Computer Aded Geometrc Desgn 23 (2006) 125 145 www.elsever.com/locate/cagd Dscrete surface modellng usng partal dfferental equatons Guolang Xu a,1,qngpan a, Chandrajt L. Bajaj b,,2 a State Key Laboratory of Scentfc and Engneerng Computng, Insttute of Computatonal Mathematcs, Academy of Mathematcs and System Scences, Chnese Academy of Scences, Bejng, 100080, Chna b Center for Computatonal Vsualzaton and Insttute for Computatonal Engneerng & Scences, Department of Computer Scence, Unversty of Texas, Austn, TX 78712, USA Receved 29 July 2004; receved n revsed form 15 May 2005; accepted 23 May 2005 Avalable onlne 5 July 2005 Abstract We use varous nonlnear partal dfferental equatons to effcently solve several surface modellng problems, ncludng surface blendng, N-sded hole fllng and free-form surface fttng. The nonlnear equatons used nclude two second order flows, two fourth order flows and two sxth order flows. These nonlnear equatons are dscretzed based on dscrete dfferental geometry operators. The proposed approach s smple, effcent and gves very desrable results, for a range of surface models, possbly havng sharp creases and corners. 2005 Elsever B.V. All rghts reserved. Keywords: Free-form surface fttng; Blendng; N-sded hole; Dfferental equatons 1. Introducton We use varous partal dfferental equatons (PDE) to solve several surface modellng problems. The PDEs we use nclude the mean curvature flow, the averaged mean curvature flow, two fourth order (surface dffuson flow and quas surface dffuson flow) and even hgher order flows. All these equatons are * Correspondng author. E-mal address: bajaj@cs.utexas.edu (C.L. Bajaj). 1 Supported n part by Natural Scence Foundaton of Chna (10371130) and Natonal Key Basc Research Project of Chna (2004CB318000). 2 Supported n part by NSF grant ACI-9982297, NSF-ITR grants ACI-0220037 and EIA-03-25550, and a grant from NIH R01 GM074258-021. 0167-8396/$ see front matter 2005 Elsever B.V. All rghts reserved. do:10.1016/j.cagd.2005.05.004
126 G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 nonlnear and the geometry s ntrnsc,.e., the PDEs do not depend upon any partcular parameterzaton. The problems we solve nclude surface blendng, N-sded hole fllng and free-form surface fttng wth hgh order boundary contnuty. For the problems of surface blendng and N-sded hole fllng, we are gven trangular surface meshes of the surroundng area. Trangular surface patches need to be constructed to fll the openngs enclosed by the surroundng surface mesh and nterpolate the hole boundary wth some specfed order of contnuty. For the free-form surface fttng problem, we are possbly gven a set of ponts, or a wre frame of curves that defnes an outlne of the desred shape, or even some surface patches. We construct a surface whch nterpolates the ponts or curves or the boundares of the patches wth specfed order of contnuty. The free-form surface fttng problem s the most general, ncludng the surface blendng and N-sded hole fllng problems, as ts specal cases. Our twofold strategy for solvng these problems s as follows: Frst we construct an ntal trangular surface mesh ( fller ) usng any of a number of automatc or sem-automatc free-form modellng technques (see (Bajaj and Ihm, 1992; Bajaj et al., 1993; Grener, 1994; Peters and Wttman, 1996; Xu et al., 2001)). One may also nteractvely edt ths fller to meet the weak assumptons for an ntal soluton shape. Ths fller may be bumpy or nosy, and n general ths fller does not satsfy the smoothness boundary condtons, though t may roughly characterze the shape of the surface to be constructed. Second we deform the ntal mesh by solvng a sutable flow PDE. Unlke most of the prevous free-form modellng technques, our approach solves hgh-order boundary contnuty constrants wthout any pror estmaton of normals or dervatve jets along the boundary. The soluton of the PDE s tme dependent. We consder two possbltes for the tme span of the evoluton. One s a short tme evoluton, where we requre the soluton to respect to the ntal shape or geometry (see Fg. 7). The other s a long tme evoluton, where the ntal fller provdes a topologcal structure, and what we look for s a stable soluton state of the flow (see Fgs. 1 and 4). In ths paper, we focus our attenton on these twofold solutons of PDEs wth boundary contnuty constrants, rather than the constructon of ntal fller mesh. In Secton 3.4, we present automatc approaches for constructng the ntal fller mesh, and our preferred choce. Prevous work. Earler research on usng PDEs to handle surface modellng problems trace back to Bloor et al. s papers at the end of the 1980s (Bloor and Wlson, 1989, 1990). The basc dea of these Fg. 1. (a) shows a head mesh wth a hole around the nose. (b) shows an ntal fller constructon of the nose wth a pece of mnmal surface. (c) the fller surface, after 30 teraton, generated usng fourth order flow (k = 2 n (2.9)) wth tme step sze 0.0002. (d) the fller surface, after 20 teraton, generated usng sxth order flow (k = 3 n (2.9)) wth tme step sze 0.00002.
