Hex-dominant mesh generation has recently received increasing attention from researchers and the simulation community due to its robustness compared to pure hex-mesh generation techniques. In this work, we present a first structure-informed simplification framework, aiming to reduce the number of non-hex cells from the hex-dominant meshes. Our framework extracts individual sub-structures via parallel relations from the input hex-dominant mesh, decomposes self-tangent and self-intersecting sub-structures for structure complexity control, and collapses certain sets of edges that are adjacent to particular non-hex cells to remove them. We design a filtering and ranking strategy to select edges and sub-structures for collapsing. To better understand the complexity of sub-structure configurations, we introduced a novel relation graph that captures the connections between edges and between edges and sub-structures. Additionally, we designed a smoothing algorithm for hex-dominant meshes that enhances hex cell quality, even in meshes containing various cell types. We evaluate the effectiveness of our framework by applying it to many hex-dominant meshes produced by three state-of-the-art hex-dominant meshing techniques. Our results achieve various levels of reduction in the number of non-hex cells of the input meshes, affirming the applicability of our framework for improving the hex-dominant meshes.

Hairpin vortices are fundamental structures within turbulent boundary layers, playing a crucial role in energy dissipation, mixing, and momentum transport. However, accurately extracting these structures remains challenging due to their irregular shapes, varying scales, and entanglement with surrounding vortical structures. This article presents a novel framework for the extraction of hairpin vortices from turbulent boundary layers. The method begins by identifying vortical regions and decomposing them into smaller segments using merge tree-based segmentation. A novel bottom-up rejoining approach is then introduced to group candidate segments according to the geometric and physical characteristics of hairpin vortices, resulting in regions that encompass complete hairpin vortex structures. These regions are subsequently refined and validated through skeleton analysis to detect the characteristic hairpin shape and are further confirmed using additional scalar-based criteria. Finally, smooth enclosing surfaces are generated for effective visualization. To enable quantitative evaluation, reference hairpin vortices are extracted from several flow datasets and used as ground truth. Compared with existing approaches, the proposed method eliminates manual parameter tuning, reduces under- and over-segmentation, and significantly improves both accuracy and computational efficiency. Demonstrations on multiple turbulent flow cases show that the method is robust and effective for hairpin vortex extraction under varying boundary layer conditions.

Hex-dominant mesh generation has received significant attention in recent research due to its superior robustness compared to pure hex-mesh generation techniques. In this work, we introduce the first structure for analyzing hex-dominant meshes. This structure builds on the base complex of pure hex-meshes but incorporates the non-hex elements for a more comprehensive and complete representation. We provide its definition and describe its construction steps. Based on this structure, we present an extraction and categorization of sheets using advanced graph matching techniques to handle the non-hex elements. This enables us to develop an enhanced visual analysis of the structure for any hex-dominant meshes. We apply this structure-based visual analysis to compare hex-dominant meshes generated by different methods to study their advantages and disadvantages. This complements the standard quality metric based on the non-hex element percentage for hex-dominant meshes. Moreover, we propose a strategy to extract a cleaned (optimized) valence-based singularity graph wireframe to analyze the structure for both mesh and sheets. Our results demonstrate that the proposed hybrid base complex provides a coarse representation for mesh element, and the proposed valence singularity graph wireframe provides a better internal visualization of hex-dominant meshes.

Integral curves have been widely used to represent and analyze various vector fields. In this paper, we propose a Curve Segment Neighborhood Graph (CSNG) to capture the relationships between neighboring curve segments. This graph representation enables us to adapt the fast community detection algorithm, i.e., the Louvain algorithm, to identify individual graph communities from CSNG. Our results show that these communities often correspond to the features of the flow. To achieve a multi-level interactive exploration of the detected communities, we adapt a force-directed layout that allows users to refine and re-group communities based on their domain knowledge. We incorporate the proposed techniques into an interactive system to enable effective analysis and interpretation of complex patterns in large-scale integral curve datasets.

Vortices and their analysis play a critical role in the understanding of complex phenomena in turbulent flows. Traditional vortex extraction methods, notably region-based techniques, often overlook the entanglement phenomenon, resulting in the inclusion of multiple vortices within a single extracted region. Their separation is necessary for quantifying different types of vortices and their statistics. In this study, we propose a novel vortex separation method that extends the conventional contour tree-based segmentation approach with an additional step termed "layering". Upon extracting a vortical region using specified vortex criteria (e.g., λ2), we initially establish topological segmentation based on the contour tree, followed by the layering process to allocate appropriate segmentation IDs to unsegmented cells, thus separating individual vortices within the region. However, these regions may still suffer from inaccurate splits, which we address statistically by leveraging the continuity of vorticity lines across the split boundaries. Our findings demonstrate a significant improvement in both the separation of vortices and the mitigation of inaccurate splits compared to prior methods.

