Visualization of Time-Varying Volumetric Data using Differential Time-Histogram Table

Transcription

1 Visuliztion of Time-Vrying Volumetric Dt using Differentil Time-Histogrm Tble Hmid Younesy Simon Frser University Torsten Möller Simon Frser University Hmish Crr University College, Dublin ABSTRACT We introduce novel dt structure clled Differentil Time- Histogrm Tble (DTHT) for visuliztion of time-vrying sclr dt. This dt structure only stores voxels tht re chnging between time-steps or during trnsfer function updtes. It llows efficient updtes of dt necessry for rendering during sequence of queries common during dt explortion nd visuliztion. The tble is used to updte the vlues held in memory so tht efficient visuliztion is supported while gurnteeing tht the sclr field visulized is within given error tolernce of the sclr field smpled. Our dt structure llows updtes of time-steps in the order of tens of frmes per second for volumes of sizes of.5gb, enbling rel-time time-sliders. CR Ctegories: I.3. [Computing Methodologies]: Computer Grphics Methodology nd TechniquesE.1 [Dt]: Dt Structures Keywords: time-vrying dt, volume rendering 1 INTRODUCTION As computing power nd scnning precision increse, scientific pplictions generte time-vrying volumetric dt with thousnds of time steps nd billions of voxels, chllenging our bility to explore nd visulize interctively. Although rendering single time step cn generlly be chieved efficiently, rendering speed is hmpered by the bility to trnsfer dt efficiently from externl storge (i.e. disks) to min memory to grphics hrdwre. In this pper we use the temporl coherence of sequentil time frmes, the sptil distribution of dt vlues, nd the histogrm distribution of the dt to permit efficient visuliztion. Temporl coherence [1] is used to prevent loding unchnged smple vlues, sptil distribution [1] is used to preserve loclity of rendering between imges, while histogrm distribution is used for incrementl updtes during the dt clssifiction. Of these, temporl coherence nd histogrm distribution re encoded in differentil tble computed in preprocessing step. This differentil tble is then used during visuliztion to minimize the mount of dt tht needs to be loded from disk between ny two successive imges during dt explortion. Sptil distribution is then used to ccelerte rendering by retining unchnged geometric, topologicl nd combintoril informtion between djcent imges rendered, bsed on the differentil informtion loded from disk. Specificlly, the contributions of this pper re to identify nd nlyze sttisticl chrcteristics of lrge scientific dt sets tht emil: hyounesy@cs.sfu.c emil: torsten@cs.sfu.c hmish.crr@ucd.ie permit efficient error-bounded visuliztion nd to unify the sttisticl coherence of the dt with the known sptil nd temporl coherence of the dt in the form of single differentil tble to be used in updting memory-resident version of the dt. We define coherence nd distribution more formlly in Section, review relted work in Section 3 nd support our definition of coherence nd distribution in Section with cse study. In Section 5 we show how to exploit dt chrcteristics to build binned dt structure representing the temporl coherence nd sptil nd histogrm distribution of the dt, nd how to use this structure to ccelerte interctive explortion of such dt. We then present some results in Section nd some conclusions in Section 7, followed by comments on possible future work in Section 7. DATA CHARACTERISTICS In order to discuss both previous work nd our own contributions, we strt by defining our ssumed input nd the sttisticl properties of the dt tht we will exploit for ccelerted visuliztion: we will support these ssertions empiriclly in our cse study in Section. Formlly, time-vrying sclr field is function f : R R. Such dt is often generted by smpling f t fixed loctions V = {v 1,...,v n } nd fixed times T = {t 1,...,t τ }. In this pper, we will exploit this smpling regulrity to ccelerte visuliztion, so we ssume tht our dt is of the form F = {(v,t, f (v,t)) v V,t T}, where ech triple (v,t, f (v,t)) is smple. Previous reserchers hve reported [] tht only smll frction of F is needed for ny given imge generted during visuliztion. Thus, ny imge cn be thought of s the result of visulizing subset of F defined by query q. More formlly, given F nd condition q t time t (e.g. trnsfer function), find the set F q of ll smples (v,t, f (v,t)) for