Direct Volume Visualization of Three-Dimensional Vector Fields
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1 Direc Volume Visualizaion of Three-Dimensional Vecor Fields Roger Crawfis Nelson Max Lawrence Livermore Naional Laboraory PO. Box 808 / L 301 Livermore, CA (crawfis@llnl.gov) (max2@llnl.gov) Absrac Curren echniques for direc volume visualizaion offer only he abiliy o examine scalar fields. However mos scienific exploraions require he examinaion of vecor and possibly ensor fields as well as numerous scalar fields. This paper describes an algorihm o direcly render hree-dimensional scalar and vecor fields. The algorihm uses a combinaion of sampling and splaing echniques, ha are exended o inegrae he display of vecor field daa wihin he image. Addiional Keywords: vecor field, flow field, volume rendering, vecor filer, composiing, scalar field, climae modeling. Inroducion The rendering of hree-dimensional scalar fields has received much aenion over he pas several years. These 3D scalar fields can be represened using eiher isoconour surface reconsrucion algorihms, or as semi-ransparen densiy clouds. Wih isoconour surfaces, inermediae geomery is produced and processed using he normal geomeric pipelines developed during he las few decades. Thus, addiional geomeric objecs such as axes, vecors or addiional isoconours (from possibly differen scalar fields) can be easily added o he display of he isoconour. Unforunaely, his is no he case for volume densiy clouds. Shirley and Neeman [Shirley89] and Levoy [Levoy89] discuss he inegraion of separae geomeric objecs using rayracing. The soring required a each sample poin makes his algorihm infeasible for a large number of geomeric objecs, such as ha produced by he display of many iny vecors. Max, Hanrahan and Crawfis [Max90] demonsrae how o incorporae geomeric surfaces ino heir back-o-fron composiing of volume polyhedra. This was limied o isoconour surfaces, and also required spliing he polyhedron up ino several pieces and shipping each individual piece o he volume renderer. Laely, projecive echniques have been developed ha use a geomeric descripion of he cloud densiy wihin each voxel [Wesover90], [Wilhelms91], [Laur91]. While hese echniques are geomerically based, hey require passing he polygons o he geomery pipeline in a back-o-fron order. Research ino he display of hree-dimensional flow has also been explored over he pas few years. Various algorihms o represen he flow via ribbons have been developed. Helman and Hesselink [Helman91] and Globus e. al. [Globus91] have developed algorihms o display criical poins wihin he flow field. These algorihms and he sandard vecor or hedgehog plos have no direc way of being combined wih he direc volume visualizaion mehods recenly developed. Paricle sysems [Reeves85] can be used o represen boh scalar ([Max90], [Sabella88]) and vecor fields ([van Wijk91], [van Wijk90]). Vecor fields require an advecion of he paricles for each ime sep, and usually involve creaing and deleing paricles as ime progresses. Unforunaely, hese algorihms are quie compuaionally inensive, and he number of paricles required for his purpose would be prohibiive. Kajiya [Kajiya89] also made allusions o represening vecor fields using algorihms developed for he display of hairy surfaces. This echnique is limied o he display of he vecor flow upon a surface and is very compuaionally expensive, requiring several hours of CPU ime per image. Spo noise [van Wijk91] offers an ineresing echnique for visualizing flow fields, bu is again limied o he flow over a surface. and is also compuaionally expensive. Our goal was o render he relaionship beween urbulen flow fields and scalar densiy fields hroughou a hree-dimensional volume. This was driven by a requiremen o visualize he cause and effec relaionship of clouds and winds wihin global climae models [Poer91]. Global climae modeling produces a ime hisory of daa, each ime sep of which needs o be rendered. We have invesigaed he use of high frequency exures o represen vecor fields in wo-dimensions [Crawfis91]. Here, an anisoropic exure is derived from he vecor field. Time dynamics are hen creaed by simply regeneraing an anisoropic exure a each ime poin. Since we recognize frequency, bu no phase, in paerns and exures, a smooh flow is creaed ha provides he illusion of moion in an animaion, wihou requiring any advecion. A mehod ha does use advecion on he same climae daa is described in [Max92]. We have exended our research of wodimensional vecor filers ino hree-dimensions, and incorporaed he inegraion of a scalar field wih he composiing of he vecor field o accomplish our goals. Our echnique for represening vecor fields is o creae a very fine-grained exure represenaion of he flow. Individual vecors, insignifican individually, combine o form a useful picure of he
