Visualizing Planar Vector Fields with Normal ComponentUsing Line Integral Convolution

Gerik Scheuermann� Holger Burbachy Hans Hagenz

Department of Computer ScienceUniversity of Kaiserslautern, Germany

Figure 1: Visualization of the deformation of an intake manifold’s flange due to axial compression using our combined LIC and deformationstrategy. This intake manifold is designed and produced by Mann & Hummel and is used in DaimlerChrysler’s Mercedes-Benz C250 andE290 passenger cars.

Abstract

We present a method for visualizing three dimensional vector fieldswhich are defined on a two dimensional manifold only. These vec-tor fields do exist in real application, as we show by an example ofan optical measuring instrument which can gauge the displacementat the surface of a mechanical part. The general idea is to com-pute LIC textures in the manifold’s tangent space and to deformthe manifold according to the normal information. The resultingLIC texture is mapped onto the deformed manifold and is renderedas a three dimensional scene. Due to the light’s reflection on thedeformed manifold, one can interactively explore the result of thedeformation.

Keywords: LIC, vector field visualization, deformation

�E-Mail: scheuer@informatik.uni-kl.deyE-Mail: burbach@informatik.uni-kl.dezE-Mail: hagen@informatik.uni-kl.de

1 Introduction

Vector fields play a substantial role in mechanical engineering andphysics. In recent years, many different techniques for visualizingtwo and three-dimensional vector fields have been developed. Tra-ditional approaches in vector field visualization comprise symbolicrepresentations with glyphs, streamlines, and stream surfaces. Al-though important features of the vector field can be extracted withall these methods, in general they require detailed knowledge aboutwhere to place symbols or seed points. Depending on their place-ment, eddies or other singularities can be missed. Texture basedmethods do not suffer from these problems and will be introducednext.

A continuous visualization of a two-dimensional vector field canbe presented by a texture. The directional information is depictedby line structures in the direction of the vector field. These lines canbe “seen” due to the higher coherency between neighboring pixelsin the field direction. Spot noise [16] and line integral convolution[3] are based on this principle (see [4] for a detailed comparison).In computational fluid dynamics (CFD)the flow direction is oftenvisualized by a gray-scale LIC image. The image can be coloredby the velocity of the flow. This can be easily achieved by usingthe HSV color model. The velocity is mapped into hue and thegray-scale LIC image is used as value.

We have focused our attention to a special case of vector fields:

v :M�! R3 (1)

whereM is an open subset ofR2 . The idea is to split the vectorfield into tangential and normal components. Then we can computean LIC texture in tangential space and deform the plane or manifoldaccording to the given vector field. Subsequently, we can use stan-dard texture mapping technique to project the LIC image onto the

0-7803-5897-X/99/$10.00 Copyright 1999 IEEE

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deformed plane. The resulting three-dimensional scene is lit by par-allel light and its diffuse reflection on the deformed plane enablesthe viewer to experience the three-dimensional deformation. Thisprocess is supported either by interactively changing the point ofview or by automatically moving the light source around the objecton a predefined path. The movement of the shadows on the de-formed surface makes it possible to detect the three-dimensionalityof the deformation.

It should be noted that the extension of our strategy is straightforward, if M is a two-dimensional manifold inR3 instead, al-though it might be hard to distinguish between the initial deforma-tion of the surface and the actual deformation itself. One idea is toenable the viewer to interactively change the scaling of the defor-mation normal to the surface, so it is possible to find the areas oflarge normal deformations.

In the following, we will first present, in Section 2, some back-ground information and cite related works. Section 3 describes ouridea and strategy in detail. In Section 4 we introduce an applicationfrom mechanical engineering to prove the relevance for examiningplanar vector fields with normal components. Some implementa-tion issues and results are discussed in Section 5. Section 6 con-cludes the paper with some final remarks and future work.

