Discrete Smooth Interpolation: Constrained Discrete Fairing for Arbitrary Meshes

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1 Dscrete Smooth Interpolaton: Constraned Dscrete Farng for rbtrary Meshes runo Lévy Jean-Laurent Mallet IS-GOCD (Inra Lorrane/CNRS ENSG, rue du doyen Marcel Roubeault, Vandoeuvre bstract In ths paper, t s shown how the Dscrete Smooth Interpolaton method (D.S.I. may be used as a new framework for creatng, farng and edtng trangulated surfaces. Gven an arbtrary mesh wth an arbtrary set of vertces fxed by the user, D.S.I. assgns coordnates to the other nodes of the mesh, enablng the fxed vertces to be nterpolated smoothly. The squared dscrete Laplacan crteron mnmzed by D.S.I. s an objectve functon smlar to the bendng energy of a thn-plate. Ths approach fulflls the requrements of subdvson methods, n that t provdes arbtrary topology, smplcty, the possblty to defne creases of varable sharpness, as well as the convergence of recursve subdvsons to a smooth surface. It does not suffer from the lmtatons nherent to more classc subdvson methods, such as the subdvson connectvty requrement. Moreover, D.S.I. offers a hgh degree of flexblty. It then becomes possble to defne the surface zones to be smoothed n order of preference. Furthermore, constrants lnearly combnng the coordnates at the vertces of the surface may be taken nto account n a least square sense. s a result, a surface can be ftted to an arbtrary set of ponts wth, or wthout, specfed normals. The method mght also have mportant mplcatons for mult-resoluton edtng and mesh compresson. CR Categores: I.3.5 [Computatonal Geometry and Object Modelng]: curve, surface, sold and object representatons Keywords: Geometrc Modelng, Curves and Surfaces, Mesh Generaton, Shape lendng, Level of Detals lgortms 1 INTRODUCTION Over the last few years, much nterest has been aroused by methods actng drectly on polygonal meshes. The famly of methods known as subdvson enables a surface to be modeled as the result of recursve refnements appled to an ntal surface [DS78, CC78]. s far as parametrc surfaces are concerned, the classc varatonal levy@ensg.u-nancy.fr, IS/Gocad (Inra Lorrane/CNRS mallet@ensg.u-nancy.fr, drector of the GOCD consortum desgn methods [Gre94, MS9, Se98] make t possble to construct ncely shaped surfaces whch satsfy a set of constrants. hybrd feld known as dscrete farng [Kob97] has recently appeared. It shares features of both subdvson and varatonal desgn. Whereas subdvson conssts of teratvely refnng a gven mesh usng a systematc rule, dscrete farng acts on an arbtrary mesh. Ths mesh has a set of vertces fxed by the user, referred to as control nodes, and the coordnates at the other vertces are assgned to smoothly nterpolate the control nodes. However, dscrete farng stll lacks several functonaltes from both felds, such as modelng creases of varable sharpness or fttng a surface to a set of ponts wth, or wthout, specfed normals. Dscrete Smooth Interpolaton (D.S.I. [Mal89, Mal9, LM98] combned wth the developments presented n ths paper enables these latter requrements to be met. From a farng pont of vew, D.S.I. s made to mnmze a farness crteron known as the squared dscrete Laplacan. Whereas such functonals are studed n prevous work usng a partcular parameterzaton [Kob97, KCVS98, WW94], t s studed here for any arbtrary parameterzaton. In addton, an teratve solver has been desgned to obtan a real tme response of the algorthm n an nteractve envronment. s wth any other dscrete farng method, D.S.I. may be used as a subdvson method for teratvely refnng meshes. These methods have been mplemented as the kernel of a modeler wdely used n the ndustry. The artcle s organzed as follows. Secton 1 ntroduces subdvson, varatonal desgn and dscrete farng. In secton, Dscrete Smooth Interpolaton s presented as a dscrete farng method for teratvely mnmzng the squared Laplacan of an arbtrary mesh. The ssues of varable sharpness creases and blendng are also addressed here. The D.S.I. method s extended n Secton 3 to take nto account lnear constrants n a least square sense. For nstance, how to ft a surface to a scattered set of ponts s demonstrated. The last secton concludes wth suggeston for further research.

