A New Solid Subdivision Scheme based on Box Splines

Transcription

1 A New Sold Subdvson Scheme based on Box Splnes Yu-Sung Chang Kevn T McDonnell Hong Qn Department of Computer Scence State Unversty of New York at Stony Brook ABSTRACT Durng the past twenty years, much research has been undertaken to study surface representatons based on B-splnes and box splnes In contrast, volumetrc splnes have receved much less attenton as an effectve and powerful sold modelng tool In ths paper, we propose a novel sold subdvson scheme based on tr-varate box splnes over tetrahedral tessellatons n 3D A new data structure s devsed to facltate the straghtforward mplementaton of our smple, yet powerful sold subdvson scheme The subdvson herarchy can be easly constructed by calculatng new vertex, edge, and cell ponts at each level as affne combnatons of neghborng control ponts at the prevous level The masks for our new sold subdvson approach are unquely obtaned from tr-varate box splnes, thereby ensurng hgh-order contnuty Because of rapd convergence rate, we acqure a hgh fdelty model after only a few levels of subdvson Through the use of specal rules over boundary cells, the B-rep of our subdvson sold reduces to a subdvson surface To further demonstrate the modelng potental of our subdvson sold, we conduct several sold modelng experments ncludng free-form deformaton We hope to demonstrate that our box-splne subdvson sold (based on tetrahedral geometry) advances the current state-of-the-art n sold modelng n the followng aspects: (1) unfyng CSG, B-rep, and cell decomposton wthn a popular subdvson framework; (2) overcomng the shortfalls of tensor-product splne models; (3) generalzng both subdvson surfaces and free-form splne surfaces to a sold representaton of arbtrary topology; and (4) takng advantage of trangle-drven, accelerated graphcs hardware Categores and Subject Descrptors I35 [Computer Graphcs]: Computatonal Geometry and Object Modelng Curve, surface, sold, and object representatons General Terms Algorthms, Desgn Emal: {yusung ktm qn}@cssunysbedu Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee SM 02, June 17-21, 2002, Saarbrucken, Germany Copyrght 2002 ACM /02/0006 $500 Keywords Representaton converson, Blends, sweeps, offsets & deformatons, Mult resoluton models, Geometrc and topologcal representatons, Reverse engneerng, User nteracton technques 1 INTRODUCTION Although sold modelng s more desrable n many engneerng and manufacturng applcatons, t has not yet ganed popularty untl recently due to both a lack of wdespread standards and ts strong need for more powerful computng resources The past two decades have wtnessed a sgnfcant growth n sold modelng, especally n the development of new sold representatons One such class of approaches employs mplct functons, such as CSG and blobby models [21] They represent a sold as the soluton set of an mplct functon, In general, a level set w = f(x, y, z) w = f(x, y, z), w= w 0 represents the boundary of the sold, and portons where w<w 0 comprse the nteror In mplct functon representaton, t s easy to dffer the nteror from the exteror and the boundary Hence, they are very well sutable for algebrac operatons on models [18] However, there are a few dsadvantages For nstance, there s no smple way to evaluate them n general cases [1] for renderng purpose In addton, drectly manpulatng the level-set geometry s very challengng because the sold boundary s mplctly defned Parametrc representatons also have been developed and are wellstuded because of ther mathematcal aspects, ncludng Bernsten- Bézer solds [13], B-splne solds, and other tensor-product based [10, 16] approaches Unlke mplct functons, parametrc representatons defne a sold by S(x, y, z) = v N (x, y, z), (1) where the N s are bass functons that satsfy certan propertes Tensor-product based solds n partcular have bass functons of the form N (x, y, z) =N 0 (x) N 1 (y) N 2 (z) where = ( 0, 1, 2) Due to ther tensor-product nature, ther domans are restrcted to a rectangular structure In general, n contrast wth mplct functons, the bass functons can be evaluated effcently and robustly Snce the poneerng work of Catmull and Clark [5] and Doo and Sabn [8] n the late 1970s, much research related to subdvson [9,

