Feature Flow Fields in Out-of-Core Settings

Transcription

1 Feaure Flow Fields in Ou-of-Core Seings Tino Weinkauf 1, Holger Theisel 2, Hans-Chrisian Hege 3, Hans-Peer Seidel 4 1 Zuse Insiue Berlin (ZIB), German. weinkauf@zib.de 2 Bielefeld Universi, German. heisel@echfak.uni-bielefeld.de 3 Zuse Insiue Berlin (ZIB), German. hege@zib.de 4 MPI Informaik Saarbrücken, German. hpseidel@mpi-inf.mpg.de Summar: Feaure Flow Fields (FFF) are an approach o racking feaures in a ime-dependen vecor field v. The main idea is o inroduce an appropriae vecor field f in space-ime, such ha a feaure racking in v corresponds o a sream line inegraion in f. The original approach of feaure racking using FFF requesed ha he complee vecor field v is kep in main memor. Especiall for 3D vecor fields his ma be a serious resricion, since he size of ime-dependen vecor fields can eceed he main memor of even high-end worksaions. We presen a modificaion of he FFF-based racking approach which works in an ou-of-core manner. For an imporan subclass of all possible FFF-based racking algorihms we ensure o analze he daa in one sweep while holding onl wo consecuive ime seps in main memor a once. Similar o he original approach, he new modificaion guaranees he complee feaure skeleon o be found. We appl he approach o racking of criical poins in 2D and 3D ime-dependen vecor fields. 1 Inroducion The resoluion of numerical simulaions as well as eperimenal measuremens like PIV have evolved significanl in he las ears. The challenge of undersanding he inricae flow srucures wihin heir massive resul daa ses has made auomaic feaure eracion schemes popular. Feaure-based analsis can be seen as a kind of daa reducion since i brings he raw daa mass down o a small number of graphical primiives ha ough o give insigh ino he flow srucures. While he oucome of mos feaure eracion algorihms has a raher small memor fooprin, he inpu ofen eceeds he main memor of high-end worksaions. This is especiall rue for 3D ime-dependen daa. Thus, feaure eracion algorihms should be compaible o an ou-of-core daa handling, i.e., reaing onl a small par of he inpu a once. A number of algorihms alread work in an ou-of-core manner. Tricoche e al. [16] and Garh e.al. [3] show how o rack criical poins in piecewise

2 2 Weinkauf, Theisel, Hege, Seidel linear vecor fields b analzing he daa in one sweep and holding onl wo ime slices a once. Their approaches eploi he lineari of non-changing piecewise linear grids and are probabl he bes wa o go for his imporan class of daa. Anoher wa of racking feaures which are defined b he parallel vecor operaor [8] is inroduced in [1]. This approach is based on a 4-dimensional isosurface eracion and herefore compaible o ou-of-core daa handling. Theisel e al. [13] propose a general approach o feaure racking b capuring he emporal evoluion of a feaure using a sream objec 5 inegraion in a derived vecor field he feaure flow field (FFF). This basic idea has been applied no onl o racking criical poins [13] and derived applicaions like simplificaion [11] and comparison [12], bu also o eracing Galilean invarian vore core lines [10] and racking closed sream lines [14]. The FFF approach is independen of an underling grid, i.e., i is enirel based on he descripion of he daa as a coninuous field. A firs glance, his concep seems o conradic he principle of ou-of-core daa handling: reaing onl a small par of he daa a once. In his paper we show ha hose wo conceps do no conradic. In fac, we show how all FFF-based racking algorihms can be formulaed in an ou-of-core manner. This will be used o re-formulae he algorihm for racking criical poins from [13] o make i compaible o ou-of-core daa handling. The resuling algorihm enables o analze he daa in one sweep while holding onl wo ime slices a once. The res of he paper is organized as follows: secion 2 surves he FFF approach as described in [13] and he basics of ou-of-core daa handling. Secion 3 describes he new ou-of-core version of he FFF approach. Secion 4 applies his knowledge o FFF-based racking of criical poins, while conclusions are drawn in secion 5. 2 Background and Problem In his secion we briefl discuss he basics behind ou-of-core daa handling and feaure flow fields. While heir main ideas are no anagonisic, a pical algorihm based on FFF is incompaible wih an ou-of-core daa handling. 2.1 Ou-Of-Core Daa Handling Ou-of-core refers o he daa handling sraeg of algorihms, which process daa oo large o fi ino main memor. Thus, onl pars of he daa can be loaded a once and aced upon. Since loading he daa from a mass sorage device is ver ime-consuming, he number of hose operaions should be reduced o a minimum. This resricion mus alread be considered when formulaing he algorihm. 5 This refers o a sream line, sream surface, sream volume, ec. depending on he dimensionali of he feaure.

