流体特征描述、检测理论总结

流场特征描述、检测理论总结报告

1.绪论

特征定义为数据中不同成分间相关性的表现模式，具有两方面的含义：“（1）矢量场中有意义的形状、结构、变化和现象，如涡流、激波等；（2）从数据场中分离出来的用户感兴趣的区域。” 。目前流场特征的分析方法有多种，大致可以归结为三类：基于图像处理、基于拓扑分析（即基于矢量场分析）以及基于物理特征方法[1]，但主要是基于拓扑分析方法。 原因分析：流场特征结构概念的提出最原始就是基于拓扑分析方法，因此每个流场特征结构一般具有严格的数学定义，而不是像一般图像处理中的模式识别方法针对的目标对象，如人脸、指纹等不存在严格定义。因此拓扑分析仍然是最有效的流场特征提取方法。 缺点：并不是所有的流场结构都有严格的数学定义，譬如涡旋结构，在2D 空间中，源点、汇点以及中心点等对应的流场周围都符合涡旋的特征。

2.基本概念

为更好的描述矢量场，首先介绍如下基本概念[1]。

2.1 梯度

三维空间中，标量p 有三个偏微分算子。标量场的梯度这三个偏微分算子表示：

{,,}p p p gradp p x y z

???=?=??? （1） 向量场v 的梯度可以通过应用梯度算子在其分量{u,v,w}上而计算。结果是 一个3*3矩阵，称作Jacobian 矩阵，或者一阶算子。

u u u x y z v v v v x y z w w w x y z ??

??? ???????

??????= ???????

????? ??????

? （2） 该矩阵可用来计算一些导数场，譬如散度（divergence ），旋度(curl)，螺旋性(helicity)，加速(acceleration)，曲率(curvature)。速度场的旋度称作涡度（vorticity ）。这些导数场暗示了流绕旋转轴旋转的强度。这些数量已被用于不同的特征抽取技术方面，将在下面介绍。

2.2 本征分析(Eigenanalysis)

另一种独立的数据技术是本征分析。一个3*3矩阵M 的特征值可以通过求解特征向量方程M x x λ= 求解得到。相应的x 称作特征向量。J 矩阵的特征向量和特征值暗示了流的正切曲线方向，被用来定义矢量拓扑，见下面定义。

2.3属性计算

作为特征抽取过程的一部分，特征属性需要计算。一种简单技术是计算一个椭圆体（ellipsoid fitting）。椭圆体给定了一个对象方向的一阶估计。轴可以被扩展来给定对象的尺寸或体积的精确描述。相关文献参见[1]。

另一种技术是中心线提取(center line extraction)。譬如，一个梗概，或者中轴变换，将对象缩减为一个单一中心线或图，同时保持了对象的原始拓扑关系。利用该图，可以构建圆柱或半球来近似表示原始拓扑对象的近似【1】。

3.向量场与动力系统

3.1 二维平面流场

略，参见文献[6]第二章。以及文献[7]相关章节。

总结：相对比较成熟、目前主要研究的是复杂流场的拓扑简化、拓扑扩张以及一些复杂不稳定现象如分岔等现象。

4. 三维流场[8,9]

拓扑方法是可视化2D向量场的标准方法。由于其能够通过有限数目的图形原语来表示复杂的流体行为，因此获得了广泛的应用。其基本思想将向量场分割为不同的流场行为区域，一般通过如分割线，主要是从临界点起始的流线，来进行抽取和可视化。一些相关参考文献见[9]中相关文献。

但对于3D向量场，在向量场的可视化简化表达需求、拓扑概念知识和作为可视分析工具之间的具有巨大的差距。

本文的目的在于研究应用拓扑方法的局限性所在，并提供在两方面扩展这些局限的方法。对于上限（即是说，如果数据太复杂，则拓扑方法会失败），存在许多技术与感知原因。这里我们论述了存在一个另外的理论原因，会强烈限制3D拓扑只能应用到相对简单的数据集(??)。原因在于不同流体行为分割区域数的迅速增长。我们证明了，相对于2D向量场，在最坏情况下不同流畅行为分割区域数目与现有特征成二次方关系增长。结果导致经典拓扑方法（焦点集中在抽取临界点与分割平面）与拓扑复杂的向量场并不相关。另外，我们证明了对于简化费那个发如鞍点连接子[]，上限超出了当前考虑的拓扑复杂性。实际上，我们应用了拓扑方法的应用比目前可视化领域考虑的向量场在拓扑结构上更为复杂。