2 CKGROUND efore gong nto the heart of the matter, t mght be useful to ntroduce subdvson, varatonal desgn and dscrete farng, snce D.S.I. shares features from all three of these rapdly evolvng domans. The purpose of ths secton s to show how these dfferent domans have evolved and how the D.S.I. method we ntroduce relates to them..1 Subdvson To create complcated shapes such as those nvolved n character anmatons, polynomal patches and NURs are often used. Several well known lmtatons appear when creatng complex models usng ths knd of representaton: objects of complcated topology cannot be represented by a sngle patch; connectng several patches to create objects of arbtrary topology whle mantanng smoothness at the seams s tedous; modelng objects wth arbtrary shaped borders may requre trmmng the patches, known to cause numercal nstabltes. To overcome these problems, a famly of approaches known as subdvson has been studed. They recently proved to be effcent n character anmaton, by makng t possble to realze the short flm Ger s game [DKT98]. s suggested n Fgure 1, startng from a gven mesh M 0, referred to as the control mesh (see Fgure 1-, rules are defned to systematcally refne t. Thus, the mesh M +1 s obtaned from M by modfyng ts topology n a smple way. The coordnates at the vertces of M +1 are computed as a smple combnaton of the vertces of M, whch defnes a seres of meshes M 0, M 1,... M. The rules to compute M +1 from M are defned so that the seres of meshes M converge to a smooth surface M (see Fgure 1-D. Hstorcally, the frst two subdvson schemes dscovered by Doo-Sabn [DS78] and Catmull-Clark [CC78] converge, to bquadratc and bcubc -Splnes, respectvely. The exact evaluaton of Catmull-Clark surfaces,.e. the evaluaton of a pont of M from ts parameter values, s addressed n [Sta98]. Subdvson surfaces present the followng usefull propertes: the control mesh M 0 can have an arbtrary topology; the refnement rules are smple, easy to mplement and very effcent; these methods work drectly on polygonal meshes: a natural representaton for computer graphcs. Moreover, the seres of generated meshes M provde an easy way to mplement levels of detal n real tme envronments. Further, tme and effort have been devoted to subdvson schemes convergng to varous smooth surface famles. The mddle-edge subdvson [PR97] and Loop s scheme [Loo87] converge to box splnes. The utterfly scheme [DLG90], actng on trangulated surfaces, provdes an nterpolatng rather than an approxmatng scheme,.e. the vertces of the control mesh M 0 are preserved. n nterpolatng scheme for surfaces based on quadrlaterals rather than trangles was studed n [Kob96b]. In [CG91], the connectons wth free-form desgn are treated. The smple subdvson rules mpled n ths famly of methods generate meshes of regular topology. However, snce the mesh M 0 can be arbtrary, the connectvty of an orgnal vertex may be dfferent from that of the generated vertces. For ths reason, the orgnal vertces are referred to as extraordnary ponts. The specal C Fgure 1: The D.S.I. method as an nterpolatng subdvson scheme for quadmeshes. : control mesh M 0 ; : after one refnement step ; C: after two steps; D: lmt surface M. connectvtes of extraordnary vertces prevent the utterfly scheme [DLG90] from convergng to a G 1 surface at all ponts. Ths problem was resolved n [ZSS96]. s far as approxmatng schemes are concerned, the analyss of the neghborhoods at extraordnary ponts was studed n [Re95, Sab91]. Several mprovements of subdvson surfaces have been made to provde the user wth more functonaltes. Thus, the ablty to tag certan edges of the control mesh M 0 as creases, and vertces as corners, was studed n [HDD + 94], makng t possble to create Catmull-Clark surfaces wth sharp edges. Ths was generalzed n [DKT98] to model fllets and blends as sem-sharp creases. The ssue of sem-sharp creases was also addressed n [SZSS98], where NURSSes (Non Unform Recursve Subdvson Surfaces are ntroduced as a non-unform generalzaton of Doo-Sabn and Catmull-Clark surfaces.. Varatonal Desgn dfferent ssue n geometrc modelng s the problem of generatng far surfaces [Gre94, MS9],.e. ncely shaped surfaces honorng a set of boundary condtons and constrants such as: nterpolatng data ponts wth or wthout specfed tangent planes; generatng surfaces smoothly nterpolatng the borders of specfed gven surfaces (blendng; smoothng an exstng surface n a specfed zone. revew of exstng methods n ths feld s gven n [Se98]. The ssue of generatng far surfaces s often referred to as varatonal desgn. Ths conssts n choosng a class of parametrc functons (u, v ϕ(u, v = {ϕ x (u, v, ϕ y (u, v, ϕ z (u, v, wth more degrees of freedom than necessary to honor the constrants, and D

