myjournal manuscrpt No. (wll be nserted by the edtor) 3D As-Rgd-As-Possble Deformatons Usng MLS Alvaro Cuno, Claudo Esperança, Antono Olvera, Paulo Roma Cavalcant Systems Engneerng and Computer Scence Program - COPPE Federal Unversty of Ro de Janero Tel.: +55-21-25628572 Fax: +55-21-25628676 e-mal: {alvaro,esperanc,olvera,roma}@lcg.ufrj.br Abstract We present a formulaton for achevng as rgd as possble deformatons of 3D models usng a Movng Least Squares (MLS) approach. Ths research was nspred by the work of Schaefer et al.[25] whch descrbes an approach solvng a 2D verson of the same problem. Our man contrbuton s showng how the problem may be effcently handled n 3D for both pont and lne segment constrants. Our prototype mplementaton s capable of performng close to real-tme deformatons of models wth a few thousand vertces. Key words Mesh deformaton Movng Least Squares (MLS) Rgd transformatons Anmaton (a) (b) (c) (d) Fg. 1 Deformatons usng MLS. (a) Orgnal model 1 and control ponts. Deformaton gven by (a) general affne transformatons, (c) smlarty transformatons, and (d) rgd transformatons. 1 Introducton The problem of deformng 3D models has undergone close scrutny n recent years. Operatons of ths knd fnd applcaton n modelng, anmaton, shape nterpolaton, llustratve vsualzaton, surgcal smulaton and many others. In general, the problem conssts of alterng the shape of a gven model n a smooth way usng some deformaton paradgm. Ths modfcaton should yeld predctable results and, n many cases, conform to certan restrctons such as mantanng the overall volume of the model, preservng detals or avodng selfntersectons. Addtonally, t s generally requred that the deformaton process s performed at nteractve rates. In ths work, we nvestgate the use of deformaton for shape posng, that s, tryng to obtan several varatons of the same model wthout overly affectng ts recognzable features. Among the many approaches proposed n the last few years, those whch try to obtan so-called as rgd as possble deformatons are of specal nterest, snce they tend to produce more physcally plausble results by avodng unnatural shearng and non-unform scalng of the model. In partcular, by makng use of a Movng Least Squares approach, Schaefer et al. [25] have recently presented a way of computng such deformatons for 2D mages. The deformaton s splt nto two components: a translaton and a rotaton transformaton for whch the authors derve closed-form equatons. These depend only on a set of pont to pont or lne segment to lne segment constrants whch are used to model the overall deformaton. The extenson of ths technque to 3D s farly smple for the translaton component, but solvng the rotaton transformaton leads to an egenvector problem. Actually, the deformaton problem can be seen as the regstraton problem between two correspondng 3D pont sets, for whch several solutons have been proposed, ether teratve [24] and closed-form [10]. We are nterested, however, n a smple closed-form soluton that can be easly adapted for use n nteractve deformatons of medum sze models. In ths paper we show how ths can be acheved by dervng a closed-form soluton for the problem. The problem s studed as an optmzaton problem, that mnmzes a sum of weghted squared errors, wth solutons defned by a rotaton axs and angular parameters. These are found by computng the 1 Thanks to the AIM@SHAPE shape repostory.
