Physically based morphing of point-sampled surfaces

Transcription

1 COMPUTER ANIMATION AND VIRTUAL WORLDS Comp. Anm. Vrtual Worlds 005; 16: Publshed onlne n Wley InterScence ( DOI: /cav.100 Anmatng Geometrcal Models Physcally based morphng of pont-sampled surfaces By Yunfan Bao*, Xaohu Guo and Hong Qn Ths paper presents an nnovatve method for naturally and smoothly morphng pontsampled surfaces va dynamc meshless smulaton on pont-sampled surfaces. Whle most exstng lterature on shape morphng emphaszes the ssue of fndng a good correspondence map between two object representatons, ths research prmarly nvestgates the challengng problem of how to fnd a smooth, physcally-meanngful transton path between two homeomorphc pont-set surfaces. We analyze the deformaton of surface nvolved n the morphng process usng concepts n dfferental geometry and contnuum mechancs. The morphng paths can be determned by optmzng an energy functonal, whch characterzes the ntrnsc deformaton of the surface away from ts rest shape. As demonstrated n the examples, our method automatcally produces a seres of natural and physcally-plausble n-between shapes, whch greatly allevates the shrnkng, stretchng, and self-ntersecton problems that often occur when lnear nterpolaton s employed for the morphng of two objects. We envson that our new technque wll contnue to broaden the applcaton scope of pont-set surfaces and ther dynamc anmaton. Copyrght # 005 John Wley & Sons, Ltd. KEY WORDS: physcally based anmaton; dynamc shape modelng; morphng; meshless method; pont-based geometry Introducton Pont-based representaton for surfaces has ganed strength and become an attractve, powerful modelng, and renderng alternatve n computer graphcs n recent years. The rapd development of the 3D scannng devces has greatly facltated the acquston of the pont samples from real-world physcal objects. Snce pont-sampled geometry can be represented by a set of dscrete ponts, no explct connectvty nformaton needs to be mantaned. Ths leads to a more compact and flexble representaton of geometry n terms of data *Correspondence to: Yunfan Bao, Computer Scence Department, State Unversty of New York at Stony Brook, Stony Brook, NY, 11794, U.S.A. E-mal: ybao@cs.sunysb.edu; xguo@cs.sunysb.edu; qn@cs.sunysb.edu Contract/grant sponsor: NSF; contract/grant number: ACI Contract/grant sponsor: ITR; contract/grant number: IIS storage and transfer. Many researchers have devoted consderable amount of effort to the accurate representaton, effcent processng, and renderng of pontsampled geometry. However, rapd and accurate anmaton/smulaton of pont-sampled geometry s stll a challengng area that demands a great deal of research endeavor wthn pont-based graphcs. As a popular anmaton technque n dgtal entertanment, shape morphng (or blendng) has been a very actve research topc n computer graphcs. Gven a source object and a target object, morphng technques can be employed to create a seres of n-between shapes that transform the source object nto the target one. In ths paper, we systematcally develop a physcally based meshless method to morph two objects represented only by ponts. To our best knowledge, ths s a frst attempt to conduct a dynamc, physcallymeanngful shape morphng between two pontsampled surfaces. In contrast, most exstng morphng technques nowadays are based on the polygonal mesh representaton, owng to the popularty and long hstory Copyrght # 005 John Wley & Sons, Ltd.