G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 127 papers s the use of bharmonc equatons on a rectangular doman to solve the blendng and hole fllng problems. One of the advantages of usng the bharmonc equaton s that t s lnear, and therefore easer to solve. However, the equaton s not geometry ntrnsc and the soluton of the equaton (the geometry of the surface) depends on the concrete parameterzaton used. Furthermore, these methods are napproprate to model surfaces wth arbtrary shaped boundares. The evoluton technque, based on the heat equaton t p p = 0, has been extensvely used n the area of mage processng (see (Preußer and Rumpf, 1999; Weckert, 1998). In (Weckert, 1998), there are 453 relevant references lsted), where s a 2D Laplace operator. Ths was extended lately to smoothng or farng nosy surfaces (see (Clarenz et al., 2000; Desbrun et al., 1999; Meyer et al., 2002)). For a surface M, the counterpart of the Laplacan s the Laplace Beltram operator M (see (do Carmo, 1992)). One then obtans the geometrc dffuson equaton t p M p = 0 for a surface pont p(t) on the surface. Taubn (1995) dscussed the dscretzed operator of the Laplacan and related approaches n the context of generalzed frequences on meshes. Kobbelt (1996) consdered dscrete approxmatons of the Laplacan n the constructon of far nterpolatory subdvson schemes. Ths work was extended n (Kobbelt et al., 1998) to arbtrary connectvty for purposes of mult-resoluton nteractve edtng. Desbrun et al. (1999) used an mplct dscretzaton of geometrc dffuson to obtan a strongly stable numercal smoothng scheme. The same strategy of dscretzaton s also adopted and analyzed by Deckelnck and Dzuk (2002) wth the concluson that ths scheme s uncondtonally stable. Clarenz et al. (2000) ntroduced ansotropc geometrc dffuson to enhance features whle smoothng. Ohtake et al. (2000) combned an nner farness mechansm n ther farng process to ncrease the mesh regularty. Bajaj and Xu (2003) smooth both surfaces and functons on surfaces, n a C 2 smooth functon space defned by the lmt of trangular subdvson surfaces (quartc Box splnes). Smlar to the surface dffuson usng the Laplacan, a more general class of PDE based methods called flow surface technques have been developed whch smulate dfferent knds of flows on surfaces (see (Westermann et al., 2000) for references) usng the equaton t p V(p,t)= 0, where V(p,t)represents the nstantaneous statonary velocty feld. Level set methods were also used n surface farng and surface reconstructon (see (Bajaj et al., 2003; Bertalmo et al., 2000; Chopp and Sethan, 1999; Museth et al., 2002; Osher and Fedkw, 2000; Whtaker and Breen, 1998; Zhao et al., 2000)). In these methods, surfaces are formulated as so-surfaces (level surfaces) of 3D functons, whch are usually defned from the sgned dstance over Cartesan grds of a volume. An evoluton PDE on the volume governs the behavor of the level surface. These levelset methods have several attractve features ncludng, ease of mplementaton, arbtrary topology (see (Breen and Whtaker, 2001)) and a growng body of theoretcal results. Often, fne surface structures are not captured by level sets, although t s possble to use adaptve (see (Preußer and Rumpf, 1999)) and trangulated grds as well as Hermte data (see (Kobbelt et al., 2001)). To reduce the computatonal complexty, Bertalmo et al. (2000) solve the PDE n a narrow band for deformng vectoral functons on surfaces (wth a fxed surface represented by the level surface). Recently, surface dffuson flow has been used to solve the surface blendng problem and free-form surface fttng problem (Schneder and Kobbelt, 2000; Schneder and Kobbelt, 2001). In (Schneder and Kobbelt, 2000), far meshes wth G 1 condtons are created n the specal case where the meshes are assumed to have subdvson connectvty. In ths paper, local surface parameterzaton s stll used to estmate the surface curvatures. The later paper (Schneder and Kobbelt, 2001) uses the same equaton for smoothng meshes whle satsfyng G 1 boundary condtons. Outer farness (the smoothness n the classcal sense) and nner farness (the regularty of the vertex dstrbuton) crtera are used n ther farng
128 G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 process. The fnte element method s used by Clarenz et al. (2004) to solve the Wllmore flow equaton, based on a new varatonal formulaton of the flow, for the am of surface restoraton. Wllmore flow s also used to smooth trangular mesh n (Yoshzawa and Belyaev, 2002). Man results. We use second order flows (mean curvature flow and averaged mean curvature flow) for G 0 contnuty, fourth order flows for G 1 contnuty and sxth order flows for G 2 contnuty n each of several surface modellng problems. The proposed approach s smple and easy to mplement. It s general, solves several surface modellng problems n the same manner, and gves very desrable results for a range of complcated free-form surface models, possbly havng sharp features and corners. Furthermore, t avods the estmaton of normals or tangents or curvatures on the boundares. The rest of the paper s organzed as follows: Secton 2 descrbes several nonlnear PDEs used n ths paper. In Secton 3, we gve detals of the dscretzaton and the numercal computaton for the solutons of the PDEs. Examples to llustrate the dfferent effects achevable from the soluton of the PDEs are gvennsecton4. 2. Partal dfferental equaton models Let M be a smooth surface and p M be the surface pont. The general form of the geometrc flows we consder s n the followng form (see (Westermann et al., 2000)) p t = V(p,t), where V(p,t) R 3 represents a velocty feld. We shall focus our attenton on usng two classes velocty felds, one s curvature drven velocty feld n the normal drecton, the other s the hgher order Laplace Beltram operators actng on surface pont p. 2.1. Geometrc partal dfferental equatons We now descrbe several geometrc PDE models we use n ths paper. More detals on the exstence and unqueness of the solutons, the numercal computatons of the solutons and evoluton behavors can be found n a seres of papers by Mayer, Smonett, Escher (Escher et al., 1998; Escher and Smonett, 1998; Smonett, 2001) and Huskens (1987) paper. Let M 0 be a compact closed mmersed orentable surface n R 3. A curvature drven geometrc evoluton conssts of fndng a famly {: t 0} of smooth closed mmersed orentable surfaces n R 3 whch evolve accordng to the flow equaton p t = N(p)V n(k 1,k 2,p), M(0) = M 0. (2.1) Here p(t) s a surface pont on, V n (k 1,k 2,p)denotes the normal velocty of, whch depends on the prncpal curvatures k 1,k 2 of, N(p) stands for the unt normal of the surface at p(t). In ths paper we dentfy the surface pont p and surface normal N(p) as 3 1 matrces (column vectors). Hence, the arthmetc operatons of these quanttes are regarded matrx operatons. The product of a scalar a R and a matrx M s wrtten as ether am or Ma.
