Unsteady Flow Visualization by Animating Evenly-Spaced Streamlines

Transcription

1 EUROGRAPHICS 2000 / M. Gross an F.R.A. Hopgoo Volume 19, (2000), Number 3 (Guest Eitors) Unsteay Flow Visualization by Animating Evenly-Space Bruno Jobar an Wilfri Lefer Université u Littoral Côte Opale, France Abstract In recent years the work on vector fiel visualization has been concentrate on LIC-base methos. In this paper we propose an alternative solution for the visualization of unsteay flow fiels. Our approach is base on the computation of temporal series of correlate images. While other methos are base on pathlines an try to correlate successive images at the pixel level, our approach consists in correlating instantaneous visualizations of the vector fiel at the streamline level. For each frame a fee forwar algorithm computes a set of evenly-space streamlines as a function of the streamlines generate for the previous frame. This is achieve by establishing a corresponence between streamlines at successive time steps. A cyclical texture is mappe onto every streamline an textures of corresponing streamlines at ifferent time steps are correlate together so that, uring the animation, they move along the streamlines, giving the illusion that the flow is moving in the irection efine by the streamline. Our metho gives full control on the image ensity so that we are able to prouce smooth animations of arbitrary ensity, covering the fiel of representations from sparse, that is classical streamline-base images, to ense, that is texture-like images. 1. Introuction Vector fiel ata are prouce by scientific experimentations an numerical simulations, which are now wiely use to stuy complex ynamic phenomena, with various areas of applicability, such as global climate moelling an computational flui ynamics. Large-scale, time-varying simulations are able to prouce large amounts of ata in a short time an raises the nee for effective techniques to get insight in the ata an to extract meaningful information. While several effective techniques have been propose to visualize steay flow fiels, only a few methos exist for visualizing unsteay vector fiels. Due to its important impact on a wie application area, visualization of timevarying vector fiels is a critical issue of toay research. This paper presents a new approach for the visualization of two-imensional unsteay vector fiels. Several techniques have been propose to visualize steay flow fiels, incluing icon plots, line representations, an textures. A streamline is a line tangential to the vector fiel at any point. Covering an image with a set of streamlines is a goo way to visualize the flow features. Image quality enhancement can be achieve by using streamline placement algorithms, which optimize the placement of a set of streamlines accoring to an image-base criterion 10,4,7. For an unsteay flow, streamlines can be viewe as an instantaneous representation of the vector fiel because their computation occurs inepenently for each time step, an thus streamlines o not truly epict the time-varying physical phenomenon over time. Pathlines, streaklines an timelines are particularly evote to the visualization of unsteay flows 6, but they o not allow to obtain a global representation of the vector fiel. Texture-like representations 12,1 allow to increase the spatial resolution an to epict small etails accurately. The LIC metho is base on the convolution of an input noise LIL - B.P Calais FRANCE {jobar,lefer}@lil.univ-littoral.fr Publishe by Blackwell Publishers, 108 Cowley Roa, Oxfor OX4 1JF, UK an 350 Main Street, Malen, MA 02148, USA.