对于拓扑方法的下限（即是说，拓扑方法对于简单拓扑失败），一种简单而众所周之的解决方法是将拓扑复杂性上移到一个范围，在该范围内拓扑结构用可视化方法能有效表示。该方法一般通过移除流体大的周围环境部分。我们证明了这种方法能揭示一些类型流体的重要结构。

下面小节组织如下，首先介绍3D向量场及其可视化的拓扑概念。然后研究上限问题。接下来用一个实例证明拓扑方法如何应用到更为复杂的流场。再下来，给出并讨论解决一个具有非常低拓扑复杂度的数据集的解决方法。最后，给出结论。

4.1 3D向量场拓扑与可视化

3D向量场拓扑结构一直是可视化研究领域的重点之一[8-13]。本节中，我们收集了最重要的概念和属性，以及相关可视化方法。

4.1.1 临界点

考虑一个3D 向量场

(1 )

一个一阶临界点x0可以通过其偏导数矩阵特征分类。偏导数矩阵v J 的特征值，i λ的实数部

分代表了流的方向，正值表示从临界点流出，而负值则表示流向临界点，假设123R e()R e()R e()λλλ≤≤，则根据特征值的大小，3D 流场临界点可以分为以下四类：

源（Sources ）： 1230R e ()R e ()R e

()λλλ<≤≤ 排斥马鞍点（Repelling saddles ）： 123Re()0Re()Re()λλλ<<≤

吸引马鞍点(Attracting saddles)： 123R e ()R e ()0R e

()λλλ≤<< 汇（Sinks ）： 123R e ()R e ()R e ()0

λλλ≤≤< 因此，源和汇是一个完全的出流/入流,而鞍点则结合了两者。排斥马鞍点表示一个方向的入流行为(称作inflow direction)和一个平面上的出流行为（称作outflow plane ）。于此类似，吸引马鞍点由一个outflow direction 和一个inflow plane 组成。

上述的四个分类中的每个根据其三个特征值的两个值的复数部分是否为0可以进一步划分为两个稳定的子类。（123,,λλλ没有按照顺序排列）

对于临界点，一般给定一个图标化的标识符来表示，因为向量场通常包含有限个临界点。相关文献中提到了不同的图标表示[12-18]，这里遵循[18]中的表示方法：吸引部分(入流)用蓝色表示，而排斥部分(出流)用红色表示。另外，本文不考虑高阶临界点。

4.1.2 分隔符

分隔符是流线或流面，它将向量场分割成不同流行为的区域。可能有不同种类的分隔符：他们能从临界点后边界转换曲线(boundary switch curves )发出，或者是封闭的分隔符而没有任何一个特定的发出结构。然而，在本文中，我们只考虑从临界点出发的分隔符。

由于围绕源和汇的流的一致性，它们对分界符没有贡献。每个马鞍点创造两个分界符：考虑一个排斥马鞍Xr,其创建一个分割曲线（是一条流线，起始于Xr ，沿入流方向后向传播）和一个分割曲面（是一个流面，起始于出流平面前向传播）；吸引马鞍情况类似。由于仅仅是分割曲面（而不是分割线）能将向量场分割为不同的流行为区域，因此在下面中我们仅仅考虑分割曲面。

These regions are separated by particular stream lines called separatrices. There exist three kinds of regions (Figure 1):

? Parabolic sectors: all stream lines either start or end in the critical point.

? Elliptic sectors: all stream lines start and end in the point.

? Hyperbolic sectors: all stream lines pass by the critical point except for the separatrices themselves.

Given a 3D vector field , a point x0 is a critical point if v(x0) = (0,0,0)T , and

for any other point in a certain neighborhood of x0. Although there exist profound mathematical studies of the behavior of higher order critical points in 3D ([Tak74, BD86, DI98]), up to now the Visualization and Computer Graphics community only considered the index of such a critical point ([MR02]). In the following we introduce a partition approach for 3D critical points.

A 3D critical point can be classified into sectors of different flow behavior. As in the 2D case, there are parabolic, elliptic and hyperbolic sectors which are separated by separation surfaces. Consider a small sphere s around a 3D critical point x0:We describe the different sectors of x0 as certain regions on s. This way an inflow surface of x0 can be described by a closed curve cin on s with the property that a stream line starting from any point of c in ends in x0 by forward integration. Similarly, an outflow surface is described by a closed curve c out with stream lines ending in x0 by backward integration. Figure 2 illustrates inflow and outflow surfaces of a critical point. Note that they may collapse to a line: this happens when c in or c out collapse to a point.