2 (a) (b) (c) Fgure 1: An example of our sold subdvson algorthm run on a smple model of 16 control ponts (a) Intal control lattce (b) Lattce after one level of subdvson (c) Lattce after three levels of subdvson The three lattces contan 24, 140, and 7280 cells, respectvely Faces are colored transparently to enhance the vsblty of nsde structures 15, 12] and ts analyss [19, 22] has been done However, most of such work has been focused on surface representatons, rather than solds One excepton s the work by MacCraken and Joy [16], whch generalzes tr-cubc B-splnes to solds of arbtrary topology Bajaj and Warren [2] also suggested alternatve sold subdvson rules for hexahedral meshes Subdvson algorthms nherently have several key propertes that make them very attractve n computer graphcs, engneerng, and manufacturng Some of these advantages nclude: Unformty of representaton, Multresoluton analyss and levels of detal, Numercal effcency and stablty, Arbtrary topology or genus, Smplcty n mplementaton, and Herarchcal structure Our motvaton s to combne the benefts of varous exstng sold modelng representatons by employng subdvson as ts foundaton We base our new subdvson scheme on volumetrc box splnes and take advantage of ts strong mathematcal foundaton In addton, we utlze non-tensor-product based tr-varate splnes to ensure topologcal freedom We ntroduce a new 3D regular structure that conssts of two dstnct types of polyhedra, whch has been brefly addressed n [3] and utlzed n [14] and [11], but has never been ntensvely employed prevously n sold modelng feld The new mesh provdes us wth the subdvson masks for our scheme that share regular topology and smplcty Smple affne combnatons over the mesh enable an effcent and robust evaluaton of the sold that can be expressed n the form of (1) By combnng sold representatons wth subdvson technques, our algorthm elegantly addresses many ssues that current sold modelng technques are confronted wth Its box splne foundaton provdes the advantage of hgh-order contnuty wthout the need to have hgh-order degree bass functons We also employ the well-known box splne based surface subdvson scheme [15] as our B-rep to facltate the data exchange wth current desgn and modelng requrements We demonstrate several applcaton examples to llustrate the sold modelng potental of our novel subdvson scheme Freeform deformaton [20, 10, 16] s one of examples through whch the advantage of our novel scheme s exhbted The underlyng tetrahedral mesh make t very easy to devse an effcent and accurate computaton of the coordnates n the correspondng parametrc doman In addton, a few drect modelng sessons are presented to demonstrate how easly and effectvely our approach can represent complex models of arbtrary topology 2 TRI-VARIATE BO SPLINE VOLUMES 21 3D Box Splnes There are many ways to defne box splnes, but one constructve way s by consderng a shadow (or mage) of a hgher dmensonal box n a lower dmensonal space [4, 7] More precsely, box splnes are defned by the projecton of n-dmensonal hypercubes onto m-dmensonal (m <n) affne space Analytcally, a box B of an n-dmensonal affne space A n s defned by B (p,p1,,p n) = {v A n v = p + c jp j,c j [0, 1]}, j where p s a box vertex and p j are lnearly ndependent vectors n A n that are representng n edges of an