2 A. Cuno, C. Esperança, A. Olvera, P. Cavalcant largest real root of a depressed quartc equaton. Our approach has been mplemented and we show a few examples of deformatons obtaned wth our prototype. 2 Related work Snce the 80s, many researchers have nvestgated algorthms and technques for applyng meanngful geometrc deformatons to 3D models [4,6]. A recent survey [1] classfes deformaton technques nto freeform methods, whch are mostly amed at producng global smooth deformatons, and detal-preservng methods. The former class s further dvded nto surface-based and spacebased methods, whle the latter class ncludes, among others, multresoluton technques and methods based on dfferental coordnates. Freeform surface-based technques try to obtan a dsplacement functon of the form f : S R 3, whch maps surface S nto a deformed verson S = {p + f(p) p S}. Usually, f s modelled by means of some energy mnmzaton process defned on the surface subject to a set of condtons on ts border. The deformaton s controlled by deformaton handles, most commonly a set of ponts p such that f(p ) = q, where q refer to the new postons of p [7,17,23,29]. Whle surface-based methods are qute flexble and support dfferent smoothness crtera, ther computatonal complexty and numercal stablty are strongly related to the sze and qualty of the nput mesh. In contrast, space-based technques deform the 3D space as a whole, thus affectng the shape of models contaned theren ndrectly [8, 14, 26]. Such methods employ a deformaton functon of the form f : R 3 R 3 to transform all ponts of the orgnal surface S to a new, deformed surface S = {f(p) p S}. Dfferental methods are characterzed by applyng modfcatons to dfferental rather than spatal coordnates of the models. Once the new values for handle postons or normals are specfed, a deformed model s reconstructed by consderng the desred dfferental propertes and mnmzng the model dstorton. Commonly used dfferental representatons are: gradent felds [30, 31], Laplacan coordnates [19, 27, 30 32], and frst/second fundamental forms of a surface [18,20]. Another mportant category of deformaton technques are the so-called multresoluton methods [17, 21, 22]. Here, the key dea s to use mesh-based sgnal processng technques n order to apply the deformaton to a low-frequency verson of the model and later reconstructng the hghfrequency detals. Many of the deformaton technques proposed n the past rely on some knd of energy mnmzaton process, where the energy functon s defned n such a way as to measure the dstorton ntroduced by the deformaton. A related concept characterzes the so-called as rgd as possble approaches, n whch the deformaton functon s defned as a smoothly varyng affne transformaton appled on the model n such a way as to mnmze local scalng and shearng. Ths dea has recently been appled to the problem of deformng mages and meshes n 2D [2,15,25]. In ths paper, we focus on obtanng an nteractve soluton for producng as-rgd-as-possble deformatons n 3D by adaptng the Movng Least Squares approach of Schaefer et al. [25]. Thus, the man problem s that of obtanng a rgd transformaton whch best approxmates a mappng n R 3 for a gven set of ponts. Ths problem s also known as pont regstraton, or, more specfcally, the Absolute Orentaton Problem. Analytcal solutons have been proposed, for nstance, based on SVD (Sngular Value Decomposton) [3], quaternons [12,16], orthonormal matrces [13] and dual quaternons [28]. Another related group of technques ncludes those based on teratve schemes such as the ICP (Iteratve Closest Pont) algorthm [5] and ts varants. Dfferently from these approaches, however, the soluton descrbed below drectly obtans the axs and angular parameters of the rotaton. Our method requres a smaller number of computatons than matrcal and even quaternon-based approaches. Notce that, n our applcaton, a new rgd transformaton must be computed for every vertex of the mesh, and thus t must be as effcent as possble. 3 Movng least squares deformaton The Movng Least Squares (MLS) formulaton can be thought of as an extenson of the tradtonal Least Squares mnmzaton technque. Rather than fndng a global optmum soluton for the problem, MLS tres to fnd contnuously varyng solutons for all ponts of the doman. Let us defne the deformaton operaton as a transformaton whch maps a set of ponts {p } of the doman onto new postons {q }. Thus, solvng the problem for a gven pont v = [x y z] of the doman can be reduced to fndng the best transformaton l v (x) that mnmzes w l v (p ) q 2, (1) where w are weghts of the form w = p v u for some nteger constant u > 0. Let us defne the deformaton functon f as f(v) = l v (v). We observe that when v s close to some constrant p, then w tends to nfnty, whch means that f s nterpolatng wth respect to the constrant ponts,.e., f(p ) = q. Further, f q = p, then f(v) = v, for all v, meanng that, n ths case, f s the dentty functon. Fnally, t can be shown that f s smooth everywhere for u 2. Ths defnes the Movng Least Squares mnmzaton n whch the sought transformaton l v depends on the pont of evaluaton v.