Hairpin vortices are one of the most important vortical structures in turbulent flows. Extracting and characterizing hairpin vortices provides useful insight into many behaviors in turbulent flows. However, hairpin vortices have complex configurations and might be entangled with other vortices, making their extraction difficult. In this work, we introduce a framework to extract and separate hairpin vortices in shear driven turbulent flows for their study. Our method first extracts general vortical regions with a region-growing strategy based on certain vortex criteria (e.g., λ2) and then separates those vortices with the help of progressive extraction of (λ2) iso-surfaces in a top-down fashion. This leads to a hierarchical tree representing the spatial proximity and merging relation of vortices. After separating individual vortices, their shape and orientation information is extracted. Candidate hairpin vortices are identified based on their shape and orientation information as well as their physical characteristics. An interactive visualization system is developed to aid the exploration, classification, and analysis of hairpin vortices based on their geometric and physical attributes. We also present additional use cases of the proposed system for the analysis and study of general vortices in other types of flows.

Mastery Learning, a pedagogical strategy in which students are allowed to prove mastery of the skills acquired in a course over multiple attempts (and used failed attempts as feedback) is becoming increasingly popular in higher education. Large introductory programming courses can use it to strengthen students' preparation for later courses, but some challenges to its adoption remain, such as how to scale this format to hundreds of students, or how to ensure that students do not fall behind on the material. In Spring 2021, the instructors at the Anonymous University transformed the structure of their CS1 course using a Mastery Learning format, reorganizing the material in units focused on the different course topics. Students were allowed to prove mastery of each unit separately and over multiple times, without penalties for missed or failed attempts. In this experience report, we will describe the strategies adopted to cater to a large cohort of novice students. We will compare the students' learning experience with a cohort of students who took the course in a more traditional format, and show that the students benefited from the new format in terms of quantity of skills mastered. Students also exhibited signs of increased motivation to practice and complete tests without grade incentives. Finally, we will discuss some pitfalls in our design and address some of the concerns of instructors interested in trying a Mastery Learning approach in their CS1 courses.

Large-scale structures have been observed in many shear flows which are the fluid generated between two surfaces moving with different velocity. A better understanding of the physics of the structures (especially large-scale structures) in shear flows will help explain a diverse range of physical phenomena and improve our capability of modeling more complex turbulence flows. Many efforts have been made in order to capture such structures; however, conventional methods have their limitations, such as arbitrariness in parameter choice or specificity to certain setups. To address this challenge, we propose to use Multi-Resolution Dynamic Mode Decomposition (mrDMD), for large-scale structure extraction in shear flows. In particular, we show that the slow motion DMD modes are able to reveal large-scale structures in shear flows that also have slow dynamics. In most cases, we find that the slowest DMD mode and its reconstructed flow can sufficiently capture the large-scale dynamics in the shear flows, which leads to a parameter-free strategy for large-scale structure extraction. Effective visualization of the large-scale structures can then be produced with the aid of the slowest DMD mode. To speed up the computation of mrDMD, we provide a fast GPU-based implementation. We also apply our method to some non-shear flows that need not behave quasi-linearly to demonstrate the limitation of our strategy of using the slowest DMD mode. For non-shear flows, we show that multiple modes from different levels of mrDMD may be needed to sufficiently characterize the flow behavior.

Incremental learning is used to continuously update and tune an existing model, as more data points come in. However, there is a tradeoff between speed and accuracy as the model becomes stable. From a big data angle, computing machine learning models is challenging when data sets cannot fit in main memory or when they exceed CPU capacity. On the other hand, data summarization is a fundamental technique that has promise to accelerate data science computations and compress a data set. Keeping these motivations in mind, we present an innovative system, VIM, that computes machine learning models in an incremental manner, visualizing continuous learning of model parameters as the data set is scanned. Our system is fast, it works for a wide spectrum of machine learning models and it can handle data sets larger than main memory. We developed an intuitive GUI which: (1) guides the user to upload data sets and choose a machine learning model, (2) offers interactive visualization of model parameters, and (3) helps getting an approximate model, stopping early, without reading the whole data set.

Abstract 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.

Abstract Most existing unsteady flow visualization techniques concentrate on the depiction of geometric patterns in flow, assuming the geometry information provides sufficient representation of the underlying physical characteristics, which is not always the case. To address this challenge, this work proposes to analyse the time‐dependent characteristics of the physical attributes measured along pathlines which can be represented as a series of time activity curves (TAC). We demonstrate that the temporal trends of these TACs can convey the relation between pathlines and certain well‐known flow features (e.g. vortices and shearing layers), which enables us to select pathlines that can effectively represent the physical characteristics of interest and their temporal behaviour in the unsteady flow. Inspired by this observation, a new TAC‐based unsteady flow visualization and analysis framework is proposed. The centre of this framework is a new similarity measure that compares the similarity of two TACs, from which a new spatio‐temporal, hierarchical clustering that classifies pathlines based on their physical attributes, and a TAC‐based pathline exploration and selection strategy are proposed. A visual analytic system incorporating the TAC‐based pathline clustering and exploration is developed, which also provides new visualizations to support the user exploration of unsteady flow using TACs. This visual analytic system is applied to a number of unsteady flow in 2D and 3D to demonstrate its utility. The new system successfully reveals the detailed structure of vortices, the relation between shear layer and vortex formation, and vortex breakdown, which are difficult to convey with conventional methods.