n-dmensonal box If all of the p j s are of unt length, we call t a cube or hypercube An affne map π : A n A m denotes a projecton onto an m- dmensonal affne space A m Consder the mage of B (p,p1,p n) wth respect to the map π We have π B = {w A m w = q + j c jq j,c j [0, 1]}, where q = π(p) and q j = π(p j ) Snce m<n, t s obvous that the q j s are not lnearly ndependent Hence, the pre-mage of each pont w A m forms a non-trval affne subspace: π 1 (w) ={v A n π(v) =w}, (2) whch s called a fbre of the map π at w We can also derve the fbre by solvng the lnear system c 1q 1 + c 2q c nq n = w q, whch has dmenson d = n m We now defne a box splne as M B(w) = vol π 1 (w) B vol U(w), where U s a fxed, d-dmensonal unt box parallel to the fbre Note that M B has local support and satsfes C d contnuty nsde the support Fnally, the normalzed verson N B s obtaned so

3 that t forms a partton of unty over the lattce Z m It satsfes C d l contnuty over the space where l s the largest dmenson of collapsng faces under π [6] For nstance, Loop s surface subdvson scheme [15] s based on the case of n =6, m =2 Our sold approach s the case of n =8, m =3 However, the lack of regular tessellaton over the lattce Z m (except cubc-grd) n 3D makes an analogy complcated Ths aspect s addressed n the followng sectons n detals 22 3D Regular Mesh Snce q j does not form a lnearly ndependent set on the m- dmensonal space, we are consderng a regular mesh on the range space and allowng some of the edges to overlap Unlke 2D space, 3D space does not have a regular mesh whch conssts of smplces, e, tetrahedra Nonetheless, t s possble to fll the space wth two types of polyhedra rather than a sngle type of equlateral tetrahedra The ntal form of the mesh can be seen by projectng a 4D hypercube onto 3D space through ts longest dagonal, or the man dagonal, (0, 0, 0, 0) (1, 1, 1, 1) Let the mappng be π d Then we have (0,0,1,1) π d ((1, 0, 0, 0)) = (1, 0, 0) = u 1, π d ((0, 1, 0, 0)) = (0, 1, 0) = u 2, π d ((0, 0, 1, 0)) = (0, 0, 1) = u 3, π d ((0, 0, 0, 1)) = ( 1, 1, 1) = u 4 (0,1,1,1) (0,1,0,1) (0,1,1,0) u (0,0,1,0) 3 u 2 (0,1,0,0) (1,0,1,1) (0,0,0,0) (1,0,1,0) (1,1,1,0) (1,1,0,0) u 1 (1,0,0,0) z + y + Fgure 3: A typcal example of the octet-truss M OT(Z 3 ) n 3D space It conssts of octahedral grds wth tetrahedra n between ages of sub-cells also form box splnes, but wth smaller supports The orgnal box splne can be wrtten as an affne combnaton of box splnes of sub-cells Snce every cube has edge vectors of the same orentaton, we represent a cube as C where M OT(Z 3 ), whch corresponds to the mage of the projectonal axs, e, the man dagonal of a cube We wrte smply S = S C Also, we use a hat notaton to represent sub-structures, e, ˆMî means a box splne of a sub-cell Ĉî where î M OT ( 1 2 Z3 ) Frst, n the lnear case (n =4, m =3), t s clear to notce where α,ĵ = 8 < : N = 1 2 ĵ α,ĵ ˆNĵ, (3) 2 f = ĵ 1 f s adjacent to ĵ 0 otherwse The adjacent nformaton among the s s controlled by the presence of edges n M OT Furthermore, f our sold s gven by S = (4) v N, (5) u (0,0,0,1) 4 (1,0,0,1) (1,1,0,1) Fgure 2: 4D hypercube projected on 3D space Fgure 2 shows an mage of 4D hypercube by π d The problem s that the mage does not form a polyhedral structure n 3D space We have to ntroduce addtonal edges to make t complete At the same tme, however, we do not want to ntroduce too many auxlary edges whch can cause unwanted complexty and rregularty The smplest regular structure that conssts of 4 mesh drectons {u 1,u 2,u 3,u 4} s called an octet-truss, whch s shown n Fgure 3 It may be noted that, ths structure serves as a mesh for our subdvson scheme In a nutshell, t conssts of two dstnct types of polyhedra tetrahedra and octahedra Each vertex has a valence of 14 n the regular case and shares