3 mnmzng a gven crteron to determne a unque functon n the class. farness functonal F(ϕ s then defned as a scalar characterzng the qualty of the surface yelded by a functon ϕ. smplfed expresson of the bendng energy of a thn plate F T hnp late s often used (see, for nstance [KHD93, Kal93]. Other possble quadratc functonals are studed n [WW9, CG91]. s shown n Equaton 1, we propose to use the even more smplfed approxmaton of the thn plate energy F, where ϕ ν (u, v denotes the Laplacan of a component of the functon ϕ(u, v = {ϕ x (u, v, ϕ y (u, v, ϕ z (u, v. The postve functon µ(u, v denotes a stffness varyng all over the surface. Ths stffness enables the user to gve more mportance to the smoothng of certan zones. It s shown further n ths artcle how ths latter functonal may be used to smooth a polygonal surface. F (ϕ = µ(u, v. { ϕ ν (u, v.dudv where: ϕ ν (u, v = ϕ ν (u, v + ϕ ν (u, v u v (1 The surfaces that can be represented usng varatonal desgn are often defned as pecewse polynomals. Recent research has been devoted to the drect applcaton of varatonal desgn methods to polygonal meshes of arbtrary topology. Subdvson schemes have been studed to enable convergence to a smooth surface whch mnmzes a farng functonal. For nstance, thn plate splnes are addressed n [WW98]. Kobbelt [Kob96a] has shown how farng functonals mght be dscretzed by usng fnte dfferences, and how new subdvson schemes mght be derved from a so-dscretzed farng functonal..3 Dscrete Farng Despte the obvous advances referred to above, the statonary schemes used to construct subdvson surfaces do not offer much user nteracton. For nstance, the resultng smooth surface M may present unwanted oscllatons where the control mesh M 0 s too dense and rregular. Even when usng the extensons enablng creases to be modeled [HDD + 94, DKT98, SZSS98], often, the only way to correct these oscllatons s to modfy the control mesh. nother famly of methods, known as dscrete farng [Kob97, Kob98, KCVS98, WW94, Mal89, Mal9], offers more flexblty. Whereas subdvson conssts of teratvely refnng a control mesh M 0 usng a systematc rule, dscrete farng acts on an arbtrary mesh. Ths mesh has a set of anchored nodes, referred to as control nodes, and the other nodes are moved to smoothly nterpolate these control nodes. Clearly, ths scheme encompasses nterpolatng subdvson, snce a subdvson mesh M mght be thought of as a dscrete farng mesh where the orgnal ponts (the vertces shared wth the orgnal mesh M 0 are tagged as control nodes. Unwanted oscllatons may then be easly removed by unlockng control nodes n the zones concerned. Ths artcle ntroduces extensons of D.S.I. [Mal89, Mal9, LM98], makng t act as a new dscrete farng method. The somodfed D.S.I. method enables dscrete farng to be enhanced wth functonaltes from both the varatonal desgn and the subdvson feld, such as: defnng an teratve solver enablng a polygonal surface of arbtrary topology to be nteractvely edted; specfyng a varable stffness µ all over the surface, makng t possble to select n order of preference the surface zones to be smoothed; modelng creases of varable sharpness; honorng a set of lnear constrants such as fttng the surface to a scattered set of ponts, wth, or wthout, specfed normals. 3 DISCRETE SMOOTH INTERPOLTION Fgure : Smoothng arbtrary meshes. : The red cubes correspond to control nodes, all the other nodes have been nterpolated. Note that the topology of the mesh and the repartton of the control nodes are completely arbtrary ; : Smoothng a volumc mesh. The nodes on the border have been tagged as control nodes and nteror nodes have been nterpolated. trangulated mesh M s defned to be a trplet {Ω, E, T where Ω = {α 1,... α N denotes the vertces of the trangulaton, E denotes the edges and T the trangles. The geometrc locaton at the vertces of the trangulaton may be thought of as the samplng of an unknown contnuous parametrc functon ϕ whch puts Ω n correspondence wth a set of ponts of R 3 {ϕ(α = ϕ(u α, v α, α Ω. Gven a set of nodes L Ω, referred to as control nodes, where the value of ϕ s gven, and the set of nodes I = Ω L where ϕ s unknown, the dscrete farng problem n ts smplest form conssts n assgnng the values of ϕ at the nodes of I whle mnmzng a functonal F(ϕ. Snce ϕ s only represented at the vertces of the mesh M, the functonal F(ϕ s estmated usng the values of ϕ at the vertces of M. In what follows, t s then shown how the ntegral of the squared Laplacan may be approxmated and mnmzed over a trangulated surface. ased on ths approxmaton, the teratve farng algorthm wll be ntroduced. In addton, the next secton shows how to take nto account lnear constrants n a least square sense. v u k D(k T α N T α 1 D(k Fgure 3: pproxmatng the Laplacan on a trangulated surface. v u α L k T L α T R α R

