Isosurface Extraction in Time-varying Fields Using a Temporal Hierarchical Index Tree

Transcription

1 Isosurface Extracton n Tme-varyng Felds Usng a Temporal Herarchcal Index Tree Han-We Shen MRJ Technology Solutons / NASA Ames Research Center Abstract Many hgh-performance sosurface extracton algorthms have been proposed n the past several years as a result of ntensve research efforts. When applyng these algorthms to large-scale tme-varyng felds, the storage overhead ncurred from storng the search ndex often becomes overwhelmng. Ths paper proposes an algorthm for locatng sosurface cells n tme-varyng felds. We devse a new data structure, called Temporal Herarchcal Index Tree, whch utlzes the temporal coherence that exsts n a tme-varyng feld and adaptvely coalesces the cells extreme values over tme; the resultng extreme values are then used to create the sosurface cell search ndex. For a typcal tme-varyng scalar data set, not only does ths temporal herarchcal ndex tree requre much less storage space, but also the amount of I/O requred to access the ndces from the dsk at dfferent tme steps s substantally reduced. We llustrate the utlty and speed of our algorthm wth data from several large-scale tme-varyng CFD smulatons. Our algorthm can acheve more than 80% of dsk-space savngs when compared wth the exstng technques, whle the sosurface extracton tme s nearly optmal. Keywords: scalar feld vsualzaton, volume vsualzaton, sosurface extracton, tme-varyng felds, marchng cubes, span space. 1 Introducton An sosurface represents regons that have a constant value n a three-dmensonal scalar feld. Dsplayng sosurfaces s a useful technque for analyzng scalar data due to ts effectveness n revealng the spatal structures of the feld s value dstrbuton. To compute the sosurface, Lorensen and Clne [1] proposed a Marchng Cubes algorthm whch extracts small polygon patches from ndvdual cells n the feld. The Marchng Cubes algorthm s smple and robust. However, the process of lnear search for sosurface cells can be expensve. To mprove the performance, researchers have proposed varous schemes that can accelerate the search process. Examples nclude Wlhelm and Van Gelder s Octrees[], Lvnat et al. s NOISE method[3], Shen et al. s ISSUE algorthm[4], Itoh and Koyamada s Extrema Graph method [, 6], Bajaj et al. s Fast Isocontourng method [7, 8], and Cgnon et al. s Interval Tree[9] algorthm. Inevtably, these acceleraton algorthms ncur overhead for storng extra search ndces. For a steady scalar feld,.e., only a sngle tme step of data s present, ths extra space s often affordable, and the hghly nteractve speed of extractng sosurfaces can compensate for the overhead. However, for tme-varyng smulatons, a typcal soluton can contan a large number of tme steps, and every smulaton step can produce a great amount of data. The overall storage requrement for the search ndex structures can be overwhelmng. Furthermore, when analyzng a tme-varyng scalar NASA Ames Research Center, Mal Stop T7A-, Moffett Feld, CA 9403 (hwshen@nas.nasa.gov) feld, a user may want to explore the data back and forth n tme, wth the same or dfferent sovalues. Ths wll requre a sgnfcant amount of dsk I/O for accessng the ndces for data at dfferent tme steps when there s not enough memory space for the entre tme sequence. As a result, the performance gan from the effcent sosurface extracton algorthm could be offset by the I/O overhead. Ths paper presents an effcent sosurface extracton algorthm for tme-varyng scalar felds. The man focus s to devse a new search ndex structure for a tme-varyng feld so that the storage overhead s kept small, whle the performance of the sosurface extracton s stll hgh. In addton, our algorthm allows flexble control of the tradeoff between performance and storage space and, thus, can be used for data wth dfferent characterstcs n dfferent computng envronments. To acheve these goals, we characterze each cell n the feld based on ts extreme values and the varaton of the extreme values over tme. Consder a cell that has a hgh temporal coherence and, thus, a small scalar varaton over several tme steps. Such a cell, n a perod of several tme steps, may be referenced by a sngle ndex entry based on that cell s overall extreme values n tme. On the other hand, for a cell that has lttle coherence and, thus, a hgh scalar varaton, the cell s ndexed ndvdually at every tme step by ts correspondng extreme values. Our algorthm creates an sosurface cell search ndex for the tme-varyng feld, called Temporal Herarchcal Index Tree. Cells that have a small amount of varaton over tme are placed n a sngle node of the tree that covers the entre tme span. Cells wth a larger varaton are placed n multple nodes of the tree multple tmes, each for a short tme span. When generatng an sosurface, a smple traversal wll retreve the