the same topology as ts neghbors The mesh s denoted by M OT(Z 3 ) 3 SOLID SUBDIVISION RULES 31 Subdvson One of the attractve propertes of box splnes s that they can be decomposed nto an affne combnaton of splnes wth smaller support [3, 7] We frst consder a subdvson of the n-dmensonal hypercube nto 2 n cubes, or sub-cells wth a half edge length The m- x + where {v } are control ponts n A 3 and M OT(Z 3 ), then, by applyng (3) to (5), we obtan 1 ˆNĵ 2 S = where = ĵ v 1 2 wĵ ˆNĵ ĵ α,ĵ wĵ = 1 2 = ĵ α,ĵ v α,ĵ v ˆNĵ Therefore, control ponts wĵ for the refned mesh M OT( 1 2 Z3 ) can be expressed as affne combnatons of the orgnal control ponts, namely, the v s By conventon, N s denoted by N 1,1,1,1, snce each mesh drecton has only one edge from the hypercube projected onto t In the nterest of generatng hgh-order contnuous solds, our proposed scheme s based on n =8,orN 2,2,2,2 In ths case, each mesh drecton s projected twce by 8D hypercube edges Unlke n the lnear case, we need to count the mage of sub-cells twce Therefore, the N can be wrtten as N = 1 2 ĵ α,ĵ 1 c ˆk αĵ,ˆk ˆNˆk

4 Once agan, f we have a sold expressed by S = v N, then, S = j wĵ ˆNĵ, where wĵ = 1 2c = 1 2c ˆk ˆk α α,ĵ ĵ,ˆk v (6) β,ĵ,ˆk v (7) The coeffcent α s defned by (4) The varable c s a normalzaton factor to keep the summaton equal to one The value s 2 4 =16n our case Refned control ponts wĵ can be specfed as vertex, edge, or cell ponts, dependng on dfferent cases We call them vĵ,eĵ, and cĵ, respectvely : Vertex Ponts : Edge Ponts Fgure 4: A tetrahedron s subdvded nto an octahedron and 4 tetrahedra surroundng t : Vertex Ponts : Edge Ponts : Cell Pont Fgure 5: An octahedron s subdvded nto 6 octahedra wth 8 tetrahedra n between 32 Subdvson Mask We employ M OT(Z 3 ) as our regular mesh In the subdvson process, elements n the mesh are subdvded nto smaller ones Fgure 4 and Fgure 5 explan the way n whch the mesh elements are dvded The subdvded mesh agan forms an octet-truss of half the sze Therefore, after the frst level, any local rregulartes are contaned and only regular cases occur nsde Even though the ntroducton of octahedra offers the beneft of smplcty and regularty n structure, we need to choose an orentaton of the mesh due to asymmetry of our mask, whch s based on 4D hypercube projecton (Fgure 6) We devse ths orentaton by choosng one of the dagonals nsde an octahedron, whch s denoted by a major dagonal For each vertex pont vĵ, we defne v as an adjacent vertex f and only f there exsts an edge or a major dagonal between j and Also, an edge neghbor of eĵ s defned by a vertex of a tetrahedral cell that shares an edge ( 0, j) on whch j les, or a vertex of an octahedron that shares the edge and and whose major dagonal jons the vertex and one of the edge s end-ponts For cell ponts, cell neghbors are easly defned usng the vertces that comprse the cell Fgure 6: A regular subdvson mask It s a projected mage of a 4D hypercube that s vsualzed as an octet-truss Major dagonals (red dotted lnes) are ntroduced due to the asymmetrcal aspect of the mask 33 Subdvson Rules As mentoned prevously, each new control pont wĵ n M OT( 1 2 Z3 ) s obtaned by an affne combnaton of the v s n M OT(Z 3 ), whose coeffcents are defned by (6) The evaluaton of coeffcents could be a tedous process n general For nstance, n the ĵ = case, the coeffcent of v s 18, because β 32,ĵ,ˆk s equal to 4 f ĵ = ˆk and equal to 1 for all other adjacent ˆk s, whose number s 14 We classfy them nto the followng cases 331 Vertex Ponts Ths s the case when ĵ = for some s In the regular case, a new vertex pont s computed by vĵ = v + 1 v, (8) 32 where = ĵ and s an adjacent ndex of ĵ n M OT(Z 3 ) The number of adjacent vertces, { }, s equal to 14 n the regular case For the general case wth valence k, we can