which q is true (e.g. the opcity ssigned to f (v,t) is non-zero). Thus, we cn tret F q s set of smples, clled the ctive set of q: the set of smple vlues needed to render q. To construct query q for isosurfce extrction t isovlue h, we note tht F q does not consist solely of smples with isovlue h. Insted, F q consists of ll smples belonging to cells whose isovlues spn h. For volume rendering, we bse trnsfer functions on the isovlue, s shown in Figure : the ctive set therefore consists of smples with rnge of isovlues. We now reformulte the observtion stted bove s follows: we expect F q F. Moreover, F q is rrely rndom, nor is it evenly distributed throughout the dt set. Insted, F q tends to consist of reltively lrge contiguous blocks of smples. This reflects the ssumed continuity of the physicl system underlying the dt, nd the fct tht the physicl system being studied usully involves n orgnized phenomenon rther thn rndom noise. Thus, the continuity of physicl systems, nd the nture of physicl systems being studied, both lend structure to the dt: we exploit this structure to ccelerte visuliztion. For interctive visuliztion, we lso exploit nother phenomenon: tht humn explortion of dt usully involves continuous (i.e. grdul) vrition of queries. Formlly, we re interested, not in single query q, but in sequence q 1,...,q k of closely-relted queries, with occsionl brupt chnges to new

2 query sequence q 1,...,q m. Becuse of the grdul vrition between queries, we expect tht the ctive sets for ny two sequentil queries will be nerly identicl, i.e. tht F q i+1 F q i. The crucil dt chrcteristics we exploit re coherence nd distribution. Let us denote two smples σ 1 = (v 1,t 1, f (v 1,t 1 )) nd σ = (v,t, f (v,t )), nd choose smll vlues δ,λ >. Sptil coherence is coherence with respect to the sptil dimensions x, y, z of the domin of f, nd is bsed on the sptil continuity of the physicl system. Smll sptil chnges imply smll functionl chnges, i.e. if v 1 v < δ nd t 1 = t, sptil coherence implies tht f 1 f < λ for smll λ. Temporl coherence is coherence with respect to the time dimension t of the domin of f nd is bsed on the temporl continuity of the physicl system. Hence, if t 1 t < δ nd v 1 = v, temporl coherence implies, tht f 1 f < λ for smll λ. Given two sequentil queries q i,q i+1 which differ only slightly in spce or time, F q i nd F q i+1 will therefore overlp significntly. We exploit this to ccelerte the tsk of updting the ctive set for rendering purposes, focusing principlly on temporl explortion. We lso exploit sttisticl properties of the dt sets in question, which we describe in terms of dt distribution: Histogrm distribution is property of the functionl dimension (i.e. the rnge of f ) nd depends lrgely on the dt being studied. However, it is usully the cse tht extrcted fetures re chosen t vlues where the rnge of f vries gretly: thus we expect f to be firly well distributed, lthough smpling my ffect this. Sptil distribution is lrge scle form of sptil coherence, nd is bsed (gin) on the physicl phenomen scientists nd engineers study. Although dt is commonly generted over lrge domins, it is often the cse tht nothing interesting hppens in most of the domin. As result, the ctive sets for queries mde by the user re often clustered in subset of the spce. Histogrm distribution is crucil where two sequentil queries q i nd q i+1 differ only slightly in the functionl dimension. For well-distributed dt vlues, we expect F q i nd F q i+1 to be substntilly identicl: i.e. tht we cn chnge trnsfer functions more rpidly by loding only the new dt vlues required. Moreover, the sptil distribution of the dt llows us to hve sptilly sprse dt structures. 3 RELATED WORK The difficulty of rendering lrge dt sets is well-recognized in the literture, with successive solutions ddressing progressively lrger dt sets by exploiting vrious dt chrcteristics. Brodly speking, however, the solutions proposed hve delt with isosurfce extrction nd volume rendering s seprte topics, wheres our solution is pplicble to either with suitble modifictions. Modern isosurfce extrction nd volume rendering methods re bsed on the recognition tht the problem is decomposble into smller sub-problems. Lorenson & Cline [] recognized this nd decomposed the extrction of n isosurfce over n entire dt set by extrcting the surfce for ech cell in cubic mesh independently. Almost simultneously, Wyvill, McPheeters & Wyvill [17] exploited sptil coherence for efficient isosurfce extrction in their continution method. Sptil coherence hs lso been exploited for volume rendering [15]. Other reserchers [3,, 5, 1] exploit sptil distribution for efficient extrction of isosurfce ctive sets. In prticulr, Wilhelms & vn Gelder [1] introduced the Brnch-On-Need Octree (BONO), spce efficient vrition of the trditionl octree. Octrees, however do not lwys represent the sptil distribution of the dt, nd re inpplicble to irregulr or unstructured grids. Livnt et l. [5] introduced the notion of spn spce - plotting the isovlues spnned by ech cell nd building serch structure to find only cells tht spn desired isovlue. Since this represents the rnge of isovlues in cell by single entry in dt structure, it exploits histogrm distribution. Histogrm distribution hs lso been exploited to ccelerte rendering for isosurfces [1] by computing ctive set chnges with respect to isovlue. Ching et l. [] clustered cells in irregulr meshes into metcells of roughly equl numbers of cells, then built n I/O- efficient spn spce serch structure on these met cells, gin exploiting histogrm distribution. For time-vrying dt, it is impossible to lod n entire dt set plus serch structures into min memory, so reserchers hve reduced the impct of slow disk ccess by building memory-resident serch structures nd loding dt from disk s needed. Sutton & Hnsen [1] extended the brnch-on-need octree to time-vrying dt with Temporl Brnch-on-Need Octree (T- BON), which uses sptil nd temporl coherence to minimize disk ccess by ccessing only those portions of serch structure nd dt which re necessry to updte the ctive set. Similrly, Shen [1] noted tht seprte spn spce structures for ech time-step re inefficient, nd proposed Temporl Hierrchicl Index (THI) Tree to suppress redundncy between serch structures for sequentil time-steps. This exploits both histogrm distribution nd temporl coherence. Reinhrd et l. [9] binned smples by isovlue for ech time step, loding only those bins required for given query: this form of binning exploits histogrm distribution but not temporl coherence. For volume rendering, Shen & Johnson [1] introduced Differentil Volume Rendering, in which they precomputed the voxels tht chnged between djcent time-steps, nd loded only those voxels tht chnged, principlly exploiting temporl coherence. Shen [11] extended Differentil Volume Rendering with Time- Spce Prtitioning (TSP) Trees, which exploit both sptil distribution nd temporl coherence, bsed on n octree representtion of the volume. Binry trees re used to store sptil nd temporl vritions for ech sub-block, nd this informtion is used for erly termintion of ry-trcing lgorithm, bsed on visul error metric. Shen lso optimized rendering by cching imges for ech sub-block in cse little or no chnge occurred between frmes. M & Shen [7] further extended this by quntizing the isovlues of the input dt nd storing them in octrees before computing differentils. Finlly, Sohn nd Bjj [13] use wvelets to compress timevrying dt to desired error bound, exploiting temporl coherence s well s sptil nd histogrm distribution. Our contribution in this pper is to unify temporl coherence in the form of differentil visuliztion with sttisticl distributions in the form of isovlue binning. Before introducing our dt structures, however, we demonstrte the vlidity of coherence nd distribution with some cse studies. CASE STUDIES: DATA CHARACTERISTICS OF ISABEL & TURBULENCE As noted in the previous sections, our pproch to ccelerting visuliztion depends on mthemticl, sttisticl nd physicl properties of the dt being studied. Before progressing further, it is therefore necessry to demonstrte tht these properties re the cse. We do so by mens of cse studies of three different dt sets. The first nd lrgest dt set is the Hurricne Isbel dt set provided by the Ntionl Center for Atmospheric Reserch (NCAR) for the IEEE Visuliztion Contest nd hs time steps of dimension 5x5x1. It contins 13 different floting point sclr