2 Background and Related Work

To visualize three dimensional vectors on a manifold, placingglyphs (e.g. directed lines, arrows, cones, etc.) at selected pointsof the vector field, is an idea which immediately springs to mind.The directed line, for example, starts at the point with which thevector is associated and is oriented in the direction of the vectorcomponents. The length of this line should be equal to the vectornorm, but typically, scaling by a constant is necessary to improvevisual results. This method is often referred to as ahedgehog. Ren-dering techniques for displaying deformation surfaces have beenpreviously investigated and are a standard feature in some visual-ization systems such as FAST [1].

One more sophisticated method was presented by Max, Crawfisand Grant [10]. Their idea was to grow little hairs out of the surfaceand then have them bend in the flow field. Since we have no vectorfield defined outside of the plane, this method is in our case ratherequivalent to the above hedgehog.

Another method has recently gained very much popularity:LineIntegral Convolution (LIC). It is based on van Wijk’sspot noise[16] and was introduced by Cabral and Leedom [3]. Although itis able to deal with two dimensional vector fields only, it providesmaximum insight into the directional structure of the vector field.Such an vector fieldv : R2 �! R

2 can be graphically depicted byits integral curves, also known as streamlines. The tangent vectorsof an integral curve coincide with the vector field:

d

du�(u) = v(�(u)) (2)

Using arc length reparametrization and chain rule yields

d

ds�(s) =

d�

du

du

ds=

v

jvj(3)

Note, that this reparametrization is only valid ifjvj 6= 0 along�(u).Now we can compute the streamline atx0 by solving the differentialequation (3) with initial condition�(0) = x0.

Let � be an arbitrary streamline. Then theline integral convolu-

sub sampling

triangulation

triangle reduction

creating triangle strips

computing normals

texture mapping

OpenGL rendering

vector field

FastLIC

light, camera, material properties

warping geometry

Figure 2: Data-flow diagram of our visualization pipeline

tion atx0 = �(0) is defined by

I(x0) =

LZ

�L

k(s)T (�(s))ds (4)

whereT is a white noise texture andk is some filter kernel withR L�L

k(s)ds = 1. The resulting image consists of highly correlatedpixel intensities along the streamlines, but independent in perpen-dicular directions. This original LIC method tends to be slow, dueto the fact that a new streamline is computed for every pixel ofthe output image. Premising a constant filter kernelk, Stalling andHege [15] developed a very fast method for computing LIC images,calledFastLIC. This method computes many pixel intensities alongeach individual streamline by updating the pixel intensities with justtwo small correction terms.

The above standard LIC algorithm is applicable only to vec-tor fields over regular two dimensional Cartesian grids. Forssell[5] presented an extension to the LIC method to visualize vec-tor fields over curvilinear grid surfaces, i.e. surfaces which can beparametrized globally using 2D-coordinates. Battke, Stalling, andHege [2,7] have used an approach in which the surfaces are tessel-lated with triangles. Therefore they are able to handle the case ofgeneral, possibly multiply connected surfaces.

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A related technique for our approach was presented by Okadaand Kao [11] in which LIC textures were colored based on thenormal-component of the 3D vector field to reveal flow separationfrom a manifold in fluid flows.

3 Visualizing Planar Vector Fields withNormal Component

Let us assume, that the three dimensional vector fieldv : R2 �!R3 is discretely defined on a planar and rectangular Cartesian grid.

3.1 LIC Computation

First, we project the given vector field into the tangent space. Sincethe tangent space is here the parameter space (the plane) itself, weget a two dimensional vector fieldv : R2 �! R

2 which simplyconsists of thex- andy-components ofv.

Then we compute an LIC texture of the vector fieldv. The in-put texture is a white noise image with 256 different gray shades.We implemented the FastLIC algorithm using a fifth-order Runge-Kutta integrator with adaptive step size adjustment (see [12] fordetails). As already mentioned by Stalling and Hege [15], the orderof pixels, where streamlines are being computed, is relevant for theexecution time of the FastLIC algorithm. We verified in our testcases that quasi-random sequences are superior to simple box orscanline sequences, but results may vary depending on the vectorfield topology.

