Quadrilateral and Tetrahedral Mesh Stripification Using 2-Factor Partitioning of the Dual Graph

Transcription

1 The Visul omputer mnusript No. (will e inserted y the editor) Plo Diz-Gutierrez M. Gopi Qudrilterl nd Tetrhedrl Mesh Stripifition Using 2-Ftor Prtitioning of the Dul Grph strt In order to find 2-ftor of grph, there exist O(n 1.5 ) deterministi lgorithm [7] nd O(n 3 ) rndomized lgorithm [14]. In this pper, we propose novel O(n log 3 n log log n) lgorithms to find 2-ftor, if one exists, of grph in whih ll n verties hve degree four or less. Suh grphs re tully dul grphs of qudrilterl nd tetrhedrl meshes. 2-ftor of suh grphs impliitly defines liner ordering of the mesh primitives in the form of strips. Further, y introduing few dditionl primitives, we redue the numer of tetrhedrl strips to represent the entire tetrhedrl mesh, nd represent the entire qud-surfe using single qud-strip. Keywords Grph mthing, 2-ftor, qudrilterl stripifition, tetrhedrl stripifition. Fig. 1 ue nd its dul grph. 2-ftor of the grph is 2-regulr spnning grph. Two possile 2-ftors of the dul grph re shown. They define disjoint qudrilterl strip yles on mnifold qudrngultion. Note tht the edges tht elong to the omplement of 2-ftor lso define disjoint yles in the dul of qudrngulted mnifold. 1 Introdution Qudrilterl nd tetrhedrl meshes re fundmentl geometri strutures in mny mehnil nd sientifi simultions, nd visuliztion. Due to the importne of these primitives, numer of lgorithms hve een designed to rete these primitive representtions of the given mesh from other representtions [1, 3, 4, 9]. Given mesh with qudrilterl or tetrhedrl representtion, liner ordering of these primitives n e used in geometri proessing pplitions inluding rendering nd ompression of lrge dt sets [10]. In ft, mny ompression tehniques yield primitive strips s yprodut [12,11,10,19,13,16,22]. Most of the ompression sed stripifition lgorithms re greedy lgorithms tht ollet primitives while wlking long the mesh elements. Plo Diz-Gutierrez Deprtment of omputer Siene University of liforni, Irvine E-mil: plo@is.ui.edu M. Gopi Deprtment of omputer Siene University of liforni, Irvine E-mil: gopi@is.ui.edu Further, there re speifi lgorithms for tetrhedrl nd qudrilterl stripifition. For regulr tetrhedrl meshes, [15] suggests spe-filling urve pproh for stripifition. Heuristis for strip genertion from retngulr pthes of qudrilterls ws suggested y [6] nd ws used y [18]. Further, given polygonl mesh, lgorithm to effiiently deompose them to rete tringulr strips ws suggested y [24]. In this ontext, there re lso works on reting tringle strips speifilly from qudrilterl meshes [20, 23]. In this pper, given qud or tetrhedrl mesh, we propose grph sed lgorithm tht performs glol nlysis of the mesh to find qud nd tetr strips. This lgorithm tkes dvntge of the similrity in the dul grph strutures of qud nd tetrhedrl meshes nd presents unified solution for the prolem in meshes with either of these primitives. Our work is losely relted to the tringle strip genertion lgorithm y Gopi nd Eppstein [8] tht finds strips using the 1-ftor in the dul grph of mnifold tringulted mesh. We use the 2-ftor in the dul grph of tetrhedrl meshes nd mnifold qudrilterl meshes.