2 texture with a one-imensional filter kernel 1,9. An extension to hanle time-varying vector fiels has been propose in 3, where the convolution is performe on pathlines rather than on streamlines. This approach suffers several rawbacks, such as lack of spatial coherence, non accurate time stepping an problems in establishing a goo temporal coherence. Another approach base on LIC is the UFLIC metho 8, which uses a time-accurate value epositing scheme, which performs a temporal convolution of an initial white noise texture, an a successive fee forwar algorithm to maintain a high temporal coherence between successive frames. This approach achieves goo spatial an temporal coherence but uring the animation it is ifficult to unerstan the motion of the flow. Because each image is obtaine by a temporal convolution over a number of time steps, regions with low turbulence give rise to highly contraste areas with zebra stripes of arbitrary thickness, while highly turbulent regions ten to prouce blurre zones. Inee the zebra stripes obtaine in regions of low turbulence are oriente in the irection of the flow, they look like thick streamlines an uring the animation their shapes are slowly moifie so that they actually seem to move in a irection orthogonal to the irection of the flow. While previous approaches try to correlate successive frames at the pixel level, we make it at the streamline level. This is important because it allows us to state the conitions for a goo animation as conitions on a set of streamlines. For instance we are able to maintain a goo streamline placement at any time, which is impossible by just ealing with pixels. Inee a goo correlation at the streamline level yiels a goo correlation at the pixel level because all information is conveye by streamlines, i.e. by pixels lying on them. Our algorithm prouces a correlate sequence of streamlinebase representations of the flow. Its main features are: Optimize streamline placement for each frame using our streamline placement algorithm presente in 4. The image ensity can be controlle easily by setting the separating istance between streamlines. Motion encoing along each streamline by mapping a set of one-imensional cyclical textures epicting the orientation of the flow. We propose an extension of the Motion-Map metho 5,2 suitable for unsteay flows. Animating the texture gives the illusion that the flow moves along the streamlines. Correlation between successive frames using a streamline-oriente correlation algorithm. The basic principle of the algorithm consists in fining the best matching streamline at the next time step an by maximizing the number of correlate streamlines from one frame to the next. Both shape an texture of a streamline are correlate. As a result our metho prouces smooth an accurate animations of the structure of the flow over an arbitrary number of frames. Direction an orientation of the flow are visualize at any time. Our correlation algorithm is robust enough so that goo results are obtaine both for sparse an ense representations. Since our streamline placement algorithm gives a total control on the ensity of the representation, we have a full control of the image ensity at any time. The remaining of this paper is organize as follow. Section 2 escribes our streamline placement algorithm presente in 4. Section 3 explains how to encoe motion along streamlines an escribes our streamline correlation algorithm in etails. Section 5 explains how streamline textures are correlate together. In Section 6 results are presente an we compare our metho with previous approaches. Section 7 conclues an gives irections for future research. Figure 1: Decreasing the separating istance allows us to prouce ense representations of the flow fiel. In this paper we propose to use the streamline representation to visualize a two-imensional time-varying flow fiel. The goal is to show the evolution of the instantaneous representation of the vector fiel, which is visualize by way of a set of streamlines. Our approach gives full control on the ensity of the representation, that is we are able to prouce frames of arbitrary ensity, incluing traitional streamline-base sparse representations an LIC-like ense representations. To obtain a ense representation, we just have to make the separating istance be the size of a pixel so that every pixel will be covere by a streamline (see Figure 1). 2. Streamline Placement are a goo way to visualize a vector fiel. By covering a omain on which a vector fiel is efine by a set of streamlines, we are able to show the global flow structure, its egree of turbulence, an all critical points, such as sinks an sale points. To make this information accurate an smooth it is necessary to select only a subset of all possible streamlines, accoring to some criterion. We can show that the image quality is a function of the streamline length. For a sparse representation it woul be ifficult to unerstan the real topology of the flow if the streamlines are too short (see Figure 2 left) because the eye ten to follow the path

