Intersection-free Dual Contouring on Uniform Grids: An Approach Based on Convex/Concave Analysis

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1 Intrsction-fr Dual Contouring on Uniform Grids: An Approach Basd on Convx/Concav Analysis Charli C. L. Wang Dpartmnt of Mchanical and Automation Enginring, Th Chins Univrsity of Hong Kong Figur 1. Intrsction-fr dual contouring in uniform grids: (a) triangls intrsct ach othr on th rsultant msh from dual contouring (DC) [2], and (b) our intrsction-fr dual contouring. 1. Introduction Th purpos of this rsarch is to invstigat a mor fficint approach than [1] for gnrating intrsctionfr iso-surfac from uniformly sampld grids using dual contouring (DC). Crack-fr msh surfac can b gnratd by DC, and DC is abl to rconstruct sharp faturs whn Hrmit data is availabl. Howvr, th surfac producd by Dual Contouring is rarly intrsction-fr. Th only work xisting in litratur to addrss th problm of gomtric intrsction on th msh surfac gnratd by DC is [1], which combins th primal and dual contouring. Som ncssary conditions hav bn drivd in [1] to try to rduc th numbr of triangls gnratd in DC. Howvr, although th conditions in [1] ar sufficint, th dtction involvs th dtction of polygon-dg intrsction which is nithr as robust nor as fficint as our nw approach basd on convx/concav dtction. Also, a qustion is lft at th nd of [1]: whthr it is possibl to apply Rul 1 along to an intrsction-fr msh surfac without introducing th dg vrtics. Our convx/concav analysis givs th ngativ answr for this qustion in most cass, it is not ncssary to introduc th dg vrtics so that w can hav an intrsction-fr msh surfac with th sam numbr of triangls as th original DC algorithm (rf.[2]) whn working on uniform grids. Howvr, thr ar som xtrm cass whr dg vrtics ar ndd. Traditional contouring tchniqus, such as th Marching Cubs (MC) mthod [3] and its variants [4-7], fulfill th intrsction-fr rquirmnt as thy gnrat triangls nclosd in grid clls and, within ach cll, th triangls ar not intrscting. Howvr, ths mthods typically produc much mor triangls than DC whn contouring th volumtric data sampld at th sam grid rsolution. In [1], Ju and Udshi borrowd th ida of nclosing triangls in non-ovrlapping volums (calld nvlops) from MC to a modifid DC approach that avoids potntial intrsctions. By thir mthod, intrsction-fr triangular msh surfacs can b gnratd but with mor triangls than th msh surfacs gnratd by th original DC approach [2]. Hr, w driv a convx/concav analysis basd approach, which is mor robust and fficint. Morovr, huristic 1

2 ruls ar dvlopd in this work for prsrving sharp-dgs bttr on th rsultant msh surfacs. Figur 1 shows a rsult of our approach vrsus th original DC. 2. Prliminary To b slf-containd, th principl of intrsction-fr DC will b brifd in this sction. Mor dtails can b found in [1, 2]. Basically, whn applying th dual contouring algorithm to volumtric data sampld on uniform grids, th isosurfac can b gnratd by linking vrtics in boundary grid clls. A grid cll with its ight grid nods hav inconsistnt insid (or outsid) configurations is a boundary grid cll. In ach boundary cll c, a vrtx v c on th rsultant msh surfac is cratd and locatd at th position minimizing th Quadratic Error Function (QEF) dfind by th sampls on th grid dgs of c, whr th sampls ar quippd with normal vctors (i.., Hrmit data). For ach cll dg th contains a sign chang with on nd insid but th othr outsid, two triangls will b constructd to linking four vrtics in th clls around it. Howvr, th msh surfac constructd in this way is rarly intrsction-fr. A hybrid approach of primal/dual mthods is dvlopd in [1]. As illustratd in Fig.2, a triangl fan with ight triangls is gnratd around a cll dg with sign chang in th hybrid approach, whr ach triangl conncts th dg vrtx v on, th fac vrtx v f on a fac f sharing and th cll vrtx v c in a cll c sharing f. Ju and Udshi thn provd that th hybrid approach gnrats intrsction-fr triangls if ach cll vrtx, fac vrtx and dg vrtx lis intrior to th corrsponding cll, fac or dg. Th nvlop E at a grid dg is dfind as th union of ight ttrahdra, ach formd by th dg, th fac vrtx v f and th cll vrtx v c (s Fig.2). It has provn that th nvlops E of diffrnt dgs ar disjoint, and th triangls gnratd by th hybrid mthod around ar containd in sparat ttrahdral of th nvlop E. Thrfor, th triangulatd surfac is guarantd to b intrsction-fr. v E v c v f v f v c v f (a) (b) Figur 2. Triangl fans gnratd by th hybrid mthod at a grid dg with sign chang (a), and th nvlop E of triangls formd by th union of ttrahdra (b). To rduc th numbr of triangls gnratd by th hybrid mthod, thr ruls ar dfind in [1] for contouring an Octr grid. Th ruls ar basd on th dtction of polygon-dg intrsctions. Howvr, whn contouring uniform grids, a simplr dtction mthod basd on convx/concav analysis can b dvlopd, which is th major contribution of our work blow. 3. Convx/Concav Condition for Intrsction-fr Triangulation A simplr convx/concav condition for intrsction-fr triangulation can b drivd. First of all, as long as all cll vrtics ar locatd in thir corrsponding cll boxs, th intrsction btwn th cll fac f pq of two nighboring clls c p and c q (containing cll vrtics and ) and th lin sgmnt - is always in th fac f pq whn contouring uniform grids. Thus, th fac vrtics in th hybrid mthod of [1] ar not ndd. By rmoving fac vrtics, th nvlop E around is thn formd by four ttrahdra. 2

