Designing 2D Vector Fields of Arbitrary Topology

We introduce a scheme of control polygons to design topological skeletons for vector fields of arbitrary topology. Based on this we construct piecewise linear vector fields of exactly the topology specified by the control polygons. This way a controlled construction of vector fields of any topology is possible. Finally we apply this method for topology‐preserving compression of vector fields consisting of a simple topology.     

Interactive Exploratory Visualization of 2D Vector Fields

In this paper we present several techniques to interactively explore representations of 2D vector fields. Through a set of simple hand postures used on large, touch‐sensitive displays, our approach allows individuals to custom‐design glyphs (arrows, lines, etc.) that best reveal patterns of the underlying dataset. Interactive exploration of vector fields is facilitated through freedom of glyph placement, glyph density control, and animation. The custom glyphs can be applied individually to probe specific areas of the data but can also be applied in groups to explore larger regions of a vector field. Re‐positionable sources from which glyphs—animated according to the local vector field—continue to emerge are used to examine the vector field dynamically. The combination of these techniques results in an engaging visualization with which the user can rapidly explore and analyze varying types of 2D vector fields, using a virtually infinite number of custom‐designed glyphs.     

Compression of 2D Vector Fields Under Guaranteed Topology Preservation

In this paper we introduce a new compression technique for 2D vector fields which preserves the complete topology, i.e., the critical points and the connectivity of the separatrices. As the theoretical foundation of the algorithm, we show in a theorem that for local modifications of a vector field, it is possible to decide entirely by a local analysis whether or not the global topology is preserved. This result is applied in a compression algorithm which is based on a repeated local modification of the vector field ‐ namely a repeated edge collapse of the underlying piecewise linear domain. We apply the compression technique to a number of data sets with a complex topology and obtain significantly improved compression ratios in comparison to pre‐existing topology‐preserving techniques.     

与物理特征相关的平面向量场的拓扑简化及压缩

针对现有拓扑简化方法忽略物理特征保持的缺陷,提出一种对物理特征敏感的平面向量场拓扑简化算法,其中心思想是使用为应用定制的物理判据分类向量场区域,将向量场的特征检出与拓扑简化关联起来.通过合并次要物理特征所在区域上的网格及在新网格布局下重新提取向量场拓扑,该算法不仅能完好地保持场内的重要特征,还能同时实现向量场的数据压缩.实验结果表明,该算法在复杂流场的特征提取方面可发挥重要的作用.     

An Illustrative Visualization Framework for 3D Vector Fields

Most 3D vector field visualization techniques suffer from the problem of visual clutter, and it remains a challenging task to effectively convey both directional and structural information of 3D vector fields. In this paper, we present a novel visualization framework that combines the advantages of clustering methods and illustrative rendering techniques to generate a concise and informative depiction of complex flow structures. Given a 3D vector field, we first generate a number of streamlines covering the important regions based on an entropy measurement. Then we decompose the streamlines into different groups based on a categorization of vector information, wherein the streamline pattern in each group is ensured to be coherent or nearly coherent. For each group, we select a set of representative streamlines and render them in an illustrative fashion to enhance depth cues and succinctly show local flow characteristics. The results demonstrate that our approach can generate a visualization that is relatively free of visual clutter while facilitating perception of salient information of complex vector fields.     

三维矢量场的流动拓扑结构抽取

提出一种基于关键点分类的三维矢量场流动拓扑结构抽取算法,可应用于三维曲线网格、结构化网格和分块网格中.在许多计算流体力学计算中,存在非滑移边界,这种边界上流体的速度为0.通过分析流场边界的表面摩擦场的拓扑,展示绕壁面流体的流动结构;使用图标定位关键点,可交互式地标记和显示涡核区域,并通过选择暗示螺旋流动的图标,沿着该关键点的实特征值对应的特征矢量方向积分流线来完成.测试结果清晰地展示了关键特征区域的流体流动特征.     

Robust Reference Frame Extraction from Unsteady 2D Vector Fields with Convolutional Neural Networks

Robust feature extraction is an integral part of scientific visualization. In unsteady vector field analysis, researchers recently directed their attention towards the computation of near‐steady reference frames for vortex extraction, which is a numerically challenging endeavor. In this paper, we utilize a convolutional neural network to combine two steps of the visualization pipeline in an end‐to‐end manner: the filtering and the feature extraction. We use neural networks for the extraction of a steady reference frame for a given unsteady 2D vector field. By conditioning the neural network to noisy inputs and resampling artifacts, we obtain numerically stabler results than existing optimization‐based approaches. Supervised deep learning typically requires a large amount of training data. Thus, our second contribution is the creation of a vector field benchmark data set, which is generally useful for any local deep learning‐based feature extraction. Based on Vatistas velocity profile, we formulate a parametric vector field mixture model that we parameterize based on numerically‐computed example vector fields in near‐steady reference frames. Given the parametric model, we can efficiently synthesize thousands of vector fields that serve as input to our deep learning architecture. The proposed network is evaluated on an unseen numerical fluid flow simulation.     