4 3.1 Dscrete Laplacan on Trangulated Surfaces The Laplacan ϕ ν (u, v of a gven functon ϕ ν at parameters (u, v s gven n Equaton below. Our goal s now to approxmate ths Laplacan on a trangulated surface. s can be seen, the Laplacan characterzes parametrc functons, and for ths reason, a parameterzaton of the trangulated surface has to be chosen. Constructng a global parameterzaton of a trangulated surface s possble [LM98], but ths clearly lmts the scope of the study to open surfaces homeomorphc to dsks. { ϕ ν ϕ ν (u, v = u + ϕ ν (u, v ( v To overcome ths problem, a common practce s to use local parameterzatons. Gven a vertex k, ts neghborhood N(k s defned to be the set of nodes drectly connected to k, ncludng k. local parameterzaton conssts n assgnng (u α, v α coordnates to the vertces α N(k. The doman D(k shown n Fgure 3 s defned to be the polygon n parameter space defned by N(k Projectng on an estmated tangent plane to defne a local parameterzaton s clearly not a good dea, snce t may result n overlaps n the hghly curved regons. The symmetrc parameterzaton used n [KCVS98] conssts n evenly postonng the neghbors of the consdered vertex k on a crcle. nother possblty s gven by the exponental map (also referred to as geodesc polar map used n [WW9]. In what follows, the Laplacan s computed for an arbtrary parameterzaton. It should be mentoned that the symmetrc parameterzaton can be computed very quckly, snce t solely depends on the number of vertces, whereas the exponental map s less senstve to the heterogenetes of the szes of the trangles. Gven one of these three possble parameterzaton (global, unform or exponental map, how to estmate the Laplacan on a trangulated surface and mnmze t s explaned below. To estmate the Laplacan at a gven vertex k, the trangulated surface s consdered as a samplng of an unknown C functon ϕ(u, v = {ϕ x (u, v, ϕ y (u, v, ϕ z (u, v defned over the doman D(k of the vertex k (see Fgure 3. Then, consderng a component ν {x, y, z, the dscrete Laplacan Dϕ ν (k at the vertex k s defned to be an estmaton of the Laplacan of ϕ ν, usng the known values of ϕ ν at the vertces α N(k. The frst approxmaton we do s to consder the Laplacan to be approxmately equal to ts average value on the doman D(k, where D(k denotes the area of D(k: ϕ ν 1 (u k, v k ϕ ν (u, vdudv (3 D(u k, v k D(k n expresson of the Laplacan equvalent to Equaton s gven by ϕ ν = dv(gradϕ ν, whch enables the prevous expresson to be smplfed, by applyng the Green Gauss theorem (see Equaton 4. The vector N denotes the normal to the border D(k of D(k whch ponts outwards D(k. The ntegral on the doman D(k has been replaced by the smpler ntegral along the curve D(k. It should be mentoned that n our case, applyng ths theorem ntroduces an approxmaton snce the polygon D(k s not a C 1 curve. ϕ ν 1 (u k, v k D(k D(k 1 D(k D(k dv(gradϕ ν (u, vdudv gradϕ ν (s.n(s.ds Snce the value of ϕ ν s only known at the vertces of N(k, t s natural to use on each trangle the pecewse lnear nterpolaton Φ ν to approxmate ϕ ν on the doman D(k. The gradent (4 gradφ ν (u, v s thus constant over each trangle T (k, α 1, α of D(k and s denoted gradφ ν (T. The normal vector N s also constant for each trangle T and s denoted N T. s suggested n Fgure 3-, the curvlnear ntegral along the border of D(k can then be replaced by the followng sum on the trangles of D(k. The length of the edge E(α 1, α n parameter space s denoted by E(α 1, α. ϕ ν (k 1 D(k. T (k,α 1,α D(k E(α 1, α.gradφ ν (T.N T (5 It can be shown that the expresson E(α 1, α.gradφ ν (T.N T can be rewrtten as a lnear combnaton of the values of ϕ ν at the three vertces of the trangle T (k, α 1, α (see Equaton 6: E(α 1, α.gradφ ν (T.N T = ϕ ν (α 1.τ( ˆα T + ϕ(α.τ( ˆα 1 T + ϕ ν (k.{ τ( ˆα T τ( ˆα 1 T The coeffcents τ(ˆα T denote the absolute value of the cotangent n parameter space of the angle at the vertex α of the trangle T : τ(ˆα T (k, α, β = cotg(ˆα (7 Gven these coeffcents, t s possble to sort them by the vertex they are assocated wth. Equaton 8 gves the so-obtaned v α (k coeffcents enablng the dscrete Laplacan Dϕ ν (k to be computed. Note the arbtrary multplcaton by the term 3/ D(k to help smplfy later expressons. In ths equaton, N(k denotes the set of vertces drectly connected to k, ncludng k. s shown n Fgure 3-, the trangles T L and T R are defned as the two trangles sharng the edge E(k, α. The vertces α L and α R are the vertces opposte to E(k, α n T L and T R, respectvely. The remanng coeffcents α (k are border correcton coeffcents, defned below. α + v u T + β + k D(k Fgure 4: Computaton of the border correcton coeffcents α (k to be taken nto account when the vertex k les on the border of the surface. Dϕ ν 3 (k =. D(k where: α N(k {k, α N(k v α (k.ϕ ν (α T - v α (k = {τ( αˆ R T R + τ( αˆ L T L + α (k D(k v k (k = α N(k {k v α (k β - α - (6 (8