set of nodes that contans all of the cell ndex entres needed for a gven tme step. The cells are organzed at each node usng a data structure that was developed for generatng sosurfaces from a steady data set. For a typcal tme-varyng scalar feld, not only does ths temporal herarchcal ndex tree requre much less storage space, but also the amount of I/O requred to access the ndces at dfferent tme steps from the secondary storage s greatly reduced. We begn ths paper by gvng an overvew of the sosurface extracton problem and some exstng technques. We then present our algorthm on buldng the temporal herarchcal ndex tree and the sosurface extracton method for tme-varyng felds. Fnally, we present expermental results to demonstrate the effectveness of our algorthm and provde concludng remarks and future research plans. Background and Related Work Gven an sovalue, cells that have mnmum value lower, and maxmum value hgher, than the sovalue are ntersected by the sosurface. We call these cells sosurface cells. To expedte the sosurface cell search process, researchers have proposed varous technques for creatng search ndces by parttonng the cells based on ther spatal and/or value nformaton. An example of the space-partton methods s Wlhelm and Van Gelder s octrees algorthm [], whch parttons the data herarchcally and coalesces the extreme values,

2 .e., the mnmum and maxmum values, of cells wthn each local cluster. The octrees algorthm s prmarly for structured grd data. The effcency of the method s reported to be O(k +log(n=k)) [3], where k s thenumber of sosurface cells, and n s the total number of cells. There are many value-partton methods proposed n the past years [10, 11, 1, 3, 4, 9]. Among those methods, Lvnat et al. [3] proposed span space, a two-dmensonal space where every cell n the feld s represented by a pont. The pont s x coordnate represents the correspondng cell s mnmum value, and the y coordnate represents the cell s maxmum value. Lvnat et al. use a Kd-Tree, and subsequently Shen et al. [4] use a lattce subdvson, to subdvde the cells n span space based on ther value ranges. Cgnon et al. [9] proposed the use of an nterval tree as the search ndex, whch has an optmal effcency of O(log(n)). Recently, Chang and Slva [13] proposed I/O optmal technques to buld the nterval tree on dsk, and the access of the nterval tree s drven by demand. Chang, Slva, and Schroeder also expanded the I/O-optmal technques for out-of-core sosurface extracton [14]. In addton to the space- and value-partton methods, Itoh et al. [, 6] and Bajaj et al. [7, 8] proposed algorthms usng a surface propagaton scheme. In ther methods, a small set of seed cells s frst extracted; and sosurfaces of any gven sovalue can then be computed by propagatng surfaces from certan seeds through adjacences. Bajaj et al. s algorthm s able to create only a small number of seeds and has an optmal effcency of O(log(n)). The acceleraton algorthms descrbed above nevtably ncur overhead for storng extra search ndces. For nstance, the BON octrees proposed n [] ncrease the orgnal data by 16%, whch s the rato of the number of tree nodes to the orgnal data ponts. Ths overhead does not yet nclude the mnmum and maxmum scalar values assocated wth each node necessary nformaton for sosurface extracton. In addton, the leaf node n the BON octrees s a cluster of eght cells,.e., ndvdual cells are not ndexed. The value-partton methods ndex down to ndvdual cells so that hgher nteractvty can be provded. However, each cell ndex entry needs to store the cell s mnmum and maxmum values and the cell dentfcaton. As a result, the total space requred for the ndex can be larger than the sze of the orgnal data. Bajaj et al. s method creates seed sets that ncur the least amount of space overhead. However, for unstructured grd data, the requred adjacency nformaton s often not avalable and, thus, the space overhead can be comparable to, or even hgher than, the value-partton methods f the adjacences need to be computed and stored. To our knowledge, to date there s no sosurface extracton algorthm that s optmzed for tme-varyng data. Although t s possble to extend the octrees to the fourth dmenson,.e., tme, t can only be used for structured grd data. In addton, the fourdmensonal octrees couple together the temporal and the spatal dmensons, whch makes cell parttonng awkward because the underlyng data may have very dfferent resolutons n tme and space. Furthermore, treatng temporal and spatal domans as equals mpedes the utlzaton of the temporal coherence exstng n the data. In the followng, we propose an optmzaton algorthm for sosurface extracton n tme-varyng felds. The value-partton paradgm s used because of ts nteractvty and ts equal effectveness for both structured and unstructured grd data. We assume that the tme-varyng feld has a steady grd, or has a grd that s transformed, but not redefned, over tme. Our goal s to reduce the overall sze of the search ndex for data n a tme-varyng feld, whle stll provdng hgh-performance sosurface extracton. 3 Isosurface Extracton from Tmevaryng Felds Gven a tme nterval [; j] and a tme-varyng feld, we defne a cell s temporal extreme values, that s, the extreme values over tme, n ths nterval as: mn j = MIN(mnt);t= ::j max j = MAX(maxt);t = ::j where MIN and MAX are the functons that compute the mnmum and the maxmum values, and mn t and max t are the cell s extreme values at the t th tme step; we call them the cell s tmespecfc extreme values. To locate the sosurface cells n the tmevaryng feld, one can approxmate a cell s extreme values at any tme step wthn the tme span [; j] by the cell s temporal extreme values, mn j and maxj, and use them to create a sngle search ndex. Usng ths approxmated search ndex, an sosurface at a tme step t; t [; j], can be computed by frst fndng the cells that have mn j smaller and maxj larger than the sovalue. The actual scalar data of these cells at the specfc tme t are then used to compute the geometry of the sosurface. Usng the approxmated search ndex can greatly reduce the storage space requred snce only one ndex s used for all the j, +1tme steps. It also guarantees to fnd all the sosurface cells because: f t [; j] and mn t <V so and max t >V so =) mn j <Vso and maxj >Vso where V so s the sovalue and t s the tme step at whch the query s ssued. The algorthm just descrbed can be neffcent because the temporal extreme values only provde a necessary but not a suffcent condton to qualfy a cell as an sosurface cell. As a result, many non-sosurface cells are vsted as well. In the followng, we propose an adaptve scheme that enables hgh performance sosurface extracton, whle t also reduces the storage overhead ncurred by the search ndex for sosurface extracton n tme-varyng felds. We devse a new search ndex structure, called Temporal Herarchcal Index Tree. Ths tree s bult by classfyng the cells accordng to the amount of varaton n the cell s values over tme. Cells that have a small amount of varaton are placed n a sngle node of the tree that covers the entre tme span. Cells wth a larger varaton are placed n multple nodes of the tree multple tmes, each for a short tme span. When generatng an sosurface, a smple traversal wll retreve the set of nodes that contans all cell ndex entres needed for a gven tme step. The cells n each node can be organzed usng exstng algorthms developed for generatng sosurfaces from a steady data set. It s noteworthy that a smlar concept ndependently developed by Fnkelsten et al. [1] on buldng a herarchcal representaton of multresoluton vdeo has been recently brought to our attenton. The paper proposes a Tme Tree whch s a bnary tree of sparse quadtrees. Each node n the tme tree corresponds to a sngle frame at some temporal resoluton. The tree can grow to dfferent depths for dfferent regons of the frame to support a vdeo sequence wth dfferent temporal resolutons. 3.1 Temporal Herarchcal Index Tree In ths secton, the temporal herarchcal ndex tree data structure s descrbed. We frst dscuss how to characterze a cell by the temporal varaton of ts extreme values. We then present the tree constructon algorthm usng the results of cell characterzaton. The span space [3] s useful for analyzng the temporal varaton of a cell s extreme values. In the span space, each cell s represented by a pont whose x coordnate represents ts mnmum value

3 max N 4 N mn Fgure 1: In ths example, the span space s subdvded nto 9 9 lattce elements. Each lattce element s assgned an nteger coordnate based on ts row and column number. The shaded lattce element n ths fgure has a coordnate (; 4). 0 1 N 3 4 N Fgure : Cells n a tme-varyng feld are classfed nto a temporal herarchcal ndex tree based on the temporal varatons of ther extreme values. In ths fgure, the tree s bult from a tme-varyng feld wth a tme nterval [0,]. and whose y coordnate represents t maxmum value. For a tmevaryng feld, a cell has multple correspondng ponts n the span space, and each pont represents the cell s extreme values at one tme step. To characterze a cell s scalar varaton over tme, the area over whch the correspondng ponts spread n the span space provdes a good measure the wder these ponts spread, the hgher s the cell s temporal varaton. Ths varaton can be quantfed by usng the lattce subdvson scheme of the span space [4], whch subdvdes the span space nto L L non-unformly spaced rectangles, called lattce elements. To perform the subdvson, we frst sort, n ascendng order, all the dstnct extreme values of the cells n the tme-varyng feld wthn the gven tme nterval and establsh a lst. We then fnd L +1scalar values, fd 0;d 1;:::;d Lg, n the lst that can evenly separate the lst