2 overall flow of he field. We have developed a filer which can be used o sweep hrough a volume image in back-o-fron order. The kernel of his filer can be used o represen boh scalar and vecor quaniies for wo- and hree-dimensional daa ses. The basic algorihm o render a wo-dimensional vecor field passes a vecor kernel filer across he resulan image. The kernel deposis an ani-aliased line over he widh of is domain. Each individual line is composied in using he OR operaor, and is orienaion is based on a sampling of he vecor field. By carefully conrolling he movemen of he filer, highly dynamic flow fields can be represened (Figure 5). Filer Movemen A key crierion for good exure generaion of he vecor field is o avoid paerns caused by he regular movemen of he filer, or he regular spacing of he daa. Three opions o overcome hese paerns are available. Wihin he filer kernel, he cener poin (P in Figure 2) hrough which he vecor passes is randomly chosen (Figure 3a). The major reducion in paerns comes from conrolling he movemen of he enire filer. The filer is moved wih random jiers in is incremen o preven he regular spacing apparen from he clipped edges of he vecors (Figure 3b) Finally, he filer is moved in incremens smaller han is widh (Figure 3c). This allows vecors o overlap, and blurs he exen of each individual vecor. While he differences beween hese hree images may no be subsanial, when we animae he images, very differen resuls appear. Wha we are afer is he illusion ha he paricles or he exure as a whole is moving, no individual flags waving in he wind. Wheher all of hese jierings are necessary has ye o be deermined, however none of hem require addiional resources of any significance. These jierings were developed for wo-dimensional filers and side or op views of hree-dimensional daa ses. Oblique views in hree-dimensions will naurally break up regular paerns o some exen. Vecor Kernels Vecors are represened on he image as line segmens wih varying color and opaciy. As he filer moves along he oupu image, a vecor kernel performs hree asks: deermine he projecion of he vecor, calculae he color and opaciies of he vecor line segmen and he scalar funcion, and composie his informaion ino he image. The vecor is projeced ono he viewing plane by aking he do produc of he vecor wih wo basis vecors defining he viewing plane: ux = r V r i uy = r V r j where he projeced vecor, r u = (ux uy) T This projeced vecor, r u, is hen normalized for use in fuure operaions. Once we have he projecion of he vecor ono he screen, we hen need o deermine for each pixel wha fracion of he vecor lies j i P V u n p Figure 2. Deermining Pixel values wihin he pixel and wha he various aribues of he vecor are a ha pixel. Assuming circular pixels, he area of overlap wih a hick line segmen can be esimaed by aking he absolue value of he do produc of he vecor, r n p, perpendicular o r u wih he vecor from he cener poin o he curren pixel (Figure 2). This gives us he perpendicular disance from he axis of he vecor o he pixel. The funcion: 1.0 r r1 f(r) = ar + b r1 r r2 0.0 r r2 can hen be used o produce smooh ani-aliased lines. The values of r1 and r2 conrol he hickness of line segmens, and are specified by he user. These ani-aliased lines work beer and are compuaionally easier han using cylinders. The area is used as an opaciy and color scaling facor in composiing he vecor ino he image. Several conrols over he represenaion of he vecors are available. Depending on he kernel, an arbirary color mapping scheme is offered. Curren kernels will map he eiher he world z-coordinaes or he screen z-coordinaes (z-heigh) o a color in a user specified color able. The z-heigh can include boh he relaive posiion in he daa se and he heigh increase of he vecor across he filer. By heavily weighing his laer erm, color can be mapped o show he verical velociy componen. Oher color mapping schemes such as hose proposed by Van Gelder and Wilhelms [VanGelder92], could easily be incorporaed. This color is used as he base color or hue. By desauraing one end of he vecor, we can add an indicaion of he signed direcion of he vecor (i.e., a vecor head). Here, if we simply ake he do produc of he projeced vecor and he vecor o he pixel cener, we will ge a measure of where we are along he axis of he vecor. Since we are only concerned wih he pixels along he vecor axis a his poin, we can use his measure direcly. Finally, we adjus he vecor's inensiy by is magniude. A deph cue can also be applied by adjusing he inensiy based on he linear disance from he view poin.