3D As-Rgd-As-Possble Deformatons Usng MLS 3 4 As rgd as possble MLS By mposng dfferent addtonal requrements on the form of l v, we may obtan dfferent results. We may requre, for nstance, that l v s a general affne transformaton, n whch case the classcal normal equatons soluton can be appled drectly [25]. For obtanng deformatons whch are as rgd as possble, l v must be constraned to be a rgd transformaton,.e., l v must be of the form: l v (x) = xr + T, where R s a rotaton matrx and T s a translaton vector. Solvng for T yelds T = q p R, where q and p are the weghted centrods of {q } and {p } respectvely: p = w p w, q = w q w. (2) Ths permts us to factor out the translaton from (1) by rewrtng t as w ˆp R ˆq 2, (3) where ˆq = q q and ˆp = p p. Expandng (3) then yelds 2 w ˆq Rˆp T + w ˆq 2 + w ˆp 2. Snce the last 2 terms are constant, we nfer that R mnmzes (3) f and only f t maxmzes w ˆq Rˆp T. (4) 4.1 3D rgd transformatons In 3D space, R may be defned as a rotaton of an angle α around an axs e. Applyng such a rotaton on a vector v yelds: R e,α (v T ) =e T ev T + cos(α)(i e T e)v T + 0 e z e y sn(α) e z 0 e x v T. (5) e y e x 0 By replacng ths defnton of R n (4) we obtan w ˆq Rˆp T = e Me T +cos(α)(e e Me T )+sn(α)ve T, where M= w ˆq xˆp x w ˆq xˆp y w ˆq T ˆp w ˆq xˆp z = w ˆq yˆp x w ˆq yˆp y w ˆq yˆp z w ˆq zˆp x w, ˆq zˆp y w ˆq zˆp z E = w ˆq ˆp = Trace(M), V= w ˆp ˆq = (M 23 M 32 M 31 M 13 M 12 M 21 ). (6) 4.2 Optmzaton problem Thus, the optmzaton problem can be wrtten as max F(e,α) = e Me T + cos(α)(e e Me T ) + sn(α)ve T s.t. e = 1, cos(α) 2 + sn(α) 2 = 1. (7) By consderng the optmalty condtons (Kuhn-Tucker) for ths problem, the solutons must satsfy (1 cos(α))e(m + M T ) + sn(α)v = k 1 e, (8) ( ) ( ) E e Me T cos(α) Ve T = k 2.(9) sn(α) If these condtons are satsfed wth α = 0 or k 2 = 0 then F(e,α) = E. Whle searchng for (e,α) such that F(e, α) > E, we can, therefore, assume that both these condtons do not hold. If that search does not succeed, the null rotaton s a soluton of (7). From equaton (9), we have sn(α) = Ve T /k 2, and, defnng N = M + M T, we may rewrte equaton (8) as where a = 1 k 2 (1 cos(α)) (N + av T V)e T = λe T, (10) and λ = k 1 (1 cos(α)). (11) In summary, the optmal rotaton axs wll correspond to the egenvector of matrx (N + av T V) assocated wth egenvalue λ. We notce, however, that the values of a and λ depend on k 1 and k 2, makng ths a non-standard egenvalue problem. To reduce the ndetermnaton of the problem, t s possble to show that a and λ are related by a = 4.3 Egenvalue determnaton 1 λ 2E. (12) An analyss of equaton (10) reveals that snce λ s an egenvalue of (N + av T V), t must also be a root of the characterstc polynomal [ P(λ) = λ 3 (Trace(N) + a V 2 )λ ] 2 + ( 1 2 (Trace(N)2 N 2 ) + a( V 2 Trace(N) VNV T ) λ det(n)(1 + VN 1 V T a). By substtutng equaton (12) and notng that Trace(N) = 2Trace(M) = 2E and that 2 M 2 = 1 2 N 2 + V 2, then equaton P(λ) = 0 becomes λ 4 4Eλ 3 + (6E 2 2 M 2 )λ 2 +(4( M 2 E 2 )E 2VMV T det(n))λ +det(n)(2e VN 1 V T ) = 0.
4 A. Cuno, C. Esperança, A. Olvera, P. Cavalcant The thrd degree term may be elmnated by effectng a varable change of the form y = λ E. Thus, we obtan the followng equaton n y whch allows for a closed-form soluton: y 4 2 M 2 y 2 8det(M)y E 4 +2 M 2 E 2 8det(M)E+det(N)(2E VN 1 V T ) = 0. (13) The ndependent term n (13) s more easly obtaned by the expresson M 4 4 M M j 2, j< where M k s the k th column of M. It can be shown that f we use the optmalty condtons (8) and (9) n the objectve functon (7), the latter becomes λ E = y, meanng that a value of e whch satsfes the optmalty condtons s gven by the value of y tself. Snce our ultmate goal s to satsfy the objectve functon, we must look for a value for y whch s the largest real soluton of the polynomal (13). 