use vĵ = v + v, (9) 32 32k wthout much degeneraton It should be mentoned that the (9) only ensures convergence around rregular vertces (k 14) However, the dfference s hard to notce Therefore, t s used n our applcatons because of ts smplcty 18/32 : vertexpont( v ˆj = v ) : adjacent vertces( v ) Fgure 7: A vertex pont and ts mask n the regular case Only the top half s shown 4 more adjacent vertces are placed below the vertex pont Red dotted lnes ndcate major dagonals Gray areas ndcate faces whch belong to tetrahedra 332 Edge Ponts For the case ĵ for any, there are two sub-cases The frst case s an edge pont, whch les on an edge by 0 and 1 n M OT(Z 3 ) that s not a major dagonal of an octahedron In the

5 regular case, we have eĵ = (v 0 + v 1 ) v, (10) where denotes an edge neghbor of ĵ There are 6 edge neghbors, as shown n Fgure 8 In general case, we may use eĵ = (v 0 + v 1 )+ 12 v, 32k (11) where k s a number of edge neghbors 10/32 10/32 : edge pont ( eˆ) j : edge end ponts ( v, v ) 0 1 : edge neghbors ( v ) Fgure 8: An edge pont and ts mask Note that vertces on major dagonals (red dotted lnes) are ncluded Gray areas ndcate faces of tetrahedra 333 Cell Ponts When ĵ les on a major dagonal of an octahedron formed by 0 and 1, we use cĵ = 8 32 (v 0 + v 1 ) v, (12) where s a cell neghbor that belongs to the same octahedron (Fgure 9) The number of neghbors s always 4 8/32 4/32 4/32 4/32 4/32 8/32 : cellpont ( cˆ) j : major dagonal vertces ( v, v ) 0 1 : cell vertces ( v ) Fgure 9: A cell pont and ts neghbors The vertces on the major dagonal have dfferent weghts 34 Splttng and Reconnectng Because our algorthm keeps octahedra that are created durng the splttng of cells, t s obvous what the connectvty of subcells should be Each tetrahedron s splt nto 4 tetrahedra, each of whch conssts of one vertex pont and 3 edge ponts adjacent to t, and one octahedron that consst of 6 edge ponts that are generated by the 6 edges of the tetrahedron (Fgure 4) An octahedron s dvded nto 6 octahedra and 8 tetrahedra Each of the sub-octahedra comprses a vertex pont that s from each vertex of the octahedra, 4 edge ponts from an adjacent edges, and one cell pont Each new tetrahedron s made by connectng 3 edge ponts from one face and the cell pont (Fgure 5) 35 Boundary Surface The boundary requres specal treatment Snce our sold subdvson scheme s based on box splnes, t s natural to choose a box splne surface as ts B-rep We employ Loop s scheme [15] for ths purpose The weght used n the general case of valence k s based on a modfed verson by Warren [23] For surface vertces, masks nclude only neghbors that are on the surface Ths guarantees hgh-order contnuty not only n the nteror, but on the boundary as well Attenton also must be pad to the nterface between the boundary and nteror of a sold object For surface subdvson algorthms, one can ntroduce modfed rules [23] to acqure open (rather than closed) surfaces over 2D parametrc spaces However, the modfed masks are often very complex In sold subdvson, boundares and nterfaces are nevtable There could be some vsual rregulartes n nterfaces whch have to be addressed n future research Nonetheless, we can stll assure convergence and C 0 contnuty 4 IMPLEMENTATION 41 Data Structure Our subdvson scheme requres a data structure to handle two types of cells, even though our nput data consst of only tetrahedra We have mplemented a flexble structure whch can handle arbtrary type of faces and cells Edges (e, adjacency nformaton) and faces are reconstructed each tme the subdvson s nvoked In the nterest of memory effcency, ponters are used to record adjacency nformaton In each step, new vertces are generated by takng affne combnatons of vertces from the prevous level, formulated n (6) Ths can be expressed n the form