5 s shown n Fgure 4, the computaton of the dscrete Laplacan Dϕ ν (k at a node k on the border of the surface requres addtonal coeffcents α (k, snce the border D(k of the ntegraton doman comprses two addtonal edges (k, α + and (k, α. These two edges have to be taken nto account when ntegratng along the border of D(k, (see Equaton 4. Ths yelds the followng border correcton coeffcents α (k. α (k = τ( β ˆ T ; α+ (k = τ( β ˆ+ T + { β (k = τ( α ˆ T + τ(ˆα T { β+ (k = τ( α ˆ+ T + + τ(ˆα T + α (k = 0 everywhere else (9 Remark: Usng the symmetrc parameterzaton on a regular mesh where all vertces are of degree 6 (the degree of a vertex denotes the number of vertces connected to t, all the trangles n parameter space are equlateral. The known result [KCVS98] to wthn a constant multplyng factor s then obtaned,.e. v α (k = 1 for k N(k {k and v k (k = degree(k. 3. The Dscrete Smooth Interpolaton Method Gven a trangulated surface {Ω, E, T and a subset of the vertces L Ω tagged as control nodes, our purpose s now to make the other vertces of I = Ω L smoothly nterpolate the control nodes. Ths can be acheved by mnmzng the dscrete functonal F D approxmatng the functonal F. The coeffcent 1/3 comes from the fact that each trangle s taken nto account three tmes when ntegratng (once per vertex of each trangle. The gven postve coeffcents µ(k corresponds to the stffness of the surface and enable the user to modulate the farng all over the surface. The functon ϕ ν denotes one of the three components of ϕ = {ϕ x, ϕ y, ϕ z. F (ϕ = F D (ϕ = ν {α,β,γ ν {α,β,γ k Ω µ(u, v. { ϕ ν (u, v.dudv µ(k. {Dϕ ν (k. 1 3 D(k (10 y usng the expresson of the v α (k coeffcents (see Equatons 8 and 9, enablng the Laplacan to be approxmated, t s possble to gve the followng expresson of the dscrete functonal F D. The coeffcent 3/ D(k arbtrarly ntroduced n the Equaton 8, gvng the v α (k coeffcents, are smplfed when ntroduced nto Equaton 10. F D (ϕ = k Ω µ k. { α N(k v α (k.ϕ ν (k The mnmum of ths functonal F D (ϕ n functon of ϕ ν (α s reached f F D / ϕ ν (α = 0 for all ν {x, y, z. Ths yelds the followng equaton (see Equaton 1. (11 C Fgure 5: The classc monkey saddle problem conssts n generatng a blendng surface whch smoothly connects sx planes radatng n dfferent drectons. : Control mesh M 0. The red cubes are control nodes, the other nodes are free to move ; : Result after applyng D.S.I. ; C: Result after one subdvson step ; D: Lmt surface M. ϕ ν (α = Gν (α g ν (α where: G ν (α = g ν (α = k N(α α N(α { µ(k.v α (k. µ(k. {v α (k D β N(k {α v β (k.ϕ ν (β ased on Equaton 1, the followng algorthm teratvely computes the assgnments of (ϕ x, ϕ y, ϕ z coordnates whle mnmzng the dscrete functonal F D (ϕ gven n Equaton 11. Mallet [Mal89, Mal9] proved that ths algorthm converges to a unque soluton, provded that the followng condtons are satsfed (as when usng the v α (k coeffcents prevously ntroduced. 1. The set I of anchored control nodes s non-empty.. k Ω, α N(k v α (k > 0 3. k Ω, v k (k = α N(k {k vα (k let I be the set of nodes where ϕ s unknown let ϕ [0] be a gven ntal approxmated soluton whle (more teratons are needed { for all( α I { for all( ν {x, y, z ϕ ν (α := Gν (α g ν (α (1

6 3.3 Subdvson Surfaces ased on D.S.I. The algorthm ntroduced n the prevous secton s teratve, whch means t can beneft a great deal from an ntal guess ϕ [0]. Wthn the framework of an nteractve modeler, the surface obtaned after the nterventon of the user, such as movng control nodes, lockng/unlockng vertces, may be consdered as an ntal guess. In practce, the D.S.I. algorthm converges n real tme after user nteractons (t takes a few seconds to update a surface of several thousands of trangles on a SGI R10000 machne. ased on ths remark, D.S.I. could be consdered as a possble refnement scheme for defnng subdvson surfaces. The mesh M +1 s obtaned by splttng the polygons of the mesh M and smoothng t usng D.S.I. t any gven step, only the vertces shared wth M 0 are set as control nodes. Ths yelds the followng algorthm: let M 0 be the ntal control mesh. set all the vertces of M 0 as control nodes. whle (more teratons are needed { M +1 M splt all the polygons of M +1 apply D.S.I. to M for α k and v k (k = degree(k, ths requrement wll be met. Moreover, these trval coeffcents may be used on meshes of arbtrary topology. The meshes shown n Fgure at the begnnng of the secton have been obtaned usng these coeffcents. However, for a trangulated surface, the best results are arrved at usng more sophstcated parameterzatons, such as the exponental map [WW94]. The proposed algorthm may be seen as a multgrd solver for the D.S.I. equaton. Usng more optmzed multgrd approaches may lead to mprove the performances. However, the smple algorthm presented here acts at a reasonable speed and allows nteractvty. What follows deals wth modelng creases of varable sharpness. Even more flexblty wll be ntroduced n the next secton by makng the surface ft a set of scattered data ponts. More generally, lnear constrants wll be taken nto account n a least square sense. 