G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 129 Let A(t) denote the area of, V(t)denote the volume of the regon enclosed by. Then t has been shown that (see (Wllmore, 1993, Theorem 4)) da(t) dv(t) = V n H dσ, = V n dσ, (2.2) dt dt where H = 1 2 (k 1 + k 2 ) s the mean curvature of. 2.1.1. Mean curvature flow (see (Dzuk, 1991; Whte, 2002)) Takng V n = H = 1 2 (k 1 + k 2 ) n (2.1), we obtan the mean curvature flow PDE: p t = N(p)H(p), M(0) = M 0. (2.3) It follows from (2.2) that da(t) = dt H 2 dσ. (2.4) (2.4) mples that the mean curvature flow s area reducng. 2.1.2. Averaged mean curvature flow (see (Escher and Smonett, 1998; Huskens, 1987; Sapro, 2001)) In (2.1), f we take V n = h(t) H(t),whereh(t) = H dσ/ dσ, then we have the averaged mean curvature flow PDE: p t = N(p)[ h(t) H(p) ], M(0) = M 0. (2.5) The exstence proof of the global solutons to ths flow can be found n Huskens (1987) paper. It follows from (2.2) that da(t) [ = (hh H 2 ) dσ = hh H 2 h(h H) ] dσ = (h H) 2 dσ 0, (2.6) dt snce obvously h(h H)= h(h dσ H dσ)= 0. On the other hand, the second equaton of (2.2) mples that dv(t) = h(t) dσ H dσ = 0. dt Hence the averaged mean curvature flow s volume preservng and area shrnkng. The area shrnkng stops f H h. 2.1.3. Surface dffuson flow (see (Schneder and Kobbelt, 2001)) If we take V n = H, we get the so-called surface dffuson flow PDE: p t = N(p) H (p), M(0) = M 0, (2.7)
130 G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 where := M s Laplace Beltram operator whch acts on functons defned on surface. The exstence and unqueness of solutons for ths flow s gven n (Escher et al., 1998). From (2.2) and Green s formula we have d dt A(t) = H H dσ = H 2 dσ 0, d dt V(t)= dv( H)dσ = H (1) dσ = 0, where stands for the (tangental) gradent operator (see (do Carmo, 1976, pp. 101 102)) actng on dfferental functons defned on the surface M. Hence, the surface dffuson flow s area shrnkng, but volume preservng. The area stops shrnkng when the gradent of H s zero. That s, M s a surface wth constant mean curvature. 2.1.4. Hgher order geometrc flows p t = ( 1)k+1 N(p) k H(p), M(0) = M 0. (2.8) Usng Green formula, we have k H dσ = ( k 1 H)dσ = ( k 1 H) (1) dσ = 0. Hence, the flow (2.8) s volume preservng f k 1 from the second equaton of (2.2). Remark 2.1. We should note that the area/volume preservng/shrnkng propertes for the flows mentoned above are vald for closed surfaces. In our applcaton of these flows, these propertes may not be true snce the surfaces always have fxed boundares. For a open surface wth fx boundary, the volume V(t) could be defned as the drectonal volume between M(0) and. It s easy to see that the volume preservng property for the averaged mean curvature flow s stll vald. But for the hgher order flow (2.8) (k 1), ths property s no longer vald, because a term related to the boundary does not vansh when Green s formula s used. For our modellng problems, volume preservaton s not a desrable property (see Fgs. 1 and 4). Remark 2.2. In (Schneder and Kobbelt, 2001), Schneder and Kobbelt use ellptc equaton N(p) H(p) = 0, whle we use several tme dependent parabolc type equatons. In our approach, we have a progressve process startng from an ntal value, so that a famly of solutons s obtaned. Such an approach s very desrable f the ntal value s an approxmaton of the requred soluton. 2.2. Quas geometrc partal dfferental equatons Now we generalze the heat equaton on a surface to the followng hgher order flows: p t = ( 1)k+1 k p, M(0) = M 0, k>0. (2.9)
G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 131 Snce p = 2H(x)N(p), t s easy to see that (2.9) s the mean curvature flow when k = 1 (up to a factor 2). But snce ( k H)N k (H N) n general, (2.9) s dfferent from the flow (2.8). To dstngush the dfference between (2.8) and (2.9), we call (2.9) as a quas geometrc PDE. The experments conducted n ths paper show that flows (2.9) sometmes behave better than the geometrc flows mentoned above for our geometry modellng problems. However, the theoretcal analyss on the exstence and stablty of ther solutons s currently unavalable. 3. Soluton of the PDEs There are bascally two classes of approaches for solvng a PDE on any doman. One approach s based on fnte dvded dfferences, the other s based on fnte elements (see (Bajaj and Xu, 2003; Clarenz et al., 2004; Deckelnck and Dzuk, 2002)). The approach we adopt n ths paper s based on fnte dvded dfferences. Snce we are dealng wth dfferental equatons over 2-manfolds n R 3, the classcal fnte dvded dfferences wll be replaced by dscretzed dfferental geometrc operators over surfaces. Secton 3.1 deals wth dscretzed geometrc dfferental operators. Next n Secton 3.2 we detal how the boundary condtons are respected. Dscretzatons of the PDEs n the spatal drecton are descrbed n Sectons 3.3 and 3.4. Sem-mplct dscretzaton n the tme doman s consdered n Secton 3.5. Other ssues, such as mesh regularzaton and ntal mesh constructon, are addressed n Secton 3.6. 