3 Feaure Flow Fields in Ou-of-Core Seings 3 o be loaded o be loaded curren posiion currenl loaded currenl loaded previousl loaded (a) Block-wise random access. previousl loaded (b) Slice-wise sequenial access. Fig. 1. Ou-of-core daa handling sraegies. There are differen pes of ou-of-core daa handling sraegies. We jus wan o menion wo here: Block-wise random access: Daa is loaded in blocks of same size. All dimensions are reaed equall. The loaded daa block wih he oldes access ime is subjec o be subsiued wih he ne block o be loaded. An applicaion for his access paern is he inegraion of a pah line, which ouches onl pars of he domain. Figure 1a gives an illusraion. Slice-wise sequenial access: Daa is loaded in slices, i.e., one dimension is fied for ever slice. Slices will be loaded as a fied sequence in ascending or descending order. The procedure o load all slices from firs o las is called a sweep. A ver common approach is he usage of ime slices, since a number of daa ses are organized such ha each ime sep is given as a separae file. An applicaion for his access paern is he eracion of fold bifurcaions, where all pars of he domain need o be eamined. Figure 1b gives an illusraion. Feaure eracion algorihms usuall do no have a-priori knowledge abou he locaion of he feaure and herefore, he need o eamine he whole domain. A slice-wise sequenial access sraeg seems o be even more preferable, if he daa is alread given in ime slices. For he res of his paper we consider his daa handling sraeg onl. Noe, ha b loading wo consecuive ime seps i and i+1 and appling a linear inerpolaion in beween hem, we obain he ime-dependen field in ha ime inerval. 2.2 Feaure Flow Fields The concep of feaure flow fields was firs inroduced in [13]. I follows a raher generic idea: Consider an arbirar poin known o be par of a feaure in a (scalar, vecor, ensor) field v. A feaure flow field f is a well-defined vecor field a

4 4 Weinkauf, Theisel, Hege, Seidel i+1 feaure of v a i+1 i feaures of v a i sream lines of f sream lines of f s i feaure of v a (a) Tracking feaures b racing sream lines. i b (b) Evens: a he ime b a new feaure is born, a s i splis ino wo feaures. Fig. 2. Feaure racking using feaure flow fields. Feaures a i+1 can be observed b inersecing hese sream lines wih he ime plane = i+1. poining ino he direcion where he feaure coninues. Thus, saring a sream line inegraion of f a ields a curve where all poins on his curve are par of he same feaure as. FFF have been used for a number of applicaions, bu mainl for racking feaures in ime-dependen fields. Here, f describes he dnamic behavior of he feaures of v: for a ime-dependen field v wih n spaial dimensions, f is a vecor field IR n+1 IR n+1. The emporal evoluion of he feaures of v is described b he sream lines of f. In fac, racking feaures over ime is now carried ou b racing sream lines. The locaion of a feaure a a cerain ime i can be obained b inersecing he sream lines wih he ime plane i. Figure 2a gives an illusraion. Depending on he dimensionali of he feaure a a cerain ime i, he feaure racking corresponds o a sream line, sream surface or even higherdimensional sream objec inegraion. The sream lines of f can also be used o deec evens of he feaures: A birh even occurs a a ime b, if he feaure a his ime is onl described b one poin of a sream objec of f, and all sream lines in he neighborhood of his poin are in he half-space b. A spli occurs a a ime s, if one of he sream lines of f describing he feaure ouches he plane = s from above. An ei even occurs if all sream lines of f describing he feaure leave he spaial domain. The condiions for he reverse evens (deah, merge, enr) can be formulaed in a similar wa. Figure 2b illusraes he differen evens. Inegraing he sream lines of f in forward direcion does no necessaril mean o move forward in ime. In general, hose direcions are unrelaed. The direcion in ime ma even change along he same sream line as i is shown