3.4 Modelng Creases of Varable Sharpness The commonly admtted requrements for subdvson surfaces are sum up below: 1. The nput mesh M 0 can be of arbtrary topology. Ths requrement s clearly satsfed by our method.. The seres of recursvely refned meshes M should converge to a smooth surface M. The exstence and the uncty of the soluton at each step M was proven n [Mal89, Mal9]. The D.S.I. method s a dscrete farng method, whch means that at each step, all the vertces (except the control nodes are free to move (whereas, n statonary subdvson schemes, all the vertces of the mesh M are locked when computng the mesh M +1. Thus, the convergence and uncty for a gven mesh M mean that t s always possble to obtan a smooth surface at an arbtrary level. The convergence of the so-obtaned seres M has always been observed n practcal applcaton, but the formal proof of ths convergence s stll an open problem. However, as can be easly checked, the dscrete functonal F D (ϕ converges to the functonal F (ϕ when, where ϕ corresponds to the geometry of the mesh M. 3. It should be possble to tag edges as creases of varable sharpness. s shown later on, creases can be ntroduced by a smple modfcaton. C D 4. The control should be local,.e. modfyng a vertex should have an nfluence on a lmted zone around that vertex. From the local update scheme (see Equaton 1, t can be seen that two rngs of control nodes act as a tght barrer for D.S.I whle provdng an Hermte-type lmt condton. Ths suggests a user nterface smlar to what was proposed n [KCVS98], where the user chooses a zone of nfluence on the surface before modfyng t. 5. The refnement rules should be smple and easy to compute. y choosng the smplest expressons for the v α (k coeffcents, yelded by a symmetrc parameterzaton: v α (k = 1 E Fgure 6: Modelng varable sharpness creases. : Control mesh M 0 representng a cube. Its eght vertces are set as control nodes ; : Lmt surface M ; C: The edges of the top and bottom faces have been set as nfntely sharp creases. The resultng surface mmcs a soap flm ; D,E: Decreasng the sharpness of the creases ; F: The lmt surface obtaned for another set of edges tagged as nfntely sharp creases. It s easy to model creases of varable sharpness by modulat- F

7 ng the v α (k coeffcents defnng the crteron to be mnmzed by D.S.I. (see Equaton 8. crease s then defned to be a set of vertces C connected by a set of edges of the trangulaton, defnng a polygonal curve. Each crease s provded wth an assocated sharpness factor. The new coeffcents v α C(k to be appled to a vertex k when ntroducng the crease C are gven below n Equaton 13. v α C(k = v k C(k = 1 1+sharpness.vα (k vc(k α α N(k {k f k C and α / C f k C vc(k α = v α (k otherwse (13 corner t then defned to be a crease C comprsng a sngle vertex. When creases are used n the subdvson algorthm prevously ntroduced, the vertces nserted nto an edge belongng to a crease C are added to C. 4 LEST SQURE FITTING ND OTHER CONSTRINTS λ 1.ϕ x (α 1 + λ.ϕ x (α + λ 3.ϕ x (α 3 = P x (c P x λ 1.ϕ y (α 1 + λ.ϕ y (α + λ 3.ϕ y (α 3 = P y (c y P λ 1.ϕ z (α 1 + λ.ϕ z (α + λ 3.ϕ z (α 3 = P z (c P z {(ϕ ν (α 3 ϕ ν (α.n ν = 0 (c 1 N {(ϕ ν (α 1 ϕ ν (α 3.N ν = 0 (c N {(ϕ ν (α ϕ ν (α 1.N ν = 0 (c 3 N Note that these constrants lnearly combne the components of ϕ at a set of surface vertces. The general form for such a constrant s then gven n Equaton 15 below, where ν denotes one of the three components x, y, z of ϕ. To balance the equatons, t should be stressed that these constrants should be normalzed,.e. they should be such that α ν {ν c (α = 1. (14 The ablty to specfy creases of varable sharpness enables the control mesh M 0 to be smplfed to a large extent, by makng t possble to model zones of hgh curvature wthout requrng that the control mesh n these zones be densfed. However, at that pont, the only way to alter the geometry of a surface s to act on the control mesh M 0 drectly. To provde more possbltes for user nteracton, ths secton shows how D.S.I. can be extended to take nto account lnear constrants. The example of honorng data ponts wth, or wthout, normals s developed here. 4.1 Fttng Trangulated Surfaces to Scattered Data α P α 1 N N Fgure 8: Fttng data ponts. : Mesh M 0 roughly approxmatng a face. The yellow lnes have been drawn by the user to refne the model ; : Lmt surface M. The surface has been attracted by the yellow lnes. ν c (α.ϕ ν (α = b c (15 p α Ω α o T Fgure 7: Postonal and normal constrants. Gven a data pont P attractng the surface at a gven pont p, the surface should be nterpolated whle movng p to P. It s also possble to specfy a normal vector N to be honored. s shown n Fgure 7, gven a set of ponts P attractng the surface at the pont p, the purpose of data fttng s to mnmze the sum of the squared dstances between the ponts P and the ponts p. In addton, each pont P may be provded wth a normal N to be honored as well (the trangle T contanng p should be made orthogonal to N. Ths yelds the followng postonal constrants c P x, c P y and c P z, and the normal constrants c1 N, c N and c 3 N. The coeffcents (λ 1, λ, λ 3 denote the barycentrc coordnates of p n T. The vertces (α 1, α, α 3 are the three vertces of T. Usng ths formalzaton, the three postonal constrants c x P, c y P and c z P to be honored n order to ft a data pont P = {P x, P y, Pz are gven by: ν {x, y, z, α {α 1, α, α 3, α / {α 1, α, α 3, b c ν = P ν /a P a = λ 1 + λ + λ 3 c ν P (α = λ /a c ν P = 0 Each normal vector N specfed at a trangle T = {α 1, α, α 3 yelds three normal constrants c 1 N, c N and c 3 N. The specfed normal s supposed to be a unt vector: (16

