FLOW VISUALIZATION USING MOVING TEXTURES * Nelson Max Lawrence Livermore National Laboratory Livermore, California

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1 FLOW VISUALIZATION USING MOVING TEXTURES * Nelson Max Lawrence Livermore Naional Laboraor Livermore, California Barr Becker Lawrence Livermore Naional Laboraor Livermore, California SUMMARY We presen a mehod for visualizing 2D and 3D flows b animaing exures on riangles, aking advanage of exure mapping hardware. We discuss he problems when he flow is ime-varing, and presen soluions. INTRODUCTION An inuiive wa o visualize a flow is o wach paricles or exures move in he flow. The earl color able animaion of [1] was an example of his echnique. More recenl, van Wijk [2] has proposed advecing and moion blurring paricles b he flow field. The LIC mehod [3, 4, 5] uses inegrals of whie noise exures along sreamlines, moving he weighing funcion in he inegrals from frame o frame o animae he exure moion. The moion blur of he paricles and he direcional exure blurring from he LIC inegraion creae anisoropic exures which indicae he flow even in a sill frame. However he are compuaionall inensive, and canno generae animaion in real ime. The exured splas of Crawfis [6] use a loop of ccling exure maps wih precompued advecing moion blurred spos, and ake advanage of exure mapping hardware. These are composied in back o fron order in a local region near each daa poin, and oriened in he direcion of he veloci vecor, so ha he precompued advecion ccle indicaes he flow. In his paper, we show how exure mapping hardware can produce near-real-ime exure moion, *This work was performed under he auspices of he U.S. Deparmen of Energ b Lawrence Livermore Naional Laboraor under conrac number W-7405-ENG-48, wih specific suppor from an inernal LDRD gran. We hank Roger Crawfis for helpful suggesions.

2 using a polgon grid, and one fixed exure. However, we make no aemp o indicae he flow direcion in a sill frame. As discussed below, an anisoropic sreching comes from he veloci gradien, no he veloci iself. The basic idea is o advec he exure b he flow field. In [7] we gave an indicaion of he wind veloci b advecing he 3D exure coordinaes on he polgon verices of a cloudiness conour surface in a climae simulaion. This was slow, because he 3D exure was rendered in sofware, and because advecing he exure was difficul for ime-varing flows. In his paper, we replace he 3D exures b 2D exure maps compaible wih hardware rendering, and give echniques for handling ime-varing flows more efficienl. The nex secion gives our echnique for he case of 2D sead flows, and he following one discusses he problems of exure disorion. Then we discuss he problems wih exending our mehod o ime-varing flows, and our wo soluions. Nex we develop composiing mehods for visualizing 3D flows. The final secion gives our resuls and conclusions. TEXTURE ADVECTION FOR STEADY 2D FLOWS We sar wih a mahemaical definiion of exure advecion, and hen show how i can be approximaed b hardware exure-mapped polgon rendering. Le F (x, ) represen he sead flow soluion of he differenial equaion df ( x, ) V F ( x, ) d (1) where V(x, ) is he veloci field being visualized. Thus poin P is carried b he flow o he poin F (P) afer a ime dela. The flow F saisfies he composiion rule F s+ P ( ) F s F ( P) (2) for boh posiive and negaive s and. Thus (F ) -1 (P) F - (P). In his paper, we will assume ha he iniial exure coordinaes a 0 are he same as he (x, ) coordinaes of he region R being rendered. In pracice, he exure is usuall defined in a differen (u, v) coordinae ssem relaed o (x, ) b ranslaion and scaling, bu for simplici we will ignore he difference. If T(x, ) is a 2D exure being adveced b he flow, hen a new exure T (x, ) is defined b