Taylor-Couette flow (TCF) is the turbulent fluid motion created between two concentric and independently rotating cylinders. It has been heavily researched in fluid mechanics thanks to the various nonlinear dynamical phenomena that are exhibited in the flow. As many dense coherent structures overlap each other in TCF, it is challenging to isolate and visualize them, especially when the cylinder rotation ratio is changing. Previous approaches rely on 2D cross sections to study TCF due to its simplicity, which cannot provide the complete information of TCF. In the meantime, standard visualization techniques, such as volume rendering / iso-surfacing of certain attributes and the placement of integral curves/surfaces, usually produce cluttered visualization. To address this challenge and to support domain experts in the analysis of TCF, we developed a visualization framework to separate large-scale structures from the dense, small-scale structures and provide an effective visual representation of these structures. Instead of using a single physical attribute as the standard approach which cannot efficiently separate structures in different scales for TCF, we adapt the feature level-set method to combine multiple attributes and use them as a filter to separate large- and small-scale structures. To visualize these structures, we apply the iso-surface extraction on the kernel density estimate of the distance field generated from the feature level-set. The proposed methods successfully reveal 3D large-scale coherent structures of TCF with different control parameter settings, which are difficult to achieve with the conventional methods.

Unsupervised clustering techniques have been widely applied to flow simulation data to alleviate clutter and occlusion in the resulting visualization. However, there is an absence of systematic guidelines for users to evaluate (both quantitatively and visually) the appropriate clustering technique and similarity measures for streamline and pathline curves. In this work, we provide an overview of a number of prevailing curve clustering techniques. We then perform a comprehensive experimental study to qualitatively and quantitatively compare these clustering techniques coupled with popular similarity measures used in the flow visualization literature. Based on our experimental results, we derive empirical guidelines for selecting the appropriate clustering technique and similarity measure given the requirements of the visualization task. We believe our work will inform the task of generating meaningful reduced representations for large-scale flow data and inspire the continuous investigation of a more refined guidance on clustering technique selection.

Despite significant advances in the analysis and visualization of unsteady flow, the interpretation of it's behavior still remains a challenge. In this work, we focus on the linear correlation and non-linear dependency of different physical attributes of unsteady flows to aid their study from a new perspective. Specifically, we extend the existing spatial correlation quantification, i.e. the Local Correlation Coefficient (LCC), to the spatio-temporal domain to study the correlation of attribute-pairs from both the Eulerian and Lagrangian views. To study the dependency among attributes, which need not be linear, we extend and compute the mutual information (MI) among attributes over time. To help visualize and interpret the derived correlation and dependency among attributes associated with a particle, we encode the correlation and dependency values on individual pathlines. Finally, to utilize the correlation and MI computation results to identify regions with interesting flow behavior, we propose a segmentation strategy of the flow domain based on the ranking of the strength of the attributes relations. We have applied our correlation and dependency metrics to a number of 2D and 3D unsteady flows with varying spatio-temporal kernel sizes to demonstrate and assess their effectiveness.

Advances in high-throughput imaging allow researchers to collect three-dimensional images of whole organ microvascular networks. These extremely large images contain networks that are highly complex, time consuming to segment, and difficult to visualize. In this paper, we present a framework for segmenting and visualizing vascular networks from terabyte-sized three-dimensional images collected using high-throughput microscopy. While these images require terabytes of storage, the volume devoted to the fiber network is ≈ 4 percent of the total volume size. While the networks themselves are sparse, they are tremendously complex, interconnected, and vary widely in diameter. We describe a parallel GPU-based predictor-corrector method for tracing filaments that is robust to noise and sampling errors common in these data sets. We also propose a number of visualization techniques designed to convey the complex statistical descriptions of fibers across large tissue sections-including commonly studied microvascular characteristics, such as orientation and volume.

Understanding hexahedral (hex-) mesh structures is important for a number of hex-mesh generation and optimization tasks. However, due to various configurations of the singularities in a valid pure hex-mesh, the structure (or base complex) of the mesh can be arbitrarily complex. In this work, we present a first and effective method to help meshing practitioners understand the possible configurations in a valid 3D base complex for the characterization of their complexity. In particular, we propose a strategy to decompose the complex hex-mesh structure into multi-level sub-structures so that they can be studied separately, from which we identify a small set of the sub-structures that can most efficiently represent the whole mesh structure. Furthermore, from this set of sub-structures, we attempt to define the first metric for the quantification of the complexity of hex-mesh structure. To aid the exploration of the extracted multi-level structure information, we devise a visual exploration system coupled with a matrix view to help alleviate the common challenge of 3D data exploration (e.g., clutter and occlusion). We have applied our tool and metric to a large number of hex-meshes generated with different approaches to reveal different characteristics of these methods in terms of the mesh structures they can produce. We also use our metric to assess the existing structure simplification techniques in terms of their effectiveness.