3.1. Dscretzed Laplace Beltram operator One of the fundamental problems n solvng our PDEs s the dscretzaton of the Laplace Beltram operator. On a trangular surface mesh, several dscretzed approxmatons of the operator have been proposed (see (Desbrun et al., 1999; Guskov et al., 1999; Taubn, 2000; Xu, 2004b)). In ths paper we adopt the dscretzaton developed by Meyer et al. n (Meyer et al., 2002). A comparatve research about the varous dscretzed Laplace Beltram operators s conducted n (Xu, 2004a). It has been shown that the scheme of Meyer et al. s s better for dscretzng our PDEs. Let f be a smooth functon on a surface, then f s approxmated over a trangular mesh M by f (p ) 1 A M (p ) j N 1 () cot α j + cot β j 2 [ f(pj ) f(p ) ], (3.1) where N 1 () s the ndex set of 1-rng of neghbor vertces of vertex p, α j and β j are the trangle angles shown n Fg. 2 (left). A M (p ) s the area for vertex p as shown n Fg. 2 (rght), where q j s the crcumcenter pont for the trangle [p j 1 p j p ] f the trangle s non-obtuse. If the trangle s obtuse, q j s chosen to be the mdpont of the edge opposte to the obtuse angle. Snce p = 2H(p)N(p) (see (Wllmore, 1993, p. 151)), we have ( p) p=p = 2H(p )N(p ) 1 A M (p ) j N 1 () cot α j + cot β j 2 (p j p ). (3.2) Ths gves an approxmaton of the mean curvature normal (see (Meyer et al., 2002)). The hgher order Laplace Beltram operators are dscretzed recursvely as k f(p ) = ( k 1 1 cot α j + cot β j [ f )(p ) = k 1 f(p j ) k 1 f(p ) ] (3.3) A M (p ) 2 j N 1 ()
132 G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 Fg. 2. Left: The defnton of the angles α j and β j. Rght: The defnton of the area A M (p ). Fg. 3. Left: The nvolved vertces of the outer mesh for a G 0 boundary condton. The outer mesh s just the boundary of the hole. Mddle: The nvolved vertces of the outer mesh for a G 1 boundary condton. Rght: The nvolved vertces of the outer mesh for a G 2 boundary condton. wth 0 f(p ) = f(p ). Note that k f(p ) nvolves functon values on a k-rng of neghborng vertces of p. 3.2. Handlng of boundary condtons 3.2.1. Natural boundary condtons for blendng and hole fllng By the natural boundary condtons, we mean that no contnuty condtons are specfed at the boundary ponts, but the contnuty s mpled by the outer mesh ncdent to the boundary of the hole (see Fg. 3). Such a treatment for boundary condton s sutable for both the blendng problem and the N-sded hole fllng problem, snce the outer mesh always exsts n such problems. Let g be the order of contnuty at a boundary pont p, g = max g. Then we can use the order 2g contnuty at the boundary vertex p. g H s dscretzed recursvely: g H = ( g 1 H). At a boundary vertex p, k H(p ) s evaluated accordng to the followng rule: flow p t = ( 1) g+1 g H(p)N(p) for constructng the trangular surface patch wth G g Evaluaton rule at boundary. k H(p ) s evaluated recursvely by formulas (3.6) and (3.7) f k g, otherwse k H(p ) s set to zero and the recurson stops.
G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 133 Note that even for an nner vertex p j, the recursve defnton may make k H(p j ) nvolve the evaluaton of a lower order Laplace Beltram operator on the boundary. In general, the recursve evaluaton of k H(p ) at p (for p ether beng an nner or an outer vertex) nvolves (k + 1)-rng neghbor vertces of p. Some of them may be nner vertces, and the remanng are outer vertces. The nner vertces are treated as unknowns n the dscretzed equatons and the outers are ncorporated nto the rght-hand sde. 3.2.2. Natural boundary condtons for free-form surface fllng In the free-form surface fllng problem, we are gven a wreframe of curves (edges) and we wsh to flesh the wreframe wth surface patches that contan the curves as boundary wth pre-specfed order of contnuty. At each of the ntersecton ponts of the patches, an order of contnuty s pre-specfed and the evaluaton rule mentoned above s appled. For each nner pont, a dscretzed lnear equaton s generated usng the operator dscretzaton (3.7). These lnear equatons for dfferent patches are collected together and solved smultaneously. Note that one lnear equaton may nvolve nner vertces of several patches. However, f the contnuty order at each boundary pont s zero, any equaton correspondng to an nner vertex does not nvolve nner vertces of other patches. Remark 3.1. Schneder and Kobbelt (2001) use Moreton and Sequn s least square fttng of the second fundamental form relatve to a local parameterzaton to estmate the requred data on the boundary. These estmatons of the boundary dervatve data are based on ncomplete nformaton. Hence, the estmated data maybe not relable. Our approach s based on the dentty M p = 2H(p)N(p). Hence, we do not need to estmate boundary dervatve data, such as normals, tangents or curvatures. Furthermore, the boundary condtons are treated n the same way for equatons wth dfferent orders. 3.3. Spatal dscretzaton of quas geometrc flows Let us consder frst the dscretzaton of (2.9) n the spatal drecton for k = 1, 2, 3. Let P = [p 1,...,p m ] T R m 3, P =[ p 1,..., p m ] T R m 3,wherep 1,...,p m are all the unknown vertces to be determned n each of our modellng problems. Then (3.2) could be wrtten n matrx form: P = (DW)P + B (1), (3.4) 1 where D = dag[ 2A(p 1 ),..., 1 2A(p m ) ] s a dagonal matrx, W ={w j } m,j=1 wth k N 1 () cot α k + cot β k, = j, w j = (cot α j + cot β j ), j, N 1 (j), j N 1 (), 0, otherwse. Furthermore, W s a sparse, symmetrc and postve defnte matrx (see (Schneder and Kobbelt, 2001)). The constant term B (1) R m 3 s obtaned from the boundary condtons. It follows from (3.4) that 2 P = (DWDW)P + B (2), (3.5) where B (2) R m 3 s obtaned from the boundary condtons. Agan, WDW s a sparse, symmetrc and postve defnte matrx. In general, k P = ( 1) k (DW) k P + B (k), and the matrx for D 1 (DW) k s also sparse, symmetrc and postve defnte.