3 chnged voxels (%) chnged voxels (%) 1 λ=% λ=.3% λ=1.% λ=3.% Time step 1 () Isbel - temperture Time step (c) Turbulent Jets λ=% λ=.3% λ=1.% λ=3.% chnged voxels (%) chnged voxels (%) 1 λ=% λ=.3% λ=1.% λ=3.% Time step 1 (b) Isbel - pressure λ=% λ=.3% λ=1.% λ=3.% 1 Time step temperture isosurfces in the Isbel dt set never hve ctive sets of more thn 5% of the size of single slice, nd tht less thn hlf of the ctive set typiclly requires replcement between one time step nd the next. For pressure, lthough nerly 5% of the smples re in the ctive set in the worst cse, these smples chnge little between timesteps, indicting tht the physicl phenomenon mrked by this pressure feture does not move much in spce between time steps. In the jets dt set (Figure (c)), the worst-cse behviour is exhibited: t n isovlue of roughly % of the rnge, nerly 1% of the smples re in the ctive set. Although this hs implictions for the ctul rendering step, we see tht this set chnges little with respect to time. However, s it turns out, these dt vlues constitute most of the ir surrounding the flow under study nd re likely to be ignored during dt explortion. In the vortex dt set (Figure (d)), we see gin tht dt bins involve reltively few smples, nd tht these smples hve lrge degree of temporl coherence. Figure 1: Temporl Coherence: Number of smples tht chnge by more thn n error bound of λ = %,.3%,1%,3% for the temperture nd pressure sclrs of the Isbel dt set s well s the jets nd vortex dt sets. vribles of which we study the temperture nd pressure. We hve lso studied the Turbulent Jets (jets) nd Turbulent Vortex Flow (vortex) dt sets provided by Kwn-Liu M. The jets dt contins 15 time steps of 1x19x19 flots, while the vortex dt contins 1 time steps of 1 3 floting point sclrs..1 Temporl Coherence We demonstrte temporl coherence by computing the difference between djcent time steps, nd grphing the percentge of differences tht exceed n error bound expressed s percentge of the overll dt rnge. As we see in Figure 1(), t n error bound of %, ll of the voxels differ from one time step to the next. This my be due to numericl inccurcies in the floting point dt. It therefore follows tht 1% of the smples will need to be replced for the memory-resident version of the dt to be ccurte. However, s we see from the second line on the grph, between % nd % of the smples differ by more thn.3%. Thus, for.3% error in the isovlues displyed, we cn sve s much s % of the lod time between queries. As the error bound is incresed further, the number of smples to be loded decreses further: t 1% error, between % nd 9% of the smples need not be reloded, while t 3% error, over 95% of the smples need not be reloded. Similr effects cn be seen for pressure in Isbel (Figure 1(b)) nd for the jets nd vortex dt sets (Figure 1(c) & (d)). In Figure 1(d), we believe tht the shrp periodic dips indicte tht certin time steps were indvertently duplicted. This side, the overll conclusion is tht reltively smll error bounds permit us to void loding lrge numbers of smples s time chnges.. Histogrm Distribution In the lst section, we sw tht the sttistics of temporl coherence permit efficient differentil visuliztion for given error bound. It is nturl to sk whether similr sttistics re true for smll chnges in isovlues defining query. We show in Figure tht this is true by exmining the histogrm distribution for ech dt set. For this figure, we set our error bound to 1% nd computed the size of the ctive set for individul dt rnges (bins) nd for different times, nd lso the number of smples whose vlues chnged in the temporl dimension. For exmple, we see in Figure () tht.3 Sptil Distribution Unlike histogrm distribution, sptil distribution is hrder to test, s it is best mesured in terms of the sptil dt structure used to store the dt. We therefore defer considertion of sptil distribution to the discussion of Tble 1 in Section 5. 5 DIFFERENTIAL TIME HISTOGRAM TABLE In the previous section, we demonstrted tht temporl coherence nd histogrm distribution offer the opportunity to reduce lod times drsticlly when visulizing lrge time-vrying dt sets. To chieve the hypothesized reduction in lod-time we introduce new dt structure clled the Differentil Time Histogrm Tble (DTHT) long with n efficient lgorithm for performing regulr nd differentil queries in order to visulize the dt set. To exploit temporl coherence nd histogrm distribution, we pply differentil visuliztion in the temporl direction nd binning in the isovlue direction. Since we expect user explortion of the dt to consist mostly of grdul vritions in queries, we precompute the differences between successive queries. We modify temporl coherence by including in the differentil set only those smples for which the error exceeds chosen bound, further reducing