Abstract Hexahedral (hex‐) meshes are important for solving partial differential equations (PDEs) in applications of scientific computing and mechanical engineering. Many methods have been proposed aiming to generate hex‐meshes with high scaled Jacobians. While it is well established that a hex‐mesh should be inversion‐free (i.e. have a positive Jacobian measured at every corner of its hexahedron), it is not well‐studied that whether the scaled Jacobian is the most effective indicator of the quality of simulations performed on inversion‐free hex‐meshes given the existing dozens of quality metrics for hex‐meshes. Due to the challenge of precisely defining the relations among metrics, studying the correlations among different quality metrics and their correlations with the stability and accuracy of the simulations is a first and effective approach to address the above question. In this work, we propose a correlation analysis framework to systematically study these correlations. Specifically, given a large hex‐mesh dataset, we classify the existing quality metrics into groups based on their correlations, which characterizes their similarity in measuring the quality of hex‐elements. In addition, we rank the individual metrics based on their correlations with the accuracy and stability metrics for simulations that solve a number of elliptic PDE problems. Our preliminary experiments suggest that metrics that assess the conditioning of the elements are more correlated to the quality of solving elliptic PDEs than the others. Furthermore, an inversion‐free hex‐mesh with higher average quality (measured by any quality metrics) usually leads to a more accurate and stable computation of elliptic PDEs. To support our correlation study and address the lack of a publicly available large hex‐mesh dataset with sufficiently varying quality metric values, we also propose a two‐level perturbation strategy to generate the desired dataset from a small number of meshes to exclude the influences of element numbers, vertex connectivity, and volume sizes to our study.

Abstract Hexahedral (or Hex‐) meshes are preferred in a number of scientific and engineering simulations and analyses due to their desired numerical properties. Recent state‐of‐the‐art techniques can generate high‐quality hex‐meshes. However, they typically produce hex‐meshes with uniform element sizes and thus may fail to preserve small‐scale features on the boundary surface. In this work, we present a new framework that enables users to generate hex‐meshes with varying element sizes so that small features will be filled with smaller and denser elements, while the transition from smaller elements to larger ones is smooth, compared to the octree‐based approach. This is achieved by first detecting regions of interest (ROIs) of small‐scale features. These ROIs are then magnified using the as‐rigid‐as‐possible deformation with either an automatically determined or a user‐specified scale factor. A hex‐mesh is then generated from the deformed mesh using existing approaches that produce hex‐meshes with uniform‐sized elements. This initial hex‐mesh is then mapped back to the original volume before magnification to adjust the element sizes in those ROIs. We have applied this framework to a variety of man‐made and natural models to demonstrate its effectiveness.

We introduce a robust and automatic algorithm to simplify the structure and reduce the singularities of a hexahedral mesh. Our algorithm interleaves simplification operations to collapse sheets and chords of the base complex of the input mesh with a geometric optimization, which improves the elements quality. All our operations are guaranteed not to introduce elements with negative Jacobians, ensuring that our algorithm always produces valid hex-meshes, and not to increase the Hausdorff distance from the original shape more than a user-defined threshold, ensuring a faithful approximation of the input geometry. Our algorithm can improve meshes produced with any existing hexahedral meshing algorithm --- we demonstrate its effectiveness by processing a dataset of 194 hex-meshes created with octree-based, polycube-based, and field-aligned methods.

In this paper, we introduce a volumetric partitioning strategy based on a generalized sweeping framework to seamlessly partition the volume of an input triangle mesh into a collection of deformed cuboids. This is achieved by a user-designed volumetric harmonic function that guides the decomposition of the input volume into a sequence of two-manifold level sets. A skeletal structure whose corners correspond to corner vertices of a 2D parameterization is extracted for each level set. Corners are placed so that the skeletal structure aligns with features of the input object. Then, a skeletal surface is constructed by matching the skeletal structures of adjacent level sets. The surface sheets of this skeletal surface partition the input volume into the deformed cuboids. The collection of cuboids does not exhibit T-junctions, significantly simplifying the hexahedral mesh generation process, and in particular, it simplifies fitting trivariate B-splines to the deformed cuboids. Intersections of the surface sheets of the skeletal surface correspond to the singular edges of the generated hex-meshes. We apply our technique to a variety of 3D objects and demonstrate the benefit of the structure decomposition in data fitting.

Vector field topology has been successfully applied to represent the structure of steady vector fields. Critical points, one of the essential components of vector field topology, play an important role in describing the complexity of the extracted structure. Simplifying vector fields via critical point cancellation has practical merit for interpreting the behaviors of complex vector fields such as turbulence. However, there is no effective technique that allows direct cancellation of critical points in 3D. This work fills this gap and introduces the first framework to directly cancel pairs or groups of 3D critical points in a hierarchical manner with a guaranteed minimum amount of perturbation based on their robustness, a quantitative measure of their stability. In addition, our framework does not require the extraction of the entire 3D topology, which contains non-trivial separation structures, and thus is computationally effective. Furthermore, our algorithm can remove critical points in any subregion of the domain whose degree is zero and handle complex boundary configurations, making it capable of addressing challenging scenarios that may not be resolved otherwise. We apply our method to synthetic and simulation datasets to demonstrate its effectiveness.