5 Feaure Flow Fields in Ou-of-Core Seings 5 in figure 2b. This siuaion is alwas linked o eiher a birh and a spli even, or a merge and a deah even. Even hough we reaed he concep of FFF in a raher absrac wa, we can alread formulae he basics of an algorihm o rack all occurrences of a cerain feaure in a ime-dependen field: Algorihm 1 General FFF-based racking 1. Ge seeding poins/lines/srucures such ha he sream objec inegraion of f guaranees o cover all pahs of all feaures of v. 2. From he seeding srucures: appl a numerical sream objec inegraion of f in boh forward and/or backward direcion unil i leaves he space-ime domain. 3. If necessar: remove mulipl inegraed sream objecs. Algorihm 1 is more or less an absrac emplae for a specific FFF-based racking algorihm like e.g. racking of criical poins. However, i shows a vial conradicion o ou-of-core daa handling: i gives no guaranee on how he daa is processed. We would end up loading differen daa slices more han once, since boh forward and backward inegraion of f are allowed, and as alread said, he direcion in ime ma even change along he same sream line. 3 Feaure Flow Fields and Ou-Of-Core Algorihms In his secion we wan o modif algorihm 1 such ha i becomes compaible o ou-of-core daa handling. This will allow o formulae all FFF-based algorihms in an ou-of-core manner. Before we formulae he algorihm, we collec some conceps and properies of he FFF inegraion on which he new algorihm is based upon. 3.1 Direcion of Inegraion Regarding Time The FFF approach is based on a sream line inegraion of f. Given a saring poin 0 = ( s 0, 0 ), f can be inegraed in forward or backward direcion. Assuming an Euler inegraion 6, he forward inegraion goes o he ne poin 1 = 0 + ε f( 0 ), while he backward inegraion gives he ne poin 1 = 0 ε f( 0 ) for a cerain small posiive ε. In addiion o his disincion of he inegraion orienaion, we can also disinguish a -forward and -backward orienaion. We call an inegraion -forward if he ne poin 1 = ( s 1, 1 ) is ahead in ime, i.e., if 1 > 0. For 1 < 0, we have a -backward inegraion. This proper can be decided locall for a poin 0 b looking a he sign of he -componen of f( 0 ), or if his componen is zero, b looking a he sign of he parial derivaive f ( 0 ). Figure 3 illusraes some of hese cases. 6 For he acual inegraion we used a fourh order Runge Kua mehod, he Euler inegraion is onl for eplaining he conceps.

6 6 Weinkauf, Theisel, Hege, Seidel a) b) c) d) 0 f( 0 ) 0 f( 0 ) f( 0 ) 0 f( 0 ) 0 Fig. 3. Orienaion of inegraing f: a) forward and -forward; b) forward and - backward; c) backward and -forward; d) backward and -backward; he curves are he inegraed sream lines saring from Classificaion of Seeding Srucures The FFF approach is also based on finding appropriae saring srucures for he inegraion. The definiion of a complee se of seeding srucures is up o he specific FFF-based applicaion. However, we can give he following classificaion of hose srucures: -forward srucures: all inegraions sared here are -forward onl. -backward srucures: all inegraions sared here are -backward onl. inermediae srucures: a -forward and a -backward inegraion will be sared here. This classificaion is independen of a specific FFF-based applicaion, hough i migh be ha in cerain cases a class of srucures is emp, e.g. here are onl -forward and inermediae srucures bu no -backward srucures. As alread discussed in secion 2.2, a -forward inegraion ma change o a -backward inegraion even along he same sream line. This siuaion is alwas linked o eiher a birh or a deah even, which perfecl fi ino he classificaion: a birh even is a -forward poin, and a deah even a -backward poin. 3.3 Ou-Of-Core FFF-based Tracking Algorihm The spli of he inegraion ino differen direcions regarding he ime is he concepual ke o an ou-of-core version of algorihm 1: Algorihm 2 Ou-of-core FFF-based racking 1. Load he daa in a slice-wise sequenial manner from min o ma. For each ime inerval beween he ime slices i and i+1: a) Ge seeding srucures such ha he sream objec inegraion of f guaranees o cover all pahs of all feaures of v. b) Classif he seeding srucures ino -forward, -backward and inermediae srucures. c) Sar a -forward inegraion a all -forward and inermediae srucures. Sop he inegraion, if i. he spaial domain was lef.