3 Dfinition 1 Envlop E around an dg that dos not hav volum ovrlap with othr dg nvlops is formd by four ttrahdra: - --, - --, - -- and - --, whr,, and ar cll vrtics in clls c p, c q, c r and c s around, and and ar two vrtics at th nd of. Figur 3 givs an illustration of th four ttrahdra that form th intrsction-fr nvlop E around th dg. Not that th indics p, q, s and r ar circularly usd hr. E Figur 3. Four ttrahdra form th intrsction-fr nvlop E around th dg. Dfinition 2 For th cll vrtx v c forming th intrsction-fr nvlop E around an dg, if any of th dgs v c - or v c - is concav, it is dfind as a concav nvlop vrtx of E ; othrwis, it is namd as a convx nvlop vrtx. Rmark 1 Th dg - is concav if is blow th plan dfind by th orintd triangl - -, and th dg - is concav if is blow th plan dfin by th orintd triangl - -. Figur 4 givs an illustration for Rmark 1. Figur 4. Two cass that bcoms a concav nvlop vrtx: (lft) th dg - is concav and (right) th dg - is concav. Proposition 1 For two cll vrtics and in two non-nighboring clls c s and c q, th ttrahdron T sq formd by its prvious cll vrtx, its lattr cll vrtx and th dg can only intrsct thr of th four clls around. Proof: As illustratd in Fig.5, without loss of th gnrality, four clls ar locatd in th four rgions split by th x-o-z and y-o-z plans. If th ttrahdron T sq has intrsctd with c r and p r c r is a point insid T sq, a point in T sq must at th sam sid of plan -- as p r. Sinc th plan -- passing through th origin and z- axis which sparat th volums of c r and c p into two sid of it, it is impossibl to find a point from th cll c p at th sam sid of th plan as p r. By Rmark 1 and Proposition 1, w can conclud th following rmark sinc all points in c p will b abov th orintd plans - - and - - whn th ttrahdron T sq intrscts c r (s Fig.5). 3