134 G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 3.4. Spatal drecton dscretzaton of geometrc flows Let cot α k +cot β k k N 1 () 2A M (p, = j, ) ω j = cot α j +cot β j 2A M (p, j, N ) 1 (j), j N 1 (), 0, otherwse, and N()= N 1 () {}. Then we have N(p )H (p ) 1 ω j p j. 2 j N() The hgher order Laplace Beltram operators actng on H are dscretzed recursvely as k H(p ) = ( k 1 H )(p ) ω j k 1 H(p j ) (3.7) wth 0 H(p ) = H(p ) 1 2 j N() j N() (3.6) ω j N(p ) T p j. (3.8) Note that k H(p ) nvolves values of the mean curvature on a k-rng of neghborng vertces of p. Usng (3.6) (3.8) and the evaluaton rule at the boundary, we can wrte N(p ) k H(p ) as the followng form: N(p ) k H(p ) ( 1) k ω (k) j p j + B (k), ω (k) j R 3 3,B (k) R 3, j J 0 where J 0 s the ndex set of the (unknown) vertces to be determned, B (k) comes from boundary condton. To be more specfc, let J denote the ndex set of the mesh M, J k be the unon of J 0 and the ndex set of the boundary vertces where C k condton s specfed. Then N(p )H (p ) 1 ω j p j = 1 ω j p j + 1 ω j p j 2 2 2 j N() j N() J 0 j N() {J \J 0 } = j J 0 ω (0) j p j + B (0), (3.9) where ω (0) j = 1 2 ω j I 3 for j N() J 0, ω (0) j = 0otherwse,B (0) = 1 2 j N() {J \J 0 } ω j p j. Smlarly, N(p ) H (p ) N(p ) ω j H(p j ) = N(p ) ω j H(p j ) j N() j N() J 1 = N(p ) ω j N(p j ) T N(p j )H (p j ) j N() J 1 [ ] ω j N(p )N(p j ) T m (0) jk p k + B (0) j j N() J 1 k J 0
G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 135 = ω (1) j p j + B (1), (3.10) j J 0 N(p ) 2 H(p ) N(p ) ω j H (p j ) = N(p ) ω j H (p j ) j N() j N() J 2 N(p ) ω j ω jk H(p k ) j N() J 2 k N(j) J 1 [ ] ω j ω jk N(p )N(p k ) T m (0) kl p l + B (0) k k N(j) J 1 l J 0 j N() J 2 = j J 0 ω (2) j p j + B (2). (3.11) (3.9) (3.11) are used to dscretze the rght-handed sde of (2.8) for k = 0, 1, 2. The dscretzaton of N(p ) k H(p ) for k>2 s recursvely calculated usng (3.7) and boundary condtons. 3.5. Tme dscretzaton Gven an approxmate soluton {p (n) } m =1 of the order 2k PDE at t n for all the nner vertces, we construct an approxmate soluton {p (n+1) } m =1 for the next tme step t n+1 = t n + τ (n) by usng a sem-mplct Euler scheme. That s, we replace the dervatve p wth [p(t t n+1 ) p(t n )]/τ (n), and the quanttes w j n (3.4), ω j and N(p ) n (3.6) (3.8), h(t) n (2.5) are computed usng the prevous result at t n.normalsn(p ) are computed from Loop s subdvson surface (see (Bajaj and Xu, 2003) for detal). Such a treatment yelds a lnear system of equatons wth the nner vertces as unknowns. Let P (n+1) =[(p (n+1) 1 ) T,...,(p m (n+1) ) T ] T R 3m. The lnear system for the geometrc flows can be wrtten as the matrx form [ I + τ (n) W (k)] P (n+1) = B (k), W (k) = { ω (k) } j, B (k) R 3m. (3.12) The matrx W (k) R 3m 3m s hghly sparse, hence an teratve method for solvng such a lnear system s desrable. We use Saad s teratve method (Saad, 2000), named GMRES, to solve the system. The experment shows that ths teratve method works very well. Let P (n+1) =[p (n+1) 1,...,p m (n+1) ] T R m 3. The lnear system for the flows (2.9) can be wrtten as the matrx form [ I + τ (n) (DW) k+1] P (n+1) = B (k), or W (k) P (n+1) = D 1 B (k) (3.13) where B (k) R m 3, W (k) = D 1 + τ (n) W(DW) k R m m s a hghly sparse, symmetrc and postve defnte matrx, and hence we use a conjugate gradent teratve method wth dagonal precondtonng to solve the system. Note that for the same sze problem, the sze of coeffcent matrx n (3.12) s three tmes larger than that of coeffcent matrx n (3.13). Furthermore, the matrx W (k) n (3.13) s symmetrc and postve defnte. The matrx n (3.12) s not. We also note that the dscretzaton of (2.9) does not nvolve the computaton of the surface normals. Remark 3.2. It s well known that the condton of the lnear system arsng from the proposed semmplct dscretzaton behaves lke O(1 + τ (n) h 2k ),whereh s the mnmal edge length of the mesh.