3 T (x, ) T F 1 ( x, ) T F ( x, ). Thus, o compue T a a poin P, we go backwards along he sreamline hrough P, o find he poin Q such ha F (Q) P, and hen evaluae he exure funcion T a Q. When animaed, his will give he appearance ha he iniial exure T is being carried along b he flow. B equaion (2) above, F ( + ) ( P) F F ( P). Thus he sreamlines F - (P) needed for he exure coordinaes can be compued incremenall. There are wo problems wih his formulaion when he domain of definiion for V(x, ) or T(x, ) is limied o a finie region R in which he veloci daa or exure is available. Firs of all, if he sreamline F - (P) leaves he region R, he necessar velociies are no available o coninue he inegraion. One mus eiher exrapolae he known velociies ouside R, or coninue he sreamline as a sraigh line using he las valid veloci. Forunael, eiher of hese exrapolaion mehods will give a correcl moving exure in animaion. This is because he visible exure moion a a poin P inside R is deermined onl b he veloci a P, and he exrapolaion of he sreamline beond R serves onl o deermine wha exure will be brough in from off screen. Second, even if F - (P) is exended ouside R, he exure ma no be known here. The sandard soluion o his is o ake T(x, ) o be a periodic funcion in boh x and, so ha i is defined for all (x, ). Mos exure mapping hardware is capable of generaing his sor of wraparound exure, b using modular arihmeic (or runcaion of high order bis) o compue he appropriae exure map address from he x and values. There are also ools o generae exures which wrap around wihou apparen seams [8]. To adap his echnique o hardware polgon rendering, he 2D region R is divided up ino a regular grid of riangles, and he exure coordinaes F - (P i ) are onl compued for he verices P i of he grid. During he hardware scan conversion, exuring, and shading process, he exure coordinaes a each pixel are inerpolaed from hose a he verices, and he appropriae exure pixels are accessed. For riangles, he sandard bilinear inerpolaion, which is no roaion invarian, reduces o linear inerpolaion, which is. For ani-aliasing, he hardware can use he higher order bis of he exure coordinaes o weigh an average of four adjacen exure map values (or four values in each of he wo mos-nearl-appropriae-resoluion versions of he exure, if MIP mapping [9] is emploed.) TEXTURE DISTORTION The flow F - (P) can change he shape of a riangle, so ha i becomes long and hin in exure space, as shown in figure 1. In he direcion where he riangle is sreched b F -, he exure will be compressed b F. This disorion will no be presen if he veloci is consan, so ha F - and F are boh ranslaions. The disorion insead indicaes anisoropies in he derivaives of V. For incompressible 2D flows, sreching

4 in one direcion will be compensaed b compression in a perpendicular direcion. For compressible flows, here ma be sreching in all direcions a some posiions, and shrinking in all direcions a ohers. Figure 1. The riangle on he righ is mapped o he exure on he lef, which ends up being compressed vericall when he riangle is rendered. During he animaion of he exure advecion, his disorion coninues o build up, so ha evenuall he visualizaion will become useless. Therefore we periodicall resar he exure coordinaes back a heir original posiions in he regular grid. To avoid he sudden jump his would cause in he animaion, we graduall fade up he new exure and fade down he old one, according o he weighing curves in figure 2. Each exure sars wih weigh zero, fades up over he old exure unil i alone is presen, and hen fades Figure 2. Three ccles of he weighing curves for fading he exures up and down. down as an even newer exure akes is place. This cross dissolve can be done in hardware, using α com-