Exploring multivariate spatial data attracts much attention in the visualization community. The main challenge lies in that automatic analysis techniques is insufficient in discovering complicated patterns with the perspective of human beings, while visualization techniques are incapable of accurately identifying the features of interest. This paper addresses this contradiction by enhancing automatic analysis techniques with human intelligence in an iterative visual exploration process. The integrated system, called EasyXplorer, provides a suite of intuitive clustering, dimension reduction, visual encoding and filtering widgets within 2D and 3D views, allowing an inexperienced user to visually explore and reason undiscovered features with several simple interactions. Case studies show the quality and scalability of our approach in quite challenging examples.

Recently, generating a high quality all-hex mesh of a given volume has gained much attention. However, little, if any, effort has been put into the optimization of the hex-mesh structure, which is equally important to the local element quality of a hex-mesh that may influence the performance and accuracy of subsequent computations. In this paper, we present a first and complete pipeline to optimize the global structure of a hex-mesh. Specifically, we first extract the base-complex of a hex-mesh and study the misalignments among its singularities by adapting the previously introduced hexahedral sheets to the base-complex. Second, we identify the valid removal base-complex sheets from the base-complex that contain misaligned singularities. We then propose an effective algorithm to remove these valid removal sheets in order. Finally, we present a structure-aware optimization strategy to improve the geometric quality of the resulting hex-mesh after fixing the misalignments. Our experimental results demonstrate that our pipeline can significantly reduce the number of components of a variety of hex-meshes generated by state-of-the-art methods, while maintaining high geometric quality.

Vector field simplification aims to reduce the complexity of the flow by removing features in order of their relevance and importance, to reveal prominent behavior and obtain a compact representation for interpretation. Most existing simplification techniques based on the topological skeleton successively remove pairs of critical points connected by separatrices, using distance or area-based relevance measures. These methods rely on the stable extraction of the topological skeleton, which can be difficult due to instability in numerical integration, especially when processing highly rotational flows. In this paper, we propose a novel simplification scheme derived from the recently introduced topological notion of robustness which enables the pruning of sets of critical points according to a quantitative measure of their stability, that is, the minimum amount of vector field perturbation required to remove them. This leads to a hierarchical simplification scheme that encodes flow magnitude in its perturbation metric. Our novel simplification algorithm is based on degree theory and has minimal boundary restrictions. Finally, we provide an implementation under the piecewise-linear setting and apply it to both synthetic and real-world datasets. We show local and complete hierarchical simplifications for steady as well as unsteady vector fields.

Vector field simplification aims to reduce the complexity of the flow by removing features in order of their relevance and importance, to reveal prominent behavior and obtain a compact representation for interpretation. Most existing simplification techniques based on the topological skeleton successively remove pairs of critical points connected by separatrices, using distance or area-based relevance measures. These methods rely on the stable extraction of the topological skeleton, which can be difficult due to instability in numerical integration, especially when processing highly rotational flows. These geometric metrics do not consider the flow magnitude, an important physical property of the flow. In this paper, we propose a novel simplification scheme derived from the recently introduced topological notion of robustness, which provides a complementary view on flow structure compared to the traditional topological-skeleton-based approaches. Robustness enables the pruning of sets of critical points according to a quantitative measure of their stability, that is, the minimum amount of vector field perturbation required to remove them. This leads to a hierarchical simplification scheme that encodes flow magnitude in its perturbation metric. Our novel simplification algorithm is based on degree theory, has fewer boundary restrictions, and so can handle more general cases. Finally, we provide an implementation under the piecewise-linear setting and apply it to both synthetic and real-world datasets.

We present a novel integrated visualization system that enables interactive visual analysis of ensemble simulations of the sea surface height that is used in ocean forecasting. The position of eddies can be derived directly from the sea surface height and our visualization approach enables their interactive exploration and analysis.The behavior of eddies is important in different application settings of which we present two in this paper. First, we show an application for interactive planning of placement as well as operation of off-shore structures using real-world ensemble simulation data of the Gulf of Mexico. Off-shore structures, such as those used for oil exploration, are vulnerable to hazards caused by eddies, and the oil and gas industry relies on ocean forecasts for efficient operations. We enable analysis of the spatial domain, as well as the temporal evolution, for planning the placement and operation of structures.Eddies are also important for marine life. They transport water over large distances and with it also heat and other physical properties as well as biological organisms. In the second application we present the usefulness of our tool, which could be used for planning the paths of autonomous underwater vehicles, so called gliders, for marine scientists to study simulation data of the largely unexplored Red Sea.