136 G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 Hence, f the mesh to be evolved s very rregular, the resultng system wll be ll-condtoned. In such a case, a small tme step sze s requred to make an teratve solver converge. Such a problem s releved by the mesh regularzaton treatment (see Secton 3.6). On the other hand, more advanced teratve method, such as mult-grd technques based on a herarchcal mesh representaton (see (Lang, 2001)) or algebrac mult-grd technques, could be used to accelerate the teraton process. In the current mplementaton, these technques are not ncorporated. Upper-bound of tme step. It s known that several surface evolutons (e.g., the mean curvature flow (see (Dzuk, 1991; Whte, 2002)) and the surface dffuson flow (see (Bansch et al., 2002))) may develop sngulartes. For our geometrc modellng problems, suppose we have a topologcally correct ntal surface mesh constructon and we look for solutons that have the same topology as the ntal mesh. Hence, we requre that our soluton s wthn the tme perod n that no sngularty occurs. Therefore, we shall determne the tme step τ (n) so that t n should not go beyond the tme moment when the sngularty occurs. Let L(p (n),m(t n )) be the spatal dscretzaton of V(p,t) at vertex p (n) over the mesh M(t n ). Then from the approxmate equalty p (n+1) τ (n) B n := 1 2 mn 1 m = τ (n) L(p (n) p (n),m(t n )) and the requrement p (n+1) p (n) 1 2 mn p (n) j j N 1 () we determne an upper-bound for τ (n) as follows { mnj N1 () p (n) j p (n) p (n) (3.14) L(p (n),m(t n )) Requrement (3.14) guarantees that no vertex-collson happens. When the sngularty s nearly to occur, the upper-bound B n wll approach to zero. Hence the evoluton cannot move beyond the sngular pont for tme. Remark 3.3. When the sngularty s nearly to occur, the upper-bound B n wll approach to zero. Ths wll be a very low effcency process. So a threshold value ɛ 0 should be put on the mnmal B n.ifthe determned B n s smaller than the threshold value, we termnate the evoluton process (see (3.17) (3.18)). 3.6. Other mportant ssues 3.6.1. Mesh regularzaton The surface moton by the geometrc PDEs descrbed n Secton 2 may cause a very rregular (nonunform) dstrbuton of the mesh vertces. Hence, ntroducng a regularzaton mechansm n the evoluton process s necessary. Snce the tangental dsplacement does not nfluence the geometry of the deformaton, just ts parameterzaton (see (Epsten and Gage, 1987)), we also add a tangental dsplacement to the moton. Hence, the general form of our geometrc evoluton problem could be wrtten as p t = V(p,t)+ V t(p)t (p), M(0) = M 0, (3.15) }.
G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 137 where T(p)s a tangent drecton at the surface pont p, V t (p) s the tangental velocty. In the process of numercal soluton of Eq. (3.15), V t (p)t (p) s chosen as U 0 (p (n) ) ( U 0 (p (n) ), N(p (n) j N 1 () (p(n) ) ) N(p (n) ) (3.16) p (n) ), N s the surface normal computed from the lmt sur- where U 0 (p (n) 1 ) = card(n 1 ()) j face of Loop s subdvson. Ths dscretzaton of V t (p)t (p) s very smlar to the one gven by Ohtake et al. (2000), whch s U 0 (p (n) ) (U 0 (p (n) ), N(p (n) ))U 0 (p (n) ). The dfference s that our dsplacement s n the tangent plane. In (3.16), U 0 (p (n) ) could be replaced by U 0 (p (n+1) ) to use as many of the new values as possble, and stll yeld a lnear system. However, such a treatment destroys the symmetrc property of the coeffcent matrx. The tangental moton (3.16) s also used by Wood et al. (2000) and Ohtake et al. (2001). 3.6.2. Stoppng crtera We need to determne the mnmal teraton number n, so that the evoluton procedure stops at t = t n. The followng two crtera are used M(tn ) M(0) ɛ1 or B n <ɛ 0 (3.17) M(t n+1 ) M(t n ) /τ (n) ɛ 2 or B n <ɛ 0 (3.18) where ɛ are gven control constants, B n s the determned upper-bound for τ (n). Crteron (3.17) s for short tme evoluton, where we requre M(nτ (n) ) near M(0). Crteron (3.18) s for long tme evoluton, where we are lookng for a stable status of the soluton. Condton B n <ɛ 0 s mposed for avodng dead-loop around the sngular pont of tme. 3.6.3. Constructon of ntal surface mesh To provde an ntal soluton to the geometrc evoluton problem, we need to construct an ntal trangular surface mesh ( fller ) for each openng usng any of a number of automatc or sem-automatc free-form surface constructon technques (Bajaj and Ihm, 1992; Bajaj et al., 1993; Davs et al., 2002; Grener, 1994; Peters and Wttman, 1996; Xu et al., 2001). One can also nteractvely edt ths fller to meet the weak assumptons for an ntal soluton shape. Snce the openng to be flled could be topologcally complcated, we solve the problem n two steps. In the frst step we ft each openng by an mplct algebrac surface or splne whch nterpolates or approxmates the boundary data (Bajaj et al., 1993; Bajaj and Xu, 1994; Peters and Wttman, 1996). The approach we used s the one developed by Bajaj et al. (1992, 1993, 1994). In ths approach, the data to be nterpolated or approxmated could be ponts or curves (even wth normals). For ours, the boundary data are always ponts. Of course, ths approach may not guarantee to produce topologcally correct surfaces. If ths happens, we break the openng nto several parts by nsertng a few curves (polygons) and then repeat the surface fttng for each part untl we acheve a reasonable shape for the fller. After the algebrac surface s obtaned, a trangulaton step s employed. Snce ths trangulaton should be consstent wth the boundary polygon of the openng, we adopted the expanson technque developed n (Bajaj and Xu, 1994). Usng ths approach, we trangulate the surfaces startng from the boundary of the openng. Remark 3.4. Comparng wth fnte element approach, the fnte dfference approach descrbed above s easy to mplement and t treats the equatons wth dfferent orders n a unform fashon. In the fnte
138 G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 element approach, one has to make efforts to derve a varatonal form for each of the PDEs. For hgher order flows, hybrd method s used n general, such an approach wll ntroduce much more unknowns, and therefore the resulted lnear system s much larger. For example, n order to use fnte element method (lnear element) for the surface dffuson flow, Bänsch et al. (2002) splt the PDE nto a system of four equatons. 4. Comparatve examples In ths secton, we gve several examples to show how the PDEs are used to solve dfferent problems n a unform fashon. We also compare the effects of flows (2.8) and (2.9). All the fgures produced by the fourth and sxth flows are generated usng (2.9), except for the fgures of the second row of Fg. 4 and thrd row of Fg. 6. These fgures are produced usng the flow (2.8). When we compare the effects of (2.8) and (2.9), we use the same number of teratons but double tme step sze for (2.8) because the factor 2 n the relaton p = 2HN. 4.1. Comparson of the flows The frst three fgures of the frst row of Fg. 4 show the long tme evoluton solutons of the mean curvature flow, the fourth order flow, and the sxth order flow (2.9) for the nput sem-sphere wth an ntal constructon of the openng, a trangulated dsk. The mean curvature flow does not change the dsk. (b) and (c) are the results after 10 teratons wth τ (n) = 0.1 andτ (n) = 0.001, respectvely. Further teratons do not have a sgnfcant change on the shape of the soluton surface. The fourth and sxth order flows yeld convex surfaces and the smoothness s clearly observed. Also notce that the sxth order flow Fg. 4. The frst and second row show the results of (2.9) and (2.8), respectvely. (a) (same as (g)) The nput sem-sphere (left part) wth an ntal planar trangulaton of the dsk openng. The mean curvature flow does not change the dsk (ntal mesh). (b) The result of fourth order flow after 10 teraton wth τ (n) = 0.1. (c) The result of the sxth order flow after 10 teraton wth τ (n) = 0.01. (d), (e) and (f) show three ntermedate results of the sxth order flow wth τ (n) = 0.001, and 1, 6 and 10 teratons, respectvely. (h) The result of the surface dffuson flow after 10 teraton wth τ (n) = 0.2. () The result of the sxth order flow (2.8) after 10 teraton wth τ (n) = 0.02. (j), (k) and (l) show three ntermedate results of the sxth order flow (2.8) wth τ (n) = 0.002, and 1, 6 and 10 teratons, respectvely.