2 2 Plo Diz-Gutierrez, M. Gopi 1.1 Min ontriutions The fundmentl ontriution of this pper re the lgorithms to find 2-ftor of grph with degree four or less. 2-ftor of grph hs mny more pplitions thn just in qudrilterl nd tetrhedrl stripifition, like network flow nlysis, sequenil storge in dtses, et. Further, we present unified lgorithm to find qudrilterl nd tetrhedrl strips from qud mnifold surfe meshes nd tetrhedrl volume meshes using the ove 2-ftor finding lgorithm. Effiient mngement of strips n e done if the numer of strips is less. We propose novel nd generlized sudivision tehniques for qud nd tetrhedrl elements to merge strip-yles nd minimize the numer of strip loops. In the proess, we n hieve single Hmiltonin qudrilterl strip overing the entire mnifold qudrilterl mesh. We disuss the theory of k-ftor of the grph in Setion 2 nd propose lgorithms to ompute 2-ftor of the grph. 2-ftor of the dul grph of qud or tetrhedrl mesh impliitly defines qud nd tetr strips. In Setion 3 we present sudivision tehniques to merge these disjoint strips. Finlly, we disuss implementtion nd results of our stripifition lgorithms in Setion 4. 2 Qudrilterl nd Tetrhedrl Stripifition The dul of qudrilterl mesh representtion of mnifold, dul qud-grph, is 4-regulr grph (every node hs degree 4). For ll prtil purposes, tetrhedrl volume mesh form 3-mnifold surfes with oundries. Hene its dul grph, dul tetr-grph, will hve nodes with less thn degree four on the oundries. ut for this differene, these dul grphs hve very similr strutures nd hene most of the grph lgorithms tht re pplile to one dul grph n e pplied to the other lso. In this pper, we propose lgorithms tht re pplile to oth dul qud- nd tetr-grphs to find disjoint yles in the grph tht trnsltes to disjoint qud/tetr strip loops in the priml mesh. Post-proessing of these strips re dependent on mnfoldness of the mesh nd hene independent tehniques re developed for qud nd tetr strips. The fundmentl onept we use to develop qudnd tetr-strips is the 2-ftor of grph. In the next setion, we disuss the lssil definition of 2-ftor of grph. In Setion 2.2 we present two lgorithms to find 2-ftor in the dul grph whih impliitly defines prtition of the qudrilterl nd tetrhedrl priml meshes into qud nd tetrhedrl strips. Fig. 2 (left) The dul degree three grph of the tringultion of genus 0 mnifold nd perfet mthing shown y drk edges. (enter) The set of unmthed edges form disjoint yles. Two suh yles re shown. These disjoint yles re onneted to eh other y mthed edges. The lgorithm onstruts spnning tree of these disjoint yles nd hene hoose mthed edges tht onnet these yles. (right) The tringle pirs orresponding to hosen mthed edges in the tree re split reting two new tringles for every pir. Mthing is toggled round the new (nodl) verties resulting in tringultion with Hmiltonin yle of unmthed edges. Reprodued from [8]. 2.1 The 2-ftor of Grph Definition 1 k-ftor of grph G is spnning k- regulr sugrph of G. For exmple, 1-ftor mthes every node of the grph with one nd extly one of its neighors. 1-ftor is lso lled perfet mthing. y Peterson s theorem [17], 3-regulr, ridgeless grph lwys hs perfet mthing. Using the ft tht the dul grphs of tringulted two mnifolds re 3-regulr, ridgeless grphs nd tht perfet mthing exists for suh grphs, Gopi nd Eppstein [8] onstrut tringle strip loop prtitions nd eventully single tringle strip loop of tringulted two mnifold of ny ritrry genus. Speifilly, sine every node hs degree three nd extly one of the edges is hosen y the 1-ftor grph, the rest of the two edges form disjoint loops tht prtitions in the vertex set of the dul grph nd hene the input mesh. These disjoint loops were merged y sudividing two djent tringles elonging to two different yles into four tringles y inserting new vertex in the midpoint of the shred edge. This retes single strip yle overing ll the tringles in the input mesh. This lgorithm is illustrted in Figure 2. We use similr pproh to onstrut qud nd tetr strips using the following lssil result ttriuted to Peterson [17]. Theorem 1 Every regulr grph of even degree hs 2-ftor. The implition of the ove theorem is tht the dul qud-grph of two mnifold, whih is 4-regulr grph, hs 2-ftor; tht is, there is sugrph in whih every vertex hs degree two. Hene ll the nodes in the dul grph long with the hosen edges in the 2-ftor form disjoint loops nd thus qud-strip loops in the priml qudrngultion. Sine the dul tetr-grph is not regulr grph, the ove theorem does not pply to tetrhedrl meshes.