2 Y. BAO, X. GUO AND H. QIN of mesh geometry (as the domnant shape representaton) n Graphcs. Despte the wdespread use of meshbased shape morphng technques, certan drawbacks persst. For example, the map overlay algorthm needed to generate any ntermedate mesh wth consstent connectvty from source and target surface meshes are notorously dffcult to mplement. Ths remeshng process can be crcumvented by usng pont sampled surfaces snce no connectvty nformaton s necessary durng shape morphng. Gven two shapes, there are nfnte possble morphng paths. Our method focuses on fndng an deal transformaton va physcs-based energy optmzaton. We approach the morphng path problem from the vewpont of both dfferental geometry and contnuum mechancs. Our method s completely mesh free and only the vcnty nformaton of the pont set s utlzed n the morphng procedure. In our framework, the pont-sampled geometry s modeled as a meshless thn shell surface. The energy functonal s defned usng the ntrnsc measurement of surface deformaton derved from dfferental geometry. The morphng path s then automatcally derved by mnmzng the energy functonal. One key beneft of our method s that t mnmzes unnecessary dstorton such as shrnkng, stretchng, and self-ntersecton of the surface as demonstrated n Fgure 1, whch often exsts n lnearnterpolaton-based methods. Furthermore, surface crack problem assocated wth pont-based morphng 1 can be avoded n our physcally based system. Therefore, the ntermedate shapes generated by our method are both physcally plausble and vsually natural as shown n Fgure. Fgure 1. The undesrable artfact caused by lnear blendng. (a) Intal shape. (b) 50 % morph. (c) Fnal shape. Fgure. Our method leads to a physcally-plausble ntermedate shape. (a) Intal shape. (b) 50% morph. (c) Fnal shape. Related Work Shape Morphng Shape morphng s an actve and nterestng research area n computer anmaton. A complete survey s beyond the scope of ths paper and we shall only brefly revew several lteratures that are most relevant to our work here. The readers are referred to References [,3] for more detals. The key to mesh-based morphng s to establsh a good mappng between the source object and the target object, whch s known as the correspondence problem. Defnng such a mappng s far from trval, snce t usually nvolves parameterzaton 4 of surfaces of arbtrary genus and the problem becomes much more challengng when the two surfaces have dfferent topology. After the parameterzaton step, a map overlay algorthm s used to generate the common connectvty. Whle the correspondence problem has been extensvely studed by researchers, the path problem s relatvely less-explored, hence demandng more research effort towards further mprovement. The goal of the path problem s to fnd an deal transton path between the source and target shapes. However, the fundamental queston of What consttutes an deal path? s rather subjectve. A nave choce for most morphng applcatons s lnear nterpolaton because of ts smplcty. However, a straghtforward applcaton of lnear nterpolaton can somehow lead to dspleasng vsual results as shown n Fgure 1, especally n cases that nvolve dramatc, near-rgd-body transformaton between the end shapes. To address ths knd of problems, Sederberg et al. 5 ntroduced a technque that mnmzes the deformaton of the boundares of D shapes. Alexa et al. 6 took the nteror of shapes nto account and decomposed an affne transformaton to mprove the morphng qualty. One smlar work to ours s that of Yan et al., 7 whch s based on nonlnear stran feld nterpolaton derved from physcs. Whle ther method can only morph two planar polygons, our novel Copyrght # 005 John Wley & Sons, Ltd. 510 Comp. Anm. Vrtual Worlds 005; 16:

3 MORPHING OF POINT-SAMPLED SURFACES method handles complcated 3D pont-sampled surfaces. Furthermore, we derve the formulaton for meshless, thn-shell dynamcs as lnear systems, whch s then solved to reconstruct the n-between shapes. PhyscallyBasedAnmatonandMeshless Methods Followng the poneerng work of Terzopoulos et al., 8 a large amount of research has been carred out n the feld of physcally based anmaton. For example, many mesh-based methods for physcal smulaton of deformable objects have been proposed n computer graphcs based on ether the boundary element method 9 or the fnte element method. 10 Cloth, whch can be modeled as a thn-plate, was recently studed. 11,1 Compared wth cloth, thn-shell objects are naturally curved and can not be modeled usng plate formulatons. 13 Grnspun et al. 14 proposed a smple dscrete shell model that can be derved geometrcally for trangle meshes. Most of the exstng physcal smulaton approaches are based on mesh structures. For pont-sampled surfaces, however, t would be extremely challengng to smulate the contnuum mechancs wthout the explct nformaton of mesh connectvty. In recent years, consderable research has been devoted to the effcent modelng and anmaton of pontsampled geometry. In Reference [15], the Pontshop 3D system was presented for nteractve shape and appearance edtng of 3D pont sampled geometry. Later, Pauly et al. 16 presented a free-form shape modelng framework for pont-sampled geometry usng the mplct surface defnton of the movng least squares (MLS) approxmaton. More recently, Guo et al. 17 developed a framework for local and global edtng of pont set surfaces based on level-set method. Mueller et al. 18 developed a method for anmatng elastc, plastc, and meltng pontbased volumetrc objects based on the MLS approxmaton of the gradent of the dsplacement feld. Xao et al. 1 proposed an approach for nteractve morphng of pont surfaces based on Floater s 19 meshless parameterzaton. Most recently, Pauly et al. 0 presented a new meshless anmaton framework for elastc and plastc materals that fracture. Guo and Qn 1 combned the meshless method wth the modal warpng technque to acheve real-tme deformaton. In ths paper, we wll demonstrate that the meshless method s a natural and ntutve soluton to performng physcal smulaton drectly on pont-sampled surfaces, hence leadng to the automatc, physcally-plausble shape morphng of pont-set geometry. Deformaton Analyss Deformaton of Surface The deformaton of surfaces and solds has been wellstuded n elastcty theory and contnuum mechancs. In ths paper, we defne the deformaton of a surface usng concepts n dfferental geometry []. A surface n R 3 can be defned parametrcally as a vector functon Xð 1 ; Þ, where 1 and are the parametrc coordnates of the surface. The quadratc form (n Ensten summaton conventon) ds ¼ g j d d j measures the length of an arc element on the surface, and s known as the frst fundamental form. The coeffcents g j are components of a covarant tensor whch s called the metrc tensor or fundamental tensor: g j ðxþ ¼X X j where X Intutvely, the frst fundamental form measures the stretchng and shearng of the underlyng surface. The frst fundamental form alone does not completely determne the shape of a surface, because the curvature can be altered wthout affectng the metrc. Hence the second fundamental form whch takes the form b j d d j needs to be ntroduced. The second fundamental form measures the curvature of a surface and the coeffcents b j are components of a tensor called the curvature tensor: b j ðxþ ¼X j n ¼ X n j where X j j, n and n s the unt normal vector. The second fundamental form, together wth the frst fundamental form, can entrely determne the shape of a surface and therefore s an ntrnsc measurement of the shape of the surface. Deformaton Energy Havng defned the metrc tensor and the curvature tensor, the elastc stran energy for a deformable surface s gven by 8 Z ¼ ðjjg n g 0 jj þ jjb n b 0 jj Þd where g 0, b 0 are the metrc tensor and curvature tensor assocated wth the rest shape of the surface; g n, b n are correspondng tensors assocated wth the deformed Copyrght # 005 John Wley & Sons, Ltd. 511 Comp. Anm. Vrtual Worlds 005; 16:

4 Y. BAO, X. GUO AND H. QIN shape of the surface and jj jj s the Frobenus norm. The frst term n the stran energy formulaton s the membrane energy, whch ressts stretchng and shearng, and the second term s the bendng energy, whch ressts bendng and twstng. The membrane and bendng energy dstrbuton of a deformng torus s shown n Fgure 7. The above formulaton of the deformaton energy s purely motvated by dfferental geometry and apples to any surface n R 3. For any pont-sampled surfaces, f we assume that one dmenson,.e., the thckness, of the surface body s sgnfcantly smaller than the other two dmensons, we can consder the pont-sampled surface as a thn shell. The generc confguraton of the shell can be descrbed as S ¼ x R 3 jx ¼ Xð 1 ; Þþ 3 X 3 ð 1 ; Þ; 1 ; and h 3 h where X 3 s a unt drector vector normal to the mddle surface of the shell both n the reference and deformed confguraton under the Krchhoff Love hypothess. In the Krchhoff Love thn shell framework, the deformaton of the surface body s fully characterzed by the deformaton of the mddle surface. The Green Lagrange stran tensor can therefore be derved from the frst and second fundamental forms of the mddle surface of the shell. The membrane and bendng stran tensors are related to the deformaton of the shell surface as follows j ¼ 1 X X j X 0 X 0 j j ¼ 1 X X 3j X 0 X 0 3j Here, we use the superscrpt 0 to denote the measurement n the orgnal (reference) confguraton. Assumng lnearzed knematcs, the dsplacement feld of the mddle surface s ntroduced as uð 1 ; Þ¼ Xð 1 ; Þ X 0 ð 1 ; Þ. Thus, the lnearzed membrane and bendng stran can be wrtten as j ¼ 1 j ¼ 1 X0 u j þ X 0 j u X0 X 3j þ X 0 j X 3 þ u X 0 3j þ u j X 0 3 ð1þ ðþ Introducng the Krchhoff Love hypothess explctly here, the deformed shell drector vector s constraned to concde wth the unt normal vector to the deformed mddle surface of the shell,.e., X 3 ¼ J 1 ðx 1 X Þ where J ¼jX 1 X j. Therefore, the dervatves of the drector vector n the reference confguraton are X 0 3 ¼ðJ0 Þ 1 X 0 1 X0 þ X0 1 X0 ð3þ The ncrement X 3 ¼ X 3 X 0 3 can also be derved straghtforwardly by only keepng lnear terms. X 3 ðj 0 Þ 1 u 1 X 0 þ X0 1 u Smlar to the dervaton of Equaton (3), we can wrte the dervatves of X 3 as follows X 3 ¼ðJ 0 Þ 1 u 1 X 0 þ X 1 X 0 þ X0 1 u þ X 0 1 u Fnally, the bendng stran can be expressed n terms of the dsplacement feld as h j ¼ u j X 0 3 þðj0 Þ 1 u 1 X 0 j X0 þ u X 0 1 X0 j ð4þ Note that the dervaton of the membrane stran s ndependent of the ntroducton of the Krchhoff Love hypothess. Gven two homeomorphc pont-set surfaces, we can measure the metrc tensor and curvature tensor pontwsely on the two surfaces. Through the use of nterpolaton, we can obtan the tensor feld for any ntermedate tme step t ½0; 1Š. The dfference between the nterpolated tensor feld and the tensor feld of the rest confguraton gves us the stran feld that characterzes the ntrnsc deformaton of the surface away from ts rest, ntal shape. We can then apply the lnear membrane (Equaton 1) and bendng stran (Equaton 4), whch are functons of the dsplacement to mnmze the followng energy W ¼ 1 Z jjðuþ t jj þ jjðuþ t jj d ð5þ where t and t are the membrane and bendng stran obtaned by nterpolaton at tme step t. In our examples, we use smple lnear nterpolaton for the membrane and bendng stran: t ¼ t ðg n g 0 Þ and t ¼ t ðb n b 0 Þ. More sophstcated nterpolaton scheme can be used for smoother vsual qualty. Copyrght # 005 John Wley & Sons, Ltd. 51 Comp. Anm. Vrtual Worlds 005; 16:

5 MORPHING OF POINT-SAMPLED SURFACES MeshlessTechnques Snce the nventon of the fnte element method (FEM) n the 1950s, FEM has become the most popular and wdely used method n engneerng, scentfc computng, and computer anmaton. In FEM, the ndvdual elements are connected together by a topologcal map, whch s usually called a mesh. The fnte element nterpolaton functons are then bult upon the mesh, whch ensures the compatblty of the nterpolaton. However, ths procedure s not always advantageous, because the numercal compatblty condton s not the same as the physcal compatblty condton of a contnuum. For nstance, n a Lagrangan type of computatons, one may experence mesh dstorton, whch can ether end the computaton altogether or result n drastc deteroraton of accuracy. Therefore, t would be computatonally effcacous to dscretze