the number of smples to be loded t ech time step. 5.1 Computing the DTHT Our DTHT, shown in Figure 5 stores smples in two-dimensionl rry of bins defined by isovlue rnge nd time step. In ech bin, we store the ctive set (), nd set of differences (rrows) between the ctive sets of djcent bins. For given dt set with τ time steps, we crete DTHT with τ bins in the temporl direction nd b bins in the isovlue direction. Lrge vlues of b will increse the size of the tble, but decrese the rnge of vlues in ech bin, mking ctive set queries more efficient for tht bin, but not necessrily fster, since lrge numbers of bins result in more storge overhed. Initilly, we choose b, the number of isovlue bins nd divide the isovlue rnge into b bins. We lso choose λ, the mount of error we will tolerte in isovlues for single smple over time. We compute the DTHT by loding the first time step into memory nd binning the smples. Once the first time step hs been processed, we lod the second time step into memory longside it, nd compute the difference between the two time steps. Any smple whose difference exceeds the error bound is dded to the differentil set in the time direction. Any smple whose difference does not exceed the error bound is modified so tht the vlue in the second

4 Isovlue Bin Active Set Differentils Time () Isbel - temperture Figure 3: The Differentil Time Histogrm Tble (DTHT). In ech bin, we store the ctive set (), nd the differentil between the bin nd ech djcent bin (rrows). Components of the DTHT not used for our experiments re shown in lighter colour. (b) Isbel - pressure (c) Turbulent Jets Figure : Temporl Coherence + Histogrm distribution: Number of smples tht chnge by more thn n error bound of λ = 1% for the temperture nd pressure sclrs of the Isbel dt set s well s the jets nd vortex dt sets. Ech dt sets is divided up into 1 iso-vlue bins. time step is identicl to the vlue in the first time step: this ensures tht, over time, errors do not ccumulte beyond our chosen bound. At the sme time, differentils re computed in the isovlue direction if these re desired. In our experiments, we hve chosen to render using spltting: in this cse ech smple belongs to only one bin nd the ctive sets in djcent bins re disjoint. It follows tht the isovlue differentils re equl to the ctive sets, nd cn be omitted: we hve ccordingly shown them in lighter shde in Figure 5. The first time step is then discrded, nd the next time step is loded nd compred to the second time step. This process continues until the entire dt set hs been processed. For ech smple, the informtion stored in given bin consists of the loction of the smple, the isovlue of the smple, nd its normlized grdient vector. Although this informtion could be stored s list of voxels, we chose to store it in brnch-on-need octree structures to exploit the sptil distribution of the dt. At ll times, the current ctive set is stored in brnch-on-need octree, s re the ctive sets nd the differentils for ech bin. Smples re dded or removed from the current ctive octree by tndem trversl with the octrees for the bins. This lso sves spce, s the loction of the smple is stored reltive to the octree cell insted of bsolutely, requiring fewer bits of precision. Moreover, octrees mke the run-time updting process much more efficient, since updting liner lists will be bound by the complexity of serches. The trde-off is tht the octrees increse the memory requirements, s shown in Tble 1. Here we note tht octrees re more efficient thn simple list structure. We lso note tht the size of the octree implementtion is principlly driven by the size of the octrees for the ctive sets. Further reductions in storge re possible depending on the nture of the queries to be executed. We ssume tht brupt queries re few nd fr between, nd dispense with storing ctive sets explicitly except for the first time step. If n brupt query is mde t time t, we construct the ctive set by strting with the corresponding ctive set t time nd pplying ll differentil sets up to time t. Doing so reduces the mount of storge required even further, s shown in the third column of Tble 1. As with the isovlue differentils, we indicte this in Figure 5 by displying the unused ctive sets in lighter shde. 5. Queries in the DTHT For ny query, we first clssify the query s grdul ( smll chnge) or brupt ( lrge chnge): s noted in Section, we expect