3 escribe by the streamlines in orer to unerstan the structure of the flow. Inee streamlines ens ten to act as artifacts an to isturb the observer. In case of a ense representation, pixels along a streamline are correlate together, thus the longer the streamline, the higher egree of correlation. A streamline is efine as a sequence of so-calle sample points. Each sample point is obtaine by integrating its position as a function of the position of the previous sample point. Sample points are compute so that they are equally space in space. Once a see point has been etermine (see Section 2.2), integration occurs backwar an forwar until either the separating istance with the closest streamline falls uner a fixe threshol, or the bounary of the image has been reache, or a source or a sink point has been encountere. To spee up the process, instea of really computing the separating istance between a caniate sample point with all streamlines, we use a gri as an acceleration structure (see 4 ). In orer to maintain the esire level of correlation of the image pixels, only the streamlines whose length is above a fixe threshol are consiere as vali. Figure 2: Left an mile figures have been obtaine by placing see points at the intersection of a regular gri. Left: short streamlines. Mile: long streamlines. Right: image obtaine by our streamline placement algorithm. Another important criterion to evaluate the quality of such an image is the istribution of white an black pixels in the image. Regions with higher ensity of black pixels ten to appear as more important, concentrating more flow features an catching the eye, while this is generally a visualization artifact (see Figure 2 mile). To aress this issue Turk an Banks propose an image-guie streamline placement algorithm base on a progressive refinement scheme 10. In a previous paper we propose a more efficient an irect approach to obtain similar results 4 (see Figure 2 right), an recently Mao et al. extene Turk s metho to eal with curvilinear gris 7. The metho propose in this paper for time-varying vector fiels uses our streamline placement algorithm, which is escribe in etails in 4. The following subsections recall the main steps of the former metho Streamline Integration an Density Control Figure 3: are erive from the first (rawn as thick here) one by selecting see points at the separating istance See Point Selection an Domain Coverage Our algorithm starts by selecting an initial see point from which a first streamline is integrate an put into a stack. Then the main loop of the algorithm is as follow. A streamline is extracte from the stack an all see points that are vali at the separating istance from the current streamline are etermine. From each of these see points a new streamline is compute (see Figure 3) an if the streamline is vali it is pushe into the stack. The algorithm finishes when the stack becomes empty. A etaile algorithm is given in 4. It has to be note that, as compare to streaklines, pathlines, an timelines, we obtain a complete an uniform coverage of the image, thus information about the vector fiel is available at any point. For unsteay flows, in orer to achieve goo correlation between successive frames, a new see point selection algorithm has been esigne. 3. Encoing Motion Along In aition to the irection, orientation is an important feature to help unerstaning the behavior of the flow. On a static image orientation can be visualize by mapping an oriente texture onto the streamline accoring to the orientation of the flow, such as a sawteeth texture for instance. By stretching the texture, the velocity of the flow can be visualize, although results obtaine with this technique for static images are not convincing. Visualizing the velocity properly requires to animate the flow, which can be achieve by moving a texture along each streamline 11. In case of a steay flow the same set of streamlines can be use for every time step an only the texture correlation issue has to be aresse. In 5 we propose the Motion Map, an original ata structure an algorithm to compute a ense set of streamlines with textures mappe on them. By using a cyclical set of one-imensional textures, which are shifte from one frame to the next, we are able to prouce smooth animations of the flow.

4 In case of a time-varying vector fiel, the structure of the flow changes at each time step, hence the set of streamlines use for the previous frame is no longer vali an new streamlines have to be compute. If we simply compute a new set of streamlines, without any correlation between streamlines at ifferent time steps, it is easy to figure that we woul not obtain a smooth animation but rather a superposition of blinking effects that woul make it impossible to unerstan the behavior of the flow. Thus it is necessary to highly correlate streamlines at consecutive time steps together. In this paper we present an original metho to compute a sequence of correlate streamline sets that are able to properly epict the behavior of the flow over time. The correlation is performe at the streamline level an both the shapes of the streamlines an the textures that are mappe on them are correlate. streamlines at time step t+1. The first frame is generate using the algorithm escribe in Section 2. For subsequent frames our algorithm processes in two steps. First we compute all streamlines at time step t+1 that match a streamline at time step t accoring to a efine criterion, which measures the ifference between two streamlines in terms of position an shape. In the remaining of this ocument we will call reference streamline the streamline at time step t an corresponing streamline the streamline at time t+1 that matches the reference streamline. The secon step of our algorithm consists in aing new streamlines to fill the image in orer to obtain a uniform representation with the esire ensity. These new streamlines are compute at time step t+1. Figure 4 shows the ifferent stages of our algorithm. 