5 posiing [10]. If he exures are random, and conain an adequae range of spaial frequencies, his cross dissolve will no disurb he percepion of coninuousl flowing moion. Since each exure is used for onl a shor ime, he disorion does no become exreme. For a sead flow, one cross dissolve ccle ends wih he same image a which i began, so an animaion loop ma be creaed which can be ccled rapidl and repeaedl on a worksaion screen. Similar precompued loops are possible wih he surface paricle [2], LIC [3], and exured spla [6] echniques. TEXTURE ADVECTION FOR UNSTEADY 2D FLOWS If he veloci V depends on, he differenial equaion df ( x, ) V F ( x, ), d (3) defines a flow which no longer saisfies equaion (2). For a fixed iniial posiion Q, he curve F (Q) is a paricle race C() as in [11], raher han a sreamline. To find he exure coordinaes for P a ime 0 we need o find he poin Q such ha differenial equaion F 0 (Q) P. We mus go backwards along he paricle race, and hus solve he dc () V( C(), ) d (4) for he range 0 0, wih final condiion C( 0 ) P, and hen se Q C(0). Wih he change of variables u 0 -, his is equivalen o he differenial equaion dc ( u) V ( C( u), du 0 u) (5) for he u range 0 u 0, wih iniial condiion C(0) P. Then Q C( 0 ). In he case of unsead flow, he differenial equaions (5) for differen 0 are no relaed and define compleel differen paricle races, so incremenal mehods can no longer be used. In [7] we inegraed equaion (5) anew for each frame ime 0. To find he exure coordinaes for frame 0, we had o access he ime varing veloci daa for he whole range 0 0, which is ver inefficien for large daa ses. Here we propose wo more pracical mehods. The firs mehod is o derive a differenial equaion for he flow G ( x, ) F 1 ( x, ). This flow maps a poin P o he exure coordinae poin Q needed a frame ime, ha is, he poin wih F (Q) P. Thus we have

6 F G ( P) P. (6) Le G x and G be he x and componens of he vecor-valued funcion G (x, ), and similarl le F x and F be he componens of F. Then b differeniaing he componens of equaion (6) wih respec o b he chain rule, we ge he pair of equaions F x F G x x x F x F F G x x F , 0. F Now b equaion (3), x F V and, where V x and V are he componens of he veloci field x V a posiion F G ( P ) P and ime. Therefore we have M G where M is he Jacobian marix for he flow F (x, ): M Thus G x. M 1 V x G V Bu since G (x, ) F 1 ( x, ), he marix M 1 is he Jacobian marix J for G (x, ): G x F x V x V F x x F F x. J G x Thus G (x, ) saisfies he parial differenial equaions: G x x G G x.

7 ( x, ) G ( x, ) G x G x ( x, ) G x ( x, ) V x x V G ( x, ) G ( x, ) V. (7) x x V These differenial equaions esseniall sa ha he flow G (x, ) is deermined from he negaive of he veloci V, as ransformed ino he exure coordinae ssem appropriae for 0, so he deermine he exure flow necessar o give he desired apparen veloci a ime. The iniial condiion for G a 0 is ha G 0 (P) P, ha is, G 0 is he ideni map. Equaions (7) can be inegraed incremenall in ime b Euler s mehod. If G (P i ) is known a ime for all verices on a regular grid, he parials in he Jacobian marix J(P i ) can be esimaed from cenral differences beween he G values a adjacen grid verices. (For verices a he boundar of R, one-sided differences mus be used.) Then, using he curren veloci V(G G x G (P i ), ), incremens G x and G are found for he componens of G. If necessar, can be a fracion of he ime sep beween frames, and/or he verex grid used for solving equaions (7) can be finer han he riangle grid used in rendering he exure, in order o make he soluion more accurae. The verex grid spacing will affec he accurac of he finie difference approximaions o he parial G derivaives like This accurac is criical, because small errors in hese parials will cause errors in x posiion in he nex frame, which ma compound he errors in he parials, and cause hem o grow exponeniall from frame o frame. Here again, i is useful o fade ou he curren adveced exure and fade in a new exure whose coordinaes are reiniialized o he ideni map, so ha he inegraion errors canno accumulae for oo long. The second mehod for handling unsead flows is o move he riangle verices b he flow F (x, ), keeping heir exure coordinaes consan. This advecs he exure direcl, b moving he riangles, and carring he exure along wih hem. To do his, we incremenall inegrae equaion (3), and no parial derivaive esimaes are needed for a Jacobian. However we again have a problem a he edges of he region R. The boundar verices ma move inside R, leaving gaps a he edges, or ma move ouside, causing oo much exure o be rendered. The excess rendering is easil prevened b clipping all riangles o he boundar of R. The gaps can be eliminaed b creaing exra guard polgons around he edges of R, widening i o a larger region S. Whenever an verex on he boundar of S crosses ino R, a new row of guard polgons is added o he affeced side of S. Again i is useful o inegrae onl over a limied ime inerval before reiniializing he exure coordinaes, o avoid creaing oo man exra polgons. FLOWS IN 3D In hree dimensions, one could advec 3D exure coordinaes, bu 3D exuring is no widel available.