of the subdvson matrx: 0 wĵ 1 C A = A 0 v 1 C A We need to mantan the matrx A for several reasons Even though dong so mposes a heavy memory requrement, ths data s crtcal for our Free-Form Deformaton (FFD) and other shape modelng applcatons to sustan real-tme performance It may be noted that matrces are requred only when we are performng free-form deformaton or drect manpulatng on the models Fortunately, snce most of the matrx conssts of zeros, sparse matrx storage schemes can be used to dramatcally reduce memory consumpton 42 Dagonal Orentaton One problem wth the mesh s that we need both to choose major dagonals for octahedra and to mantan ther orentatons In the frst level of subdvson, we can select dagonals for sub-octahedra arbtrarly snce we make no assumptons about the nput lattce However, each tme we choose dagonals, addtonal nformaton must be recorded to recover orentatons n the next level Each sub-tetrahedron remembers sub-octahedra n the same parent cell as orentaton references Hence, the next tme we subdvde the sub-cell, t refers to the lnked cell to extract orentaton nformaton In ths way, major dagonals are mantaned regularly durng the process

6 5 APPLICATIONS AND RESULTS 51 Free-Form Deformaton Free-form deformaton (FFD) s one of the mportant applcatons to whch our subdvson s drectly appled It plays a crucal role n graphcs, desgn, and manufacturng Usually, FFD nvolves generatng parametrc solds and translatng model coordnates back to parametrc space [20], so that changes to the solds can be reflected n the models Our approach s smlar to that of MacCracken and et al [16] However, unlke the tensor-product nature of Catmull-Clark solds and volumetrc splnes, our sold s much more flexble due to the tetrahedral structure of the mesh The followng s an overvew of our mplementaton of FFD: 1 Generate an approprate mesh that contans the model to be deformed 2 Subdvde the mesh up to the user-specfed level usng our new sold subdvson scheme 3 Calculate barycentrc coordnates for each vertex n the model on the fnal level 4 The user then nteractvely moves control ponts n any coarser level 5 Recalculate the coordnates by followng the subdvson matrces The barycentrc coordnates are easly computed Suppose a model vertex p les wthn a tetrahedron (v 0,v 1,v 2,v 3) The coordnate (c 1,c 2,c 3) s gven by p = v 0 + c 1 u 1 + c 2 u 2 + c 3 u 3, (13) where u = v v 0 for =1, 2, 3 Condtons 0 c 1,c 2,c 3 1 and 0 c 1 + c 2 + c 3 1 are requred to be n the tetrahedron We can solve the lnear system c1 c 2 c A = u 1 u 2 u 3 1 C A 1 0 p v 0 1 C A, to obtan the coordnate If a vertex s n an octahedral cell, we splt the octahedron nto 4 parts usng a major dagonal and compute the barycentrc coordnate wthn the correspondng tetrahedron Fgure 10 and Fgure 11 show two applcatons of FFD The mesh does not necessarly have a smple topology (e, genus zero), as shown n 12 It s also possble that we manpulate objects locally (Fgure 13), by smply assgnng control meshes to certan regon of an object that we want to deform Note that all processes are done n real-tme except barycentrc coordnate computaton Even the coordnate computaton can be done n a matter of seconds Most of tme s consumed by ntersecton checks between tetrahedra/octahedra and model vertces We are workng on several methods to accelerate these checks 52 Drect Manpulaton of Solds Fgure 1 and Fgure 14 demonstrate examples of varous models that can be obtaned by our sold subdvson The tetrahedral structure offers the greatest freedom to generate objects of arbtrary topology Also, by smple user nteracton, we can perform real-tme modfcatons on solds by manpulatng control ponts at arbtrary levels We can also ntroduce dscontnutes on vertces or along edges by assgnng exceptonal rules (e, smple bsecton