4. Streamline Correlation over Time When a steay flow fiel is animate using the Motion Map metho, two important features are visualize: the structure of the flow, i.e. the irection at any point, an the orientation, by the animate streamline textures. All the motion that appears on the image is ue to the real motion of the flow, that is nothing is moving but the flow itself, an the flow is moving in the right irection. When ealing with an unsteay vector fiel, the streamline sets at two consecutive time steps are ifferent, even if they have been correlate together. During the animation, replacing streamlines of a time step by corresponing streamlines of the next time step will give the illusion that streamlines are moving in a irection orthogonal to the irection of the flow. If we eal with quality of the visualization, all motion appearing uring the animation, except those ue to the texture motion along streamlines, which epicts the real motion of the flow, shoul be consiere as a visualization artifact an is a potential source of misunerstaning for the observer. For this reason we have concentrate our efforts in minimizing the impact of those effects on the image quality. Following a set of observations we have mae on several image series, we have ientifie three main sources of motion artifacts, which are liste below by orer of increasing impact on the quality of the visualization, in the sense of the ability for an observer to unerstan the behavior of the flow: big shape changes between 2 corresponing streamlines in two consecutive frames (see Section 4.1), isappearance of streamlines, appearance of new (uncorrelate) streamlines. For a single time step the number of streamlines that coul be rawn is potentially infinite an we reach the esire ensity by selecting a subset of those streamlines. The goal of our correlation algorithm is to select the best set of streamlines accoring to a correlation criterion Algorithm Overview To fin the best set of streamlines, we use a so-calle fee forwar algorithm, where streamlines at time step t are use to compute Vector Fiel at Time t=0 Vector Fiel at Time t t = t + 1 Place new Fin Best Corresponing Place new Store / Display Initial Set of Streamline Set at Time t Figure 4: Fee forwar streamline placement for unsteay flows Best Corresponing Streamline Selection For each streamline at time step t we etermine the corresponing streamline at time step t+1. This is achieve by computing a so calle caniate streamline at time step t+1 for each sample point of the reference streamline. The streamline is integrate accoring to the esire image ensity, that is we check the separating istance between this streamline an every other streamline in the current frame. As for the first frame only streamlines whose length is

5 above a fixe threshol are kept as caniates. Our experiences showe that a goo threshol is a function of the integration step an the flow structure since a trae-off has to be foun between having long streamlines, which increases the spatial correlation between pixels of the image, an having a goo correlation between corresponing streamlines in successive frames. In practice we use a threshol of 10 sample points. Once all caniate streamlines have been compute, a correlation criterion between each caniate an the reference streamline is evaluate an the streamline with the highest score is electe as the corresponing streamline. Corresponing Sample Point Reference an Caniate Corresponing Portions ist( sp 1, sp 2 ) ( sp correlation 1, sp 2 ) C( S 1, S 2 ) = Car( C( S 1, S 2 )) where C( S 1, S 2 ) is the corresponing portion of both streamlines. Figure 5: Corresponing portions are efine in orer to evaluate the corresponing criterion between reference an caniate streamlines. The correlation criterion has been efine as the average istance between all pairs of corresponing sample points in the corresponing portion of each streamline. The corresponing portion is the portion of each streamline so that for each sample point a corresponing sample point exists in the other streamline. To etermine the corresponing portion we start from the common sample point of both streamlines (remember that the streamline at the current time step has been generate by integrating backwar an forwar from a sample point of the reference streamline) an then sample points on both streamlines are associate to each other in orer in both irections (see Figure 5). This criterion allows us to consier the longest corresponing portions of both streamlines. for which their portions ten to go away from the reference streamline will obtain lower scores while those that keep close to their reference streamline will obtain higher scores. Once the best corresponing streamline has been etermine, it is inserte in the list of streamlines of the current frame. The orer use for processing reference streamlines is the same at each time step, so that a corresponing streamline will be constructe in the same conitions as its reference streamline. For instance if some streamlines were alreay create at the time the reference streamline was built, their corresponing streamlines will be create before the current corresponing streamline be compute. In this way the constraints uner which a corresponing streamline is constructe are the same as for the construction of its reference streamline. By processing in this way, the overall correlation scores between streamlines have increase substantially Image Completion with New The secon step of the algorithm consists in completing the image with new streamlines in orer to ensure an uniform image ensity. This step is necessary because the first step ensures that the separating istance between streamlines