8 ν = {x, y, z ν c 1 N (α 1 = 0 ; ν c 1 N (α = Nν ; ν c 1 N (α 3 = Nν ν c N (α 1 = Nν ; ν c N (α = 0 ; ν c N (α 3 = Nν { Γ ν c (α = ν c (α. β α x ν c (α = η ν β Ω ν c (β.ϕν (β b c + x ν c (α η c (β.ϕη (β γ ν c (α = {ν c (α (0 ν c 3 N (α 1 = Nν ; ν c 3 N (α = Nν ; ν c 3 N (α 3 = 0 α / {α 1, α, α 3, ν c 1 N (α = ν c N (α = ν c N (α = 0 b c 1 = b c = b c 3 = 0 N N N How D.S.I. may be modfed to take nto account the so-defned lnear constrants s then shown below. 4. Constraned Dscrete Smooth Interpolaton The dscrete functonal F (ϕ defned n Equaton 18 below characterzes both the farness of ϕ and the degree of volaton of the constrants. The term F D s the ntegrated squared dscrete Laplacan (see Equaton 11. The second term ρ(ϕ s added to take nto account the set of constrants C where each data pont P yelds the postonal constrants c P x, c y and c P P z and/or the normal constrants c 1 N, c N and c 3 N. s often done n data fttng methods, ths term ρ(ϕ s weghted by a φ fttng factor enablng the user to tune the mportance of the constrants relatve to the farng. In addton, each ndvdual constrant c s modulated by a coeffcent ϖ c, whch makes t possble to tune ts mportance relatve to the other constrants: F (ϕ = F D (ϕ + φ.ρ(ϕ ρ(ϕ = c C ϖ c. {( α Ω ν c (α.ϕν (α bc Mnmzng the functonal F (ϕ can be realzed wth an algorthm smlar to the one presented n the prevous secton. The local updatng scheme has to be completed wth addtonal terms correspondng to the constrants (see Equatons 19 and 0. The terms G ν (α and g ν (α, nduced by the squared dscrete Laplacan, are not modfed (see Equaton 1 above. ϕ ν (α := Gν(α+φ.Γν (α g ν (α+φ.γ ν (α Γ ν (α = c C ϖ c.γ ν c (α γ ν (α = c C ϖ c.γ ν c (α Each ndvdual constrant c yelds the terms shown n Equaton 0 below. Note that the term x ν c (α s always null for a constrant c that does not combne the components x, y, z of ϕ, such as the postonal constrants c P ν. (17 (18 (19 s shown n Fgure 8, ths constraned nterpolaton method may be ntroduced n the subdvson algorthm presented n the prevous secton. In that case, when ntalzng the mesh M +1 from the mesh M, the constrants have to be assgned to the rght vertces. They can be easly found from the barycentrc coordnates λ 1, λ, λ 3 nvolved n the postonal constrants. 5 CONCLUSION In ths paper, we have presented the D.S.I. method, a new technque for dscrete farng. s compared to smlar approaches, our method enables lnear constrants to be honored n a least square sense. For nstance, t s possble to ft a set of ponts wth, or wthout, specfed normals. It provdes a modelng nterface for edtng polygonal meshes smlar to varatonal desgn [Gre94, MS9, Se98], wthout any restrcton regardng the topology of the objects. Moreover, our method does not requre an estmate of the normals as n [WW94], and also allows for the use of an arbtrary parameterzaton for estmatng the Laplacan. When used as a subdvson method, as wth any dscrete farng approach, D.S.I. offers more flexblty than statonary schemes. Durng the process, t then becomes possble to lock,unlock or move arbtrary vertces to have a fner control on the fnal surface. Moreover, snce the updatng scheme s the same for all vertces, no specal treatment s requred for extraordnary ponts. Furthermore, when creatng the mesh M +1 from the mesh M, the fact that no subdvson connectvty s requred makes t possble to splt the polygons adaptvely,.e. n hgh curvature zones only. Mult-resolutonal aspects wll be consdered n future research. In a mult-resoluton approach, such as the one presented n [KCVS98], t would be nterestng to study the nfluence of the lnear constrants at an arbtrary resoluton level. It would then be possble to ft a surface rch n fne-scale textural detals to a set of data ponts wthout blurrng those detals. Fgure 8 suggests that an nteracton model smlar to the one provded by the Wres deformaton technque [SF98] mght be desgned as well. In the framework of mesh compresson, D.S.I. mght be used as a predctor, meanng that just the dfference between the surface nterpolated by D.S.I. and the data needs to be stored. When used as a progressve descrpton wthn a networked envronment, the teratve D.S.I. algorthm enables a smooth transton between the dfferent levels of detals as they are loaded. Future research wll nclude the defnton of new lnear constrants, the nvestgaton of other functonals to be approxmated and optmzed, and applcatons of the method to volumc meshes. KNOWLEDGEMENTS Ths research has been performed n the frame of the G CD project, and the authors want to thank here the sponsors of the consortum, especally Gaz de France for supportng the Ph.D. of. Lévy. Specal thanks to Mchel Chpot for hs help concernng