4 Rmark 2 A cll vrtics in th cll c p will NOT b a concav nvlop dg if th ttrahdron T sq formd by its prvious cll vrtx, its lattr cll vrtx and th dg dos not intrsct th volum of cll c p. y z c q c p c q y T sq () x c r T sq c p x c r c s c s Figur 5. Th ttrahdron formd by two cll vrtics in diagonal clls and th dg can only intrsct thr of th four clls around. Proposition 2 For four cll vrtics,, and around an dg, if no concav nvlop vrtx is found, all th four triangls: - -, - -, - - and - - ar all nclosd by E. Proof: As shown in Fig.5, th vrtx is always blow th triangls: - -, - -, - - and - -, and th vrtx is always abov th triangls: - -, - -, - - and - -. Thrfor, th dgs -, -, - and - ar all convx. Sinc all dgs ar convx, th nvlop E is a convx hull. Linar combination of thr vrtics on th convx hull lads to a point insid E. Thus, th triangls - -, - -, - - and - - ar all nclosd by E. Proposition 3 For four cll vrtics,, and around an dg, if only is a concav nvlop vrtx, triangls - - and - - will b nclosd by th nvlop E, but th triangls - - and - - will hav som part NOT nclosd by E. Proof: If only is a concav nvlop vrtx and th dg - has no intrsction with th triangl - -, th dg - will b a convx dg for th rgion nclosd by two ttrahdra - -- and This rgion bcoms a convx hull. Thus th triangl - - is insid E. Th rgion formd by th ttrahdra and - -- can b split into othr two ttrahdra and - - -, which ncloss th triangl - -. If th dg - has an intrsction with th triangl - -, th dg - will b a convx dg for th rgion nclosd by two ttrahdra - -- and Thus, th rgion is a convx, and th triangl - - is insid. Th rgion formd by th ttrahdra - -- and - -- can b split into othr two ttrahdra and - - -, which ncloss th triangl - -. Sinc is a concav nvlop vrtx, th dg - is outsid E. Thrfor, th triangls - - and - - will hav part NOT nclosd by E. Rmark 3 For four cll vrtics,, and around an dg, only on or two vrtics can b concav nvlop vrtics, and ths two vrtics must b in th adjacnt clls. If is a concav nvlop vrtx, th ttrahdron T sq formd by, and th dg can only intrsct c p, c s and c q but not th cll c r (by Proposition 1). Thrfor, cannot b a concav nvlop vrtx (by Rmark 2). Also, if is a concav nvlop vrtx, cannot b a concav nvlop vrtx. Rmark 4 if two vrtics ar concav nvlop vrtics, non of th triangls - -, - -, - - and - - is nclosd by E. 4

5 If is a concav nvlop vrtx (or is a concav nvlop vrtx), th dg - is outsid E ; and th dg - is outsid E whn (or ) is outsid E. Figur 6 shows an xampl for that. If two vrtics ar concav nvlop vrtics, w hav to furthr split thn into four triangls by adding an dg vrtx v on to nsur that E ncloss all th triangls. v Figur 6. An xampl with two concav nvlop vrtics: (lft) location of vrtics around dg and (right) th triangulation by insrting an dg vrtx v. Analysis of Robust and Efficint Implmntation: Our convx/concav analysis basd approach is mor robust and fficint comparing th intrsction-fr approach prsntd in [1], which is basd on th dgpolygon intrsction tsts. In gnral, 4 to 6 tsts of dg-triangl intrsction ar ndd for th triangls around a sign chang dg. By th fastst algorithm of dg-triangl intrsction tst in litratur (rf. [8]), ach dg-triangl intrsction tst nds 3 cross-product and 3 (or 4) dot-products. Thrfor, totally 18 cross-product and 24 dot-products ar rquird in th worst cas. In our convx/concav analysis, w only nd to dtct whthr th four vrtics ar concav nvlop vrtics. Spcifically, whn dtcting whthr a vrtx vp is blow th orintd triangl - - can b fficintly dtctd by chcking if th scalar tripl product blow satisfis ( ) (( - ) ( - )) < 0. In th worst cas, 8 cross-products and 8 dot-products ar ndd. Whn computing scalar tripl product by dtrmination, th computation can b furthr rducd. Morovr, whn find is a concav nvlop vrtx, th dtction of can b nglctd (by Rmark 3). Th dtction on and can b simplifid in th sam way. In th aspct of robustnss, th dtction basd on dg-polygon intrsction tst is suffrd by th xtrm cass (.g., th intrsction occurs on th dg or vrtx of th triangl). Obviously, our dtction basd on convx/concav analysis dos not hav such problm. Figur 7. Sharp fatur prsrvation: (a) th rsult without applying th huristic ruls and (b) th rsult aftr applying th huristic ruls for sharp fatur prsrvation. 5