8 The differen pes of moving exures discussed were implemened as a class hierarch in C++. Inven- We have insead used 2D exures on parallel secion planes. We made he exured planes semi-ransparen, and composied hem from back o fron using he α composiing hardware in our worksaion. (This is how 3D exure mapping is usuall implemened in hardware.) For he mehods which change onl he exure coordinaes, we used he 2D projecion of he veloci ono he secion plane. For he mehod which moves he riangle verices, we used he rue 3D veloci, allowing he secion surfaces o warp ou of planari. Combining he composiing for he cross-dissolve of figure 2 wih he composiing of he separae exure planes can lead o problems in he accumulaed opaci. Given wo objecs wih opaciies α 1 and α 2, he resuling opaci from composiing boh objecs is α 1 + α 2 - α 1 α 2. (See [10] or mulipl he ransparencies.) Suppose f 1 () and f 2 () are he wo weighing curves shown in figure 2, wih f 1 + f 2 1, and α is he desired secion plane opaci. If we jus ake he wo componen opaciies o be α 1 αf 1 and α 2 αf 2, he resul is a composie opaci α C αf 1 + αf 2 α 2 f 1 f 2 α αf 1 f 2 The unwaned las erm causes a periodic pulsaion in α C. A soluion is o use exponenials, which have beer muliplicaive properies. Define an opical deph l - ln(1 - α), so ha α 1 - e - l, and le α and α The resuling composie opaci is hen as desired. α C α 1 + α 2 α 1 α 2 1 e f 1 l 1 e f 2 l + 1 e f ( f 1 + f 2 )l e f 1 l e f 2 l Anoher problem wih composiing exure planes of consan ransparenc is ha he fronmos planes will evenuall obscure he ones o he rear if he daa volume ges large. One soluion is o use variableransparenc exures, so ha some regions of he exure are compleel ransparen. Anoher is o specif he ransparenc on riangle verices using a separae scalar daa variable which can selec ou regions of ineres where he exure moion should be visible. In [7] we used percen cloudiness conour surfaces o specif he locaion of he advecing sofware-rendered exure. Wih our new hardware based echnique, his cloudiness variable is used o specif he verex ransparenc, and produces similar realism in much less ime. 1 l 1 e 1 e l α 1 e f 2 l IMPLEMENTATION AND RESULTS