wthout weghts) to desred parts of objects 53 Performance All of our results have been run on a wde-range of consumer level PCs whch do not have any specal hardware for volume vsualzaton We were able to perform some of the free-form deformatons on a relatvely low-end system (Intel Celeron 700MHz wthout hardware accelerated OpenGL renderng) It clearly demonstrates how effcent our subdvson algorthm s n many aspects In most cases, the only requrement s a large amount of memory (desrably more than 256 MB), whch could be further allevated by optmzng the data structure Even volume vsualzaton can be done on PCs by usng the OpenGL 3D texture mplementaton avalable n some recent vdeo cards On a Pentum III 1 GHz machne wth 1 GB RAM, the car model (Fgure 10) requred only 0701 seconds to subdvde up to level 3 (7280 cells) The coordnate update, trggered by user nput, takes roughly 003 seconds n each tme, whch guarantees realtme nteracton The flter model (Fgure 11) conssts of vertces and faces, takes 05 seconds to subdvde meshes (5200 cells) The nteracton cost was comparable to that of the car model Even though t s possble to use hgher subdvson levels wthout much addtonal computatonal cost, the results are vsually ndstngushable 6 CONCLUSION AND FUTURE WORK We have developed a novel sold subdvson scheme based on powerful box splnes The new sold subdvson scheme has a lot of potental, especally n sold modelng Snce t s founded on well-defned box splnes, t s easy to ntegrate wth current ndustral standard models based on splnes It can also acheve hghorder contnuty wth relatvely low degree bass functons and numercal stablty Its B-rep, whch s crtcal n renderng, s the well-known box splne surface The tetrahedral mesh affords users much freedom n modelng and deformaton, n such way that other tensor-product based or parametrc solds can not represent Moreover, the subdvson nature of the entre process offers fast evaluaton wthout much numercal cost, provdes a multresoluton soluton, and facltates real-tme manpulaton Our free-form deformaton applcaton demonstrates the robustness and effcency of our novel scheme The structural smplcty of our meshes gves rse to the crtcal robustness ssue that s requred to manpulate hghly complcated models, and the rapd convergence rate makes real-tme user nteracton possble Levels of detals can be easly computed smply by choosng the subdvson level, and effects from one level are propagated by means of the subdvson matrx wthout any extra numercal cost Therefore, our subdvson algorthm could offer a number of advantages to nteractve sold modelng, for nstance, Vrtual Clay by McDonnell et al [17] Fnally, even though the process does not generate any new extraordnary ponts, orgnal control ponts may nclude non-regular cases whch are not yet fully analyzed and addressed Many researchers have analyzed extraordnary cases for surface subdvsons [19, 22], and these methods can be appled to our sold subdvson We have already analyzed cases usng subdvson matrx [8] and numercal technques, and the values currently used n our experments assure convergence and certan level of vsual contnuty around extraordnary vertces A complete analyss s another topc for future research