nto L sublsts wth an equal length. These L +1scalar values are used to draw L +1vertcal lnes and L +1horzontal lnes to subdvde the span space. The lst d s chosen n ths way to ensure that cells can be more evenly dstrbuted among the lattce elements. Fg. 1 s an example of the lattce subdvson. Usng the lattce subdvson, we propose a bnary tree data structure, called Temporal Herarchcal Index Tree, to classfy the cells n a tme-varyng feld based on the temporal varatons of ther ex- Fgure 3: In ths example, tree nodes that are nsde the rectangular boxes are on the traversal path for an sosurface query at tme step 1. treme values. Gven a tme nterval [; j] n the tme-varyng feld, the root node n the temporal herarchcal ndex tree, denoted as N j, contans cells that have low scalar varatons n the tme nterval [; j]. We determne that a cell has a low temporal varaton by nspectng the locatons of the cell s j, +1correspondng ponts n the span space. If all of the cell s correspondng ponts are located wthn an area of lattce elements, we characterze the cell as a cell of low temporal varaton. Ths cell s then placed nto the node N j, and s represented by ts temporal extreme values mn j and maxj. On the other hand, for cells that do not satsfy the crteron, we splt the tme nterval [; j] n half, that s, nto [; +(j, +1)=, 1] and [ +(j, +1)=;j], and contnue to classfy the cells recursvely nto each of N j s two subtrees that have roots N +(j,+1)=,1 and N j. The temporal herarchcal tree has leaf nodes N t t ;t = ::j. The leaf nodes contan +(j,+1)= cells that have the hghest scalar varatons n tme so that the cells tme-specfc extreme values are used. Cells that are classfed nto non-leaf nodes are represented by ther temporal extreme values. The use of the temporal extreme values drectly contrbutes to the reducton of the overall ndex sze because the temporal extreme values are used to refer to a cell for more than one tme step. Fg. shows an example of the temporal herarchcal ndex tree wth a tme nterval [0; ]. To facltate an effcent search for sosurface cells, a search ndex for each node of the temporal herarchcal tree s created. Ths can be done by usng any exstng sosurface extracton algorthm based on the value-partton paradgm. Here we propose to use a modfed ISSUE algorthm [4] whch can provde optmal performance. For every node N j n the temporal herarchcal ndex tree, cells contaned n the node are represented by ther extreme values (mn j ;maxj ). To create the search ndex, we use the lattce subdvson descrbed prevously and sort cells that belong to the lattce elements of each row, excludng the lattce element at the dagonal lne, nto a lst based on the cells representatve mnmum values n ascendng order. Another lst n each row s created by sortng the cells representatve maxmum values n descendng order. For those lattce elements at the dagonal lne, the nterval tree method [9] s used to create one nterval tree for each element. 3. Isosurface Extracton Gven the temporal herarchcal ndex tree, ths secton descrbes the algorthm that s used to locate the sosurface cells at run tme. We frst descrbe a smple traversal method to retreve the sets of

4 max max (V, V ) so so mn mn Fgure 4: In ths case, lattce element (4; 4) contans the pont (V so;v so). Isosurface cells are located n the shaded area. nodes that contan all cell ndex entres needed for a gven tme step. We than descrbe the sosurface cell search algorthm used for the lattce search ndex bult n each node. Gven an sosurface query at tme step t, we compute the sosurface by frst locatng the nodes n the tree that may contan the sosurface cells. Ths s done by recursvely traversng from the root node N j to one of ts two chld nodes, N b a, such that a t b untl the leaf node N t t s reached. Along the traversal path, we perform the sosurface cell search, usng a method that wll be descrbed next, at each encountered node. The tree s constructed so that every cell n the feld exsts n one of the nodes n the traversal path. These cells have ther representatve extreme values, temporal or tme-specfc, as the approxmaton of ther actual extreme values at tme step t. Fg. 3 shows an example of the traversal path. At every node along the traversal path, the lattce search ndex bult at the node s used to locate the canddate sosurface cells. GvenansovalueV so, we frst locate the lattce element wth nteger coordnates [I;I] that contans the pont (V so;v so) n the span space. The sosurface cells are then located n the upper left corner that s defned by the vertcal lne x = V so and the horzontal lne y = V so as shown n Fg. 4. The canddate sosurface cells can be collected from the followng three categores: 1. For every lst n the row R; R = I +1::L, 1 that was sorted by the cells mnmum values, we collect the cells from the begnnng of the lst untl the frst cell s reached whch has a representatve mnmum value that s greater than the sovalue.. For the lst n row I that was sorted by the maxmum values, we collect the cells from the begnnng of the lst untl the cell s reached whch has a representatve maxmum value that s smaller than the sovalue. 