9 or [12] quadmeshes were used o represen exure laers. An Iris Explorer module was hen consruced in order o make use of color maps and daa readers. Figure 3 shows wha happens when he verices hemselves are adveced. The whole surface disors, even in he direcion perpendicular o he plane. In Figure 4 he exure coordinaes are adveced backwards while he verices are held fixed. This gives he impression of moion in he direcion of flow. Unforunael he exure disors oo much over a long period of ime. Also he exure verices ma move ouside he defined domain. A soluion o he firs problem is o fade in a second exure wih he exure coordinaes rese o heir original posiions. The resuling cross dissolve is shown in Figure 5. The opaci for each exure is compued using exponenials, as discussed above, so here is no disracing variaion in he overall inensi during animaion. To avoid he problem of having o sample ouside he domain, we used he inverse flow G for he exure coordinaes, as explained above, while keeping he verices fixed (Figure 6). This mehod also gives bad resuls over ime if we do no periodicall fade in a new unadveced exure as shown figure 7. Figure 8 illusraes how flow moves hrough paricles of aerogel, a maerial wih ver low densi which is a good hermal insulaor. Figure 9 shows a frame from an animaion of global wind daa on a spherical mesh. The opaque regions represen high percen cloudiness. Alhough he vecor field is saic, he exure (bu no he colors) appear o move in he direcion of flow. Figures 10 and 11 depic sead flow near a high densi conour in an inersellar cloud collision simulaion (daa courese of Richard Klein). Figure 10 has moving verices, while figure 11 has moving exure coordinaes. The color indicaes densi. A frame from an animaion of unsead wind daa over Indonesia on a curvilinear mesh is shown in Figure 12. Percen cloudiness is mapped o color and opaci. We ran our sofware on an SGI Onx supporing hardware exure mapping. For a 32 b 32 slice of a volume (as in he aerogel example) we were able o achieve abou four frames per second. To roae a complee 50x40x10 volume, like he one shown in Figure 9, abou 15 seconds was required. REFERENCES 1. Shoup, Richard: Color Table Animaion. Compuer Graphics Vol. 13, No. 2 (Augus 1979) pp van Wijk, Jarke: Flow Visualizaion Wih Surface Paricles. IEEE Compuer Graphics and Applicaions, Vol. 13, No. 4 (Jul 1993) pp Cabral, Brian; and Leedom, Lieh: Imaging Vecor Fields Using Line Inegral Convoluion. Compuer Graphics Proceedings, Annual Conference Series (1993) pp Forssell, Lisa: Visualizing Flow over Curvilinear Grid Surfaces using Line Inegral Convoluion. Proceedings of IEEE Visualizaion 94, pp

10 5. Salling, Delev; and Hege, Hans-Chrisian: Fas and Resoluion Independen Line Inegral Convoluion. ACM Compuer Graphics Proceedings, Annual Conference Series, 1995, pp Crawfis, Roger; and Max, Nelson: Texure Splas for 3D Scalar and Vecor Field Visualizaion. Proceedings of IEEE Visualizaion 93, pp Max, Nelson; Crawfis, Roger; and Williams, Dean: Visualizing Wind Velociies b Advecing Cloud Texures. Proceedings of IEEE Visualizaion 92, pp Heeger, David; and Bergen, James: Pramid-Based Texure Analsis and Snhesis. ACM Compuer Graphics Proceedings, Annual Conference Series, 1995, pp Williams, Lance: Pramidal Paramerics. Compuer Graphics Vol. 17 No. 3 (Jul 1983) pp Porer Tom; and Duff, Tom: Composiing Digial Images. Compuer Graphics Vol. 18, No. 4 (Jul 1984) pp Lane, David: UFAT - A Paricle Tracer for Time-Dependen Flow Fields. Proceedings of IEEE Visualizaion 94, pp Wernecke, Josie: The Invenor Menor. Addison -Wesle Publ. Co., Inc., Figure 3. Acual verices are adveced in 3D. Figure 4. Texure coordinaes are adveced backwards. Figure 6. Texure coordinaes are adveced using vecors ransformed b he local jacobian marix, while verices are held fixed. Figure 5. Same mehod as figure 4, bu wih a new exure fading in as soon as he oher becomes oo disored. Figure 7. Same mehod as figure 6, bu wih a new exure fading in as soon as he oher becomes oo disored.

11 Figure 8. Mehod of figure 6 applied o a sead flow moving hrough paricles of aerogel and using a colored exure. Figure 9. Mehod of figure 5 applied o a sead flow depicing wind daa on a spherical mesh. Color and opaci from percen cloudiness. Figure 11. Mehod of figure 6 applied o a sead flow represening a field from an inersellar cloud collision simulaion. Figure 10. Several laers of exures adveced using he mehod of figure 3. The laers are colored b densi and move near a high densi solid conour surface. Figure 12. Mehod of figure 7 applied o an unsead flow represening global climae daa. Color and opaci indicae percen cloudiness. Boh he winds and percen cloudiness var in ime.