三维无旋矢量场的一种新的可视化方法

提出了三维无旋矢量场的一种新的可视化方法,即构造空间曲面,使得矢量场在曲面上任意一点处垂直于该曲面。首先找到曲面所满足的偏微分方程组,通过采用类似于经典四阶龙格―库塔方法的数值解法对其求解,得到曲面上的离散点,然后进行三角剖分,从而得到逼近于曲面的空间三角网格。论文的偏微分方程组的求解借鉴了常微分方程求解算法的设计思想,构造出的曲面与传统的点图标和线图标相比,在更大程度上揭示了矢量场本身的连续性。     

Visualization of Equivalence in 2D Bivariate Fields

In this paper, we show how the equivalence property leads to the novel concept of equivalent regions in mappings from ?ⁿ to ?ⁿ. We present a technique for obtaining these regions both in the domain and the codomain of such a mapping, and determine their correspondence. This enables effective investigation of variation equivalence within mappings, and between mappings in terms of comparative visualization. We implement our approach for n = 2, and demonstrate its utility using different examples.     

Uncertain 2D Vector Field Topology

We introduce an approach to visualize stationary 2D vector fields with global uncertainty obtained by considering the transport of local uncertainty in the flow. For this, we extend the concept of vector field topology to uncertain vector fields by considering the vector field as a density distribution function. By generalizing the concepts of stream lines and critical points we obtain a number of density fields representing an uncertain topological segmentation. Their visualization as height surfaces gives insight into both the flow behavior and its uncertainty. We present a Monte Carlo approach where we integrate probabilistic particle paths, which lead to the segmentation of topological features. Moreover, we extend our algorithms to detect saddle points and present efficient implementations. Finally, we apply our technique to a number of real and synthetic test data sets.     

Selective Visualization of Vector Fields

In this paper, we present an approach to selective vector field visualization. This selective visualization approach consists of three stages: selectdon creation, selection processing and selective visualization mapping. It is described how selected regions, called selections, can be represented and created, how selections can be processed and how they can be used in the visualization mapping. Combination of these techniques with a standard visualization pipeline improves the visualization process and offers new facilities for visualization. Examples of selective visualization of fluid flow datasets are provided.     

Hierarchical Correlation Clustering in Multiple 2D Scalar Fields

Sets of multiple scalar fields can be used to model many types of variation in data, such as uncertainty in measurements and simulations or time‐dependent behavior of scalar quantities. Many structural properties of such fields can be explained by dependencies between different points in the scalar field. Although these dependencies can be of arbitrary complexity, correlation, i.e., the linear dependency, already provides significant structural information. Existing methods for correlation analysis are usually limited to positive correlation, handle only local dependencies, or use combinatorial approximations to this continuous problem. We present a new approach for computing and visualizing correlated regions in sets of 2‐dimensional scalar fields. This paper describes the following three main contributions: (i) An algorithm for hierarchical correlation clustering resulting in a dendrogram, (ii) a generalization of topological landscapes for dendrogram visualization, and (iii) a new method for incorporating negative correlation values in the clustering and visualization. All steps are designed to preserve the special properties of correlation coefficients. The results are visualized in two linked views, one showing the cluster hierarchy as 2D landscape and the other providing a spatial context in the scalar field's domain. Different coloring and texturing schemes coupled with interactive selection support an exploratory data analysis.     

基于聚类的二维向量场可视化

已有的二维流场可视化中,鞍点等临界点是最重要的特征之一.文中从一个新的角度提出一种基于流线聚类的二维向量场可视化方法.首先生成采样流线集合,然后将流线聚类,最后引入共轭法向量场和流线密度矩阵对同一个类的流线进行加速排序.在此基础上,提出3种可视化应用:抽取每一类的代表流线进行向量场的流线简洁表达;根据流线之间距离进行多分辨率均匀流线表达;生成权值图,增强基于纹理的向量场可视化.实验结果表明,该方法具有良好的鲁棒性,可视化效果优于已有的方法.     

Inertial Steady 2D Vector Field Topology

Vector field topology is a powerful and matured tool for the study of the asymptotic behavior of tracer particles in steady flows. Yet, it does not capture the behavior of finite‐sized particles, because they develop inertia and do not move tangential to the flow. In this paper, we use the fact that the trajectories of inertial particles can be described as tangent curves of a higher dimensional vector field. Using this, we conduct a full classification of the first‐order critical points of this higher dimensional flow, and devise a method to their efficient extraction. Further, we interactively visualize the asymptotic behavior of finite‐sized particles by a glyph visualization that encodes the outcome of any initial condition of the governing ODE, i.e., for a varying initial position and/or initial velocity. With this, we present a first approach to extend traditional vector field topology to the inertial case.     