3.2 Triangulation of the Plane

The next step is the triangulation of the defining plane. If the vec-tor field v is defined on a regular grid, e.g. quadrilateral cells, thistask is quite easy. Otherwise more sophisticated triangulation algo-rithms must be employed.

3.3 Deformation of the Planar Triangle Mesh

We now deform the previously generated triangle mesh accordingto the vector fieldv. Therefore, we shift every mesh point alongthe normal vector about the normal component of the associatedvector. Typically, this length is scaled by a user defined scalingfactor� > 0. Figure 3 shows how the grid pointP is moved to itsnew positionP 0 = P + �(~v(P ) � ~n) � ~n, where~n is the surfacenormal atP .

P

P’n

v

Figure 3: Location of the grid pointP before (left) and after thedeformation (right).

3.4 Data Reduction and Smoothing

Regular grids can be stored in a very compact way: both the gridpoints and cells are represented implicitly by specifying the dimen-sion, data spacing, and origin. The deformation process destroysboth the planarity and the regularity. Therefore we have an in-creased memory consumption. Since one of our goals was to renderthe scene with interactive frame rates on non high-end computers,we decided to reduce the number of triangles in the mesh. There areseveral suitable polygon reduction methods available, which pre-serve the original topology and form a good approximation to theoriginal geometry. Lee et al. [9], for example, provide a method forcomputing a multiresolution adaptive parameterization of surfaces,which enables one to choose the resolution at runtime; possiblydepending on the hardware capabilities. On the other hand progres-sive schemes [8] allow incremental transmission and reconstructionof triangle meshes. However, we decided to use Schroeders algo-rithm “decimation of triangle meshes” [14] in our test implementa-tion, since it is an integral part of VTK [13].

Furthermore it is often necessary to smooth the decimated trian-gle mesh, to avoid unintentional light reflections in the forthcomingrendering step. Therefore we choose the very simple, but effective,Laplacian smoothing. This smoothing operation is described by thefollowing equation. Letxi be the position of pointPi, then

xi+1 = xi + �wi = xi + �Xj

(xj � xi) (5)

wherexi+1 is the point’s new position,xj is the position of pointPj , which is connected toPi, and� is some user-defined weight.Equation (5) adjustsPi by the average vectorwi multiplied by�.This equation is applied to all mesh points one after the other. Typ-ically, � is a small number (e.g. 0.01) and the smoothing process isappropriately repeated (e.g. 50–100 iterations). Its global effect isa reduction of the curvature’s high frequencies.

3.5 Texture Mapping and OpenGL Rendering

In the end we merge the results from the above visualization tech-niques by mapping the computed LIC texture onto the deformedplane. Therefore, texture coordinates and normals need to be com-puted. We chose Phong shading, also known as normal-vector in-terpolation shading, as illumination model. We defined the material

Figure 4: Out-of plane measurement principle of the ESPI

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(a) x-comp.; min = �0:28�m,max= 0:25�m

(b) y-comp.; min = �0:54�m,max= 0:22�m

(c) z-comp.; min = �1:02�m,max= 1:42�m

Figure 5: Gray-scale images of flange’s deformation

properties of the deformed plane in such a way that diffuse reflec-tion occurred mostly. Higher contributions of specular reflectionslead to poor results, because the low curvature of the deformedplane results in large highlights on the surface, which tend to bemore confusing than informative.

4 Application

In this section we give an detailed example of a planar vector fieldwith normal componentsv : R2 �! R

3 , since it is not very obvi-ous that these vector fields do exist in real-world applications.

The research group forRecyclability in Product Design and Dis-assemblyat the University of Kaiserslautern investigates the defor-mation of loaded mechanical parts. One of their goals is to replacecomposite materials by pure ones. This simplifies the disassemblyand recycling process and reduces costs; production costs as wellas disassembly costs.

Figure 6: Measurement setup

The deformation is gauged by anelectronic speckle pattern in-terferometer (ESPI)which calculates the deformation distributionsin quantity with the help of phase shifting and digital image pro-cessing. The object under test is illuminated with laser light andis viewed using a CCD camera (see Figure 4, for example). Theinterference pattern caused by the laser and observed by the CCD

camera includes the information of the deformation in each point ofthe imaged object.

We are now looking into the deformation of an intake mani-folds’s flange. This intake manifold is designed and produced byMann & Hummel GmbH, which is a german auto component com-pany. The intake manifold feeds air to the various cylinders of thefive cylinder turbo-charged diesel engine, that is built intoDaimler-Chrysler’s Mercedes-BenzC250 and E290 passenger cars (see Fig-ure 7).

Figure 7: Daimler-Chrysler Mercedes-Benz E-class.

The flange is bolted to the engine block. So the deformationdue to the axial pressure caused by the bolting is simulated by acompressive load and gauged by the ESPI (Figure 6 and 8).

The flange consists of glass-fiber reinforced polyamide. Thecharacteristics of this material depend on the glass fiber content,the fiber orientation as well as on its length and its distribution.Furthermore the material properties indicate a strong dependencyon time, temperature, and moisture content. Usually, the flange isstiffened by a carrying bracket, but has been removed for the exper-iments. The recess at the flange lower surface is just as caused byproduction as the point of molding on the lower left-hand side.

The ESPI yields the relative displacements between the unde-formed and the deformed state. Therefore, we assume the part’ssurface to be planar, which it is obviously not, but the ESPI does notknow, either. The displacements, gauged by the ESPI, are given asa 16-bit gray-scale image for eachx-, y-, andz-component, respec-tively (see Figure 5) on a regular758x572 grid (it is subsampled bya factor of five to speed up the visualization pipeline, see Fig. 2).

The absolute displacement is known for the minimum and max-

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Figure 8: One of the three intake manifolds’s flanges (tubes havebeen cut off to fit flange into instrument).

Figure 9: Visualizing the flange’s deformation using LIC

imum gray-scale only, so other gray-scales need to be interpolatedlinearly. The individual components are joined for each grid pointto a three-dimensional vector and serve then as input data for thefollowing visualization algorithms.

5 Results

TheHedgehogmethod supplies a rough overview of the deforma-tion of the flange (Figure 10). The areas, within which material ismoved forward or backward, can be clearly distinguished. How-ever, the in-plane shift remains practically invisibly, since the de-formation is larger in the normal direction than those of the othertwo tangential directions.

Critical points of the vector field are likely to be overlooked, be-cause this method relies on a ”sufficient“ chosen number and place-ment of the spines. The molding point, for example, is insufficientlyemphasized.

Let us now have a look at the standard LIC method. Thereforewe project the vector field into tangent space, that is in this specialcase thexy-plane. The LIC image shows the directional informa-

Figure 10: Visualization of the flange deformation by the hedgehogmethod (number of spines:5000; scaling:3:7 � 103).

tion only (Figure 9), but the amount of displacement, i.e. the vectorlength, can be visualized by a color coding scheme (see [11], forexample). A surprising result of the LIC visualization of the de-formed flange is, that the deformation is asymmetric, although theflange and its loading situation let us expect a symmetric deforma-tion. This is caused by the anisotropic behaviour of the glass-fiberreinforced polyamide.