5 Dt Structure List Octree Octree All Active Sets 1st Active Set Dt Set GB %ge GB %ge GB %ge λ = % Isbel Temp % % % Isbel Press % % % Turb Jet 7. 73% 5. 55% 3. 33% Turb Vortex. 731% 3.7 5%.5 39% λ =.3% Isbel Temp. 77% %.7 1% Isbel Press % %.9 1% Turb Jet 3. 31%. 1%.71 7% Turb Vortex. 3% 3. 59%. 3% λ = 1.% Isbel Temp 15. 3% 1. 5%. % Isbel Press 13. 9% % 1.9 % Turb Jet. 55% %. % Turb Vortex %. 5% 1. 5% λ = 3.% Isbel Temp 1. % 9. 9%.59 1% Isbel Press 1. % %.7 15% Turb Jet. 3% 1. 1%. % Turb Vortex. 13%.1 33%.93 15% Tble 1: Size (in GB nd percent of originl dt set) of DTHT using lists nd octrees with ll or only the first ctive set stored. Isovlue TF Opcity Time Active DTH Cells Figure : Active DTHT Cells For Given Trnsfer Function. the nture of user interction with the dt will cuse most queries to be grdul in nture rther thn brupt. We chose to implement volume rendering using point-bsed spltting, in which the ctive set consists only of those smples for which the opcity is non-zero. This ctive set my spn multiple DTHT cells, s shown in Figure, in which cse we use the union of the ctive sets for ech DTHT cell. Abrupt queries re hndled by discrding the existing ctive set nd reloding the ctive set from the corresponding bin or bins on disk. Becuse the dt ws binned, the ctive set is over-estimted for ny given trnsfer function. This is conservtive over-estimte which includes the desired ctive set, but is sensitive to the size of the bin. Too few bins leds to lrge numbers of discrds for prticulr query, while too mny bins leds to overhed in dt structures nd overhed in merging octrees. Grdul queries in the temporl direction use the differentils for ech bin spnned by the trnsfer function to edit the ctive set. Since the differentils forwrds nd bckwrds re inverse, we economize by storing on ech rrow the set of smples tht needs to be dded in tht direction. The set of smples tht needs to be removed is then the set of smples stored on the opposite-direction rrow between the sme two bins. Grdul queries in the isovlue direction re hndled by discrding ll smples in bin when the trnsfer function no longer spns Totl Rendering Cost (seconds) Totl Rendering Cost (seconds) DTHT direct loding DTHT differentil updting histogrm 1 Trnsfer function width (%) () Isbel - Temperture 1 1 DTHT direct loding DTHT differentil updting histogrm 1 Trnsfer function width (%) (c) Turbulent Jets Cumultive sum of histogrm (%) Cumultive sum of histogrm (%) Totl Rendering Cost (seconds) Totl Rendering Cost (seconds) DTHT direct loding DTHT differentil updting histogrm 1 Trnsfer function width (%) (b) Isbel - Pressure 1 1 DTHT direct loding DTHT differentil updting histogrm 1 Trnsfer function width (%) Figure 5: λ = 1.% Rendering nd dt loding performnce for DTHT using ctive sets (green) nd DTHT using differentil files (red). The reported times re verged over multiple time steps. tht bin, nd dding ll smples in bin when the trnsfer function first spns tht bin. For fster rendering, prtil imges for ech bse level block in the current ctive octree re stored, s in the TSP tree [11]. Since we updte bse level blocks in the octree only when chnges exceed the λ-bound, these prtil imges often survive two or more time steps. RESULTS AND DISCUSSION We implemented our differentil temporl histogrm tble lgorithm nd visulized different sclr time-vrying dtsets introduced in Section, using 3.GHz Intel Xeon processor, GB of min memory nd n nvidi AGP x GeForce grphics crd, with dt stored on fileserver on 1Mb/s Locl Are Network. In our experiments we used b = 1 isovlue bins, set the bselevel brick size in the octrees to 3x3x3, nd tested four different vlues for error bounds (λ = %,.3%, 1.%, 3.%). In our first experiment we controlled the number of ctive voxels by chnging the trnsfer function width (i.e. the rnge of nontrnsprent dt vlues). Strting with % width trnsfer function (covering no bins), t ech step we grdully incresed the width until ll dt vlues were clssified s non-trnsprent (i.e. ll bins were loded). The odd rows of Figure 1 show representtive visuliztions of this scenrio. In Figure 5, we compre the performnce of using DTHT ctive sets only, with the performnce of using DTHT differentil sets exclusively for the rendering for our four dt sets with n error bound of λ = 1.