2D Vector field approximation using linear neighborhoods

We present a vector field approximation for two-dimensional vector fields that preserves their topology and significantly reduces the memory footprint. This approximation is based on a segmentation. The flow within each segmentation region is approximated by an affine linear function. The implementation is driven by four aims: (1) the approximation preserves the original topology; (2) the maximal approximation error is below a user-defined threshold in all regions; (3) the number of regions is as small as possible; and (4) each point has the minimal approximation error. The generation of an optimal solution is computationally infeasible. We discuss this problem and provide a greedy strategy to efficiently compute a sensible segmentation that considers the four aims. Finally, we use the region-wise affine linear approximation to compute a simplified grid for the vector field.     

Mandatory Critical Points of 2D Uncertain Scalar Fields

This paper introduces a novel, non‐local characterization of critical points and their global relation in 2D uncertain scalar fields. The characterization is based on the analysis of the support of the probability density functions (PDF) of the input data. Given two scalar fields representing reliable estimations of the bounds of this support, our strategy identifies mandatory critical points: spatial regions and function ranges where critical points have to occur in any realization of the input. The algorithm provides a global pairing scheme for mandatory critical points which is used to construct mandatory join and split trees. These trees enable a visual exploration of the common topological structure of all possible realizations of the uncertain data. To allow multi‐scale visualization, we introduce a simplification scheme for mandatory critical point pairs revealing the most dominant features. Our technique is purely combinatorial and handles parametric distribution models and ensemble data. It does not depend on any computational parameter and does not suffer from numerical inaccuracy or global inconsistency. The algorithm exploits ideas of the established join/split tree computation. It is therefore simple to implement, and its complexity is output‐sensitive. We illustrate, evaluate, and verify our method on synthetic and real‐world data.     

Visual Analysis of Governing Topological Structures in Excitable Network Dynamics

To understand how topology shapes the dynamics in excitable networks is one of the fundamental problems in network science when applied to computational systems biology and neuroscience. Recent advances in the field discovered the influential role of two macroscopic topological structures, namely hubs and modules. We propose a visual analytics approach that allows for a systematic exploration of the role of those macroscopic topological structures on the dynamics in excitable networks. Dynamical patterns are discovered using the dynamical features of excitation ratio and co‐activation. Our approach is based on the interactive analysis of the correlation of topological and dynamical features using coordinated views. We designed suitable visual encodings for both the topological and the dynamical features. A degree map and an adjacency matrix visualization allow for the interaction with hubs and modules, respectively. A barycentric‐coordinates layout and a multi‐dimensional scaling approach allow for the analysis of excitation ratio and co‐activation, respectively. We demonstrate how the interplay of the visual encodings allows us to quickly reconstruct recent findings in the field within an interactive analysis and even discovered new patterns. We apply our approach to network models of commonly investigated topologies as well as to the structural networks representing the connectomes of different species. We evaluate our approach with domain experts in terms of its intuitiveness, expressiveness, and usefulness.     

面状矢量拓扑数据快速栅格化算法

针对GIS面状拓扑数据,提出了一种快速栅格化算法——差分边界标志与累加扫描算法,首先对所有的面状拓扑数据中的弧段进行顺序扫描,在栅格缓冲区中利用差分边界标志法进行边界标志,然后利用累加扫描线法对栅格缓冲区的各行从左至右进行累加扫描充填,该算法不仅实现简单,而且由于算法中充分利用了弧段的拓扑特征,避免了多边形区域的组织和弧段的重复处理,从而保证了海量面状拓扑数据栅格化的效率;同时还对栅格化算法中的退化问题提出了解决方案,实际应用表明,文中算法具有较高的效率和较强的实用性。     

A Linear 3D Elastic Segmentation Model for Vector Fields. Application to the Heart Segmentation in MRI

In this paper a 3D elastic model for the segmentation of vector fields has been proposed and analyzed. Elastic models for segmentation usually involve minimization of internal and external energy. A problem we observed with standard internal and external energy is that the local or the global reached minima do not force the external energy to be zero. To eliminate this difficulty, we propose introducing a constraint. The constraint problem is proved to be mathematically well posed, and a simple algorithm which avoids computing the lagrange multiplier is provided. This algorithm is proved to be convergent. Then the algorithm is applied to the segmentation of cardiac magnetic resonance imaging, and its efficiency is shown with two experiments. Martine Picq is member of the Institute of Mathematics C. Jordan in National Institute of Applied Sciences in Lyon, where she is teaching mathematics since 1997. Jerome Pousin received a Ph.D. degree in Applied Mathematics from University of Paris 6 France in 1983 and Ph.D. degree in Mathematic Sciences from EPFL Switzerland in 1992. Since 1993 he is professor of Mathematics at the National Institute of Applied Sciences in Lyon. His research interests are approximation of nonlinear Partial Differential Equations with Finite Element Method; domain decomposition methods and image segmentation with deformable models. Youssef Rouchdy received a Ph.D. degree in Applied Mathematics from the National Institute of Applied Sciences in Lyon in 2005. He is currently a Postdoc at INRIA Sophia Antipolis France.