7 (a) (b) (c) (d) Fgure 10: Free-form deformaton of a car model The model contans 2244 vertces wth 21 dfferent components (a) The orgnal model (b) The subdvson mesh (c) The deformed model (d) Underlyng subdvson sold All deformaton has been done n real-tme (a) (b) (c) (d) Fgure 11: Another example of free-form deformaton of an ndustral flter block model The model contans vertces wth more than faces, and s converted from the B-splne surface model (a) The orgnal model (b) The subdvson mesh (c) The deformed model (d) The deformed model n wreframe (a) (b) (c) (d) Fgure 12: A mesh wth non-trval topology In ths case, our deformaton mesh contans a hole (a) The orgnal model (b) The subdvson mesh n level 2 (c) The part of the model (the central cylnder) that s nsde the hole has not been changed (d) The deformed subdvson mesh Acknowledgments Ths research was supported n part by the NSF CAREER award CCR , the NSF grants IIS and IIS , Honda Intaton Award, and Alfred P Sloan Fellowshp 7 REFERENCES [1] P R Atherton A scanlne hdden surface removal procedure for constructve sold geometry Computer Graphcs (SIGGRAPH 83 Proceedngs), 17(3):73 82, July 1983 [2] C Bajaj, J Warren, and G u A smooth subdvson scheme for hexahedral meshes The Vsual Computer, 2001 To appear n the specal ssue on subdvson [3] W Boehm Subdvdng multvarate splnes Computer-Aded Desgn, 15: , Nov 1983 [4] W Boehm Calculatng wth box splnes Computer Aded Geometrc Desgn, 1(2): , 1984 [5] E Catmull and J Clark Recursvely generated B-splne surfaces on arbtrary topologcal meshes Computer-Aded Desgn, 10: , Sept 1978

8 (a) (b) (c) (d) Fgure 13: Localzed free-form deformaton We can choose any regon of the model and perform FFD (a) The orgnal model (b) Locally deformed model (c) The orgnal model (d) Locally deformed model Each model contans 3760 and 7854 vertces, respectvely (a) (b) (c) (d) Fgure 14: Examples of some models and ther manpulatons (a) The orgnal model wth genus two (b) The deformed model (c) The orgnal model wth genus one (d) The plane made from the model (c) [6] C de Boor and K Höllg B-splnes from paralleleppeds J Analyse Math, 42:99 115, 1982 [7] C de Boor, K Höllg, and S Remenschneder Box Splnes Sprnger-Verlag, New York, 1993 [8] D Doo and M Sabn Behavour of recursve dvson surfaces near extraordnary ponts Computer-Aded Desgn, 10(6): , Sept 1978 [9] N Dyn, D Levn, and J Gregory A butterfly subdvson scheme for surface nterpolaton wth tenson control ACM Transactons on Graphcs, 9(2): , Aprl 1990 [10] J Gressmar and W Purgathofer Deformaton of solds wth trvarate B-Splnes In Proceedngs of Eurographcs 89, pages , 1989 [11] M Hall and J Warren Adaptve polygonalzaton of mplctly defned surfaces IEEE Computer Graphcs and Applcatons, 10(6):33 42, Nov 1990 [12] L Kobbelt Interpolatory subdvson on open quadrlateral nets wth arbtrary topology In Computer Graphcs Forum (Proceedngs of Eurographcs 96), volume 15(3), pages , 1996 [13] D Lasser Bernsten-bezer representaton of volumes Computer Aded Geometrc Desgn, 2(1-3): , 1985 [14] E Letner and S Selberherr Mxed-element decomposton method for three-dmensonal grd adaptaton IEEE Trans on Computer-Aded Desgn of Integrated Crcuts and Systems, 17(7): , July 1998 [15] C Loop Smooth subdvson surfaces based on trangles Master s thess, Unversty of Utah, Dept of Math, 1987 [16] R MacCracken and K I Joy Free-Form deformatons wth lattces of arbtrary topology In SIGGRAPH 96 Computer Graphcs Proceedngs, Annual Conference Seres, pages , Aug 1996 [17] K T McDonnell, H Qn, and R A Wlodarczyk Vrtual clay: A real-tme sculptng system wth haptc toolkts In Proceedngs of the 2001 ACM Symposum on Interactve 3D Graphcs, pages , March 2001 [18] A Pasko and VSavchenko Algebrac sums for deformaton of constructve solds In Proceedngs of Sold Modelng 95, pages , May 1995 [19] H Prautzsch Generalzed subdvson and convergence Computer Aded Geometrc Desgn, 2(1-3):69 76, 1985 [20] T W Sederberg and S R Parry Free-form deformaton of sold geometrc models In Computer Graphcs (SIGGRAPH 86 Proceedngs), volume 20, pages , Aug 1986 [21] B Wyvll, C McPheeters, and G Wyvll Anmatng soft objects The Vsual Computer, 2(4): , 1986 [22] D Zorn Smoothness of statonary subdvson on rregular meshes Constructve Approxmaton, 16:3, 2000 [23] D Zorn and P Schröder Subdvson for modelng and anmaton In SIGGRAPH 2000 Course Notes, 2000