We present a novel integrated visualization system that enables interactive visual analysis of ensemble simulations used in ocean forecasting, i.e, simulations of sea surface elevation. Our system enables the interactive planning of both the placement and operation of off-shore structures. We illustrate this using a real-world simulation of the Gulf of Mexico. Off-shore structures, such as those used for oil exploration, are vulnerable to hazards caused by strong loop currents. The oil and gas industry therefore relies on accurate ocean forecasting systems for planning their operations. Nowadays, these forecasts are based on multiple spatio-temporal simulations resulting in multidimensional, multivariate and multivalued data, so-called ensemble data. Changes in sea surface elevation are a good indicator for the movement of loop current eddies, and our visualization approach enables their interactive exploration and analysis. We enable analysis of the spatial domain, for planning the placement of structures, as well as detailed exploration of the temporal evolution at any chosen position, for the prediction of critical ocean states that require the shutdown of rig operations.

Image-space line integral convolution (LIC) is a popular scheme for visualizing surface vector fields due to its simplicity and high efficiency. To avoid inconsistencies or color blur during the user interactions, existing approaches employ surface parameterization or 3D volume texture schemes. However, they often require expensive computation or memory cost, and cannot achieve consistent results in terms of both the granularity and color distribution on different scales. This paper introduces a novel image-space surface flow visualization approach that preserves the coherence during user interactions. To make the noise texture under different viewpoints coherent, we propose to precompute a sequence of mipmap noise textures in a coarse-to-fine manner for consistent transition, and map the textures onto each triangle with randomly assigned and constant texture coordinates. Further, a standard image-space LIC is performed to generate the flow texture. The proposed approach is simple and GPU-friendly, and can be easily combined with various texture-based flow visualization techniques. By leveraging viewpoint-dependent backward tracing and mipmap noise phase, our method can be incorporated with the image-based flow visualization (IBFV) technique for coherent visualization of unsteady flows. We demonstrate consistent and highly efficient flow visualization on a variety of data sets.

Image-space line integral convolution (LIC) is a popular approach for visualizing surface vector fields due to its simplicity and high efficiency. To avoid inconsistencies or color blur during the user interactions in the image-space approach, some methods use surface parameterization or 3D volume texture for the effect of smooth transition, which often require expensive computational or memory cost. Furthermore, those methods cannot achieve consistent LIC results in both granularity and color distribution on different scales. This paper introduces a novel image-space LIC for surface flows that preserves the texture coherence during user interactions. To make the noise textures under different viewpoints coherent, we propose a simple texture mapping technique that is local, robust and effective. Meanwhile, our approach pre-computes a sequence of mipmap noise textures in a coarse-to-fine manner, leading to consistent transition when the model is zoomed. Prior to perform LIC in the image space, the mipmap noise textures are mapped onto each triangle with randomly assigned texture coordinates. Then, a standard image-space LIC based on the projected vector fields is performed to generate the flow texture. The proposed approach is simple and very suitable for GPU acceleration. Our implementation demonstrates consistent and highly efficient LIC visualization on a variety of datasets.

Abstract The ability to capture and visualize information within the flow poses challenges for visualizing 3D flow fields. Stream surfaces are one of many useful integration based techniques for visualizing 3D flow. However seeding integral surfaces can be challenging. Previous research generally focuses on manual placement of stream surfaces. Little attention has been given to the problem of automatic stream surface seeding. This paper introduces a novel automatic stream surface seeding strategy based on vector field clustering. It is important that the user can define and target particular characteristics of the flow. Our framework provides this ability. The user is able to specify different vector clustering parameters enabling a range of abstraction for the density and placement of seeding curves and their associated stream surfaces. We demonstrate the effectiveness of this automatic stream surface approach on a range of flow simulations and incorporate illustrative visualization techniques. Domain expert evaluation of the results provides valuable insight into the users requirements and effectiveness of our approach.

Abstract We introduce a novel mid‐structure called the generalized swept mid‐structure (GSM) of a closed polygonal shape, and a framework to compute it. The GSM contains both curve and surface elements and has consistent sheet‐by‐sheet topology, versus triangle‐by‐triangle topology produced by other mid‐structure methods. To obtain this structure, a harmonic function, defined on the volume that is enclosed by the surface, is used to decompose the volume into a set of slices. A technique for computing the 1D mid‐structures of these slices is introduced. The mid‐structures of adjacent slices are then iteratively matched through a boundary similarity computation and triangulated to form the GSM. This structure respects the topology of the input surface model is a hybrid mid‐structure representation. The construction and topology of the GSM allows for local and global simplification, used in further applications such as parameterization, volumetric mesh generation and medical applications.

Robust analysis of vector fields has been established as an important tool for deriving insights from the complex systems these fields model. Traditional analysis and visualization techniques rely primarily on computing streamlines through numerical integration. The inherent numerical errors of such approaches are usually ignored, leading to inconsistencies that cause unreliable visualizations and can ultimately prevent in-depth analysis. We propose a new representation for vector fields on surfaces that replaces numerical integration through triangles with maps from the triangle boundaries to themselves. This representation, called edge maps, permits a concise description of flow behaviors and is equivalent to computing all possible streamlines at a user defined error threshold. Independent of this error streamlines computed using edge maps are guaranteed to be consistent up to floating point precision, enabling the stable extraction of features such as the topological skeleton. Furthermore, our representation explicitly stores spatial and temporal errors which we use to produce more informative visualizations. This work describes the construction of edge maps, the error quantification, and a refinement procedure to adhere to a user defined error bound. Finally, we introduce new visualizations using the additional information provided by edge maps to indicate the uncertainty involved in computing streamlines and topological structures.