3. Collect the sosurface cells from the nterval tree bult at lattce element [I;I]. The method and ts detals are presented n [9]. After the canddate sosurface cells are located, we then use the cells actual data attme step t to perform trangulaton. Our algorthm has optmal performance snce the sosurface cells n categores 1 and are collected wthout the need for any search. The number of cells n category 3 s usually small. Furthermore, the nterval tree method has an optmal effcency of O(logN), where N s the number of cells n the feld. Fgure : At every tree node, the non-sosurface cells beng unnecessarly vsted are confned wthn the two rows and two columns of the lattce elements as shown n the shaded area. Increasng the resoluton of the lattce subdvson can reduce the number of cells n ths area, for the prce of a larger temporal herarchcal ndex tree. As mentoned prevously, a canddate sosurface cell may not be an sosurface cell after all. These non-sosurface cells come from non-leaf nodes n our temporal herarchcal ndex tree snce a cell s tme-specfc extreme values, mn t and max t, may not contan the gven sovalue even though the approxmated extreme values,.e., the temporal extreme values mn j and maxj, do contan the sovalue. Although ths problem wll not cause a wrong sosurface to be generated, snce the trangulaton routne wll detect the case and create no trangles from these cells, t does ncur performance overhead. Actually, ths performance overhead s an expected consequence of usng temporal extreme values as the approxmated extreme values for cells, where we trade performance for storage space. In fact, the performance overhead s bound by the resoluton of the lattce subdvson n the span space. In our algorthm, we place a cell nto the node N j n the temporal herarchcal ndex tree n such a way that ts representng ponts at dfferent tme steps wthn tme nterval [; j] always resde wthn an area of lattce elements n the span space. Therefore, for any node N j n the tree, the worst case for the number of the non-sosurface cells beng vsted s estmated as the number of cells n the two rows and two columns of the lattce elements at the boundary layers of the lattce elements that are searched for the canddate sosurface cells, as shown n the shaded area n Fg.. Therefore, the user-specfed parameter L, n an L L lattce subdvson becomes a control parameter that s used to determne the tradeoff factor between the storage space and the sosurface extracton tme. 3.3 Node Fetchng and Replacement Ideally, f the entre temporal herarchcal ndex tree resdes n man memory, there s no I/O requred when the user randomly queres for sosurfaces at dfferent tme steps. However, the memory requrement s usually too hgh to make ths practcal. In our algorthm, the temporal herarchcal tree can be output to a fle. When an sosurface at a tme step s quered, our algorthm follows the traversal path as descrbed prevously and brngs those nodes nto man memory. Intally, all nodes on the traversal path need to be read n. Subsequently, f the user queres for an sosurface at a dfferent tme step, our algorthm traverses the search tree and brngs n only those nodes that are not already n man memory. In fact,

5 0 1 3 N 4 N Fgure 6: In ths case, f the user changes the sosurface query from tme step 1 to tme step, only the node N needs to be brought n from the dsk. Data Set F-18 Delta Wng Post # of cells 1,66,90 68,944 13,039 # of nodes 1,764, , ,07 Grd sze Soluton sze Table 1: Densty felds n three CFD smulaton data sets were used n our experments. Informaton lsted here s for one tme step, and the fle szes are n megabytes. because the non-leaf nodes contan cell ndex entres that are shared by several tme steps, they are very lkely to be n memory already. In ths case, only the dfferental nodes, a small porton of the ndex tree, need to be read n from the dsk. As a result, the amount of I/O requred for a subsequent sosurface query can be consderably smaller. Fg. 6 gves an example. Although t s always desrable to retan as many nodes n memory as possble n case that the user needs to go back and forth n tme when queryng the sosurfaces, those nodes that are not n use have to be replaced when the memory lmtaton s exceeded. To determne whch node needs to be replaced, we develop a node replacement polcy that assgns a prorty to every node, based on ts depth n the tree. The smaller the depth of a node s, the hgher s ts prorty. For example, the root of a tree has a depth of zero therefore t has the hghest prorty. The reason s that the root node contans search ndex entres to those cells that have the lowest temporal varatons, and, thus, these ndex entres are used by many tme steps. When a node has to be replaced, we select the node that has the lowest prorty. If there are more nodes than one wth the same prorty, we remove the one that s the least recently used (LRU). 