3 Figure 3. Vecor Kernel Movemen Effecs: a) Jiering of he cener poin, b) Jiering of he sride lengh, c) Overlapping srides. Scalar Rendering For overlapping filers, a splaing-like algorihm works well. The ideal reconsrucion filer described by Max [Max91] is used as he basis for he spla. This filer: 1 r2 s ( r) 2 g(r) = ( s) 0 r s s r 0.0 r wih s = 0.48 and = 1.37 for a filer sride of one, has a finie span, allowing he size of he kernel o be arbirarily large. This allows long vecors, while limiing he effec of he spla o he sride aken in he filer. This does presen he problem ha he vecor drawn wih he spla is only affeced by ha spla and no neighboring splas ha may overlap he vecor. Overcoming his would require complicaed neighborhood ess, and he handling of muliple vecor segmens wihin he kernel. Since hese discrepancies in he renderer are no noiceable for he es cases we have run, we choose o ignore hem. This implies ha a perfecly accepable soluion would be o simply spla in a vecor, and hen spla in he volume over i for each kernel insaniaion. A more accurae soluion is described in he nex secion. The addiion of scalar splaing increases he compuaional ime of he vecor kernel subsanially. The reason for his is wofold: 1) The pixels ouside he projeced vecor mus now be calculaed and composied in. 2) Kernel calculaions were skipped if he vecor lengh was less han some user specified olerance. These mus now be drawn if he scalar field conribues o he image (i.e., he scalar field is greaer han a cerain hreshold). Composiing Once we have he conribuion due o he vecor field and he conribuion due o he scalar field a each sampled voxel, we can calculae he oal conribuion o each pixel. Consider a ray, of uni lengh, from he eye passing hrough a polyhedron wihin which we wish o render a vecor (Figure 4). The segmen of his ray passing hrough he polyhedron is broken up ino hree pars: ha segmen in fron of he vecor, he segmen passing hrough he vecor, and he segmen behind he vecor. Le represen he lengh of he fron segmen. If he vecor has a hickness, dv, hen he segmens have lenghs, dv, and (1--dv). If we hen assume a homogenous opaciy and color, he inensiy can be calculaed using he equaion: 1 Ω(u)du I = Ω()e 0 d 0 ρ(u)du = ρ()e 0 d 0 ρ(u)du +e 0 +dv ν(u)du ν()e d +dv ρ(u)du ν(u)du +e 0 e 1 ρ(u)du ρ()e +dv d +dv where ρ(x) and υ(x) represen he scalar densiy and vecor densiy disribuions along he ray, and Ω(x)he oal densiy disribuion. If we assume an infiniesimal hickness in he vecor, and give i a fixed opaciy, α ν, and color, I ν, hen he equaion simplifies o:
4 ρ(u)du I = ρ()e 0 d 0 ρ(u)du +I ν e 0 ρ(u)du 1 ρ(u)du +(1 α ν )e 0 ρ()e d The value can be calculaed for each pixel ray from he plane consising of he ransformed vecor and one of he basis vecors defining he viewing plane. By using he analyical inegraion proposed by Max, Hanrahan and Crawfis [Max90], his calculaion requires only wo exponenial evaluaions, or one addiional exponenial over he sraigh volume rendering. The color imes deph approximaion, C*D, proposed by Wilhelms and Van Gelder [Wilhelms91], can be used o furher simplify his equaion. Here, hree colors and opaciies are compued for he vecor, in fron of he vecor, and in back of he vecor and composied ogeher. If I s and α s are he color and opaciy per uni lengh for he scalar field, he equivalen equaion for he simplified C*D calculaion is: I = I s + (1 α s )I v + (1 α s )(1 α v )I s (1 ) While his follows logically, i does no produce he desired resul. Consider he case where a ray jus grazes he edge of an anialiased vecor, such ha α v is almos zero. The cumulaive inensiy is hen: I = I s + (1 α s )I s (1 ) bu, he inensiy of a neighboring pixel which does no inersec he vecor is simply I s. Thus for he C*D inegraion calculaion, he formulas: I = I s + (1 α s )I v + (1 α v )I s (1 ) α = α s + (1 α s )α v + (1 α v )α s (1 ) should be used. These are hen composied ino he image. Efficiency Consideraions A leas hree possible