Morse decomposition provides a numerically stable topological representation of vector fields that is crucial for their rigorous interpretation. However, Morse decomposition is not unique, and its granularity directly impacts its computational cost. In this paper, we propose an automatic refinement scheme to construct the Morse Connection Graph (MCG) of a given vector field in a hierarchical fashion. Our framework allows a Morse set to be refined through a local update of the flow combinatorialization graph, as well as the connection regions between Morse sets. The computation is fast because the most expensive computation is concentrated on a small portion of the domain. Furthermore, the present work allows the generation of a topologically consistent hierarchy of MCGs, which cannot be obtained using a global method. The classification of the extracted Morse sets is a crucial step for the construction of the MCG, for which the Poincare´ index is inadequate. We make use of an upper bound for the Conley index, provided by the Betti numbers of an index pair for a translation along the flow, to classify the Morse sets. This upper bound is sufficiently accurate for Morse set classification and provides supportive information for the automatic refinement process. An improved visualization technique for MCG is developed to incorporate the Conley indices. Finally, we apply the proposed techniques to a number of synthetic and realworld simulation data to demonstrate their utility.

Design of time-varying vector fields, i.e., vector fields that can change over time, has a wide variety of important applications in computer graphics. Existing vector field design techniques do not address time-varying vector fields. In this paper, we present a framework for the design of time-varying vector fields, both for planar domains as well as manifold surfaces. Our system supports the creation and modification of various time-varying vector fields with desired spatial and temporal characteristics through several design metaphors, including streamlines, pathlines, singularity paths, and bifurcations. These design metaphors are integrated into an element-based design to generate the time-varying vector fields via a sequence of basis field summations or spatial constrained optimizations at the sampled times. The key-frame design and field deformation are also introduced to support other user design scenarios. Accordingly, a spatial-temporal constrained optimization and the time-varying transformation are employed to generate the desired fields for these two design scenarios, respectively. We apply the time-varying vector fields generated using our design system to a number of important computer graphics applications that require controllable dynamic effects, such as evolving surface appearance, dynamic scene design, steerable crowd movement, and painterly animation. Many of these are difficult or impossible to achieve via prior simulation-based methods. In these applications, the time-varying vector fields have been applied as either orientation fields or advection fields to control the instantaneous appearance or evolving trajectories of the dynamic effects.

Vector field visualization techniques have evolved very rapidly over the last two decades, however, visualizing vector fields on complex boundary surfaces from computational flow dynamics (CFD) still remains a challenging task. In part, this is due to the large, unstructured, adaptive resolution characteristics of the meshes used in the modeling and simulation process. Out of the wide variety of existing flow field visualization techniques, vector field clustering algorithms offer the advantage of capturing a detailed picture of important areas of the domain while presenting a simplified view of areas of less importance. This paper presents a novel, robust, automatic vector field clustering algorithm that produces intuitive and insightful images of vector fields on large, unstructured, adaptive resolution boundary meshes from CFD. Our bottom-up, hierarchical approach is the first to combine the properties of the underlying vector field and mesh into a unified error-driven representation. The motivation behind the approach is the fact that CFD engineers may increase the resolution of model meshes according to importance. The algorithm has several advantages. Clusters are generated automatically, no surface parameterization is required, and large meshes are processed efficiently. The most suggestive and important information contained in the meshes and vector fields is preserved while less important areas are simplified in the visualization. Users can interactively control the level of detail by adjusting a range of clustering distance measure parameters. We describe two data structures to accelerate the clustering process. We also introduce novel visualizations of clusters inspired by statistical methods. We apply our method to a series of synthetic and complex, real-world CFD meshes to demonstrate the clustering algorithm results.

Robust analysis of vector fields has been established as an important tool for deriving insights from the complex systems these fields model. Many analysis techniques rely on computing streamlines, a task often hampered by numerical instabilities. Approaches that ignore the resulting errors can lead to inconsistencies that may produce unreliable visualizations and ultimately prevent in-depth analysis. We propose a new representation for vector fields on surfaces that replaces numerical integration through triangles with linear maps defined on its boundary. This representation, called edge maps, is equivalent to computing all possible streamlines at a user defined error threshold. In spite of this error, all the streamlines computed using edge maps will be pairwise disjoint. Furthermore, our representation stores the error explicitly, and thus can be used to produce more informative visualizations. Given a piecewise-linear interpolated vector field, a recent result [15] shows that there are only 23 possible map classes for a triangle, permitting a concise description of flow behaviors. This work describes the details of computing edge maps, provides techniques to quantify and refine edge map error, and gives qualitative and visual comparisons to more traditional techniques.