3 Qudrilterl nd Tetrhedrl Mesh Stripifition Using 2-Ftor Prtitioning of the Dul Grph 3 E DE () () () (d) Fig. 3 Illustrtion of two-pss grph mthing lgorithm on the dul of ue. () Dul grph of ue. () First pss perfet mthing. () Remove the mthed edges from the first pss nd run the seond pss of grph mthing. (d) Union of mthed edges from () nd () gives 2-ftor. Hene the 2-ftor finding lgorithms tht re explined in the susetions elow, when pplied to the dul tetrgrphs might not produe 2-ftor not ll nodes might hve degree two. In other words, the stripifition of the priml tetrhedrl mesh my ontin oth liner strips nd strip-loops. Sine liner tetrhedrl strips lso re eptle s solutions to our prolem, we propose the following 2-ftor finding lgorithms s unified lgorithms for the stripifition of oth qudrilterl nd tetrhedrl meshes. 2.2 lgorithms for Finding 2-Ftor Most reent work [14] uses rndomized lgorithms to find 2-ftor in sprse grphs with n verties in O(n 3 ) expeted time. There re deterministi lgorithms tht find it in O(n 1.5 ) time [7]. Our grphs re speil grphs tht re dul to qud/tetr meshes whih might led to simpler solutions. Here we present two lgorithms to find the 2-ftor in the dul qud/tetr grph using 1-ftor finding lgorithm whih n e omputed in O(n log 4 n) nd n e improved to O(n log 3 n log log n) using the reent results from [21]. Though the theoretil improvement over the results is mrginl, the most importnt prtil dvntge of the presented lgorithms is tht of ode reusility: there re mny implementtions of grph mthing (the 1-ftor finding) lgorithms ville in puli domin tht we n use to find the 2-ftor Two Pss Grph Mthing lgorithm The first lgorithm, whih we ll the two-pss grph mthing method, pplies the grph mthing lgorithm twie on the input dul grph to get 2-ftor. Speifilly, we run the grph mthing lgorithm one, remove the mthed edges from the input grph, nd run the mthing lgorithm gin on the new grph. The union of mthed edges hosen from these two runs gives us 2-ftor of the originl grph, nd hene disjoint yles in the mesh. This lgorithm is illustrted in Figure 3. We use puli implementtion of rdinlity grph mthing lgorithm tht gives perfet mthing if one exists nd mximizes the mthing, otherwise. D D ED Fig. 4 Left: five node 4-regulr grph with no 1-ftor (perfet mthing) ut hs 2-ftor s shown. Hene twopss grph mthing lgorithm to find two-ftor will not work in suh grphs. Right: geometri reliztion of 3- node 4-regulr grph. Every qudrilterl shres two edges with its neighor usully n uneptle geometry for grphis nd visuliztion pplitions. simple yle in the dul grph is its 2-ftor. This simple lgorithm produes orret results for most of the prtil models ut does not work on grphs with odd numer of nodes even if they re 4-regulr (Figure 4). ut most of suh odd-node grphs either do not hve n orientle 2-mnifold geometri reliztion or hve geometry (e.g non-plnr fes) nd topology (e.g. two qudrilterls shring two edges) tht re usully uneptle in grphis nd visuliztion pplitions. We show one geometri reliztion of n odd-vertex dul grph in Figure 4. Our grphs re derived from geometri meshes used in visuliztion pplitions nd in ft, ll the models we tried with the two-pss grph mthing method yielded 2-ftor, if one existed. The run-time omplexity of the mthing lgorithm is O(n log 3 n log log n), where n is proportionl to the numer of verties or edges in the dul grph. Note tht given the peulirities of the grph, the numer of verties nd edges is equl, up to smll onstnt ftor Templte Sustitution lgorithm The seond lgorithm, the templte sustitution method, is gurnteed to find 2-ftor, if one exists, on ny grph in whih every vertex is of degree four or less. Speifilly, the grph need not e 4-regulr grph nd hene this lgorithm is suitle for the dul tetr-grphs lso. In this method, we first trnsform the input dul grph G into new grph G y sustituting every node in G with the templte shown in Figure 5. Let us ll the grph G, the inflted grph. We stte nd prove the following reltionship etween the 2-ftor in the originl grph G nd the perfet mthing (1-ftor) in the inflted grph G. In the templte tht is sustituted for eh originl node, in ddition to the qudruplites representing the originl node, there re two more nodes. During grph mthing on this templte, these two dditionl nodes n get mthed to two of the qudruplites leving extly two other nodes to e mthed outside the templte. We ll the proess of dding these extr nodes to engge suset of nodes in the templte s doping for its resemlne to similr proess in semiondutor