G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 139 Fg. 5. Comparson of dfferent flows. k represents 2k order flow (2.9) s used. AM denote the averaged mean curvature flow. The tme step szes for the second, fourth and sxth order flows are chosen to be 0.1, 0.0025, and 0.0000625, respectvely. (c), (e), (g) are the fared nterpolatng surface meshes after 6 teratons, where the contnutes at the boundary curves are set to 0, 2 and 0, respectvely. (d), (f), (h) are the mean curvature (MC) plots of (c), (e), (g), respectvely. recovers the sphere accurately. The last three fgures show three ntermedate results of the sxth order flow. The second and thrd fgures of the second row of Fg. 4 show the evoluton solutons of the surface dffuson flow and sxth order flows (2.8) for the nput sem-sphere wth an ntal constructon of the openng. (h) and () are produced usng the same number of teratons as (b) and (c), respectvely, and double tme step szes. Agan, the last three fgures show three ntermedate results of the sxth order flow. Comparng wth the fgures of the frst row, the geometrc flows change the surface shape n a much slower rate. Remark 4.1. We have ponted that the geometrc flows (2.8) have volume preservng propertes for a closed surface. However, for an open surface wth fxed boundary, the volume preservng propertes are not guaranteed. (h) and () show that the volume preservng property s not vald. Fg. 5 shows the combned use of dfferent flows. The am of ths toy example s to llustrate the dfference of these flows, especally the contnuty on the patch boundares. (a) shows four crcles to be nterpolated. Two of the crcles are n the xz-plane, the other two are n the yz-plane. (b) shows an ntal G 0 surface mesh constructed usng (Bajaj and Ihm, 1992) wth some addtonal nose added. (c), (e) and (g) are the fared nterpolatng surfaces after 6 teratons usng dfferent combnatons of the flows. The tme step szes for the second, fourth and sxth order flows are chosen to be 0.1, 0.0025, and 0.0000625, respectvely. Snce the hgher order flows evolve faster than the lower order flows, we use smaller tme step szes for hgher order flows to obtan nearly the same surface evoluton speed. Each of the meshes
140 G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 conssts of four surface patches. The left two patches are n the regons R + := {(x,y,z): x 0, y 0} and R := {(x,y,z): x 0, y 0}, respectvely, and generated by one type of flow. The rght two patches are n the regons R ++ := {(x,y,z): x 0, y 0} and R + := {(x,y,z): x 0, y 0}, respectvely, and generated by a dfferent flow. (d), (f) and (h) are the mean curvature plots of (c), (e) and (g), respectvely. The mean curvature at each vertex s computed by (3.2). The am of (c) s to show the dfference between the mean curvature flow and the averaged mean curvature flow, where the left part s generated by the averaged mean curvature flow and the rght part s produced by the mean curvature flow. The mean curvature flow shrnks the surface very fast whle the averaged mean curvature flow does not. Further evoluton usng the mean curvature flow wll yeld a pnch-off of the surface. Therefore, f we model a surface patch usng second order flows wth G 0 boundary condton, the averaged mean curvature flow s more desrable than the mean curvature flow. The patches n R + and R of (e) are produced by the sxth order flow (2.9) (wth k = 3), whle the patches n R ++ and R + are produced by the fourth order flow (2.9). As a whole, the surface looks smooth, our curvature plot reveals the smoothness dfference at the ntersecton curves, the sxth order flow gves a smoother result than the fourth order flow. (g) s produced as (e), but the contnuty order at the four crcles are set to zero. Hence G 0 contnuty s acheved there. 4.2. Surface blendng Gven a collecton surface mesh wth boundares, we construct a far surface to blend the meshes at the boundares wth specfed geometrc contnuty. Fg. 6 shows the case, where three cylnders to be blended are gven (a) wth an ntal G 0 constructon (b) usng (Bajaj and Ihm, 1992) wth some addtonal nose added. The blendng surfaces (c), (e) and (g) are the fared blendng meshes generated usng the flow (2.9) wth k = 1, 2, 3, respectvely. These fgures show the results after 32, 32 and 60 teratons wth tme step szes 0.01, 0.001, and 0.0001, respectvely. (d), (f) and (h) show the mean curvature plots correspondngly. These fgures clearly show the dfference of smoothness acheved at blendng boundares. The mean curvature flow gves G 0 contnuty results. The fourth order flow produces smooth surfaces at boundares. The sxth order flow produces even smoother surfaces as expected. () and (k) are the fared blendng meshes generated usng the flow (2.8) wth k = 1, 2, respectvely. These fgures show the results after 32 and 60 teratons wth tme step szes 0.002 and 0.0002, respectvely. (j) and (l) show the mean curvature plots of () and (k), respectvely. It should be noted that the flows (2.9) generate lttle fatter surface than the flows (2.8). 4.3. N-sded hole fllng Gven a surface mesh wth a hole, we construct a far surface to fll the hole wth specfed geometrc contnuty on the boundary. Fg. 1 shows such an example, where a head mesh wth a hole n the nose subregon s gven as nput (a). An ntal G 0 reconstructon of the nose s shown n (b) usng (Bajaj and Ihm, 1992) and then evolved wth the mean curvature flow. The blendng surfaces ((c) and (d)) are generated usng the flow (2.9) wth k = 2 and 3, respectvely. It should be observed that the sxth order flow yelds a better restoraton surface. The head mesh wth the hole n the nose subregon s avalable from http://lsec.cc.ac.cn/~xuguo/xuguo2.htm.