ess can be used o reduce compuaions and hereby improve he efficiency. The firs is on he lengh of he vecor. If he magniude of he vecor falls below a cerain hreshold, hen he calculaions needed o render i can be skipped. Wih his comes he second es, on he maximum conribuion of he spla. If he opaciy of he scalar field falls below some hreshold, hen he calculaions o render i can be skipped. Finally, he bigges win comes when boh he above condiions are rue. In his case, he enire kernel can be skipped. The size of he resuling image, he span of he filer, and he sride of he filer all have an effec on he performance of he filer. Smaller images and filer size and larger srides can improve he performance of he filer. The resoluion of he image's z-space also has a significan impac on he performance of he filer. All of hese variables are specified under user conrol. The simpliciy of a filer makes i a naural choice for vecorizaion and parallel processing. Each pixel wihin he filer requires he same arihmeic, allowing i o be compued in parallel on even a SIMD machine. For he 2D filer or a op down view wih he 3D filer, several insaniaions of he filer kernel can also operae in parallel. Finally, since he filer samples boh he vecor and he scalar field, large amouns of memory may be necessary o mainain his daa. However, he filer does process his daa in a fairly sequenial order. Resuls Figures 5 and 6 are aken from an HDTV animaion presened a he SIGGRAPH '92 Film and Video show. Figure 5 illusraes he direc volume rendering of jus he wind velociies, while Figure 6 illusraes he wind velociies and he percen cloudiness. All of his daa was calculaed from a global climae model wih grid dimensions of 320 by 160 by 19. Figure 5 required 30 seconds o generae on a SGI Personal IRIS a NTSC resoluion. Figure 6 required one minue. The simulaed daa consiss of clouds and winds a every hour for en days. Each day of he simulaion generaes 380Mb of daa for he wind and percen cloudiness fields. Figure 7 shows an oblique view of a es funcion, simulaing a ornado. Figure 8 illusraes he elecric field around an wihin a small porion of a Boeing 737 je, he avionics' bay. Fuure Work dv n Figure 4. Inegraing along he viewing direcion. The above echniques provide an effecive soluion o he simulaneous display of a single scalar field and a single vecor field. This allows he scieniss o sudy he complex relaionships beween he winds and he clouds or he winds and a specific amospheric heaing erm. However, he scieniss sill need o undersand he complex dynamics beween he winds and several scalar variables (i.e., percen cloudiness, incoming and ougoing radiaion, percen humidiy, ec.). This is a general research opic o be addressed in no only he vecor domain, bu he scalar domain as well. We have simplified he problem here by flaening he errain in he climae models and dealing only wih a regular grid. In fac
5 global climae models and many oher grand challenge problems deal wih irregular opologies which mus be deal wih. We have also concenraed our aenion on he echniques and represenaions, raher han on efficien soluions. While he echniques are fairly efficien, improvemens mus sill be made o achieve ineracive levels. The use of able lookups as described by Laur and Hanrahan [Laur91], and Wesover [Wesover90] and he use of Gouraud shaded or hardware exure mapped polygons should be evaluaed. Finally, he echnique oulined here does no ake ino accoun he overlap of he filers when drawing he vecors. This involves a rade-off decision beween hese inaccuracies and he complexiy associaed wih keeping vecors consisen across spla or voxel domains. Acknowledgmens We would like o hank Jerry Poer, Bob Mobley and Dean Williams of he Program for Climae Model Diagnosis and Inercomparison (PCMDI) a he Lawrence Livermore Naional Laboraory for providing us wih he daa and direcion. The European Cenre for Medium-range Weaher Forecasing (ECMWF) provided he analyical model used o