4 4 Plo Diz-Gutierrez, M. Gopi mnufturing. We use this onept of doping to prove the following theorem. Theorem 2 There exists 1-ftor in the inflted grph if nd only if there exists 2-ftor in the originl grph. Proof Let us ssume tht there exists 1-ftor in the inflted grph. This mens tht oth the dopes re mthed thus engging extly two of the qudruplites. Sine there exists 1-ftor, the other two qudruplites re lso mthed, nd hve to e mthed externl to the templte long the edges in the originl grph. This yields 2-ftor in the originl grph. Let us ssume tht there is 2-ftor in the originl grph. Hene extly two of the qudruplites re onneted externl to the templte struture thus leving two other nodes mong the qudruplites to e mthed internlly with the dopes. Note the importnt onnetivity struture of dopes inside the templte: ny two nodes mong the qudruplites n e mthed with the dopes. Hene, given ny 2-ftor, there exists perfet mthing in the inflted grph. Fig. 5 Left: Dul grph node of one qudrilterl or tetrhedron of qud/tetr mesh. Right: The templte tht is sustituted for every node in the dul grph. Unshded nodes re the new dope nodes dded to the dul grph. If there exists 2-ftor outside this templte, tht is, if extly two of the qudruplites re mthed externlly, the dope nodes re mthed with the remining two nodes thus produing perfet mthing for the entire grph. Note the internl onnetivity with the dope nodes tht enles perfet mthing given ny omintion of two externlly mthed edges. 1-ftor in the inflted grph gives 2-ftor in the originl grph nd hene qud/tetr strip loops in the priml mesh. Speifilly, the inflted grph dul of qudrngulted 2-mnifold mesh will lwys hve 1-ftor nd n e found using (rdinlity) grph mthing lgorithm whose puli implementtions re ville. Suh 1-ftor is not gurnteed in the inflted dul grph of tetrhedrl mesh (with oundry) sine 2-ftor is not gurnteed y Peterson s theorem. If there is no 1-ftor, the grph mthing lgorithm will mximize the rdinlity of mthings leving minimum numer of nodes unmthed. These unmthed nodes might e either the nodes of the qudruplites or the dopes. If ny of the dope nodes is unmthed, we rek the externl mthing of qudruplite nodes to mth them internlly with the free dope nodes. Thus in the originl grph every node will hve either two or fewer mthed edges. ll tetrhedr in the priml mesh orresponding to unmthed dul grph nodes (tht is, with zero mthed edges) re defined s singleton strips. From eh of the tetrhedron orresponding to the dul node with extly one mthed edge, we follow the mthed edges to form liner strip of tetrhedr till it rehes nother tetrhedron with extly one mthed edge. Rest of the tetrhedr hve two mthed edges eh nd hene they form disjoint tetrhedron strip loops. Insted of using mthed edges, n lterntive pproh is to use the unmthed edges to define strips. This pproh is similr to tht of [8] to use unmthed edges to define tringle strips. The dvntge of this pproh is tht there will e no singleton tetrhedrl strips sine every node will either hve two, three, or four unmthed edges (sine they hve two, one or zero mthed edges). suitle trversl long the unmthed edges will gin produe tetrhedrl strip loops or liner strips, Fig. 6 Left: Nodl vertex proessing on vertex with four inident qudrilterls nd two disjoint yles. The resulting merged yle is shown.similr rrngements n e relized in tetrhedrl meshes lso, with even numer of tetrhedr inident on n edge. Right: The dul grph of qud/tetr strip. Six primitives inident on n (n 2)-dim simplex. Thik edges re mthed edges nd the thin edges show the unique disjoint yles. Every node hs degree four. lternting edges re mthed nd unmthed edges. Toggling these ssignments merges the inident yles. nd every tetrhedron in the mesh will elong to one of the strips. In ft, we follow this pproh in our implementtion. The run-time omplexity of the mthing lgorithm is lso O(n log 3 n log log n) s the two-pss grph mthing lgorithm. ut this lgorithm uses the inflted grph whih hs six times more nodes nd edges thn the originl grph. Hene this lgorithm is n order of mgnitude slower thn the two-pss grph mthing method. On the other hnd, it finds two ftor in ny grph with vertex degrees four or less. 3 Merging yles Finding single simple yle tht onnets ll nodes of the grph is NP-omplete. ut we n merge the strips yielded y the ove lgorithms to form single strip y sudividing the mesh primitives. Even though multiplestrip prtitions re vlid representtions for mny grphis nd visuliztion rendering pplitions, [5] showed tht the overhed of mintining multiple strips over