oes not fall below the esire istance but it oes not ensure an uniform coverage of the image, as for the metho escribe in Section 2. Thus, especially if important changes have occurre in the structure of the flow, some areas in the image may not have the esire ensity. To complete the image we use the algorithm escribe in Section 2, except that, instea of an arbitrary initial streamline, the stack is initialize with the set of streamlines generate at step 1. The new streamlines generate at step 2 are inserte at the en of the streamline list, in orer for the future corresponing streamlines to be generate uner the same constraints as for their reference streamlines. As a consequence the rank of a streamline in the list is a function of its relative age in the list, oler streamlines being locate at the beginning of the list. Remember we process reference streamlines from the beginning of the list an that only vali streamlines, that is streamlines that are longer enough, are consiere vali. It is easy to unerstan that the chance for a streamline to be eclare as vali ecreases as the number of alreay create streamlines in the image increases. Hence reference streamlines at the beginning of the list will more likely give rise to vali corresponing streamlines, since they are processe at the beginning, when the image contains a small number of streamlines. At the beginning of Section 4 we explaine that an important streamline shape moification was visually less isturbing than the appearance or isappearance of a streamline. By orering reference streamlines from the olest to the newest, we increase the probability of long life for most of the streamlines. The results obtaine have shown that less than 5% of new streamlines are create for each frame in average. Inee those streamlines ten to be shorter than those create at the first step because they are generate when more streamlines have been alreay rawn in this image. It means that the number of pixels covere by new streamlines is far below 5% of the number of black pixels in the image. As a consequence the effect of the appearance of new streamlines is minimize. Moreover we propose in Section 4.4 a technique to attenuate the visual effect of streamline appearance Improvements of Animation Quality Although the correlating algorithm escribe above prouces animations of goo quality, a couple of aliasing effects remain. Some techniques are propose here to enhance the smoothness of the animation.

6 Priority to Circular Circular streamlines are important features in an image because they show critical points, which are generally of significant importance for analyzing the unerlying ynamic system. For instance in meteorology, circular streamlines can represent anticyclone or atmospheric epression centers. In orer to keep track of these features uring the animation it is important to avoi them appearing an isappearing several times but rather just to upate their shapes from one frame to the next. For this reason we have moifie the orering algorithm, all circular streamlines being locate at the beginning of the list at any time. This is achieve by etecting circular streamlines at the en of step 2, which are immeiately move to the beginning of the list. Circular streamlines are etecte uring the construction of the streamline by performing an aitional test, which consists in evaluating the istance between the current sample point an the first sample point of the streamline. If this istance falls uner a certain threshol (actually /2 where is the separating istance), the construction process is stoppe an the streamline is eclare as circular. Tapering Effect For sparse representations of vector fiels with streamlines, it is better to raw anti-aliasse streamlines, which implies to raw streamlines whose thickness is more than one pixel. The o effect is that streamlines ens ten to ebase the overall image quality. Tapering for streamlines is a technique introuce in 10 to attenuate the visual effect of streamlines ens. It consists in progressively ecreasing the streamline thickness as we go closer to one of its extremities. This technique is an important factor of the improvement of streamline-base image quality. While in 10 tapering requires a post-processing step, it is part of our streamline generation algorithm. Progressively Increasing Thickness In orer to attenuate the flickering effect ue to the appearance of new streamlines, the thickness of the streamline is a function of its age. A new streamline is always create with a thickness of 1 an those thickness will grow up until it reaches the ault thickness. Train Effect As for the visualization of small objects, such as particles, for which a train effect is use to keep track of the object from one frame to the next, train effect can be enable to increase the correlation between successive frames. Our experience showe that this effect is really helpful to track flow features uring the animation. At each time step, the streamlines of a couple of frames are isplaye together, each streamline being rawn with a color intensity compute as a function of its age (recent frames being rawn with a higher intensity). Train effect can be use at any ensity. It is important to note that when train effect is enable, streamlines may cut each other if they have been compute at ifferent time steps. 