1. INTRODUCTION

Vector fie ld data are produced by scient i ficexperimentations and numerical simulations, whichare now widely used to study complex dynamicphenomena, with various areas of applicability, suchas global climate modelling and computational fluiddynamics. Large-scale, time-varying simulations areable to produce large amounts of data in a short timeand raises the need for effective techniques to getinsight in the data and to extract meaningfulinformation. This paper presents a hierarchicalapproach for the visualizat ion of large two-dimensional steady vector fields.

Several techniques have been proposed to visualizesteady flow fields, including icon plots, l inerepresentations, and textures. By covering the imagewith a set of streamlines, the global structure of thevector field can be visualized, its degree of turbulencecan be estimated, and the position of all criticalpoints, such as sinks and saddle points, can be easilylocated in the domain.

To obtain good quality images requires to controlthe placement of streamlines so as to ensure a uniformdensity of colored pixels. This issue was addressed byTurk and Banks in [1], and their method was extendedto curvilinear grids later [2]. In [3] we have presenteda more effective approach, based on a direct

placement technique, which uses the separatingdistance between streamlines to control the density inthe image. Here we propose an extension of thistechnique in order to allow the computation andvisualization of streamline images at different levelsof density.The process of exploring a large vector field generallyrequires different kind of visualization techniques. Atthe beginning only a rough representation on thewhole data domain is needed. As the user refines theregion of interest more detailed visualizations arerequired, to give accurate information on specificparts of the vector field. It should also be possible tovisualize different regions with different degrees ofaccuracy on a single image. So we are interested inhow to control the density of streamlines in an image,how to compute a set of images of the same vectorfield with different densities and hence how tointeractively change the desired density of parts of theimage while maintaining a given criterion of quality,that is a uniform distribution of streamlines withinevery part of the image. The main features of themethod proposed in this paper are:

• the computation of streamline-based images of avector field with an optimized visual quality. Thisis achieved by maximizing the average length ofthe streamlines used to cover the image domainand by obtaining a uniform density of stream-

MULTIRESOLUTION FLOW VISUALIZATION

Bruno Jobard and Wilfrid Lefer

Université du Littoral Côte d'OpaleB.P. 719, 62228 Calais, France

e-mail: {jobard,lefer}@lil.univ-littoral.fr

ABSTRACT

Flow visualization has been an active research field for several years and various techniques havebeen proposed to visualize vector fields, streamlines and textures being the most effective andpopular ones. While streamlines are suitable to get rough information on the behavior of the flow,textures depict the flow properties at the pixel level. Depending on the situation the suitablerepresentation could be streamlines or texture. This paper presents a method to compute a sequenceof streamline-based images of a vector field with different densities, ranging from sparse to texture-like representations. It is based on an effective streamline placement algorithm and a productionscheme that recalls those used in the multiresolution theory. Indeed a streamline defined at level J ofthe hierarchy is defined for all levels J’>J. A viewer allows us to interactively select the desireddensity while zooming in and out in a vector field. The density of streamlines in the image can alsobe automatically computed as a function of a derived quantity, such as velocity or vorticity.

Keywords: Flow Visualization, Streamlines, Multiresolution Representation, InteractiveVisualization, Large Vector Fields.

lines, i.e. a uniform density of colored pixels inthe image. The tapering technique, which hasproved to be an efficient factor of enhancementof the quality of streamline-based images, hasbeen easily integrated in our algorithm withoutany additional computation cost.

• the production of images of an arbitrary density,ranging from sparse representations, that is clas-sical streamline-based images, to dense ones,similarly to those produced by techniques such asSpot Noise [5] and LIC [6][7].

• a strong and efficient control of the density of theimage by the user. The density is expressed as thedistance between adjacent streamlines, which isactually a criterion that can be easily determinedvisually, thus it is quite easy for the user toparameterize our algorithm.

• a representation of streamline-based images atdifferent level of density, an algorithm to producesuch a hierarchical representation, and a tech-nique to interactively select the desired level ofdensity, for instance during the process of inter-active exploration of large vector fields.

• low computation times. The principle of ouralgorithm is to place each streamline directly atits final place in the image, so that there is nosuperfluous computation. Indeed we prove thatthe computation terminates and the computationtimes for a complete hierarchy of streamlines isthose of a single image of the highest density.

The remainder of this paper is organized asfollows. Section 2 gives the algorithm of ourstreamline placement algorithm. Section 3 presentsi ts extension to produce representat ions ofstreamline-based images at different levels ofdens i ty. Sec t ion 4 shows how to use suchrepresentations for the interactive exploration oflarge vector fileds and section 5 allows us toconclude.

2. STREAMLINE PLACEMENT

Our streamline placement technique is described in[3]. Here we just give the main algorithm, which isnow part of our multiresolution method. Thealgorithm uses a FIFO list F, which contains thestreamlines that have been placed in the image butnot yet used for generating new ones. The algorithmgenerally starts by randomly selecting an initial seedpoint, from which a first streamline is integrated andput into F. But F can be initialized with any set ofstreamlines as well, which will be useful forgenerating images of different densities. In the mainloop of the algorithm, a streamline is extracted fromF and used for generating new streamlines at itsneighborhood. The IntegrateStreamline() functioncomputes a new streamline from a given seed point.A streamline is considered as valid if it has therequired minimal length. After a new validstreamline has been computed it is necessary toupdate the list of seed points so as to delete the seedpoints that are no longer valid. The algorithm finisheswhen F is empty. This method ensures that we obtain

a complete and uniform coverage of the image, thusinformation about the vector field is available at anypoint.By setting the separating distance appropriately, weare able to produce both sparse and dense (texture-like) visualizations with a unified streamline-basedapproach.

3. MULTIRESOLUTION STREAMLINE SETS

There are at least two situations in which amultiresolution streamlines set could be of greatinterest: enrichment and zoom. Enrichment consistsin getting more details from a specific area of interestby adding more streamlines in this region. Zoomingmeans the ability for the user to zoom in and outwhile maintaining the same density in the image.These two important features in the context ofinteractive exploration of vector fields can besupported through a multiresolution representation ofstreamline-based images.

The multiresolution theory is based on the conceptof nested spaces, a direct consequence of this beingthe property that a vector defined at a given level J ofthe hierarchy is defined for all levels J’>J too.Although there is no such mathematical theorybehind our multiresolution representation, this lastfeature is part of our method, that is a streamlinedefined at level J is defined for all levels J’>J too.This is a direct consequence of our multiresolutionstreamlines generation method described in section3.1. The corresponding algorithm is given in section3.2.

3.1 Generating Multiresolution StreamlinesImages

The set of images at different densities is generatedfrom the lowest density to the highest one. A first set

Function PlaceStreamlines (V, Linit, dsep): Lplaced

in: V : vector fieldLinit : initial streamlines setdsep : separating distance

out: Lplaced : final streamlines set

Lplaced := LinitForeach streamline S in Linit Do

Push(F, S)EndForeachdseed := dsep * ratioseed/sepWhile F is not empty Do

Scur := Pop(F)Lsp := seed points collection at dseed from ScurForeach seed point P in Lsp Do

Snew := IntegrateStreamline(V, Lplaced, p, dseed)If Snew is valid Then

Push(F, Snew)Lplaced := Lplaced + SnewUpdate Lsp

EndIfEndForeach

EndWhileReturn Lplaced

of evenly spaced streamlines at the lowest density iscomputed, which yields the image at level 0 of thehierarchy. Then level 1, whose density is higher thanlevel 0, is obtained by adding a set of streamlines tothe streamlines defined at level 0, as shown by figure1. Then the process is repeated until the desireddens i ty has been reached. Thus , as for themultiresolution theory, a streamline computed atlevel J exists for all levels J’>J. Now you canunderstand why the feature of our streamlineplacement algorithm, which uses an arbitrary initialset of streamlines, is important.

Figure 1: The process of generating multiresolutionstreamline-based images.

3.2 Algorithm

Our algorithm for generating multidensity imagestakes as entry parameters the number of levels to begenerated and the required densities for the images atthe first and last levels of the hierarchy. After theimage at level 0 has been computed, the generationof an image for a given level consists in completingthe streamlines set of the previous level by newstreamlines in order to reach the required density. Todo this we use the streamline placement algorithm

described in section 2. From one image to the nextthe separating distance is decreased according to ageometric series, that is dj+1 = dj * reduction_coef.It is possible to start with an arbitrary initial set ofstreamlines. In this case new streamlines will beadded to obtain the image at level 0, then thealgorithm processes as explained above. Streamlinesin the hierarchy are stored as lists of sample points,they are rasterized only during rendering, which isquite less memory expensive as if all the images atthe different levels were stored.

4. INTERACTIVE WALKTHROUGHS INCOMPLEX VECTOR FIELDS

Streamlines are not rasterized in the image as theyare computed but rather they are stored as lists ofsample points. Thus several rendering modes can beused to make images of these streamlines, dependingon which kind of representation is needed, sparse ordense, possibly with glyphs added to the streamlines.Choos ing a l a rge d i s t ance y i e ld s spa r serepresentations, which evoke traditional hand-drawnpictures. Choosing a minimal distance allows toproduce texture-like representations, such as thoseproduced by methods like Spot Noise [5] or LIC[6][7].In order to provide the user with an effective tool forexploring large-scale vector fields, we havedeveloped a multiresolution viewer to navigateinteractively within the data. The viewer uses ahierarchy of streamlines sets at different levels ofdensity and allows the user to select the level ofdetails interactively. Moving from level J to the nextlevel J’ consists in either adding the streamlinesgenerated at level J’ or deleting streamlinesgenerated at level J. Since streamlines are stored aslists of sample points in the hierarchy, drawing ordeleting a streamline involves its rasterization in theimage. This is not a time-consuming operation andindeed moving from a level to the next is achievedinstantaneously. We could also make this operationreal ly smooth by using so-cal led geomorphfunctions, whose aim is to make the apparition ordisappearance of streamlines as smooth as not to