In the previous section we described a slightly different approachto visualize the normal component of the vector field: We deformthe plane according to the vector field and then map the LIC textureonto this deformed plane. We have implemented a program, whichenables us to input the ESPI’s gray-scale images and to perform thevisualization pipeline in Section 3. The program is implementedin C++ using theVisualization Toolkit1 [13]. It runs on SGI-Irixas well as PC-Linux. We had to add only a few filters to VTK(LIC computation, for example), to construct the whole pipeline ofFigure 2. One of the various results that can be obtained by ourprogram is shown in Figure 1. Some results can be experiencedinteractively using an arbitrary VRML-2.0 viewer (CosmoPlayer2,for example). Point your favorite viewer to the URL given in thefootnote.3

Eye point

Light source

α

x

y

r h

Figure 12: Circular path of light source around object

Another idea is to fix the point of view and move the light sourcealong a predefined path. Let us assume the undefined plane, i.e. theparameter space of the vector field, lying inside thexy-plane. Thenwe can choose a radiusr in the xy-plane and heighth above the

1http://www.kitware.com/vtk.html2http://www.cosmosoftware.com3http://davinci.informatik.uni-kl.de/�burbach/flange.wrl

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Figure 11: Deformed Plane lit from eight different angles�i = i � 45Æ (i = 0; : : : ; 7). Order: left-right; top-down. Pictures are rendered bytheBlue Moon Rendering Tools (BMRT)[6] which adhere to the RenderManr interface standard (RenderMan is a registered trademark ofPixar).

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plane based on the dimensions of the bounding box of the deformedplane (Figure 12). The movement of the light source can now be pa-rameterized by the angle�. Figure 11 shows eight rendered imageswith �i = i � 45Æ (i = 0; : : : ; 7). The three dimensional defor-mation of the plane can be ”‘seen”’ due to the shadow movementon the surface. The importance of the molding point might even beunderestimated by the LIC visualization technique (Figure 9), butthe deformation peak in the normal component is clearly empha-sized here. The information gained from the deformation and theLIC texture gives engineers hints how to change parameters in themolding process to produce even more durable and loadable parts.

6 Conclusion and Future Work

We have described a simple, but effective method for visualizingplanar vector fields with normal components. Further we presentedan application from mechanical engineering, to prove the practi-cal relevance of such vector fields. This visualization method isespecially useful in structural mechanics, but can be used in otherapplication areas, e.g. fluid dynamics, too.

We still have several ideas in mind: First, the gauged informationat the surface of a mechanical part can be used to verify the resultsobtained fromfinite element analysis. In the case of the flange pre-sented in the previous section, FEA is complicated due to the highlyanisotropic behavior of the fiber-reinforced polyamide. Moreover,the stress-strain relation (the material property tensor) depends onthe position, i.e. the material is not homogeneous. Therefore, themolding process needs to be simulated to gain additional informa-tion about the fiber distribution and orientation.

The second goal is to extract the geometry of the mechanical partfrom a CAD model. Then we can visualize the exact deformationon the part’s surface. As we have already mentioned, the LIC tex-ture can then be computed directly on the triangulated surface [2].Since the ESPI gauges the deformation in a rectangular area only,one challenging problem is to automatically find the correspondingarea on the CAD model. If the area under consideration containsany feature edges, this task could probably be accomplished.

Acknowledgement

We thank Professor Renz’s research groupRecyclability in Prod-uct Design and Disassemblyat the University of Kaiserslauternfor their collaboration. We are especially grateful to Franz-JosefCasper and Jens Greis, who run several measuring experiments es-pecially for us.

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Figure 1: Visualization of the deformation of an intake manifold’s flange due to axial compression using our combined LIC and deformationstrategy. This intake manifold is designed and produced by Mann & Hummel and is used in DaimlerChrysler’s Mercedes-Benz C250 andE290 passenger cars.

Figure 10: Visualization of the flange deformation by the hedgehog method (number of spines:5000; scaling:3:7 � 103).

Figure 11: Deformed Plane lit from eight different angles�i = i � 45Æ (i = 0; : : : ; 7). Order: left-right; top-down. Pictures are rendered bytheBlue Moon Rendering Tools (BMRT)[6] which adhere to the RenderManr interface standard (RenderMan is a registered trademark ofPixar).