% nd trnsfer functions of vrying widths. The performnce in principlly driven by the width of the trnsfer function, nd it is cler tht significnt speed gin in loding the dt is chieved with the differentil sets. In the second set of experiments we were interested to investigte the performnce of our lgorithm when different rnges of dt re explored. We collected the sme mesurements s before, but this time the mesurements were done for constnt width (1%) trnsfer functions with different plcements. The even rows of Figure 1 show representtive visuliztions of this scenrio. The results for λ = 1.% nd different dt sets re shown in Figure. For the ske of stright forwrd comprison, we clculted the rtio between the differentil DTHT updtes nd direct DTHT updtes for both sets of experiments. Since the gol of this pper Cumultive sum of histogrm (%) Cumultive sum of histogrm (%)

6 Totl Rendering Cost (seconds) Totl Rendering Cost (seconds) DTHT direct loding DTHT differentil updting histogrm 1 Plcement of 1% width trnsfer function () Isbel - Temperture 1 1 DTHT direct loding DTHT differentil updting histogrm 1 Plcement of 1% width trnsfer function (c) Turbulent Jet Convolved histogrm (%) Convolved histogrm (%) Totl Rendering Cost (seconds) Totl Rendering Cost (seconds) DTHT direct loding DTHT differentil updting histogrm 1 Plcement of 1% width trnsfer function (b) Isbel - Pressure 1 1 DTHT direct loding DTHT differentil updting histogrm 1 Plcement of 1% width trnsfer function Figure : λ = 1.% Rendering nd dt loding performnce for DTHT using ctive sets (green) nd DTHT using differentil files (red). The reported times re verged over multiple time steps. Further, the mount of ctive voxels s rtio to the full dt set (yellow bckdrop) is displyed. A trnsfer function spnning 1% of the dt rnge hs been used slid cross the dt rnge. Results hve been verged over severl time steps. Convolved histogrm (%) Convolved histogrm (%) rtio of direct loding to differentil updting rtio of direct loding to differentil updting λ=% λ=.3% λ=1.% λ=3.% rtio = 1 histogrm 1 Trnsfer function width (%) 1 () Isbel - Temperture 1 1 λ=% λ=.3% λ=1.% λ=3.% rtio = 1 histogrm 1 Trnsfer function width (%) (c) Turbulent Jet Cumultive sum of histogrm (%) Cumultive sum of histogrm (%) rtio of direct loding to differentil updting rtio of direct loding to differentil updting λ=% λ=.3% λ=1.% λ=3.% rtio = 1 histogrm 1 Trnsfer function width (%) 1 (b) Isbel - Pressure 1 1 λ=% λ=.3% λ=1.% λ=3.% rtio = 1 histogrm 1 Trnsfer function width (%) Figure 7: Rtio between DTHT differentil updting time nd DTHT direct loding time for different error bounds, for different trnsfer function widths. Cumultive sum of histogrm (%) Cumultive sum of histogrm (%) ws to improve the loding times, we removed the time to ctully render n imge from the updted octree of ctive cells. The gol ws to investigte the cses in which updting the octree using differentil files were fster thn strightforwrd loding of the ctive voxels files. Figure 7 nd Figure show the rtio between the performnce of two methods, respectively for different width trnsfer function nd constnt width trnsfer function. In the third experiment we investigted the loding performnce when the time doesn t chnge, but the trnsfer function chnges. While in the brute-force method, the dt for ech time is only loded once nd it doesn t need to be loded/updted when the rendering prmeters chnge, our method tkes dvntge of the rendering prmeters to only lod the voxels prticipting in the finl rendering. So, chnging the trnsfer function cuses new bins to be dded to or removed from the dt set. The results re shown in the Figure 9. Loding time of DTHT methods depends gretly on the mount of ctive voxels. In both experiments, when smll number of voxels re ctive, loding the octree directly from the explicit files is comprtively fster thn using the differentil files. It is minly becuse of the fct tht loding the octree directly from the ctive voxels files, requires going through the octree only once per ech bin, while in the differentil method, it will be twice per ech bin; once to remove the invlid voxels, using voxels to remove files nd then to dd the new voxels using voxels to dd files. Hence, memory ccess time nd octree processing time overcomes the files trnsction time when smll mounts of dt re red from the hrd drive. As the number of ctive voxels increses, more dt need to be loded/updted from the externl storge. Hence, the memorystorge trnsctions become the bottleneck. Since there re reltively few dt in the differentil files, the performnce of updting the octree using the differentil files becomes better thn the direct loding. However, the rtio of this performnce highly depends on the temporl coherence of the dt set. For exmple since the Turbulent Vortex Flow dt set isn t s temporlly coherent s the other three dt sets, even with 1.