9 mathematcs. Thanks to Corne Schlumberger and Jérome Mallot (las Wavefront for the dfferent 3D models they have kndly provded (the dog, the hand and the face. We want also to thank all the G CD research team and the T-Surf company for ther work, especally Rchard Cognot (G CD, Jean-Claude Dulac and Therry Valentn (T-Surf. References [CC78] [CG91] [DKT98] [DLG90] [DS78] [Gre94] E. Catmull and J. Clark. Recursvely generated b-splne surfaces on arbtrary topologcal meshes. Computer ded Desgn, 10(6: , G. Celnker and D. Gossard. Deformable Curve and Surface Fnte Elements for Free-Form Shape Desgn. In SIGGRPH Comp. Graph. Proc., pages CM, T. DeRose, M. Kass, and T. Truong. Subdvson Surfaces n Character nmaton. In SIGGRPH Comp. Graph. Proc., pages CM, July N. Dyn, D. Leven, and J. Gregory. butterfly subdvson scheme for surface nterpolaton wth tenson control. CM Transactons on Graphcs, 9(: , D. Doo and M. Sabn. ehavour of recursve dvson surfaces near extraordnary ponts. Computer ded Desgn, 10(6: , G. Grener. Varatonal desgn and farng of splne surfaces. Computer Graphcs Forum, 13: , [HDD + 94] H. Hoppe, T. DeRose, T. DuChamp, M. Halstead, H. Jn, and J. McDonald. Pecewse Smooth Surface Reconstructon. In Computer Graphcs, volume 8, pages 95 30, July [Kal93] [KCVS98] [KHD93] M. Kallay. Constraned optmzaton n surface desgn. Sprnger-Verlag, L. Kobbelt, S. Campagna, J. Voratz, and H.P. Sedel. Interactve Mult- Resoluton Modelng on rbtrary Meshes. In SIGGRPH Comp. Graph. Proc., pages CM, July M. Kass, M. Halstead, and T. DeRose. Effcent, Far Interpolaton Usng Catmull-Clark Surfaces. CM Computer Graphcs, 7:35 44, [Kob96a] L. Kobbelt. Varatonal pproach to Subdvson. CGD, 13: , [Kob96b] [Kob97] L. Kobbelt. Interpolatory Subdvson on Open Quadrlateral Nets. In Computer Graphcs Forum, L. Kobbelt. Dscrete Farng. In Proceedngs of the Seventh IM Conference on the Mathematcs of Surfaces, pages , [Kob98] L. Kobbelt. Varatonal Desgn wth Parametrc Meshes of rbtrary Topology. Teubner, [LM98] [Loo87]. Lévy and J.L. Mallet. Non-Dstorted Texture Mappng for Sheared Trangulated Meshes. In SIGGRPH Comp. Graph. Proc. CM, July C. T. Loop. Smooth subdvson surfaces based on trangles. Master s thess. Department of Mathematcs, Unversty of Utah, ugust [Mal89] J.L. Mallet. Dscrete Smooth Interpolaton n Geometrc Modelng. CM-Transactons on Graphcs, 8(:11 144, [Mal9] J.L. Mallet. Dscrete Smooth Interpolaton. Computer ded Desgn Journal, 4(4:63 70, 199. [MS9] H. Moreton and C. Séqun. Functonal Optmzaton for Far Surface Desgn. CM Computer Graphcs, 6: , 199. [PR97] [Re95] J. Peters and U. Ref. The Smplest Subdvson Scheme for Smoothng Polyhedra. CM Transactons on Graphcs, 16(4, October U. Ref. Unfed pproach to Subdvson Near Extraordnary Vertces. Computer ded Geometrc Desgn, 1: , [Sab91] M.. Sabn. Cubc Recursve Subdvson wth ounded Curvature, pages cademc Press, [Se98] H. P. Sedel. 3D Geometry Compresson wth Splnes. In SIGGRPH course notes: 3D Geometry Compresson (course 1. CM, July [SF98] [Sta98] [SZSS98] [WW9] [WW94] [WW98] [ZSS96] K. Sngh and E. Fume. Wres, Geometrc Deformaton Technque. In SIGGRPH Comp. Graph. Proc., pages CM, July J. Stam. Exact Evaluaton of Catmull-Clark Subdvson Surfaces at rbtrary Parameter Values. In SIGGRPH Comp. Graph. Proc., pages CM, July T.W. Sederberg, J. Zheng, D. Sewell, and M. Sabn. Non-Unform Recursve Subdvson Surfaces. In SIGGRPH Comp. Graph. Proc., pages CM, July W. Welch and. Wtkn. Varatonal Surface Modelng. In SIGGRPH Comp. Graph. Proc., pages CM, 199. W. Welch and. Wtkn. Free-Form Modelng Usng Trangulated Surfaces. In SIGGRPH Comp. Graph. Proc., pages CM, H. Wemer and J. Warren. Subdvson Schemes for Thn Plate Splnes. In EUROGRPHICS 98, D. Zorn, P. Schröder, and W. Sweldens. Interpolatng Subdvson for Meshes wth rbtrary Topology. In SIGGRPH Comp. Graph. Proc., pages CM, Fgure 9: Modelng blendng surfaces between a gven set of cylnders. : Lmt surface M ; : Mesh smoothed by D.S.I. Fgure 10: : orgnal surface. The red cubes correspond to control nodes ; : The surface obtaned by applyng D.S.I., The edges of the cube have been tagged as nfntely sharp creases and ts vertces as corners.

10 C D E F G I H J,,C,D: Meshes M0, M1, M and lmt surface M. Infntely sharp creases are hghlghted n yellow n M0 ; E: Closeup of the thumb nal ; F,G,H: cartoon-lke character obtaned by smoothng a smple mesh M0 ; I,J: Data ponts have been honored to nflate the nose. Fgure 11: