ongoing conflict between Germany and Switzerland. The use cases, findings and expert feedback demonstrate how our approach helps in understanding the situation and to substantiate the otherwise often subjective discourse on the topic.
Interactive Procedural Modelling of Coherent Waterfall Scenes
(Copyright © 2015 The Eurographics Association and John Wiley & Sons Ltd., 2015) Emilien, Arnaud; Poulin, Pierre; Cani, Marie‐Paule; Vimont, Ulysse; Deussen, Oliver and Zhang, Hao (Richard)
Combining procedural generation and user control is a fundamental challenge for the interactive design of natural scenery. This is particularly true for modelling complex waterfall scenes where, in addition to taking charge of geometric details, an ideal tool should also provide a user with the freedom to shape the running streams and falls, while automatically maintaining physical plausibility in terms of flow network, embedding into the terrain, and visual aspects of the waterfalls. We present the first solution for the interactive procedural design of coherent waterfall scenes. Our system combines vectorial editing, where the user assembles elements to create a waterfall network over an existing terrain, with a procedural model that parametrizes these elements from hydraulic exchanges; enforces consistency between the terrain and the flow; and generates detailed geometry, animated textures and shaders for the waterfalls and their surroundings. The tool is interactive, yielding visual feedback after each edit.Combining procedural generation and user control is a fundamental challenge for the interactive design of natural scenery. This is particularly true for modelling complex waterfall scenes where, in addition to taking charge of geometric details, an ideal tool should also provide a user with the freedom to shape the running streams and falls, while automatically maintaining physical plausibility in terms of flow network, embedding into the terrain, and visual aspects of the waterfalls.
Interactive Rigging with Intuitive Tools
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Bang, Seungbae; Choi, Byungkuk; Ribera, Roger Blanco i; Kim, Meekyoung; Lee, Sung-Hee; Noh, Junyong; Stam, Jos and Mitra, Niloy J. and Xu, Kun
Rigging is a core element in the process of bringing a 3D character to life. The rig defines and delimits the motions of the character and provides an interface for an animator with which to interact with the 3D character. The quality of the rig has a key impact on the expressiveness of the character. Creating a usable, rich, production ready rig is a laborious task requiring direct intervention by a trained professional because the goal is difficult to achieve with fully automatic methods. We propose a semi-automatic rigging editing framework which eases the need for manual intervention while maintaining an important degree of control over the final rig. Starting by automatically generated base rig, we provide interactive operations which efficiently configure the skeleton structure and mesh skinning.
Specular Lobe‐Aware Filtering and Upsampling for Interactive Indirect Illumination
(Copyright © 2015 The Eurographics Association and John Wiley & Sons Ltd., 2015) Tokuyoshi, Y.; Deussen, Oliver and Zhang, Hao (Richard)
Although geometry‐aware filtering and upsampling have often been used for interactive or real‐time rendering, they are unsuitable for glossy surfaces because shading results strongly depend on the bidirectional reflectance distribution functions. This paper proposes a novel weighting function of cross bilateral filtering and upsampling to measure the similarity of specular lobes. The difficulty is that a specular lobe is represented with a distribution function in directional space, whereas conventional cross bilateral filtering evaluates similarities using the distance between two points in a Euclidean space. Therefore, this paper first generalizes cross bilateral filtering for the similarity of distribution functions in a non‐Euclidean space. Then, the weighting function is specialized for specular lobes. Our key insight is that the weighting function of bilateral filtering can be represented with the product integral of two distribution functions corresponding to two pixels. In addition, we propose spherical Gaussian‐based approximations to calculate this weighting function analytically. Our weighting function detects the edges of glossiness, and adapts to all‐frequency materials using only a camera position and G‐buffer. These features are not only suitable for path tracing, but also deferred shading and non‐ray tracing–based methods such as voxel cone tracing.Although geometry‐aware filtering and upsampling have often been used for interactive or real‐time rendering, they are unsuitable for glossy surfaces because shading results strongly depend on the bidirectional reflectance distribution functions. This paper proposes a novel weighting function of cross bilateral filtering and upsampling to measure the similarity of specular lobes. The difficulty is that a specular lobe is represented with a distribution function in directional space, whereas conventional cross bilateral filtering evaluates similarities using the distance between two points in a Euclidean space. Therefore, this paper first generalizes cross bilateral filtering for the similarity of distribution functions in a non‐Euclidean space. Then, the weighting function is specialized for specular lobes. Our key insight is that the weighting function of bilateral filtering can be represented with the product integral of two distribution functions corresponding to two pixels. In addition, we propose spherical Gaussian‐based approximations to calculate this weighting function analytically.
Structure‐Aware Mesh Decimation
(Copyright © 2015 The Eurographics Association and John Wiley & Sons Ltd., 2015) Salinas, D.; Lafarge, F.; Alliez, P.; Deussen, Oliver and Zhang, Hao (Richard)
We present a novel approach for the decimation of triangle surface meshes. Our algorithm takes as input a triangle surface mesh and a set of planar proxies detected in a pre‐processing analysis step, and structured via an adjacency graph. It then performs greedy mesh decimation through a series of edge collapse, designed to approximate the local mesh geometry as well as the geometry and structure of proxies. Such structure‐preserving approach is well suited to planar abstraction, i.e. extreme decimation approximating well the planar parts while filtering out the others. Our experiments on a variety of inputs illustrate the potential of our approach in terms of improved accuracy and preservation of structure.We present a novel approach for the decimation of triangle surface meshes. Our algorithm takes as input a triangle surface mesh and a set of planar proxies detected in a pre‐processing analysis step, and structured via an adjacency graph. It then performs greedy mesh decimation through a series of edge collapse, designed to approximate the local mesh geometry as well as the geometry and structure of proxies. Such structure‐preserving approach is well suited to planar abstraction, i.e. extreme decimation approximating well the planar parts while filtering out the others. Our experiments on a variety of inputs illustrate the potential of our approach in terms of improved accuracy and preservation of structure.
Visual Exploration of High-Dimensional Data through Subspace Analysis and Dynamic Projections
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Liu, Shusen; Wang, Bei; Thiagarajan, Jayaraman J.; Bremer, Peer-Timo; Pascucci, Valerio; H. Carr, K.-L. Ma, and G. Santucci
We introduce a novel interactive framework for visualizing and exploring high-dimensional datasets based on subspace analysis and dynamic projections. We assume the high-dimensional dataset can be represented by a mixture of low-dimensional linear subspaces with mixed dimensions, and provide a method to reliably estimate the intrinsic dimension and linear basis of each subspace extracted from the subspace clustering. Subsequently, we use these bases to define unique 2D linear projections as viewpoints from which to visualize the data. To understand the relationships among the different projections and to discover hidden patterns, we connect these projections through dynamic projections that create smooth animated transitions between pairs of projections. We introduce the view transition graph, which provides flexible navigation among these projections to facilitate an intuitive exploration. Finally, we provide detailed comparisons with related systems, and use real-world examples to demonstrate the novelty and usability of our proposed framework.
Exploring Traffic Dynamics in Urban Environments Using Vector-Valued Functions
(The Eurographics Association and John Wiley & Sons Ltd., 2015) Poco, Jorge; Doraiswamy, Harish; Vo, Huy. T.; Comba, João L. D.; Freire, Juliana; Silva, Cláudio T.; H. Carr, K.-L. Ma, and G. Santucci
The traffic infrastructure greatly impacts the quality of life in urban environments. To optimize this infrastructure, engineers and decision makers need to explore traffic data. In doing so, they face two important challenges: the sparseness of speed sensors that cover only a limited number of road segments, and the complexity of traffic patterns they need to analyze. In this paper we take a first step at addressing these challenges. We use New York City (NYC) taxi trips as sensors to capture traffic information. While taxis provide substantial coverage of the city, the data captured about taxi trips contain neither the location of taxis at frequent intervals nor their routes. We propose an efficient traffic model to derive speed and direction information from these data, and show that it provides reliable estimates. Using these estimates, we define a time-varying vector-valued function on a directed graph representing the road network, and adapt techniques used for vector fields to visualize the traffic dynamics. We demonstrate the utility of our technique in several case studies that reveal interesting mobility patterns in NYC's traffic. These patterns were validated by experts from NYC's Department of Transportation and the NYC Taxi & Limousine Commission, who also provided interesting insights into these results.

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