% error bound, the differentil up- rtio of direct loding to differentil updting rtio of direct loding to differentil updting λ=% λ=.3% λ=1.% λ=3.% rtio = 1 histogrm 1 Plcement of the 1% width trnsfer function 1 () Isbel - Temperture 1 1 λ=% λ=.3% λ=1.% λ=3.% rtio = 1 histogrm 1 Plcement of the 1% width trnsfer function (c) Turbulent Jet Convolved histogrm (%) Convolved histogrm (%) rtio of direct loding to differentil updting rtio of direct loding to differentil updting λ=% λ=.3% λ=1.% λ=3.% rtio = 1 histogrm 1 Plcement of the 1% width trnsfer function 1 (b) Isbel - Pressure 1 1 λ=% λ=.3% λ=1.% λ=3.% rtio = 1 histogrm 1 Plcement of the 1% width trnsfer function Figure : Rtio between DTHT differentil updting time nd DTHT direct loding time for different error bounds nd different plcements of trnsfer function of constnt width (1%). Convolved histogrm (%) Convolved histogrm (%)

7 7 resizing trnsfer function moving trnsfer function 35 3 resizing trnsfer function moving trnsfer function 5 Time (ms) 5 3 Time (ms) Bins () Isbel - Temperture 1 Bins (b) Isbel - Pressure 7 resizing trnsfer function moving trnsfer function resizing trnsfer function moving trnsfer function 5 15 Time (ms) 3 Time (ms) Bins (c) Turbulent Jets 1 Bins Figure 9: Loding time of incrementl trnsfer function chnges. dting is slower thn the direct loding nd tht s simply becuse the number of voxels to dd + voxels to remove is lrger thn the ctul mount of ctive voxels. Noise in the digrms re minly due to loding the dt through locl re network connection which is fst, but does not lwys provide constnt dt rte. There re lso lrge vritions in some prts of Figure 7 nd Figure. It cn be seen tht the noise is mostly in the prts in which not mny voxels re ctive. This is becuse of the fct tht, s the loding times become smller, they become more error prone to be ffected by slightest ctivities in the hrdwre or the operting system. 7 CONCLUSION AND FUTURE WORK Efficient explortion nd visuliztion of lrge time-vrying dt sets is still one of the chllenging problems in the re of scientific visuliztion. In this pper we studied the temporl coherence of sequentil times nd histogrm nd sptil distribution of dt in time-vrying dt sets nd proposed our dt structure clled differentil time histogrm tble (DTHT) to tke dvntge of these chrcteristics to efficiently lod nd render time-vrying dt sets. In preprocessing step, we encode the temporl coherence nd histogrm distribution of the dt in the DTHT dt structure. Lter, during the visuliztion step, we use this differentil informtion to ccelerte updting dt by minimizing the mount of dt tht needs to be trnsfered from disk into min memory. To store the dt in externl storge nd to keep the ctive dt in min memory, we use n octree dt structure which ccelertes loding nd processing dt by exploiting sptil distribution. The work presented in this pper ws minly focused on nlyzing the sttisticl chrcteristics of lrge dtsets to reducing the dt trnsfer between memory nd externl storge during the visuliztion process. We re plnning to further improve performnce by pplying compression schemes for the dt in lef nodes nd dding multi-level informtion to the sub-levels of the dt structure to enble multi-resolution renderings. Currently the rendering time is dominting the overll time to produce n imge. This is due to the fct, tht the rendering lgorithm hs not been optimized t this point. We pln to tke dvntge of the octree dt structure for hierrchicl rendering of our dt in our next implementtion. Imge cching nd occlusion culling will further drsticlly improve our rendering performnce. Our future works lso includes pplying proper modifictions to Figure 1: Rendered Imges with representtive trnsfer functions. Displyed re two rows of ech dt set - first Isbel temperture, followed by the Isbel pressure, followed by Turbulent Jets nd Turbulent Vortex Flow. The first row in ech group shows growing trnsfer functions, while the second row shows shifting trnsfer functions, corresponding to our experiments in this section.

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