L4

placed

L3

placed

L2

placed

L1

placed

L0

placed

L0 1→placed

L0 2→placed

L0 3→placed

L0 4→placed

Function PlaceStreamlinesMultiResolution (V, Linit,min_dsep, max_dsep, Jmax): Lplaced

in: V : vector fieldLinit : initial streamlines setmin_dsep : separating distance at J=0max_dsep : separating distance at J=JmaxJmax : number of levels minus 1

out: Lplaced : final streamlines set

reduction_coef := pow(min_dsep/max_dsep, 1.0/Jmax)Lplaced := Linitdcur := max_dsepWhile dcur > min_dsep Do

Lplaced := PlaceStreamlines(V, Lplaced, dcur)dcur := dcur * reduction_coef

EndWhileReturn Lplaced

disturb the observer. In [4] we have proposed suchkind of mechanism for handling the apparition anddisappearance of streamlines in the context ofvisualization of time-varying vector fields.

As an enrichment example (see section 3) it ispossible to generate an image with varying densities,according to the degree of interest of the differentareas or any scalar quantity.

The main problem while zooming on an image isthat the amount of details is generally not adaptedaccordingly. The consequence is that some details arenot visible at certain scales and the image lacksdetails at other scales. Actually the level of detailsshould be evaluated as a function of the image size.Figure 2 illustrates the possibilities our viewer offersfor managing the zoom function. The density ofstreamlines has been set by the user and is maintainedcontinuously during the exploration process, the «bestmatching» level of the hierarchy being selected at anytime.

5. CONCLUSION

A new method has been proposed, which generatesstreamline-based images at different levels of density.The streamlines are stored as lists of sample pointsand the right set of streamlines is selected forrasterization as a function of the desired density,which can be computed automatically, as a functionof a derived quantity, or set by the user. Thus it ispossible to zoom in and out in a complex vector fieldwhile maintaining a constant density of streamlines inthe image. The placement technique used by ourmethod has proved to be very effective, computing adense image being achieved in a couple of seconds onany today processor (see [3] for computation times).Let us remark that the number of streamlines in awhole hierarchy is the same as for an image at thehighest density, only the organization of thestreamlines differs. Hence the time necessary toproduce the hierarchy is a couple of seconds.

In certain situations it is useful to compute only asubset of s t reamlines and to visual ize themimmediately. For instance, at the beginning of an

exploration process of a large vector field, when theuser try to get insight in the data and to identifyregions of interest, only a sparse image of the wholedomain is required. Since our method computesstreamlines at increasing density levels, it perfectlyfits the requirements of interactive visualization oflarge data sets.

REFERENCES

[1] Turk, G. and D. Banks. Image-Guided StreamlinePlacement. Computer Graphics AnnualConference Series (Proceedings ofSIGGRAPH’96, August 4-9, 1996), pages 453-460, ACM Press.

[2] Mao, X., Y. Hatanaka, H. Higashida and A.Imamiya. Image-Guided Streamline Placementon Curvilinear Grid Surfaces. In D. Ebert, H.Rushmeier and H. Hagen, Editors (Proceedingsof IEEE Visualization’98, October 18-23, 1998),pages 135-142, IEEE Press.

[3] Jobard, B. and W. Lefer. Creating Evenly-SpacedStreamlines of Arbitrary Density. In W. Lefer andM. Grave, Editors, Visualization in ScientificComputing’97 (Proceedings of the 8thEurographics Workshop on Visualization inScientific Computing’97, April 28-30, 1997),pages 43-55, Springer-Wien-New-York.

[4] Jobard, B. and W. Lefer. Unsteady FlowVisualization by Animating Evenly-SpacedStreamlines, Eurographic’00, August 20-25, 2000.

[5] van Wijk, J. Spot Noise-Texture Synthesis forData Visualization. Computer Graphics 25(4)(Proccedings of SIGGRAPH’91, July 28 - August2, 1991), pages 309-318, ACM Press.

[6] Cabral, B. and L. Leedom. Imaging Vector FieldsUsing Line Convolution. In Computer Graphics(Proceedings of SIGGRAPH’93), pages 263-272.ACM Press.

[7] Stalling, D. and H-C. Hege. Fast and ResolutionIndependant Line Integral Convolution.Computer Graphics Annual Conference Series(Proceedings of SIGGRAPH’95, August 6-11,1995), pages 249-258, ACM Press.

Figure 2: Three snapshots of an interactive exploration of a vector field using our multiresolutionviewer. The suitable level of the hierarchy is selected at any time to maintain a roughly constant density

of streamlines.