Asymmetric tensor field visualization can provide important insight into fluid flows and solid deformations. Existing techniques for asymmetric tensor fields focus on the analysis, and simply use evenly-spaced hyperstreamlines on surfaces following eigenvectors and dual-eigenvectors in the tensor field. In this paper, we describe a hybrid visualization technique in which hyperstreamlines and elliptical glyphs are used in real and complex domains, respectively. This enables a more faithful representation of flow behaviors inside complex domains. In addition, we encode tensor magnitude, an important quantity in tensor field analysis, using the density of hyperstreamlines and sizes of glyphs. This allows colors to be used to encode other important tensor quantities. To facilitate quick visual exploration of the data from different viewpoints and at different resolutions, we employ an efficient image-space approach in which hyperstreamlines and glyphs are generated quickly in the image plane. The combination of these techniques leads to an efficient tensor field visualization system for domain scientists. We demonstrate the effectiveness of our visualization technique through applications to complex simulated engine fluid flow and earthquake deformation data. Feedback from domain expert scientists, who are also co-authors, is provided.

In the development of magnetic confinement fusion which will potentially be a future source for low cost power, physicists must be able to analyze the magnetic field that confines the burning plasma. While the magnetic field can be described as a vector field, traditional techniques for analyzing the field's topology cannot be used because of its Hamiltonian nature. In this paper we describe a technique developed as a collaboration between physicists and computer scientists that determines the topology of a toroidal magnetic field using fieldlines with near minimal lengths. More specifically, we analyze the Poincaré map of the sampled fieldlines in a Poincaré section including identifying critical points and other topological features of interest to physicists. The technique has been deployed into an interactive parallel visualization tool which physicists are using to gain new insight into simulations of magnetically confined burning plasmas.

Abstract We introduce a novel, automatic streamline seeding algorithm for vector fields defined on surfaces in 3D space. The algorithm generates evenly spaced streamlines fast, simply and efficiently for any general surface‐based vector field. It is general because it handles large, complex, unstructured, adaptive resolution grids with holes and discontinuities, does not require a parametrization, and can generate both sparse and dense representations of the flow. It is efficient because streamlines are only integrated for visible portions of the surface. It is simple because the image‐based approach removes the need to perform streamline tracing on a triangular mesh, a process which is complicated at best. And it is fast because it makes effective, balanced use of both the CPU and the GPU. The key to the algorithm's speed, simplicity and efficiency is its image‐based seeding strategy. We demonstrate our algorithm on complex, real‐world simulation data sets from computational fluid dynamics and compare it with object‐space streamline visualizations.

This paper addresses the problem of interactively modeling large street networks. We introduce an intuitive and flexible modeling framework in which a user can create a street network from scratch or modify an existing street network. This is achieved through designing an underlying tensor field and editing the graph representing the street network. The framework is intuitive because it uses tensor fields to guide the generation of a street network. The framework is flexible because it allows the user to combine various global and local modeling operations such as brush strokes, smoothing, constraints, noise and rotation fields. Our results will show street networks and three-dimensional urban geometry of high visual quality.

Existing topology-based vector field analysis techniques rely on the ability to extract the individual trajectories such as fixed points, periodic orbits, and separatrices that are sensitive to noise and errors introduced by simulation and interpolation. This can make such vector field analysis unsuitable for rigorous interpretations. We advocate the use of Morse decompositions, which are robust with respect to perturbations, to encode the topological structures of a vector field in the form of a directed graph, called a Morse connection graph (MCG). While an MCG exists for every vector field, it need not be unique. Previous techniques for computing MCG's, while fast, are overly conservative and usually results in MCG's that are too coarse to be useful for the applications. To address this issue, we present a new technique for performing Morse decomposition based on the concept of tau-maps, which typically provides finer MCG's than existing techniques. Furthermore, the choice of tau provides a natural tradeoff between the fineness of the MCG's and the computational costs. We provide efficient implementations of Morse decomposition based on tau-maps, which include the use of forward and backward mapping techniques and an adaptive approach in constructing better approximations of the images of the triangles in the meshes used for simulation.. Furthermore, we propose the use of spatial tau-maps in addition to the original temporal tau-maps. These techniques provide additional trade-offs between the quality of the MCGs and the speed of computation. We demonstrate the utility of our technique with various examples in the plane and on surfaces including engine simulation data sets.

Design and control of vector fields is critical for many visualization and graphics tasks such as vector field visualization, fluid simulation, and texture synthesis. The fundamental qualitative structures associated with vector fields are fixed points, periodic orbits, and separatrices. In this paper, we provide a new technique that allows for the systematic creation and cancellation of fixed points and periodic orbits. This technique enables vector field design and editing on the plane and surfaces with desired qualitative properties. The technique is based on Conley theory, which provides a unified framework that supports the cancellation of fixed points and periodic orbits. We also introduce a novel periodic orbit extraction and visualization algorithm that detects, for the first time, periodic orbits on surfaces. Furthermore, we describe the application of our periodic orbit detection and vector field simplification algorithms to engine simulation data demonstrating the utility of the approach. We apply our design system to vector field visualization by creating data sets containing periodic orbits. This helps us understand the effectiveness of existing visualization techniques. Finally, we propose a new streamline-based technique that allows vector field topology to be easily identified.