5. Correlation of Streamline Textures A one-imensional texture is mappe onto each streamline. Visualization of the orientation of the flow is achieve by moving the texture along the streamline Moving Textures The quality of the visualization can be assesse as the ability for an observer to keep track of the textures that are moving on the streamlines from one frame to the next. Thus textures on corresponing streamlines at successive time steps shoul be place so that, uring the animation, an illusion that the flow is moving in the irection escribe by the streamline will appear. This is achieve by using a cyclical set of textures, each one being a shifte version of the other, as shown on Figure 6. The same texture is use for all streamlines of a given frame. In orer to obtain the animation, we shift the texture at every time step. Each texture is cyclical, that is the first an last pixels have the same value. This property allows us to uplicate the texture on a streamline while ensuring continuity between consecutive pixels. In orer to save memory we store only one version of the texture an a pointer is use to mark the beginning of the texture. Figure 6: Example of a cyclical set of textures that can be use to a motion to streamlines. For a steay flow fiel the same set of streamlines is use for all frames an hence we just have to shift the texture for each time step (see 5 ). For an unsteay flow a streamline oes not longer exist at the next time step but rather we use the corresponing streamline, which iffers from its reference streamline in terms of position, shape, an length. If we woul map the texture of the reference streamline onto the corresponing streamline irectly, the texture woul be stretche at some places, compresse at some others. Thus it woul be ifficult to keep track of the texture uring the animation. In orer to correlate textures properly we nee to ecompose each streamline in a sequence of streamline segments that we call texture supports. The purpose of texture supports is to have supports of about the same length for texture mapping in orer to avoi stretching an compressing effects of great amplitue. Actually we efine a minimum an a maximum value for texture support lengths as 0.5 x Ls an 1.5 x Ls respectively, where Ls is the mean support length. A goo value for Ls is a function of the number of ifferent colors in the one-imensional texture, common values being 10 or 15 sample points. The texture is mappe onto each support inepenently.

7 5.2. Computation of Texture Supports Texture supports are efine as a set of successive sample points (actually the sample points of the streamline to which they are relate). We call control points, resp. reference control points an corresponing control points, the sample points that belong to two supports. A support is efine by two control points an a list of interior sample points. To correlate two streamline textures together it is necessary to make a corresponence between the texture supports of both streamlines. For this reason most of the texture supports of the corresponing streamline are compute from the texture supports of the reference streamline. To achieve this, for every reference control point, a corresponing control point is etermine. This corresponing control point is the closest sample point of the corresponing streamline. In orer to keep a goo correlation it is necessary for the reference control point an the corresponing control point to be close to each other. For this reason only sample points in the neighborhoo of the reference control point are consiere as caniate control points. It means that, after this step, some reference control points o not have any corresponing control point an hence the supports efine by the corresponing control points are not necessary the same length. In a secon step we make all supports about the same length. This is achieve by iviing supports that are too long in several supports of the right length an by collapsing too short supports. The algorithm for correlating textures of corresponing streamlines is as follows: For each reference control point Do Determine a corresponing control point if any EnFor For each support in the corresponing streamline Do Compute support length If support length is too long Then Divie support in as many supports as neee Else If support is too short Then Collapse support with the shortest ajacent support EnIf EnFor For each corresponing support Do For each sample point of the support Do Compute texture coorinate in current texture Set sample point color accoringly EnFor EnFor For the streamlines create at the current time step (see Section 4.1), no corresponence with any reference streamline exists. In this case the streamline is partitione in as many supports as necessary an the current texture is mappe onto every support. Reference Streamline Reference Control Point an its Neighborhoo Corresponing Streamline Corresponing Control Point Newly Ae Control Point Figure 7: Computation of texture supports. Figure 7 shows a reference streamline, its control points an their neighborhoos, an a corresponing streamline an its control points. On the corresponing streamline, circle control points have been etermine from the reference control points, while square control points have been create in the secon step. It shoul be note that inserting or collapsing texture supports ecreases the correlation egree between textures at ifferent time steps, but this moification is local an only a few supports are generate in this way. When all texture supports of the corresponing streamline have been compute, the current texture is mappe inepenently onto each support. Figure 8: Visualization of win spee preictions over Europe. In orer to show the irection of the flow, streamlines are shae with an oriente one-imensional texture.