4 Results and Dscusson In ths secton, we present expermental results of sosurface extracton for tme-varyng scalar felds usng the temporal herarchcal ndex tree. Three curvlnear grdded tme-varyng data sets generated from computatonal flud dynamcs (CFD) smulatons were used [16, 17, 18], as shown n Table 1. The tme and storage space measurements shown n the followng for the Delta Wng and the Post data sets were performed on an SGI Onyx workstaton wth an R10000 mcroprocessor and 1 megabytes of memory. For the F-18 data set, the measurements were performed on an SGI Onyx RealtyMonster wth an R10000 mcroprocessor and four ggabytes Data Set F-18 Delta Wng Post T Sequence Sequence Sequence Index Sze (one tme step) ISSUE Interval Tree Index Sze (twenty tme steps) ISSUE Interval Tree Table : The tme sequences n the test data sets and the storage space (n megabytes) requred for creatng the search ndces for one tme step and for twenty tme steps of data usng the ISSUE and the Interval Tree algorthms. F-18 Lattce Resoluton Sequence % 10.% 1.4% Sequence % 1.6% 19.% Sequence % 10% 14.8% Table 3: The szes (n megabytes) of the temporal herarchcal ndex trees for the F-18 data set usng three dfferent lattce resolutons. of memory. We studed the characterstcs of our algorthm and compared these characterstcs wth the regular Marchng Cubes algorthm, the Interval Tree algorthm, and the ISSUE algorthm. All of these algorthms were mplemented by the author. In our tests, each temporal herarchcal ndex tree was bult usng twenty tme steps of data. We performed our experments at three dfferent tme sequences n each of the test data sets, as shown n Table ; and we denote these sequences as Sequence 1, Sequence, and Sequence 3. To understand the storage overhead ncurred by the exstng value-partton technques, the Interval Tree and the ISSUE algorthms were used to create search ndces for data at every tme step. Table shows the szes of search ndces for one tme step and the szes of the search ndces for twenty tme steps. It s not a surprse that the sze of the search ndex for one tme step s much larger than the soluton data tself because the cell search ndex needs to store each cell s mnmum, maxmum values, and the cell s dentfcaton. 1 For a tme-varyng feld such as the F-18 data set, more than 00 megabytes of storage were requred to ndex 0 tme steps of data. Ths overhead s rather overwhelmng. Three dfferent resolutons of lattce subdvsons were used n our experments to buld temporal herarchcal ndex trees. A coarse resoluton of lattce structure ndcates that more cells are characterzed as havng low temporal varatons. As a result, the temporal herarchcal ndex tree wll have a smaller sze snce more cells n the tme-varyng feld are placed nto the non-leaf nodes n the tree. The tradeoff s that the search ndex tree that results from a coarse lattce subdvson wll be relatvely less effcent n extractng so- 1 In our experments, we ntentonally chose not to cluster multple cells to form meta cells for buldng the ndex as n [, 14], or use the nce chessboard approach as suggested n [9], so we can more easly study the behavor of the underlyng algorthms. However, these technques can be equally well appled to all the methods, ncludng our new algorthm, dscussed n ths secton.

6 Delta Wng Lattce Resoluton Sequence % 16.9% 7.3% Sequence % 16.6% 6.6% Sequence % 17.3% 7.8% Table 4: The szes (n megabytes) of the temporal herarchcal ndex trees for the Delta Wng data set. Post Lattce Resoluton Sequence % 44% % Sequence % 30.% 4% Sequence % 30.7% 4.% Table : The szes (n megabytes) of the temporal herarchcal ndex trees for the Post data set. surfaces. Table 3 shows the szes of the temporal herarchcal ndex trees bult for the F-18 data set. The percentages shown n the table are the ratos of the tree szes to the overall space requred by the IS- SUE algorthm, n a perod of twenty tme steps, as lsted n Table. The test results from the three dfferent tme sequences consstently showed that the storage overhead was sgnfcantly reduced, namely from more than 00 megabytes to about 30 megabytes n the 1010 lattce, and to about 100 megabytes n the lattce; the dsk space savngs amount to more than 80%. Table 4 and Table lst the results for the Delta Wng and the Post data sets. The Post data set has a hgher scalar varaton n tme. However, even wth a hgh resoluton of lattce subdvson we stll had about 0% savng n storage; for the smaller resolutons of lattce subdvson, we acheved about 7%, 90% space savngs. Table 6 shows the performance