7 Feaure Flow Fields in Ou-of-Core Seings 7 ii. he emporal domain of he ime inerval was lef, i.e., he inegraion reached i+1. Add he resul of his inegraion o he lis of -forward srucures for he ne ime inerval. iii. a deah even was reached. If his poin will no be reached b an oher -forward inegraion, add he resul o he lis of -backward srucures. 2. If he lis of -backward srucures is non-emp: load he daa in a slice-wise sequenial manner from ma o min. For each ime inerval beween he ime slices i and i 1 sar a -backward inegraion a all -backward and inermediae srucures similar as above. 3. Repea he sream objec inegraions of seps 1 and 2 unil he liss of seeding srucures are emp. The basic idea of his algorihm emplae is o ensure ha he direcion of loading he daa coincides wih he direcion of inegraion, i.e., if we are loading he daa from min o ma hen we are inegraing -forward onl, and he oher wa around. This alone does no sound ver effecive, since he daa migh been loaded more han once, possibl even an unknown number of imes. This changes, if we ake a closer look a he -forward srucures, i.e., all poins where onl a -forward inegraion is inended. A hose poins he feaures appear for he firs ime. Eamples are enr poins, birh evens or all occurrences of he feaure a min. If we can find all -forward srucures while doing he firs sweep hrough he daa, hen he whole feaure skeleon can be eraced wih his one sweep. This is alwas fulfilled, if all pes of -forward srucures are locall defined, i.e., he can be eraced b a local analsis. Under hese prerequisies, we can reformulae algorihm 2 and obain he following algorihmic emplae: Algorihm 3 One-sweep Ou-of-core FFF-based racking 1. For each ime inerval [ i, i+1] from min o ma: a) Erac all -forward seeding srucures needed o cover all pahs of all feaures of v. b) Appl a -forward inegraion saring a hose srucures unil i. he spaial domain was lef. ii. he emporal domain of he ime inerval was lef, i.e., he inegraion reached i+1. Add he resul of his inegraion o he lis of -forward srucures for he ne ime inerval. iii. a deah even was reached. Algorihm 3 ensures ha ever pah of a feaure is inegraed onl once. Thus, a removal of mulipl inegraed sream objecs is no needed anmore. In comparison o algorihms 1 and 2 i is perfecl fied for large daa ses: i reads onl pars of he daa and each par onl once. 4 Applicaion o Tracking of Criical Poins Criical poins, i.e. isolaed poins wih a vanishing flow, are perhaps he mos imporan opological feaure of vecor fields. For saic fields, heir erac-

8 8 Weinkauf, Theisel, Hege, Seidel ion and classificaion is well-undersood boh in he 2D [6] and he 3D case [17]. Criical poins also serve as he saring poins of cerain separarices, i.e. sream lines/surfaces which divide he field ino areas of differen flow behavior. Topological mehods have no onl been developed for visualizaion purposes [4, 5], bu have also been applied o simplif [2, 15], smooh [18], and compress [11, 7] vecor fields. A horough overview can be found in [9]. Considering a sream line oriened opolog of ime-dependen vecor fields, criical poins smoohl change heir locaion and orienaion over ime. In addiion, cerain bifurcaions of criical poins ma occur. To capure he opological behavior of ime-dependen vecor fields, i is necessar o capure he emporal behavior of he criical poins. Theisel e al. inroduced in [13] a FFF-based approach o rack criical poins, which maches algorihm 1. In order o rack criical poins in an effecive ou-of-core manner using algorihm 3 we need o find he complee se of -forward poins, i.e., all poins in space-ime where a criical poin appears for he firs ime. This will be done in secion 4.2. Bu before ha we discuss he definiion of he feaure flow field f iself. 4.1 FFF for Tracking Criical Poins We firs consider racking criical poins in a 2D ime-dependen vecor field, which is given as ( ) u(,,) v(,,) = (1) v(,,) in he 3D space-ime domain D = [ min, ma ] [ min, ma ] [ min, ma ]. We can consruc a 3D vecor field f in D wih he following properies: for an wo poins 0 and 1 on a sream line of f, i holds v( 0 ) = v( 1 ). This means ha a sream line of f connecs locaions wih he same values of v. Figure 4 gives an illusraion. In paricular, if 0 is a criical poin in v, hen he sream line of f describes he pah of he criical poin over ime. To ge f, we search for he direcion in space-ime in which boh componens of v locall remain consan. This is he direcion perpendicular o he gradiens of he wo componens of v. We ge f(,,) = grad(u) grad(v) = u u u v v v de(v,v ) = de(v,v ). (2) de(v,v ) The FFF approach for 3D vecor fields is a sraighforward eension of he 2D case. Given he 3D ime-dependen vecor field u(,,z,) v(,,z,) = v(,,z,) (3) w(,,z,)