a contnuum by only a set of nodal ponts wthout mesh constrants. In ths paper, we utlze MLS approxmaton method whch s used n the Element Free Galerkn (EFG) method 3 to arrve at the numercal dscretzaton. MLS Approxmaton The local approxmaton u h of a feld functon uðxþ defned n a soluton doman of arbtrary dmenson,, can be expressed as the nner product of a vector of the polynomal bass, pðxþ, and a vector of the coeffcents, aðxþ u h ðxþ ¼p T ðxþaðxþ ¼ Xm j¼1 p j ðxþa j ðxþ ð6þ where m s the number of monomals n the polynomal bass. In our current work on two-manfold surfaces, a lnear bass, p T ¼ð1; x; yþ correspondng to m ¼ 3, s used. If the feld values at a set of nodes, x, ¼ 1;...; n, are known a pror, the coeffcent vector aðxþ can be determned by mnmzng a weghted, dscrete L error norm defned as: L ¼ Xn ¼1 wðx; x Þ½u h ðx Þ u Š ð7þ Fgure 3. The smaller red balls represent the smulaton nodes and the larger translucent whte hemspheres represent the support rad of two of the smulaton nodes. where wðx; x Þ s a weghtng functon defned over a compact support (also called the doman of nfluence of node ), u s the nodal value at x, and n the number of nodes whose doman of nfluence contans the evaluaton pont x. The statonary of L wth respect to aðxþ leads to the soluton of aðxþ. Substtuton of aðxþ nto Equaton (6) gves u h ðxþ ¼ Xn ¼1 ðxþu ð8þ wth ðxþ beng the MLS shape functon. More detals on the dervaton of the shape functons can be found n Reference [4]. Fgure 3 shows the smulaton nodes scattered over a thn plane and ther correspondng support regons. Meshless Dscretzaton It now sets the stage for us to address the dscretzaton ssue of the energy defned n Equaton (5) that we attempt to mnmze. To make the mathematcal formulaton concse and consstent, we arrange the elements n symmetrc tensors nto 3 1 vectors and, for membrane and bendng stran, respectvely ¼ 4 5; ¼ Usng ths notaton, the energy functonal to be mnmzed can be reformulated as Z W ¼ f½ðuþ t Š T ½ðuÞ t Š þ ½ðuÞ t Š T ½ðuÞ t Šgd Settng the dervatve wth respect to u to be 0, we @u ½ðuÞ ½ðuÞ t Š d ¼ 0 Copyrght # 005 John Wley & Sons, Ltd. 513 Comp. Anm. Vrtual Worlds 005; 16:

6 Y. BAO, X. GUO AND H. QIN Applyng Equaton (8), the dsplacement feld can be approxmated by the MLS shape functons I ð 1 ; Þ as u h ð 1 ; Þ¼ XN I¼1 I ð 1 ; Þu I ð9þ where u I are the nodal dsplacement vectors and N s the number of smulaton nodes. Substtutng Equaton (9) nto Equaton (1) and Equaton (4) gves the approxmated membrane and bendng stran n matrx form ð 1 ; Þ¼ XN I¼1 ð 1 ; Þ¼ XN I¼1 M I ð 1 ; Þu I B I ð 1 ; Þu I ð10þ ð11þ where M I and B I are the membrane and bendng stran matrces assocated wth smulaton node I. Fnally, ntroducng Equaton (10) nto Equaton (11) yelds the lnear equaton: Ku ¼ f ð1þ where K s the stffness matrx, and u s the nodal dsplacement vector, and f s the vrtual nodal force vector. The global stffness matrx K s a block matrx whch can be convenently assembled by fllng n lowlevel 3 3 stffness matrces defned as Z h K IJ ¼ ðm I Þ T M J þ ðb I Þ T B J d And the vrtual nodal force vector can be assembled smlarly Z h f I ¼ ðm I Þ T I þ ðb I Þ T I d The lnear system n Equaton (1) can be solved usng the b-conjugate gradent method. Local Parameterzaton In prevous dscusson, we have assumed that a common analyss doman exsts for both the source and target pont-set surfaces, however, ths may not be true n general. In fact, n order to obtan such a global analyss doman, global parameterzaton of the pontset surfaces s needed. In ths paper, we adopt local parameterzaton to bypass the expensve global parameterzaton step, and the physcal smulaton can be drectly performed on the pont-based surfaces usng the vcnty nformaton only. We proceed by defnng, for every surface pont p,a neghborhood of surface ponts N. The neghborhood can be obtaned by smply takng the K nearest neghborng ponts. In our mplementaton, we set K to be between 0 and 30. The local parameterzaton s defned on the local tangent space, whch can be computed usng prncple component analyss (PCA). The two egenvectors