8 Unlike the methos base on pathlines, our approach oes not require to have the complete vector fiel for several time steps in memory. Actually only the vector fiel for the current time step is require to compute the corresponing frame, in aition to the set of streamlines compute for the previous frame. This is an important feature since time-varying ata are generally memory consuming. Computation times obtaine for an image of 512x512 pixels on a SGI O2 R5000SC at 180Mz range from 5 secons per frame for sparse representations to 30 secons per frame for ense representations. As compare to LIC-base approaches for visualization of unsteay vector fiels, our solution is more effective an achieves better visual results. 7. Conclusion Figure 9: Three consecutive frames of a time-varying vector fiel at two ifferent scales. 6. Results an Discussion The animations prouce by our metho allow an easy tracking of the features of the flow while epicting information on the vector fiel with a constant ensity everywhere in the image. Figure 8 an Figure 9 show 2 ifferent visualizations, for sparse an ense representations respectively. In both cases 3 consecutive frames of the animation are shown. Now let us remark that, in the case of timevarying vector fiels, streamlines o not correspon to the paths that woul be followe by a massless particle release in the flow. Instea they are just instantaneous virtual trajectories aime at unerstaning the flow topology an its evolutions. Thus the images of Figure 8 an Figure 9 show the real topology of the flow at a given time step. It is often useful to be able to isplay an aitional quantity, for instance pressure, temperature, or vorticity. This is easily achieve with our metho since we work with a true colors visual, so that a texture can be efine as any sequence of colors. In particular we can efine a texture as a sequence of intensities or saturations an, at the time the texture is mappe onto a streamline, the hue can be erive from the value of an aitional scalar quantity at the current point. We have propose an original approach to the visualization of time-varying 2D vector fiels. While previous approaches were base on pathlines an try to correlate successive images at the pixel level, our metho consists in correlating instantaneous visualizations of the vector fiel at the streamline level. The movement of the flow is obtaine by moving a one-imensional texture along the streamlines. In orer to obtain a smooth animation, both shapes an textures of the streamline pairs at ifferent time steps are correlate together. Using an accurate streamline placement algorithm, our metho is able to prouce animate sequences of arbitrary ensity, covering the fiel of representations from sparse to ense. Inee the image ensity can be controlle easily by setting the separating istance between streamlines. Our placement algorithm ensures an uniform coverage of the entire image an oes not require any further knowlege about the structure of the vector fiel. The spectrum of applications of this work is wie an numerous scientific areas shoul be able to take benefit from this metho. For instance sparse animations coul be use to prouce animate cartography, for the wiesprea issemination of meteorological information through the Web or for eucational purpose. Dense representations are useful when a high egree of accuracy is require an coul be use for the purpose of analyzing time-varying simulations in aeronautics for instance. Future research shoul aress the generalization of this approach to 3D, which raises new issues for the placement of streamlines an for texture mapping. In aition visualizing 3D vector fiels raises perceptual issues that shoul be aresse. There is also a nee for a more accurate corresponing criterion between streamlines, particularly a mathematically efine criterion base on curve matching shoul be investigate. Acknowlegements We woul like to thank Robert van Liere an Willem e Leeuw for the vector fiel ata sets for win spee preiction over Europe.

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