G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 141 Fg. 6. (a) shows three cylnders to be blended. (b) shows the ntal constructon. (c), (e) and (g) are the fared blendng meshes generated usng the flow (2.9) wth k = 1, 2, 3, respectvely. These fgures show the results after 32, 32 and 60 teratons wth tme step szes 0.01, 0.001, and 0.0001, respectvely. (d), (f) and (h) show the mean curvature plots correspondngly. () and (k) are the blendng meshes generated usng the flow (2.8) wth k = 1, 2, respectvely. These fgures show the results after 32 and 60 teratons wth tme step szes 0.002 and 0.0002, respectvely. (j) and (l) show the mean curvature plots of () and (k), respectvely. 4.4. Free-form surface constructon For the free-form surface fttng problem, we are gven some curves, or partal patches, or ponts as nput, and we wsh to construct a far surface mesh to nterpolate ths mult-dmensonal data. Fg. 7 shows the approach of free-form surface constructon, where some nput curves wth G 0 contnuty requrement are gven to preserve the sharp edges, and also gven are some surface bands wth a G 1 contnuty requrement (see (a)). (b) shows an ntal constructon of the G 0 surface mesh usng the patch fllng scheme (Xu et al., 2001) wth added nose. (c) s the fared surfaces, after 12 teratons, generated usng the flow (2.9) wth k = 2. The tme step sze s chosen to be 0.001. (d), (e) and (f) are zoomed n vews of (a), (b) and (c), respectvely. Fg. 8 shows the free-form fttng approach from an nput trangular mesh, where (a) shows the nput surface trangular mesh wth a G 1 contnuty requrement at the vertces (see (a)). (b) shows an ntal
142 G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 Fg. 7. Interpolatng curves and patches: (a) shows some nput curves wth G 0 contnuty requrement and some bands of mesh wth G 1 contnuty requrement. (b) shows an ntal constructon of the surface mesh. (c) s the fared surfaces, after 12 teratons, generated usng the flow (2.9) wth k = 2. The tme step sze s chosen to be 0.001. (d), (e) and (f) are the zoom n results of (a), (b) and (c), respectvely. constructon of the surface mesh, where each nput trangle s approxmated wth 16 sub-trangles. The newly ntroduced vertces are treated as unknowns and the nput vertces are fxed n the farng process. (c) and (d) are the fared meshes, after 2 teratons wth τ (n) = 0.01, generated usng the mean curvature flow and the averaged mean curvature flow, respectvely. (e) s the fared mesh by fourth order flow, after 2 teratons wth τ (n) = 0.001. (f) s the mean curvature plot of (e). The area shrnkng of the mean curvature flow makes the nput vertces to be nterpolated become thorns (see (c)), whle the area shrnkng and the volume preservaton of the averaged mean curvature flow make some of nput vertces become thorns and some others become pts (see (d)). However, the fourth order flow does not suffer from ths problem (see (e)). The obtaned surface nterpolates the nput ponts and exhbts G 1 smoothness everywhere as well. 5. Conclusons We have presented a general scheme for usng PDEs to solve several surface modellng problems and wth hgh order boundary contnuty condtons. Our scheme has the followng features: It produces very far and desrable soluton surfaces. It s smple and easy to mplement. Specfcally, t solves the free-form blendng problem, the N-sded hole fllng problem and free-form surface fttng problem n a unform fashon, and solves the hgh order boundary contnuty problem n an easy and natural way and avods pror estmaton of normals or dervatve jets on the boundares. The mplementaton results
G. Xu et al. / Computer Aded Geometrc Desgn 23 (2006) 125 145 143 Fg. 8. Interpolatng ponts: (a) shows some nput ponts and ther trangulaton. (b) shows an ntal constructon of the surface mesh. (c) and (d) are the fared surfaces, after 2 teratons wth τ (n) = 0.01, usng the mean curvature flow and the averaged mean curvature flow, respectvely. (e) s fared surfaces, after 2 teratons wth τ (n) = 0.001, usng the fourth order flow (2.9). (f) s the mean curvature plot of (e). show that our soluton works well for a wde range of surface models. Note that the C 1 or hgher order contnuty nterpolatory surface blendng soluton produced by, e.g., (Bajaj and Ihm, 1992; Peters and Wttman, 1996) for complcated corners, or holes wth many boundary curve segments, are usually of very hgh algebrac degree and thereby prone to be wth unsutable for certan applcatons. The current soluton of startng wth G 0 low degree blends, coupled wth hgher order flow evoluton, yelds n general a much better alternatve for very smooth surface solutons. Both the geometrc flows and quas geometrc flows yeld smooth surfaces at the boundares. However, quas geometrc flows (2.9) have some attractve features, ncludng ease of mplementaton, smaller and better behaved coeffcent matrces and no requrement of dervatves (normal) estmaton. Acknowledgement The authors are grateful to the referees for ther carefully readng of the manuscrpt and for ther helpful comments on mprovng the fnal verson of ths paper.
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