generae he daa. Gene Cronshagen and Chris Anderson helped in generaing and recording he animaions. Seve Pennock provided he Boeing 737 daa. Becky Springmeyer provided many useful commens on he draf. Suppor for his research came from he Deparmen of Energy's High Performance Compuing and Communicaions Program hrough he effor on Visualizaion For Global Climae Modeling. This work was performed under he auspices of he US Deparmen of Energy by Lawrence Livermore Naional Laboraory under conrac No. W-7405-Eng-48. [Levoy89] [Max90] [Max91] [Max92] [Poer91] [Reeves85] [Sabella88] [Shirley89] Levoy, Marc. Design for a Real-Time High-Qualiy Volume Rendering Worksaion. In Proceedings of he Chapel Hill Workshop on Volume Visualizaion, pp Max, Nelson, Pa Hanrahan, and Roger Crawfis, Area and Volume Coherence for Efficien Visualizaion of 3D Scalar Funcions. Compuer Graphics Vol. 24 No. 5 (November 1990, Special issue on San Diego Workshop on Volume Visualizaion) pp Max, Nelson, "An Opimal Filer for Image Reconsrucion," Graphics Gems II (James Arvo, ed.). Academic Press, Boson. Max, Nelson, Roger Crawfis, and Dean Williams, Visualizing Wind Velociies by Advecing Cloud Texures. Proceedings of Visualizaion '92, IEEE. Poer, Gerald. privae communicaion. Reeves, William T. and Ricki Blau. Approximaion and Probabilisic Algorihms for Shading and Rendering Srucured Paricle Sysems Compuer Graphics Vol. 19 No. 3 (July 1985, SIGGRAPH '85) pp Sabella, Paolo. A Rendering Algorihm for Visualizing 3D Scalar Fields. Compuer Graphics Vol. 22 No. 4 (July 1988, SIGGRAPH '88) pp Shirley, Peer, and Henry Neeman Volume Visualizaion a he Cener for Supercompuing Research and Developmen. In Proceedings of he Chapel Hill Workshop on Volume Visualizaion, pp References [Crawfis91] [Globus91] [Helman91] [Kajiya89] [Laur91] Crawfis, Roger and Michael Allison. A Scienific Visualizaion Synhesizer. In Proceedings Visualizaion '91. Gregory Nielson and Larry Rosenblum eds., IEEE Los Alamios, CA, pp Globus, A. and C. Levi and T. Lasinski. A Tool for Visualizing he Topology of Three-Dimensional Vecor Fields. In Proceedings Visualizaion '91. Gregory Nielson and Larry Rosenblum eds., IEEE Los Alamios, CA, pp Helman, J. and L. Hesselink. Visualizing Vecor Field Topology of Three-Dimensional Vecor Fields. IEEE Compuer Graphics and Applicaions Vol. 11 No. 3, pp Kajiya, James T. and Timohy L. Kay. Rendering Fur Wih Three Dimensional Texures. Compuer Graphics Vol. 23 No. 3 (July 1989, SIGGRAPH '89) pp Laur, David and Pa Hanrahan, Hierarchical Splaing: A Progressive Refinemen Algorihm for Volume Rendering. Compuer Graphics Vol. 25 No. 4 (July 1991, SIGGRAPH '91) pp [Shirley90] Shirley, Peer, and Allan Tuchman, A Polygonal Approximaion o Direc Scalar Volume Rendering. Compuer Graphics Vol. 24 No. 5 (November 1990, Special issue on San Diego Workshop on Volume Visualizaion) pp [VanGelder92] Van Gelder, Allen and Jane Wilhelms. Ineracive Visualizaion of Flow Fields Volume Visualizaion Workshop (his issue), Kaufman and Lorensen (eds), ACM SIGGRAPH, NY. [Wesover90] Wesover, Lee. Fooprin Evaluaion for Volume Rendering. Compuer Graphics Vol. 24 No. 4 (July 1990, SIGGRAPH '90) pp [Wijk90] [Wijk91] Wijk, J.J. van, A Raser Graphics Approach o Flow Visualizaion. in Vandoni, C.E., and D.A. Duce (eds.), Proceedings Eurographics'90, Norh- Holland, Amserdam, 1990, pp Wijk, J.J. van, Spo Noise.: Texure Synhesis for Daa Visualizaion Compuer Graphics Vol. 25 No. 4 (July 1991, SIGGRAPH '91) pp [Wilhelms91] Wilhelms, Jane and Allen Van Gelder, A Coheren Projecion Approach for Direc Volume Rendering Visualizaion. Compuer Graphics Vol. 25 No. 4 (July 1991, SIGGRAPH '91) pp
6 Figure 5: Global CLimae Model winds color coded by aliude Figure 6: Global CLimae Model winds wih percen cloudiness Figure 7 Voriciy of Fluid Flow - no Tes Tornado Figure 8: Avionics bay of a Boeing 737. Elecric field excied by an inciden plane wave
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