9 Feaure Flow Fields in Ou-of-Core Seings 9 i ( 0, 0,0) v( 0, 0, 0 ) Fig. 4. Tracking 2D criical poins: all poins on a sream line of f have he same value for v. sream line of f in he 4D space-ime domain D = [ min, ma ] [ min, ma ] [z min,z ma ] [ min, ma ], he 4D FFF f is defined b he condiions f grad(u) = (u,u,u z,u ) T, f grad(v), f grad(w). This gives a unique soluion for f (ecep for scaling) 7 +de(v,v z,v ) f(,,z,) = de(v z,v,v ) +de(v,v,v ). (4) de(v,v,v z ) 4.2 Complee Se of -forward Poins Theisel and Weinkauf showed in [14] ha wo classes of seeding poins guaranee ha all pahs of criical poins are capured: he inersecions of he pahs wih he domain boundaries, and fold bifurcaions. However, he did no disinguish beween -forward and -backward poins. We are going o do his here in order o find he complee se of -forward poins. To find all inersecions wih he boundaries, we have o solve v(,, min) = (0, 0) T and v(,, ma) = (0, 0) T for he unknowns,, v(, min, ) = (0, 0) T and v(, ma, ) = (0, 0) T for he unknowns,, v( min,, ) = (0, 0) T and v( ma,, ) = (0, 0) T for he unknowns,. Each of he 6 soluions urns ou o be a simple eracion of criical poins of a 2D (sead) vecor field. We can make he following disincion: Boom inersecion poins are inersecions wih he plane = min Top inersecion poins are inersecions wih he plane = ma Side inersecion poins are inersecions wih he plane = min, = ma, = min, or = ma respecivel. Side inersecion poins can be furher classified ino enr and ei poins. A an enr poin, a -forward inegraion of f goes ino D, while a an ei poin 7 Noe ha he formulaion of f(,, z, ) in [13] conains an error: he alernaing signs of he componens are missing.

10 10 Weinkauf, Theisel, Hege, Seidel a) 4 ma 2 3 Fig. 5. a) inersecions of he pahs of criical poins wih he domain of D: boom inersecion ( 1), ei ( 2) and enr ( 3) side inersecion, op inersecion ( 2); b) eample consising of a boom inersecion ( 1), ei ( 2) and enr ( 3) side inersecion, deah ( 4) and birh ( 5) fold bifurcaion, op inersecion ( 6): he pahs of criical poins consis of 4 segmens which are inegraed -forward from o: 1 2, 3 4, 5 4, 5 6. min b) ma min 2 min min ma, ma 3, Jf( ) f() a) b) f() f() Jf( ) f() Fig. 6. Classifing fold bifurcaions b he las componen of J f () f(): a) birh even; b) deah even. a -forward inegraion leaves D. Figure 5a illusraes he differen kinds of inersecion poins wih he boundar of D. I is eas o see ha onl boom and enr side inersecions are -forward poins. To deec fold bifurcaions inside D, we search for locaions wih [ v() = (0,0) T, de(j v ()) = 0 ] (5) where J v is he Jabcobian mari of v. (2) shows ha he second condiion of (5) ensures ha he las componen of f vanishes, i.e., ha f is parallel o he - ais. To solve (5), we use a numerical approach similar o eracing isolaed criical poins in 3D vecor fields. There are wo kinds of fold bifurcaions: birh and deah evens. To disinguish hem, we consider he las componen of J f () f() a he fold bifurcaion. If his componen is posiive, we have a birh bifurcaion; if i is negaive, a deah bifurcaion is presen. Figure 6 illusraes his. I is eas o see ha onl birh bifurcaions are -forward poins. For 3D ime-dependen vecor fields, he eracion of he seeding poins follows he same ideas. Boundar inersecions are found as isolaed criical poins of he 3D (sead) vecor fields a he space-ime domain boundar. Again, onl boom and enr side inersecions are of ineres. Fold bifurcaions are he soluions of