Consider one inflow and one outflow surface of x0 defined by cin and cout on s. Keeping in mind that s covers a very small neighborhood of x0, cin and cout must not intersect. They divide s into three different regions: the region inside cin, the region inside cout , and the region between them (Figure 3). The region inside cin is an inflow sector: all stream lines starting there end in x0 by forward integration. Similarly, the region inside cout is an outflow sector. As in the 2D case, both inflow and outflow sectors are parabolic sectors. The region between cin and cout may be either a hyperbolic sector or an elliptic sector. In case of a hyperbolic sector, all

stream lines pass by x0, for an elliptic sector all stream line start and end in x0. Figure 3 illustrates the different sectors.

4.特征提取方法

4.1 漩涡检测方法：

可以分为两类[19]：基于点的方法和基于集合的方法；详细请参见文献19（流场特征目录下）。

4.1.1 基于点的方法

1) Pressure Magnitude

One can use an area of low pressure to define a vortex region, so the vortex covers the region

where p ≤pthreshold (1)

The rationale behind this is that a pressure gradient with a minimum of pressure at the center of the vortex produces the centripetal force of a rotating motion. An arbitrary threshold is required for this method, and viscous forces acting on the field are neglected.

2) Helicity

High helicity can be used to define a vortex as the region in which v ·ω

|v|·|ω| ≥hthreshold (2)

where, ω = ?×v is the vorticity. This method fails in some simple situations. Since helicities are all zero in 2D flows and this method does not apply to them. Another counter example is that shear flow along a curved boundary has high helicity due to shear effects, but

vortices may not exist in it.

3) Vorticity Magnitude

This commonly used method uses vorticity as the definition of rotating motion and defines a vortex as a region of high vorticity. According to this method, a vortex is a region where the vorticity is greater than or equal to some threshold ω:

|?×v| ≥ωthreshold (3)

Like the previous methods, this method demands some arbitrary value for ωthreshold. In addition, the absence of vorticity does not necessarily guarantee that there is no vortex, as can be shown in the case of a potential vortex [20].

4) Conjugate Pair Eigenvalues

Chong et al. [2] used eigenvalues of the velocity gradient tensor ?v to classify the local streamline pattern around any point in a flow in a reference frame moving with the velocity of that point. They proposed that a vortex core is a region with complex eigenvalues of the velocity gradient tensor ?v. Complex eigenvalues imply that the local streamline pattern is closed or spiral in a reference frame moving with that point. Complex eigenvalues are essentially the necessary

condition of swirling flows in the vortex area. The eigenvalues, λ, of the tensor ?v satisfy the characteristic equation:

where P ≡vi,i = 0 (for incompressible flow), Q ≡1 2 (v2i ,i ?vi, jvj,i) = ?12 vi, jvj,i and R = Det(vi, j) are the three invariants of ?v, and the summation notation is used for repeated indices. Complex eigenvalues will occur when the discriminant (?) is positive, i.e.,

This definition, derived from ?v, is the Galilean invariant [8].

5) Negative Eigenvalue λ2

The λ2 method proposed by Jeong and Hussain [8] is a widely used vortex detection method. First, the local Jacobian J = ?v is decomposed into symmetric and antisymmetric parts S and Ω:

Next, the eigenvalues of the matrix S2 +Ω2 are determined. All three eigenvalues are real because of the symmetric nature of this matrix. The λ2 criterion then defines a vortex as the region where at least two of the three values are negative, which is identical to the condition that the second largest eigenvalue is negative λ2 < 0.

6) Positive Second Invariant of Jacobian

This is based on the second invariant of a 3D matrix. A vortex is the region where Q is positive [8]. Another vortex definition proposed [6], required that the second invariant Q of the Jacobian J is positive and that local pressure is smaller than the surrounding pressure. Jeong and Hussain [8] have observed that the second condition is always true if the first one is, thus reducing the condition to only demanding a positive second invariant Q > 0.

7）总结：

The point-based methods have been used in many applications. As long as the vortices have strong rotation speed around vortex cores and different vortices are sufficiently isolated from each other, point-based methods can be used and fairly good results will be achieved. But none of these point-based methods is able to reliably detect all vortices in large-scale real ocean flows. Here are some typical problems: In some cases, although stronger vortices which have high angular velocity are usually found, weaker vortices with low angular velocity remain undetected by point based methods. In other cases, some flows with seemingly swirling properties are detected. However, they are not necessarily vortices. They could be shear flows near the coastlines. Thus, false positives are introduced in the results. Moreover, if two vortical structures are close together, they will often be lumped together by point-based techniques, even if they for example have opposite direction of rotation. In fact, a vortex structure has only one vortex core. If detection methods indicate that two vortices merge into one, it should be further examined whether this is actually the case of two vortices simply moving too close to each other to be clearly distinguished, but still having two different vortex cores.

Last but not least, arbitrary thresholds may need to be used in point-based techniques. Thresholds are based on field values, which make thresholds hard to be determined. In different cases, threshold values need to vary in order to get good results. Thus, user interaction is required, and the whole process is not fully automated. In a nutshell, a vortex is essentially a macroscopic or regional phenomenon, yet the point values underlying the point-based criteria do not always translate into regional characteristics. A review of point-based methods can be found in Portela [14] and Roth[19].

4.1.2 基于集合的方法：

参考文献：

[1] F. H. Post, B. V rolijk, H. Hauser, R. S. Laramee and H. Doleisch. The state of the art in flow visualization:

feature extraction and tracking. Computer Graphics Forum, 22(4):775–792, 2003.

[2]Helman J L, Hesselink L. Visualizing V ector Field Topology in Fluid Flows[J]. IEEE Computer Graphics and

Applications. 1991, 11(3): 36-46.

[3]Filip Sadlo, Ronald Peikert and Etienne Parkinson. V orticity Based Flow Analysis and Visualization for

Pelton Turbine Design Optimization[C] // Proceedings of IEEE V isualization, Austin, 2004: 179-186.

[4] F. Reinders. Feature-Based V isualization of Time-Dependent Data. PhD thesis, Delft University of

Technology, The Netherlands, 2001.

[5]32. F. Reinders, M. E. D. Jacobson, and F. H. Post. Skeleton graph generation for feature shape description. In

Data V isualization 2000. Proc. V isSym’00, pages 73– 82, 2000.

[6]G Scheuermann, X Tricoche. Topological Methods for Flow V isualization. 2004

[7]Xavier Tricoche, PhP thesis: V ector and Tensor Field Topology Simplification, Tracking, and V isualization.

2004.

[8]吴克勤, 复杂时变流场中不稳定结构的检出及可视化——极限环的抽取及向量场拓扑简化研究, 中国

海洋大学硕士学位论文, 20040701.

[9]T. Weinkauf, H. Theisel, H.-C. Hege1 and H.-P. Seidel. Topological Construction and V isualization of

Higher Order 3D V ector Fields, EUROGRAPHICS 2004, V olume 23 (2004), Number 3.

[10]Holger Theisel1, Tino Weinkauf2, Hans-Christian Hege3, Hans-Peter Seidel4. On the Applicability of

Topological Methods for Complex Flow Data, 2004.

[11]D. Asimov. Notes on the topology of vector fields and flows. Technical report, NASA Ames Research Center,

1993. RNR-93-003.

[12]M. S. Chong, A. E. Perry, and B. J. Cantwell. A general classification of threedimensional flow fields. Physics

of Fluids A, 2(5):765–777, 1990.

[13]J. Helman and L. Hesselink. V isualizing vector field topology in fluid flows. IEEE Computer Graphics and

Applications, 11:36–46, May 1991.

[14]G. Scheuermann, T. Bobach, H. Hagen K. Mahrous, B. Hamann, K. Joy, and W. Kollmann. A

tetrahedra-based stream surface algorithm. In Proc. Visual-ization 01, pages 151 – 158, 2001.

[15]A. Globus, C. Levit, and T. Lasinski. A tool for visualizing the topology of three-dimensional vector fields. In

Proc. IEEE V isualization ’91, pages 33–40, 1991.

[16]H. Hauser and E. Gr¨oller. Thorough insights by enhanced visualization of flow topology. In 9th

international symposium on flow visualization, 2000.

[17]H. L¨offelmann, H. Doleisch, and E. Gr¨oller. V isualizing dynamical systems near critical points. In Spring

Conference on Computer Graphics and its Applications, pages 175–184, Budmerice, Slovakia, 1998.

[18]H. Theisel, T. Weinkauf, H.-C. Hege, and H.-P. Seidel. Saddle connectors – an approach to visualizing the

topological skeleton of complex 3D vector fields. In Proc. IEEE Visualization 2003, pages 225–232, 2003. [19]T. Weinkauf, H. Theisel, H.-C. Hege, and H.-P. Seidel. Boundary switch connectors for topological

visualization of complex 3D vector fields. In Data V isualization 2004. Proc. V isSym 04, pages 183–192, 2004.

[20]Da Guo Constantinos Evangelinos Nicholas M. Patrikalakis. Flow Feature Extraction in Oceanographic

V isualization. Pro ceedings of the Computer Graphics International (CGI’04).

[21]