of sosurface extracton usng the temporal herarchcal ndex tree for the F-18 data set. We also show the performance of the regular Marchng Cubes algorthm (denoted as MCs), the Interval Tree method (denoted as Int. Tree), and the ISSUE algorthm. We chose two representatve sovalues at each of the three representatve tme steps. Among the technques, the Interval Tree and the ISSUE algorthms have optmal performance, whch can save about 80%, 9% sosurface extracton tme compared wth the regular Marchng Cubes algorthm. Usng the temporal herarchcal ndex tree, t can be seen that when a hgh resoluton lattce such as the subdvson was used, the performance of sosurface extractons was very close to the optmal performance ganed from usng the Interval Tree or the ISSUE algorthms, whle only about 0% of the storage space used by the Interval Tree or the ISSUE algorthm was needed for storng the temporal herarchcal ndex tree. For the low resoluton lattce such as the subdvson, although the performance was slghtly lower, t was stll sgnfcantly faster than the regular Marchng Cubes algorthm. Consderng that less than 10% of space was requred to store the search ndex compared wth a full set of ISSUE or Interval Tree ndces, ths tradeoff can be very benefcal for certan applcatons. Table 7 and Table 8 show the results for the Delta Wng and the Post data sets, whch had very smlar characterstcs. Table 9 shows the number of non-sosurface cells that were vsted wth lat- I/O Tme (n msecs) Node Fetch Tme Tme Step Fgure 7: The tme (n mllseconds) for restorng tree nodes from the dsk when the user sequentally queres the sosurface n tme. The F-18 data set was used. tce subdvsons of dfferent resolutons. The percentage numbers are the ratos to the total number of cells n the feld. It can be seen that even wth a low resoluton subdvson such as 10 10, the overhead s farly small. In our algorthm, the nodes n the temporal herarchcal ndex tree are read nto man memory only when necessary. In the case when a user roams a tme-varyng data set back and forth n tme, many non-leaf nodes contanng search ndces that are shared by consecutve tme steps can be retaned n memory. As a result, only nodes that are specfc to the tme step for the current sosurface query need to be brought nto man memory and placed nto the tree. Ths can result n a substantally smaller amount of I/O. Fg. 7 shows our expermental results. In our tests, we used the F-18 data set and quered the sosurfaces for a fxed sovalue of 0:99 from tme step to n ascendng order. As shown n the fgure, at the frst tme step, no node n the traversal path was n man memory, so a hgher amount of I/O was requred. However, n the subsequent tme steps, only the nodes that are not resdent n man memory needed to be brought n. The amount of tme for fetchng the nodes shown n the fgure s proportonal to the number of nodes specfc to each tme step. Fnally, the color plate shows mages of sosurfaces extracted from the test data sets. Conclusons and Future Work We have presented a new sosurface extracton algorthm for tmevaryng scalar felds. In the algorthm, we characterze the cells n the feld based on ther extreme values and the extreme values varatons over tme. For a cell that has a low temporal varaton, ts extreme values at consecutve tme steps are coalesced, and the overall extreme values are used to refer to a cell at many tme steps. We adaptvely compute the representatve extreme values for every cell n the tme-varyng feld and place the cells nto a search structure called Temporal Herarchcal Index Tree. Ths ndex tree can effcently locate sosurface cells n a tme-varyng feld, whle the sze of the tree for a seres of tme steps s substantally smaller than the space requred by the search ndces of the exstng sosurface extracton algorthms. Our algorthm allows flexble control of the tradeoff between performance and storage space and, thus, can be used for data wth dfferent characterstcs n dfferent computng envronments. We have tested our algorthm usng three large-scale tme-varyng data sets from CFD smulatons. The space savngs

7 can amount to more than 80%, whle the sosurface extracton performance remans nearly optmal. In addton, usng the temporal herarchcal ndex tree, the amount of I/O for accessng the search ndces at dfferent tme steps can be greatly reduced. Future work ncludes devsng an out-of-core algorthm for creatng and accessng the temporal herarchcal ndex tree. The method we descrbed n secton 3:3 s a coarse out-of-core model snce a whole node s fetched nto man memory at a tme. In fact, t s also desrable to devse a fner grnd out-of-core algorthm for accessng the temporal herarchcal ndex tree so that only the subset of the nodes lattce needed for the current sovalue s brought nto man memory at a tme. In addton, we would lke to nvestgate a combnaton of the space- and