11 Feaure Flow Fields in Ou-of-Core Seings 11 (a) A i. (b) Enries. (c) Birhs. (d) Inegraion. (e) A i+1. Fig. 7. Applicaion of algorihm 3: criical poins racked in one sweep. [ v() = (0,0,0) T, de(j v ()) = 0 ] (6) which corresponds o numericall finding isolaed criical poins in 4D vecor fields. The disincion beween birhs and deahs follows he 2D case. We have found all poins in space-ime where a criical poin can appear for he firs ime: boom and enr side inersecions as well as birh bifurcaions. The can all be eraced using a local analsis. All prerequisies for algorihm 3 are fulfilled. Thus, we are now able o rack criical poins in 2D and 3D ime-dependen vecor fields in an effecive ou-of-core manner: in one sweep and b loading onl wo slices a once. We applied his algorihm o a random 2D ime-dependen daa se. Random vecor fields are useful ools for a proofof-concep of opological mehods, since he conain a maimal amoun of opological informaion. Figure 7 shows he eecuion of algorihm 3 beween wo consecuive ime seps i and i+1. Figure 8 shows he visualizaion of a vecor field describing he flow over a 2D cavi. This daa se was kindl provided b Mo Samim and Edgar Caraballo (boh Ohio Sae Universi) as well as Bernd R. Noack and Ivanka Pelivan (boh TU Berlin) ime seps have been simulaed using he compressible Navier-Sokes equaions; i ehibis a non-zero divergence inside he cavi, while ouside he cavi he flow ends o have a quasi-divergence-free behavior. Insead of loading onl wo ime seps a once, he daa se has been divided ino 10 secions each consising of 100 ime seps (Figure 8a). Each secion fis easil ino main memor and has been reaed according o algorihm 3. This approach reduced he overhead inroduced b he ou-of-core handling. The compuaion ime for his daa se was 20 minues. The opological srucures visualized in Figure 8b elucidae he quasi-periodic naure of he flow. The mos dominaing opological srucures originae in or near he boundaries of he cavi iself. The quasi-divergence-free behavior ouside he cavi is affirmed b he fac ha a high number of Hopf bifurcaions has been found in his area. 5 Conclusions In his paper we showed how all FFF-based racking algorihms can be formulaed in an ou-of-core manner. This has been used o re-formulae he

12 12 Weinkauf, Theisel, Hege, Seidel (a) LIC planes: 10 secions. (b) Tracked criical poins. Fig. 8. Cavi daa se consising of 1000 ime seps. Algorihm 3 has been applied ono he 10 depiced secions consising of 100 ime seps each. algorihm for racking criical poins from [13] o make i compaible o ouof-core daa handling. The resuling algorihm enables o analze he daa in one sweep while holding onl wo ime slices a once. For fuure work, we inend o appl algorihm 3 o oher pes of feaures, especiall o vore core lines. References 1. D. Bauer and R. Peiker. Vore racking in scale space. In Daa Visualizaion Proc. VisSm 02, pages , W. de Leeuw and R. van Liere. Collapsing flow opolog using area merics. In Proc. IEEE Visualizaion 99, pages , C. Garh, X. Tricoche, and G. Scheuermann. Tracking of vecor field singulariies in unsrucured 3D ime-dependen daases. In Proc. IEEE Visualizaion 2004, pages , A. Globus, C. Levi, and T. Lasinski. A ool for visualizing he opolog of hree-dimensional vecor fields. In Proc. IEEE Visualizaion 91, pages 33 40, H. Hauser and E. Gröller. Thorough insighs b enhanced visualizaion of flow opolog. In 9h inernaional smposium on flow visualizaion, J. Helman and L. Hesselink. Represenaion and displa of vecor field opolog in fluid flow daa ses. IEEE Compuer, 22(8):27 36, S. Lodha, N. Faaland, and J. Reneria. Topolog preserving op-down compression of 2d vecor fields using binree and riangular quadrees. IEEE Transacions on Visualizaion and Compuer Graphics, 9(4): , 2003.

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