5 Qudrilterl nd Tetrhedrl Mesh Stripifition Using 2-Ftor Prtitioning of the Dul Grph 5 single strip is signifint. Further, strips re used in mny other pplitions like onnetivity ompression, nd suh pplitions would enefit from single strip representtion. First, we desrie n extension of the nodl vertex proessing lgorithm tht ws used to merge disjoint tringle strip yles [8], to proess qudrilterl strip yles. Further, the sme onept n e extended to tetrhedrl mesh whih we ll nodl edge proessing. Suh nodl vertex/edge proessing does not require sudivision of primitives ut redues the numer of disjoint loops in the entire stripifition of the model. The remining yles fter the nodl vertex/edge proessing re merged through sudivision proesses. Sine the sudivisions re dependent on the geometry, even though the dul qud- nd tetr- grphs re lolly similr, we hve to pply different sudivision tehniques depending on the mesh primitive. We disuss independent lgorithms for qudrilterl nd tetrhedrl sudivisions. 3.1 Nodl Fn-Simplex Proessing Sine this proessing is pplile to oth qudrilterls nd tetrhedr, we refer to them in generi terms s primitives. Further, in mnifold (with oundries) mesh one or two primitives re inident on n (n 1)-dim simplex (one or two qudrilterls on n edge, nd one or two tetrhedr on tringulr fe) nd numer of primitives form fn round n (n 2)-dim simplex (qudrilterl fn round vertex nd tetrhedrl fn round n edge). We ll n (n 2)-dim simplex, generilly, s fn simplex nd hene the nodl vertex/edge proessing s fn-simplex proessing. The gol of this optimiztion is to inrese the length of the disjoint yles y merging mny yles without ny primitive splits. ssume tht we hve lredy onstruted 2-ftor, nd prtitioned the primitives of the input mesh into disjoint yles. We lssify mesh fnsimplex v s nodl fn-simplex if it stisfies the following onditions: q v, the numer of primitives inident on v is even, the totl numer of unique disjoint yles tht these inident primitives elong to is qv 2, nd the inident primitives do not shre more thn one (n 1)-dim simplex etween the djent neighors. n exmple of nodl fn-simplex proessing with four primitives nd two unique inident yles in eh of them is shown in Figure 6. The neighorhood of every nodl fn-simplex is modified suh tht the mthed nd unmthed primitive pirs re toggled. This merges ll the inident yles into one yle. If we use unionfind dt struture to keep trk of whih primitives elong to whih yles, we n test whether ny mesh fnsimplex is nodl using numer of union-find queries proportionl to the degree of the fn-simplex, so the totl time for the optimiztion is O(nα(n)) where α is the extremely slowly growing inverse kermnn funtion. Fig. 7 Sudividing qudrilterl pir using two verties ( nd ). Note tht in the se of strips rossing non-djent edges (left exmple), fter sudivision, pir of djent sudivided qudrilterls shre two edges. Further, depending on the onfigurtion of the strip-pth the onnetivity of the edges from nd hnges (right exmple). The nodl fn-simplex optimiztion step typilly signifintly redues the numer of disjoint strip-yles, ut we hve no theoretil gurntees on its performne. The results on typil numer of instnes when suh n optimiztion is done is given in Tles 1 nd 2. One this optimiztion is performed, we form new grph in whih every node orresponds to strip yle, nd two nodes re onneted if their orresponding yles re djent to eh other [8]. spnning tree in this new grph determines whih two yles n e merged using primitive sudivision opertions, s explined in the following setions. 3.2 Qudrilterl Sudivisions for Merging Strips In mnifold qudrilterl mesh, there re extly twie the numer of edges thn fes. Using this ft, we n prove tht in order to mintin the Euler hrteristi of the mesh during sudivision, the inrese in numer of verties nd fes hve to e the sme. In order not to ffet the results of the mthing lgorithm, the new fes should potentilly e mthed to eh other; in other words, there should e even numer of dditionl fes, nd hene we n insert only even numer of new verties during sudivision. Given the ove rgument, the minimum numer of verties we n introdue for qudrilterl mesh sudivision is two. Figure 7 shows two suh sudivisions. First, we oserve tht the sudivision of the qudrilterl itself is dependent on the onfigurtion of strip yle pth in oth of the qudrilterls. Further, routing of the strips inside the sudivision lso depends on the originl strip pth. This requires se-y-se hndling of the sudivision. Seond, we see tht under one of the strip pth ptterns two new qudrilterls shre two edges etween them n undesirle topologil onfigurtion. We n lso show tht the sudivision with four new verties lso suffers from the sme two drwks s dding two verties. The minimum numer of verties tht we n dd to qudrilterl pir sudivision without the ove two drwks is six (Figure 8). Eh qud-pir sudivision

6 6 Plo Diz-Gutierrez, M. Gopi will e nodl vertex round whih the two strips n e merged using nodl vertex proessing lgorithm explined in the previous setion. Different ses of strip pth re shown in Figure 8. Suh proessing removes the dependeny of the geometri onnetivity of the sudivided qudrilterls from the strip pth. Further, the routing n e done utomtilly without se-y-se nlysis of the strip pth onfigurtions etween the two qudrilterls of the sudivided qud-pir. Fig. 8 Qudrilterl sudivision with six verties. In the figure, leled s follows: () Internl qudrilterls. () Externl qudrilterls in the strip. () Externl qudrilterl not in the strip. Two new, shred verties re dded. The one hosen s nodl vertex is shded drk. The strip efore nd fter the sudivision is shown with thik rrow lines. Fig. 9 Sudividing qudrilterl pir with six verties. Four possile onfigurtions re shown, one in eh olumn: top showing the originl strip pth nd the ottom showing the strip fter sudivision nd strip-merging: The strip within eh qudrilterl is routed in order to trverse through ll the four qudrilterls fter sudivision s shown in Figure 8. One suh routing is done, the pproprite shred vertex (shown shded) is hosen for nodl vertex proessing to merge the strips in oth the qudrilterls. ll other onfigurtions of strip pths re mirror refletions of the ses shown. hs six externl qudrilterls (three in eh of the prent qudrilterls), two internl qudrilterls (one in eh of the prent qudrilterls), nd two new shred verties in the shred oundry of the qud-pir. We ll the four of the six externl qudrilterls tht re in the strip-pth s strip-qudrilterls. Within eh prent qudrilterl, we first route the strip suh tht it trverses through ll the new qudrilterls s follows: one externl strip-qud internl-qud externl qud tht is not strip-qud the other externl strip-qud. If suh routing is done in oth the djent sudivided qudrilterls then t lest one of the shred verties 3.3 Tetrhedrl Sudivisions for Merging Strips Tetrhedrl sudivision for strip-merging is extly similr to tringle sudivision for strip merging [8]. In tringulted models, two djent tringles elonging to different yles re divided into two tringles eh y introduing vertex in the mid-point of the shred edge (Figure 2). Then nodl vertex proessing is performed t this new vertex to merge the yles. In tetrhedrl meshes, two djent tetrhedr elonging to different yles re divided into three tetrhedr eh y introduing vertex in the entroid of the shred fe. Then nodl-edge proessing is performed round one of the three newly introdued edges on the shred fe. The proess of finding this nodl edge is detiled elow. Unlike dul-qud/tri grph, the dul tetr-grph fter sudivision is non-plnr (Figure 10). First, s in qud/tri-strip merging method, strips in eh of the tetrhedron is routed within the three sudivided tetrhedr. There is t lest one djent tetrhedron-pir in the strip mong the upper three tetrhedr, whose orresponding tetrhedr in the lower hlf lso re djent in their strip pth (tetrhedrons,,, in Figure 10). In the dul grph, they form four-yle in whih the mthed nd unmthed edges lternte. This orresponds to nodl edge in the priml the ommon edge etween these four tetrhedr. Nodl fn-simplex proessing is done round this edge to merge the upper nd lower strips. s disussed in Setion 2.1, the stripifition of tetrhedrl meshes might produe omintion of liner strips nd strip loops. We n merge two strips using sudivision if nd only if they re disjoint nd t lest one of them is strip loop. If oth re liner strips, then the sudivision nd merging proess would split nd onnet to produe two liner strips gin. 4 Implementtion nd Results In our implementtion, oth the two pss nd the templte mthing lgorithms use the LED implementtion of rdinlity grph mthing lgorithm. Sine the rdinlity mthing lgorithm is independent of its pplition (s in our se, stripifition), we do not hve metri to ompre the qulity of results of the strips

7 Qudrilterl nd Tetrhedrl Mesh Stripifition Using 2-Ftor Prtitioning of the Dul Grph 7 () () () (d) (e) (f) Fig. 10 Sudividing tetrhedron pir. (-) The dul grph efore nd fter sudivision. () The strip pth efore sudivision. (d) Strips in eh of the tetrhedron is routed within the three sudivided tetrhedr. There is t lest one edge in the upper tetrhedron strip pth, whose prllel pth in the lower tetrhedron is trversed y the its strip pth. In this exmple, edge in the upper nd in the lower tetrhedron re trversed. This forms four-yle (shown s dotted lines) long with the orresponding edges nd in whih the mthed nd unmthed edges in the dul lternte. This orresponds to nodl edge in the priml, shown in drk lue in (f). (e) Merging of yles y nodl-edge proessing. (f) Result of sudivision nd merging in the priml tetrhedrl mesh. produed y our two lgorithms. (If we hd use weighted mthing lgorithm, then we ould evlute the qulity of the strips sed on its dherene to the weighting funtion.) On the other hnd, the effiieny of these two lgorithms n e ompred. The templte mthing lgorithm runs on n inflted grph of six times the numer of edges nd six times the numer of nodes. On the other hnd, the two-pss mthing method runs the lgorithm twie (ut on trimmed edge grph in the seond pss). Hene there is n overll improvement of effiieny of n order of mgnitude from templte mthing to two-pss mthing method. Note the templte mthing lgorithm works on ll input grphs with degree four or less, even if it hd odd numer of verties. Further, while the two-pss mthing method my not work lwys work on tetrhedrl meshes, templte mthing works on the dul of ll tetrhedrl meshes. Tles 1 nd 2 give more informtion out these lgorithms on vrious inputs. The experiments were performed in Pentium 4 t 2.4GHz running Linux, with NVidi Qudro4 980 XGL grphis rd nd 512M RM. Model Fes Finl %in. Nodl #edge time fes verts splits se. Sphere Trio lo Tle 1 Qudrilterl stripifition using two pss mthing lgorithm: Note the signifint numer of nodl vertex proessing tht merges yles without sudivision. The totl numer of yles remining fter nodl vertex proessing is one more thn the numer of edge splits. In spite of introduing four more qudrilterls for every split, note tht the perentge inrese is very smll. The result fter the sudivisions nd yle mergings is one single Hmiltonin qudrilterl strip tht trverses through ll the qudrilterls in the mesh. Model Tetr/Verts I II III IV V Trio 15310/ / Fndisk 22491/ / ll 430/ /4.9 <1 Spring 24359/ / lo 31526/ / Tle 2 Tetrhedrl stripifition using templte sustitution lgorithm: For eh tetrhedrl mesh, we show: Numer of tetrhedr, numer or verties, (I) Numer of nodl edges, (II) Numer of otined yles, (III) Numer of otined liner strips, (IV) % inrese in tetrhedr/verties nd (V) Running time in seonds. The inrese in numer of tetrhedr nd verties re due to strip merging y sudivision. Note the numer of new verties (whih equls the numer of sudivisions) is extly equl to the numer of liner strips minus one to merge ll the loops, plus one to merge this loop with one liner strip, if one exists. Further, the numer of new tetrhedr is four times the numer of new verties. 5 onlusion In this pper, we presented new lgorithm to find the 2-ftor of lss of grphs in O(n log 3 n log log n) running time. We lso presented novel lgorithms tht use this 2-ftor in the dul grph to rete qudrilterl nd tetrhedrl strips. The diretion of the strips found y ove methods depends on the results of the grph mthing lgorithm. s prt of future work, we would like to use the weighted grph mthing to glolly steer the strip in order to stisfy ertin properties like norml sed lustering. Similr tehniques were used for k-fe ulling nd trnsprent vertex hing [2, 5]. Further, we n use weighted perfet mthing to hoose the oundry tringles of the tetrhedrl mesh to fore externl mthing to redue the numer of internl tetrhedrl strips. We would lso like to develop out-of-ore rdinlity mthing grph lgorithms tht n e pplied on gignti grphs to hndle lrge models used in grphis nd visuliztion pplitions.

8 8 Plo Diz-Gutierrez, M. Gopi Fig. 11 Rendering of single-qud strip (left) nd the otined tetrhedrl strips of sphere model. Fig. 12 Rendering of used models: lo, Fndisk, Spring nd Trio. These originl tringulted models were qudrngulted y omining tringles. Further, they were tetrhedrlized y inserting few extr verties inside them to get high qulity tetrhedr. knowledgements We would like to thnk Hng Si, the uthor of TetGen, the progrm we used to tetrhedrlize our models. Referenes 1. sno, T., sno, T., Imi, H.: Prtitioning polygonl region into trpezoids. J. M 33(2), (1986) 2. ogomjkov,., Gotsmn,.: Universl rendering sequenes for trnsprent vertex hing of progressive meshes. omputer Grphis Forum 21(2), (2002) 3. ose, P., Toussint, G.T.: No qudrngultion is extremely odd. In: Int. Symp. lgorithms nd omputtion, pp (1995) 4. onn, H.E., O Rourke, J.: Minimum weight qudrilterliztion in O(n 3 log n) time. In: Pro. of the 28th llerton onferene on omm. ontrol nd omputing, pp (1990) 5. Diz-Gutierrez, P., hushn,., Gopi, M., Pjrol, R.: onstrined Strip Genertion nd Mngement for Effiient Intertive 3D Rendering. In: Pro. of omputer Grphis Interntionl onferene (2005) 6. Evns, F., Skien, S., Vrshney,.: Optimizing tringle strips for fst rendering. In: Proeedings IEEE Visuliztion 96, pp omputer Soiety Press (1996) 7. Gions,.: lgorithmi grph theory. mridge Univ. Press (1985) 8. Gopi, M., Eppstein, D.: Single strip tringultion of mnifolds with ritrry topology. omputer Grphis Forum (EUROGRPHIS) 23(3), (2004) 9. Heighwy, E.: mesh genertor for utomtilly sudividing irregulr polygons into qudrilterls. IEEE Trns. Mgnetis, 19(6), (1983) 10. King, D., Wittenrink,.M., Wolters, H.J.: n rhiteture for intertive tetrhedrl volume rendering. Teh. Rep. HPL (R.3), HP Lortories Plo lto (2001) 11. Mllon, P.N., oo, M., mor, M., ruguer, J.: ompression nd on the fly rendering using tetrhedrl onentri strips. Teh. rep., University of Sntigo de ompostel, Spin 12. Mukhopdhyy,., Jing, Q.: Enoding Qudrilterl Meshes. In: 15th ndin onferene on omputtionl Geometry (2003) 13. Pjrol, R., Rossign, J., Szymzk,.: Implnt sprys: ompression of progressive tetrhedrl mesh onnetivity. In: Proeedings IEEE Visuliztion 99, pp omputer Soiety Press (1999) 14. Pndurngn, G.: On Simple Rndomized lgorithm for Finding 2-Ftor in Sprse Grphs. Informtion Proessing Letters p. epted for pulition (2005) 15. Psui, V.: Isosurfe omputtion mde simple: Hrdwre elertion, dptive refinement nd tetrhedrl stripping. In: Joint EUROGRPHIS - IEEE TVG Symposium on Visuliztion (2004) 16. Peng, J., Kim,.S., Kuo,..J.: Tehnologies for 3d mesh ompression: survey. Teh. rep., Preprint (2005) 17. Peterson, J.P..: Die theorie der regulren grphs (The Theory of Regulr Grphs). t Mthemti 15, (1891) 18. Sommer, O., Ertl, T.: Geometry nd Rendering Optimiztion for the Intertive Visuliztion of rsh- Worthiness Simultions. In: Proeedings of the Visul Dt Explortion nd nlysis onferene in IT&mp;T/SPIE Eletroni Imging, pp (2000) 19. Szymzk,., Rossign, J.: Grow & Fold: ompression of Tetrhedrl Meshes. In: Fifth Symp. on Solid Modeling, pp (1999) 20. Tuin, G.: onstruting hmiltonin tringle strips on qudrilterl meshes. In: Int. Workshop on Visuliztion nd Mthemtis nd IM Reserh Teh. Rep. R (2002) 21. Thorup, M.: Ner-optiml fully-dynmi grph onnetivity. In: STO 00: Proeedings of the thirty-seond nnul M symposium on Theory of omputing, pp M Press, New York, NY, US (2000). DOI Ueng, S.K.: Out-of-ore Enoding Of Lrge Tetrhedrl Meshes. In: Volume Grphis, pp (2003) 23. Vneek, P., Svitk, R., Kolingerov, I., Skl, V.: Qudrilterl meshes stripifition. Teh. rep., University of West ohemi, zeh Repuli (2004) 24. Xing, X., Held, M., Mithell, J.S..: Fst nd effetive stripifition of polygonl surfe models. In: Pro. Symp. on Intertive 3D Grphis, pp M Press (1999)

9 Qudrilterl nd Tetrhedrl Mesh Stripifition Using 2-Ftor Prtitioning of the Dul Grph 9 Plo Diz-Gutierrez Plo Diz-Gutierrez is Ph.D. student in the Deprtment of omputer Siene t the University of liforni, Irvine. He got his M.S. t the University of liforni, Irvine in 2005 nd his.s. in omputer Siene t the University of Grnd, Spin, in He worked in geogrphi informtion systems in Mdrid, Spin, nd his urrent reserh interests inlude mesh proessing, omputtionl geometry nd fundmentl dt strutures. M.Gopi Gopi Meenkshisundrm (M. Gopi) is n ssistnt Professor in the Deprtment of omputer Siene t the University of liforni, Irvine. He got is Ph.D from the University of North rolin t hpel Hill in 2001, M.S. from the Indin Institute of Siene, nglore in 1995, nd.e from Thigrjr ollege of Engineering, Mduri, Indi in He hs worked on vrious geometri nd topologil prolems in omputer grphis. His urrent reserh interest fousses on pplying grph lgorithms to geometry proessing prolems in omputer grphis.