value-partton algorthms. Furthermore, developng tme-varyng methods for surface-propagaton schemes s also an nterestng research subject. Acknowledgments Ths work was supported n part by NASA contract NAS We would lke to thank Ken Gee, Neal Chaderjan, and Denns Jespersen for provdng ther data sets. Specal thanks to Randy Kaemmerer and Davd Ellsworth for ther metculous proofreadng of ths manuscrpt and valuable suggestons. We also thank Tm Sandstrom and other members n the Data Analyss Group at NASA Ames Research Center for ther helpful comments and techncal support. References [1] W.E. Lorensen and H. E. Clne. Marchng cubes: A hgh resoluton 3d surface constructon algorthm. Computer Graphcs, 1(4): , July [] J. Wlhelm and A. Van Gelder. Octrees for faster sosurface generaton. ACM Transactons on Graphcs, 11(3):01 7, July 199. [3] Y. Lvnat, H.-W. Shen, and C.R. Johnson. A near optmal sosurface extracton algorthm usng the span space. IEEE Transactons on Vsualzaton and Computer Graphcs, (1), March [4] H.-W. Shen, C.D. Hansen, Y. Lvnat, and C.R. Johnson. Isosurfacng n span space wth utmost effcency (ISSUE). In Proceedngs of Vsualzaton 96, pages IEEE Computer Socety Press, Los Alamtos, CA, [] T. Itoh and K. Koyamada. Automatc sosurface propagaton usng an extrema graph and sorted boundary cell lsts. IEEE Transactons on Vsualzaton and Computer Graphcs, 1(4), Dec [6] T. Itoh, Y. Yamaguch, and K. Koyamada. Volume thnnng for automatc sosurface propagaton. In Proceedngs of Vsualzaton 96, pages IEEE Computer Socety Press, Los Alamtos, CA, [7] C.L. Bajaj, V. Pascucc, and D.R. Schkore. Fast socontourng for mproved nteractvty. In 1996 Symposum for Volume Vsualzaton, pages IEEE Computer Socety Press, Los Alamtos, CA, Oct [8] M. van Kreveld, R. van Oostrum, C.L. Bajaj, D.R. Schkore, and V. Pascucc. Contour trees and small seed sets for sosurface traversal. In Proceedngs of 13th ACM Symposum on Comp. Geom., pages 1 19, Fgure 8: Color Plate Isosurfaces of densty felds n the F-18, Delta Wng, and Post data sets. The surfaces are colored by velocty magntudes, wth red beng a hgh magntude and blue beng a low magntude. [9] P. Cgnon, P. Marno, E. Montan, E. Puppo, and R. Scopgno. Speedng up sosurface extracton usng nterval trees. IEEE Transactons on Vsualzaton and Computer Graphcs, 3(), June [10] M. Gles and R. Hames. Advanced nteractve vsualzaton for CFD. Computng Systems n Engneerng, 1(1):1 6, [11] R. S. Gallagher. Span flter: An optmzaton scheme for volume vsualzaton of large fnte element models. In Proceedngs of Vsualzaton 91, pages IEEE Computer Socety Press, Los Alamtos, CA, [1] H.-W. Shen and C.R. Johnson. Sweepng smplces: A fast sosurface extracton algorthm for unstructured grds. In Proceedngs of Vsualzaton 9, pages IEEE Computer Socety Press, Los Alamtos, CA, 199. [13] Y.-J. Chang and C.T Slva. I/O optmal sosurface extracton. In Proceedngs of Vsualzaton 97, pages IEEE Computer Socety Press, Los Alamtos, CA, [14] Y.-J. Chang, C.T Slva, and W.J. Schroeder. Interactve outof-core sosurface extracton. In Proceedngs of Vsualzaton 98. IEEE Computer Socety Press, Los Alamtos, CA, [1] A. Fnkelsten, C.E. Jacobs, and D.H. Salesn. Multresoluton vdeo. In Proceedngs of ACM SIGGRAPH 96, pages 81 90, [16] K. Gee, S. Murman, and L. Schff. Computaton of F-18 tal buffet. Journal of Arcraft, 33(6), Dec [17] D. Jespersen and C. Levt. Numercal smulaton of flow past a tapered cylnder. RNR Techncal Report RNR-90-01, October [18] N. Chaderjan and L. Schff. Naver-Stokes analyss of a delta wng n statc and dynamc roll. AIAA , 199.

8 F-18 Tme Step Isovalue # of Trangles 7,163 80,970 7,394 71,689 7,644 73,70 MCs Int. Tree ISSUE Temporal Herarchcal Index Tree Table 6: The performance of sosurface extracton (n seconds) for the F-18 data set. Delta Wng Tme Step Isovalue # of Trangles 0,96 17,88,78 16,760 47,84 17,990 MCs Int. Tree ISSUE Temporal Herarchcal Index Tree Table 7: The performance of sosurface extracton (n seconds) for the Delta Wng data set. Post Tme Step Isovalue # of Trangles 18,93 11,168 0,476 11,480 0,18 11,064 MCs Int. Tree ISSUE Temporal Herarchcal Index Tree Table 8: The performance of sosurface extracton (n seconds) for the Post data set. Lattce Resoluton F-18 Tme Step Isovalue Non-socell Vsted 6,1 0,193,97 1,031 11,973 7,186 Percentage 3.7% 1.% 1.4% 0.7% 0.7% 0.4% Delta Wng Tme Step 780 Isovalue Non-socell Vsted 48,61 9,738 11,89 4,86 4,31,918 Percentage 7.3% 1.% 1.7% 0.7% 0.7% 0.4% Post Tme Step 1300 Isovalue Non-socell Vsted 10,06 4,66 3,138 1,49 1,8 9 Percentage 8.% 3.%.6% 1.% 1.3% 0.% Table 9: Number of non-sosurface cells that were vsted wth lattce subdvsons of dfferent resolutons.

9 Color Plate: Isosurfaces of densty felds n the F 18, Delta Wng, and Post data sets. The surfaces are colored by velocty magntudes, wth red beng a hgh magntude and blue beng a low magntude.