4.4 Solvng the depressed quartc equaton If a depressed quartc polynomal y 4 + ay 2 + by + c s factored as (y 2 +py+q)(y 2 py+s), then p must satsfy p 6 +2ap 4 +(a 2 4c)p 2 b 2 = 0 whch s a cubc equaton n p 2 = z. If the depressed polnomal s (13), ths cubc polnomal becomes Q(z) = z 3 4 M 2 z 2 + 16( j< ( M M j ) 2 (M M T j ) 2 )z 64det(M) 2. Now, let Q be the characterstc polynomal of (M T M). Snce Q(z) = 64Q (z/4), then a root r of Q must be related to some egenvalue µ of (M T M) by r = 4µ. Ths mples that r 0, snce all egenvalues of the postve sem-defnte matrx (M T M) must be non-negatve. Let p = r. Havng p, y can then be obtaned as the largest value between p/2+ p 2 /4 + M 2 4det(M)/p and p/2+ p 2 /4 + M 2 + 4det(M)/p. Observe that Q(z) s essencally the same polynomal solved by a matrcal method lke Horn s n order to fnd the egenvalues of M T M. Moreover, f det(m) > 0, then the value of y obtaned by that process s Trace((M T M) 1/2 ). Otherwse, t s Trace((M T M) 1/2 ) 2 µ mn, where µ mn s the smallest egenvalue of M T M. From the reasonng above, we can realze that fndng λ demands no more effort than obtanng all the egenvalues of M T M, as requred by matrcal approaches. From ths pont on, the method descrbed here s advantageous. Whle Horn s approach determnes a bass of unt egenvectors of M T M usng t to compute M(M T M) 1/2, we need to solve a sngle lnear 3 3 system and, from the soluton of that system, easly obtan {e,cos(α),sn(α)} as explaned below. 4.5 Determnng the rotaton axs e Assume, ntally that Ve T 0. Then, φ = a(ve T ) s well defned and Equaton (10) can be rewrtten as (N λi)(e T /φ) = V T. Thus, consderng that ±(e,α) represent the same rotaton, we can nfer that e can be found by solvng (N λi)u T = V T (14) and takng the unt vector of u. For small deformatons, n specal, ths approach s more approprate than drectly fndng an unt egenvector of (N + avv T ) snce a may assume very large values. It remans to consder the case where Ve T = 0. Snce we assume that k 2 0 and α 0, then the three condtons below are equvalent: () Ve T = 0; () (14) s a non-determned system; () λ s an egenvalue of N. In ths case e s an untary vector assocated to t. Thus, f Ve T = 0, condton (b) holds and havng dentfed t we can fnd e as an untary egenvector of N assocated to the egenvalue λ. 4.6 Determnng sn(α) and cos(α) From equatons (11) and (12) we nfer that λ 2E = k 2 (1 cos(α)). Addtonally, equaton (9) asserts that E eme T = k 2 cos(α). Addng these two equatons, we obtan k 2 = λ E eme T. Ths allows the use of equaton (9) to obtan expressons for cos(α) and sn(α) whch do not contan any squared terms: cos(α) = E emet λ E eme T, sn(α) = Ve T λ E eme T. The optmalty condtons of (7), and the fact that u = (1/a)Ve T allow us to rewrte these two expressons as cos(α) = (VeT ) 2 (λ 2E) 2 (Ve T ) 2 + (λ 2E) 2 = 1 u 2 1 + u 2, sn(α) = 2(Ve T ) (Ve T ) 2 + (λ 2E) 2 = 2 u 1 + u 2, (15) whch requre very lttle computaton consderng that u 2 and u have already been determned whle obtanng e. The equatons (15) also ndcate that the obtaned values are n the range [-1,1]. Notce that up to the pont where vector u s found our approach s computatonally equvalent to the quaternonbased technque of Kanatan [16], except that, n that case, an egenvector of a 4 4 matrx would be determned. However, all t remans now s to fnd the values {e, cos(α), sn(α)} from u and drectly substtute them n equaton (5), whereas to effect a quaternon-based
3D As-Rgd-As-Possble Deformatons Usng MLS 5 transformaton, a more lengthy computaton s requred for every vertex. Fnally, we must observe that the obtaned rotaton s exactly M(M T M) 1/2 f det(m) >= 0. Otherwse t s M(M T M) 1/2 (I e mn e T mn ) where e mn s an egenvector of M T M assocated to the egenvalue µ mn. These expressons ndcate that the rotaton s a contnuous functon of M whle det(m) > 0. A dscontnuty can only occur f M s sngular or det(m) < 0 and µ mn s a multple egenvalue of M T M. At dscontnuty ponts, we mght notce a twstng effect whle nteractvely applyng a large deformaton to a model. It s mportant to stress, however, that any method for solvng (7) wll exhbt ths behavor. Table 1 Routnes and ther equatons Routne Equaton WeghtedCentrod (2) CorrelatonMatrx (6) MaxmumRootOfP (13) RotatonAxs (14) SnCos (15) 5 Algorthm A deformaton sesson uses as nput a 3D model n the form of a polygonal mesh wth n vertces and a set of control ponts {p }, = 0...k 1, not necessarly on the mesh. The user then establshes new postons for the control ponts whch are stored n {q }. Fnally, Algorthm 1 s nvoked to obtan new postons for each vertex v of the mesh. Fg. 2 Deformaton of a dolphn usng 4 control ponts. Algorthm 1: Compute deformed vertex poston Input: Vertex poston v Input: Orgnal and deformed postons of control ponts {p } and {q } Output: Deformed vertex poston v 1 p WeghtedCentrod(v, {p }); 2 q WeghtedCentrod(v, {q }); 3 M CorrelatonMatrx(p, q ); 4 mroot MaxmumRootOfP(M); 5 λ mroot + Trace(M) ; 6 e RotatonAxs(M, λ) ; 7 [sn(α), cos(α)] SnCos(M, e) ; 8 ˆv v p ; // egenvalue 9 d (ˆve T )e + cos(α)(ˆv (ˆve T )e) + sn(α)(ˆv e); 10 v q + d; 11 return v The auxlary routnes used n Algorthm 1 merely compute the values of the varous equatons presented n ths Secton, as summarzed n the Table 1. Our mplementaton of ths algorthm ncludes several tweaks to mprove the performance. For nstance, all values of ˆv, p, w and ˆp are precomputed durng a deformaton sesson where the values of q are changed nteractvely. Fgures 2 and 3 show sample results of the deformaton algorthm as appled to two dfferent models. Table 2 shows frame rates for deformatons appled on several polygonal meshes. We notce, as expected, that the performance s drectly proportonal to the num- Fg. 3 Twstng deformaton of a parallelepped. The effect s obtaned by applyng several small-angle torsons n successon. Table 2 Frame rates for the deformaton of several models. Model Vertces Constrants FPS Dolphn 2811 4 40 Homer 5103 6 22 Plane 1089 132 19.5 Dno 14050 5 6.5 Bar 6146 256 2 Cylnder 1058 322 9.5 ber of vertces of the model and to the number of constrants controllng the deformaton. The prototype has been mplemented n C++ under Lnux. We used the OpenGL API for renderng and all the experments have been performed settng the OpenGL s canvas resoluton to 600 600 pxels. Tmes have been taken on a PC equpped wth a Pentum-IV processor runnng at 3.2 GHz wth 1GB of man memory and a GeForce 7300LE NVda graphcs card.
6 A. Cuno, C. Esperança, A. Olvera, P. Cavalcant 6 Extensons 6.1 Smlarty deformaton If a more general smlarty rather than rgd transformaton s requred, then an unform scalng factor µ s R must be ntroduced n the optmzaton problem. In ths case, equaton 3 becomes w µ sˆp R ˆq 2, and to optmze t we must solve max µ s w ˆq Rˆp T µ 2 s w ˆp 2. Ths problem generates the same optmalty condtons wth respect to {e, cos(α), sn(α)} as (7). In addton, there s an optmalty condton wth respect to µ s gven by w ˆq Rˆp T µ s w ˆp 2 = 0. Snce the optmalty condtons of (7) determne that an optmal soluton satsfes w ˆq Rˆp T = y, we fnally obtan y µ s = w ˆp 2. Thus, step 10 of Algorthm 1 would read: v q +µ s d. 6.2 Deformatons usng lne segments In many stuatons, the deformaton nduced solely by movng control ponts may produce undesrable dstorton or counter-ntutve results, as shown n Fgure 4(a). Ths may be allevated by ncreasng the number of control ponts (Fgure 4(b)), but only at a cost n terms of performance and nteractvty. One possble soluton s to use other geometrc prmtves such as lne segments to control the deformaton (Fgure 4(c)). The generalzaton of MLS-based deformatons controlled by lne segments may be stated as the mnmzaton of 1 0 w p (t)r + T q (t) 2, where p (t) s the th orgnal lne segment and q (t) the correspondng deformed lne segment. In much the same way as what was done wth control ponts, ths can be reduced the maxmzaton of 1 0 w ˆq (t)rˆp T (t), where ˆq (t) = q (t) q and ˆp (t) = p (t) p. Thus, the optmzaton problem s also stated by Equaton (7), where M s defned as M= 1 0 w ˆq T (t)ˆp (t)dt. (a) (b) (c) Fg. 4 Deformng a model usng: (a) and (b) control ponts, and (c) lne segments. Lne segments ˆp (t) and ˆq (t) may be represented by the products ) ( ) (â ĉ ˆp (t) = (1 t t) and ˆq ˆb (t) = (1 t t), ˆd where â, ˆb are the endponts of ˆp (t) and ĉ, ˆd are the endponts of ˆq (t). Thus, M may be rewrtten as where M = (ĉ T ˆd T )W (â ˆb ), ( 1 W = 0 w ) (t)t 2 1 dt 0 w (t)t(1 t)dt 1 0 w (t)t(1 t)dt 1 0 w (t)(1 t) 2. dt Dependng on the choce of w (t), the ntegrals n W may have very complcated explct formulatons, as mentoned by Schaefer, or even requre an teratve numercal method to be computed. A smple formulaton for W may be obtaned f we consder that w (t) has a constant value gven by d u where d s the dstance between the vertex v beng evaluated and the segment a b, that s, W = d u ( ) 1/3 1/6. 1/6 1/3 Fnally, the values of p and q are determned by p = d u (a + b )/2, q = d u d u (c + d )/2. d u In our experments, we dd not notce sgnfcant dstortons caused by ths smplfcaton.
3D As-Rgd-As-Possble Deformatons Usng MLS 7 (a) (b) Fg. 7 Deformatons wth large dsplacements. (a) Intal model and control ponts. (b) Deformed model. (a) (b) (c) (d) Fg. 5 Large deformaton of cylnder: (a) Intal model and control ponts. Deformaton nduced by rotatng the top group of vertces usng dfferent weght functons: (b) w(d) = d 2, (c) w(d) = d 3 and (d) w(d) = d 4. The lower row of pctures show the postons of the weghted centrods q On the other hand, t s worth mentonng that MLSrgd deformatons behave ncely n the sense that t tends to preserve detals and global characterstcs of the model better than, say, RBF-based or dfferental coordnate-based technques. Fgure 7 supports ths clam when we compare t wth a smlar fgure shown n [11] (Fgure 7, to be precse). 8 Conclusons and future work (a) (b) Fg. 6 Deformaton caused by translaton. (a) Intal model and control ponts. (b) Deformed model. 7 Dscusson As noted by Botsch et al. [9], deformaton technques may lead to non-ntutve results when control handles are subject to large dsplacements. In our case, large translatons and/or rotatons may ndeed produce nonsmooth results as llustrated n Fgures 5, and 8. The key observaton that provdes an explanaton to ths behavor s that the weghted centrods p and q are, n fact, computed as convex combnatons of control ponts {p } and {q }. One mght be tempted to obtan a smoother dstrbuton of the centrods by modfyng the weghng functon but ths does not lead, n general, to satsfactory results, as llustrated n Fgure 5. Another nterestng effect may be observed when the handles are subject only to translatons as shown n Fgure 6. Notce, n partcular, that lnes whch are parallel to the handle movement are not dstorted. It must be stressed, however, that n many cases, an unexpected result s caused merely by nsuffcent data nput. Consder, for nstance, the deformaton depcted n Fgure 1 where the ntended result was clearly to make the character bend at the wast. For ths to occur naturally, the placement of a control pont near the navel s crucal. We have presented a practcal approach for computng as-rgd-as-possble deformatons of 3D models usng a Movng Least Squares (MLS) mnmzaton scheme. The rotaton soluton s defned by a rotaton axs and angular parameters {e, cos(α), sn(α)}. These are found by computng the largest real root of a depressed quartc equaton and solvng a 3 3 system. Our prelmnary experments show that ths approach s slghtly more effcent than the classcal formulaton whch computes the rotaton matrx R as M T (MM T ) 1/2 [13]. It also uses fewer operatons than the quaternon-based representaton of R as suggested n [16]. A more effcent approach to the problem may be devsed n the future n the form of an ncremental method whch approxmates the best rotaton wthout the need of computng egenvalues for every vertex at every frame of the nteracton. In fact, we consder our approach a good startng pont for dervng such a method. The expressve power of as-rgd-as-possble deformatons s hndered by some non-smooth results for some control pont confguratons, specally f the handles are subjected to large translatons or rotatons. Some deas for reducng these problems are beng nvestgated, such as nterpolatng the centrods q usng radal bass functons (RBFs). Some prelmnar results are shown n Fgure 8. References 1. Alexa, M.: Interactve shape edtng. ACM SIGGRAPH Courses (2006)
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