6 4. Huristic Ruls for Sharp-Edg Prsrvation Th position of a vrtx v c in a cll c is dtrmind by minimizing th Quadratic Error Function (QEF) dfind by th Hrmit sampls on th grid dgs of c. Although this can mov vrtics on th rsultant msh surfac to a position on th sharp faturs, th subsqunt triangulation on four vrtics,, and around th dg may dstroy th sharp dgs (s Fig.7(a) as an xampl). Som huristic ruls ar dvlopd in this sction to prsrv sharp-dgs whn triangulating th vrtics,, and around th dg with sign chang. First of all, whthr a vrtx v c is on (or nar) sharp fatur is dtctd by th normal voting tnsor constructd by th normal vctors of Hrmit sampls on th grid dgs of th cll c. For all Hrmit sampls (p i,n i ) (i=0,,m), th normal voting tnsor is F n = n i n i T which is a 3 3 matrix. Aftr computing th ign-valus ( λ 1 λ 2 λ 3 0) of th normal voting tnsor F n, if thr ar mor than two non-zro ign-valus, th vrtx v c is classifid into a vrtx nar sharp faturs namd as sharp-fatur vrtx. Idally, if v c is on a cornr, non of th ign-valus will b zro; whn v c is on a sharp dg, two ign-valus will b non-zro s th analysis in [9]. To b robust, w considr λ 1 <ε as a zro ign-valu with ε = According to obsrvation, w conclud th following ruls for triangulation four cll vrtics,, and around an dg with sign chang. Rul 1a Whn triangulating four cll vrtics,, and around an dg with sign chang, if only on vrtx among thm is sharp-fatur vrtx, th diagonal of triangulation must link to th sharp-fatur vrtx. Rul 1b Whn thr ar two sharp-fatur vrtics and thy do not blong to adjacnt clls, th diagonal of triangulation must link to ths two sharp-fatur vrtics. Rul 1c Whn thr ar thr sharp-fatur vrtics, th diagonal of triangulation must link thos two sharpfatur vrtics that ar not in th adjacnt clls. Figur 8 shows an illustration for ths ruls. By applying ths ruls, sharp faturs ar bttr prsrvd on th rsultant msh surfacs Fig.7(b) shows an improvd sharp dg on th rsultant msh surfac. As th intrsction-fr triangulation has highr priority than prsrving sharp faturs, ths ruls ar only applid whn non of th four cll vrtics,, and ar concav nvlop vrtics. Othrwis, th triangulation is dtrmind by Proposition 3. Figur 8. Illustration for th thr ruls of sharp fatur prsrvation, whr th vrtics in rd color ar sharp-fatur vrtics. 4. Discussion Th major drawback of this approach is that th sharp faturs ar not wll prsrvd aftr adding th constraints for nclosing th vrtics and triangls in thir corrsponding nvlop (spcially whn th rsolution is low). For xampl th modl in Fig.9, whn not constraining th vrtics and facs in thir corrsponding nvlops, th sharp faturs ar wll rconstructd vn whn th rsolution is 128 x 128 x 6

7 128. Aftr adding th intrsction-fr constraints, vn in th rsolution of 512 x 512 x 512 th thin-sharpfaturs do still mbd aliasing. Adaptiv sampling and contouring can b mployd to improv th quality of rconstructd shin-sharp faturs on th rsultant msh surfacs. Figur 9. Th thin-sharp faturs ar not wll prsrvd by adding th intrsction-fr constraints: (top-row) without th intrsction-fr constraints and (bottom-row) with th intrsction-fr constraints. Th rsults ar shown in diffrnt rsolutions (lft-column) 128 x 128 x 128, (middl-column) 256 x 256 x 256, and (rightcolumn) 512 x 512 x 512. Rfrncs: [1] T. Ju, and T. Udshi, Intrsction-fr contouring on an octr grid, Proc. of Pacific Graphics 2006, IEEE Computr Socity. [2] T. Ju, F. Losasso, S. Schafr, and J. Warrn, Dual contouring of hrmit data, Prof. of SIGGRAPH 2002, pp , [3] W.E. Lornsn, and H. E. Clin, Marching cubs: a high rsolution 3d surfac construction algorithm, Proc. of SIGGRAPH 87, pp , Anahim, California, July [4] J. Bloomnthal, Polygonization of implicit surfacs, Computr Aidd Gomtric Dsign, vol.5, no.4, pp , [5] G.M. Nilson, and B. Hamann, Th asymptotic dcidr: rsolving th ambiguity in marching cubs, Proc. of IEEE Visualization 1991, pp.83-93, [6] C. Andujar, P. Brunt P, A. Chica, I. Navazo, J. Rossignac, and A. Vinacua, Optimizing th topological and combinatorial complxity of isosurfacs, Computr-Aidd Dsign, vol.37, pp , [7] L.P. Kobblt, M. Botsch, U. Schwanck, and H.-P. Sidl, Fatur-snsitiv surfac xtraction from volum data, Proc. of SIGGRAPH 2001, pp.57-66, August [8] N. Chirkov, Fast 3D lin sgmnt triangl intrsction tst, Journal of Graphics, GPU, & Gam Tools, vol.10, no.3, pp.13-18, [9] H.S. Kim, H.K. Choi, and K.H. L, Fatur dtction of triangular mshs basd on tnsor voting thory, Computr-Aidd Dsign, vol.41, pp.47-58,