correspondng to the two largest egenvalues span the tangent space at pont p. The coordnate of any pont n the vcnty of p s then obtaned by projectng the pont onto ts local tangent space. In order to compute the stran matrces, partal dervatves of X up to the second order are needed. We compute the dervatves by mnmzng an MLS error functon. As an example, we consder the x-component of the poston n the neghborhood of a pont p.by Taylor expanson, we can approxmate the x-component of the surface poston on the local parameter doman to frst order as ~xðxðp ÞþXÞ xðxðp ÞÞþrxj p X where Xðp Þ s the local parameter value of pont p. Therefore, we can estmate the dervatve rxj p by mnmzng the followng weghted least square error: ~L ¼ X jn w j ½~xðXðp j ÞÞ xðxðp j ÞÞŠ Note that, these partal dervatves are only defned on the local parameter doman of p and therefore all coordnate values must be computed wth respect to the local tangent space of p. The second order partal dervatves of p can be obtaned smlarly after the frst order partal dervatves for all neghborng ponts of p have been computed. Incremental Update Due to the fact that lnear membrane and bendng strans are used to mnmze the energy functonal Equaton (5), large deformaton from the rest confguraton can lead to dstorton artfacts as shown n Fgure 4(a). We correct ths problem by ncrementally updatng the global stffness matrx K. Each updatng step only recomputes the partal dervatves and tensors of the surface wth respect to the fxed local parameter doman. Ths process of ncrementally updatng the stffness matrx can be consdered as usng small lnear Copyrght # 005 John Wley & Sons, Ltd. 514 Comp. Anm. Vrtual Worlds 005; 16:

7 MORPHING OF POINT-SAMPLED SURFACES Fgure 4. Physcally based morphng of a bendng bar: (a) wthout and (b) wth ncrementally updatng stffness matrx. steps to approxmate the nonlnear deformaton n ts lmt. The corrected shape s shown n Fgure 4(b). The updatng step sze can be adaptvely controlled by settng threshold of the resdual energy after mnmzng the energy functonal W. Once the resdual energy exceeds the threshold, the stffness matrx s dynamcally updated. Our experment shows that recomputng the stffness matrx 10 tmes wth unform step length yelds satsfactory results for most of our experments. Numercal Integraton Assemblng the stffness matrx K and the vrtual nodal force f nvolves numercal ntegraton over a global analyss doman. Ths ntegraton s typcally computed usng Gaussan quadrature. Snce our computaton s performed on local parameter doman, such ntegraton s nfeasble. Alternatvely, we approxmate the ntegraton as K IJ ¼ XNP f I ¼ XNP h w ðm I Þ T M J þ ðb I Þ T B J h w ðm I Þ T I þ ðb I Þ T I Fgure 5. Algorthmc overvew and flowchart. Model Surfels Nodes Frames Tme Bar Torus Iss Rabbt Moa Sphere Table 1. Model statstcs and computatonal cost (seconds/frame) where NP s the number of surface ponts and w s the surface area approxmaton assocated wth the surface pont p. Implementaton and Results We have mplemented our prototype pont-based morphng system on a Mcrosoft Wndows XP PC wth Xeon. GHz CPU, 1.0 GB RAM, and an nvda Quadro4 700 XGL GPU. The system s wrtten n VCþþ 6.0 and the renderer s bult upon OpenGL. The algorthmc flow of our system s descrbed n Fgure 5. The computaton tme for generatng the ntermedate morphng frames s documented n Table 1. Fgure 6. Dfferent number of smulaton nodes for the same pont-set surface. Copyrght # 005 John Wley & Sons, Ltd. 515 Comp. Anm. Vrtual Worlds 005; 16:

8 Y. BAO, X. GUO AND H. QIN Fgure 7. Temporal change of deformaton energy when morphng a standard torus to a squeezed one. Upper row: Stretchng energy. Lower row: Bendng energy. Fgure 8. Morphng frames of a bowng ss statue. Ideally, we would lke to ncorporate all the pont samples on the surface as smulaton nodes. However, ths becomes numercally prohbtve for densely sampled pont set n terms of both memory storage and computaton tme. For extremely dense pont sets, we acqure the smulaton nodes by smplfyng the source surface. In our mplementaton, we use the adaptve herarchcal clusterng method 5 to select a subset of the orgnal surface sample ponts as the smulaton nodes. The number of smulaton nodes ranges from 500 to 000 for the examples that we have demonstrated. Fgure 6 shows dfferent number of smulaton nodes for the same pont surface. Usng the smplfed pont set as smulaton nodes wll undoubtedly lose many deformaton detals snce the generated dsplacement feld s relatvely smooth based on MLS approxmaton. Ths problem can be tackled by usng the multresoluton modelng framework of Reference [6] to capture the deformaton detals. We can compute the offset dstance between the orgnal pont surface and the low-pass fltered surface based on smplfed smulaton nodes. For the source and target shape, we can compute ther offset dstance 0 and n, respectvely. The offset dstance assocated wth each frame can be smply calculated va lnear nterpolaton between 0 and n. See Fgures 7 11 for some morphng experments that we have conducted. One mportant characterstc of our system s that materal propertes can be assocated wth the surface by manpulatng the weghts assocated wth the two energy terms n Equaton (5). In our mplementaton, we adopt the weghts assocated wth sotropc materals ¼ Eh 1 v ; ¼ Eh 3 1ð1 v Þ Fgure 9. Morphng frames of a rabbt model twstng ts head. Copyrght # 005 John Wley & Sons, Ltd. 516 Comp. Anm. Vrtual Worlds 005; 16:

9 MORPHING OF POINT-SAMPLED SURFACES Fgure 10. Morphng frames of an nflatng moa model. Fgure 11. Morphng from sphere to moa. where E s Young s modulus and v s Posson s rato. In fact, the user can ether fne-tune the materal property by settng approprate values for the Young s modulus and the Posson s rato or manpulate the weghts drectly. Further extendng our system to accommodate ansotropc propertes can be easly acheved by makng both and spatally-varyng functons across the entre pont-set surface durng the morphng process. Concluson In ths paper, we have systematcally developed an automatc and novel method to apply the dynamc, meshless thn-shell smulaton prncple to the physcally-plausble morphng of pont-based models. Our nnovatve framework s nspred and derved from dfferental geometry and contnuum mechancs. The physcallyplausble, natural morphng s the drect result of physcs-based energy optmzaton, thus mnmzng user nterventon. Moreover, our method s based on local parameterzaton of the underlyng pont set surface, and therefore, s computatonally effcent. In essence, our method s equvalent to solvng a boundary-value PDE problem, where all the vrtual forces are derved from the change of the ntrnsc geometrc measurement of the surface. Our experments show that the proposed pont surface morphng technque can generate physcally-meanngful, convncng morphng sequences whle avodng most of the undesrable, mesh-based artfacts (such as trangle flp-over or pont surface cracks). Some mmedate future work to further extend our framework s functonaltes are as follows. Frst, we shall consder usng the mature technque for multresoluton analyss and herarchcal decomposton to handle extremely complcated, large-scale, pont-based objects. Second, we shall seek a practcal soluton to the problem of fndng best possble correspondence between two pont-set surfaces. ACKNOWLEDGEMENTS Ths research work was partally supported by the NSF grant ACI , the ITR grant IIS , and Alfred P. Sloan Fellowshp to Hong Qn. The Iss model and the rabbt model are courtesy of Cyberware, Inc. References 1. Xao C, Zheng W, Peng Q, Forrest AR. Robust morphng of pont-sampled geometry. Computer Anmaton and Vrtual Worlds 004; 15: Lazarus F, Verroust A. Three-dmensonal metamorphoss: a survey. The Vsual Computer 1998; 14: Alexa M. Recent advances n mesh morphng. Computer Graphc Forum 00; 1(): Floater MS, Hormann K. Surface parameterzaton: a tutoral and survey. Advances n Multresoluton for Geometrc Modellng 005; Sprnger: Copyrght # 005 John Wley & Sons, Ltd. 517 Comp. Anm. Vrtual Worlds 005; 16:

10 Y. BAO, X. GUO AND H. QIN 5. Sederberg TW, Gao P, Wang G, Mu H. -d shape blendng: an ntrnsc soluton to the vertex path problem. In SIG- GRAPH 1993, 1993, pp Alexa M, Cohen-Or D, Levn D. As-rgd-as-possble shape nterpolaton. In SIGGRAPH 000, 000, pp Yan H-B, Hu S-M, Martn R. Morphng based on stran feld nterpolaton. Computer Anmaton and Vrtual Worlds 004; 15: Terzopoulos D, Platt J, Barr A, Flescher K. Elastcally deformable models. In SIGGRAPH 1987, 1987, pp James DL, Pa DK. Artdefo: accurate real tme deformable objects. In SIGGRAPH 1999, 1999, pp Debunne G, Desbrun M, Can M-P, Barr AH. Dynamc real-tme deformatons usng space and tme adaptve samplng. In SIGGRAPH 001, 001, pp Baraff D, Wtkn A. Large steps n cloth smulaton. In SIG- GRAPH 1998, 1998, pp Brdson R, Fedkw R, Anderson J. Robust treatment of collsons, contact and frcton for cloth anmaton. In SIG- GRAPH 00, 00, pp Crak F, Ortz M, Schröder P. Subdvson surfaces: a new paradgm for thn-shell fnte element analyss. Internatonal Journal of Numercal Methods Engneerng 000; 47: Grnspun E, Hran AN, Desbrun M, Schröder P. Dscrete shells. In ACM SIGGRAPH/Eurographcs Symposum on Computer Anmaton San Dego, Calforna, Eurographcs Assocaton, Swtzerland, 003, pp Zwcker M, Pauly M, Knoll O, Gross M. Pontshop 3d: An nteractve system for pont-based surface edtng. In SIG- GRAPH 00, 00, pp Pauly M, Keser R, Kobbelt LP, Gross M. Shape modelng wth pont-sampled geometry. ACM Transactons on Graphcs 003; (3): Guo X, Hua J, Qn H. Scalar-functon-drven edtng on pont set surfaces. IEEE Computer Graphcs and Applcatons 004; 4(4): Muller M, Keser R, Nealen A, Pauly M, Gross M, Alexa M. Pont based anmaton of elastc, plastc and meltng objects. In ACM SIGGRAPH/Eurographcs Symposum on Computer Anmaton 004, pp Floater MS, Remers M. Meshless parameterzaton and surface reconstructon. Computer Aded Geometrc Desgn 001; 18(): Pauly M, Keser R, Adams B, Dutré P, Gross M, Gubas LJ. Meshless anmaton of fracturng solds. ACM Trans. Graph 005; 4(3): Guo X, Qn H. Real-tme meshless deformaton. Computer Anmaton and Vrtual Worlds 005; 16(3 4).. Kreyszg E. Dfferental Geometry. Dover Publcatons, Inc: New York, 1991; Belytschko T, Lu YY, Gu L. Element free galerkn methods. Internatonal Journal on Numercal Methods Engneerng 1994; 37: Guo X, Qn H. Pont-based dynamc deformaton and crack propagaton. Techncal Report, Stony Brook Unversty, October Pauly M, Gross M, Kobbelt LP. Effcent smplfcaton of pont-sampled surfaces. In IEEE Vsualzaton 0, 00, pp Pauly M, Gross M, Kobbelt LP. Multresoluton modelng of pont-sampled geometry. Techncal Report, ETH Zurch, 00. Authors bographes: Yunfan Bao s a Ph.D. student n the Computer Scence Department at SUNY Stony Brook. He receved a B.Sc. degree (003) n Computer Scence and Engneerng from Zhejang Unversty, Chna. Hs research nterests nclude shape modelng, physcs-based modelng, anmaton and human computer nteracton. For further nformaton, please vst Xaohu Guo s Ph.D. canddate n the Department of Computer Scence at SUNY Stony Brook. He has a B.S. n Computer Scence from the Unversty of Scence and Technology of Chna. Hs research nterests nclude geometrc and physcs-based modelng, computer anmaton and smulaton, nteractve 3D graphcs, scentfc vsualzaton, human-computer nteracton, and vrtual realty. For more nformaton, please vst Hong Qn s Assocate Professor (wth tenure) of Computer Scence at State Unversty of New York at Stony Brook. He receved hs Ph.D. (1995) degree n Computer Scence from the Unversty of Toronto. In 1997, Dr Qn was awarded NSF CAREER Award from the U.S. Natonal Scence Foundaton (NSF). In December 000, Dr Qn receved Honda Intaton Grant Award. In Aprl 001, Dr Qn was selected as an Alfred P. Sloan Research Fellow by the Sloan Foundaton. He was the conference Co-Char of Computer Graphcs Internatonal 005. At present, he s Assocate Edtor of IEEE Transactons on Vsualzaton and Computer Graphcs and The Vsual Computer (Internatonal Journal of Computer Graphcs). Hs research nterests nclude graphcs, geometrc and physcs-based modelng, CAD, anmaton, smulaton, and vrtual envronments. For further nformaton, please vst sunysb.edu/~qn. Copyrght # 005 John Wley & Sons, Ltd. 518 Comp. Anm. Vrtual Worlds 005; 16: