D shape manipulation via topology-aware rigid grid. Introduction. By Wenwu Yang and Jieqing Feng *
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1 COMPUTER ANIMATION AND VIRTUAL WORLDS Comp. Anm. Vrtual Worlds 2009; 20: Publshed onlne 1 June 2009 n Wley InterScence ( 2D shape manpulaton va topology-aware rgd grd By Wenwu Yang and Jeqng Feng *... Ths paper presents a new method whch allows user to manpulate a two-dmensonal shape n an ntutve and flexble way. The shape s dscretzed as a regular grd. User places handles on the grd and manpulates the shape by movng the handles to the desred postons. To meet the constrants of the user s manpulaton, the grd s then deformed n an as-rgd-as-possble way. However, ths straghtforward approach tends to produce unnatural deformatons when the grd resoluton s not hgh enough to capture the topologcal structure of the shape. In the proposed method, the regular grd s trmmed and only the cells that are nsde the fatty regons of the shape are preserved, namely nteror grd. When user manpulates the shape, the nteror grd and the shape boundary curve are deformed wth mnmum dstortons. To make the deformatons of the nteror grd and the boundary curve consstent, a juncton energy s ntroduced. In ths way, the unnatural deformaton effects could be effectvely removed and the physcally plausble results can be obtaned. Meanwhle, the proposed approach provdes user an ntutve and smple way to adjust the shape global and local stffnesses. The deformaton s formulated as an energy mnmzaton problem. The energy functon s non-quadratc and could be effcently solved usng an teratve solver wth the fast summaton technque that explots the nteror grd and boundary curve regulartes. In addton, the method could be easly extended to manpulate curves and stck fgures. Expermental results demonstrate the capablty and flexblty of the new method. Copyrght 2009 John Wley & Sons, Ltd. Receved: 24 March 2009; Accepted: 24 March 2009 KEY WORDS: shape manpulaton; topology-aware; tunable stffness; nonlnear optmzaton Introducton The 2D shape manpulaton provdes user an ntutve way to move, rotate, stretch, and bend a 2D mage or drawng nteractvely. It has been proven to be an versatle tool n varous applcatons, and s currently prevalent n commercal vdeo processng softwares and vector graphcs softwares such as Adobe After Effects, Adobe Illustrator, as well as cartoon anmaton softwares such as ToonBoom. Free-form deformaton (FFD) and skeleton-based technques are two prevalent methods for shape manpulaton. They manpulate the object by edtng the *Correspondence to: J. Feng, State Key Laboratory of CAD&CG, Zhejang Unversty, Hangzhou , P. R. Chna. E-mal: jqfeng@cad.zju.edu.cn; wwyang@cad.zju.edu.cn pre-defned lattce or skeleton. However, manpulatng densely dstrbuted control ponts n the lattce s a laborous work and defnng an approprate skeleton s not trval especally for the object wth an ambguous jont structure. Recently, a wde class of methods, 1 4 whch provde users an ntutve nterface, have been proposed. These methods allow drect manpulaton on an object by edtng the specfed constrants n a clck-and-drag way. Then the shape s deformed locally or globally n a physcally plausble way; meanwhle the deformed shape satsfes the constrants as closely as possble. Whlst these methods all am to mnmze the local dstortons of the shape nterors durng the deformaton (.e., as rgd as possble deformaton), one of dfferences among them s the underlyng shape tessellaton structure on whch the local dstorton measurements. Copyrght 2009 John Wley & Sons, Ltd.
2 W. YANG AND J. FENG Fgure 1. (a g) Comparson between the straghtforward regular grd based deformaton method 4 and our topology-aware method: (a) rest shape; (b) regular grd; (c) unnatural result; (d) trangulaton; (e) nteror grd; (f) deformed grd; and (g) natural result. Under the same user constrants, the results of Reference [4] and our approach are shown n (c) and (g), respectvely. (h k) Deformatons of the nteror grd and the boundary curve: () w s = 1; (j) w s = 0.01; and (k) w s = 100. (h) Wthout juncton energy. ( k) Wth juncton energy (w r = 6). The deformed grd cells n the marked squares of (j) and (k) are unmatched wth the boundary curve or stretched due to unsutable w s. are defned approprately. For example, References [1 3] dscretze the 2D shape usng trangles or quadrangles, however, Reference [4] dscretzes the shape usng regular grd cells. In the proposed method, a hybrd shape representaton s adopted. The hybrd shape representaton ncludes two tessellaton structures: one s the regular grd whch dscretzes the shape fatty regons, namely nteror grd, as shown n Fgure 1(e); the other s the constraned trangulaton of the shape nteror, as shown n Fgure 1(d). Based on ths hybrd shape representaton, we ntroduce a new deformaton method. In Reference [4], Yang et al. dscretzed the 2D shape usng a regular grd as shown n Fgure 1(b) and edted the shape by manpulatng the regular grd subject to the user constrants as shown n Fgure 1(c). Based on the grd regularty, they desgned a stffness tunable deformaton model and adopted the fast summaton technque to speed up the algorthm such that the user could manpulate both soft and stff objects nteractvely. But ths straghtforward method tends to produce unnatural deformaton results as shown n Fgure 1(c) when the grd resoluton s not hgh enough to capture the shape topologcal structure (see the parts marked by crcles n Fgure 1b). One soluton s to ncrease the grd resoluton, however, n ths way the regular grd scale wll ncrease quadratcally and may lead to the nteractve edtng prohbtve. Another soluton s to use the adaptve grd such as Reference [5]. However the adaptve grd structure may make the shape global stffness adjustment unntutve snce ts cells are not n the unform sze. In addton, although the fast summaton technque could be extended to the adaptve grd case, t s not a trval work yet. 6. Copyrght 2009 John Wley & Sons, Ltd. 176 Comp. Anm. Vrtual Worlds 2009; 20:
3 2D SHAPE MANIPULATION On the other hand, the trangular mesh based methods such as References [1,3] do not suffer the resoluton problem mentoned above because the trangular mesh, whch dscretzes the shape nteror, captures the shape topologcal structure fully. In ths paper, the proposed method combnes the advantages of both the regular grd and the trangular mesh. It constructs a well stffness-tunable deformaton model based on the nteror regular grd and elmnates the unnatural results by consderng the topologcal structure descrbed by the trangular mesh, as shown n Fgure 1(f) and (g). Meanwhle, by explotng the nteror grd regularty, the proposed method stll could be speeded up for several tmes wth the fast summaton technque, 7 especally for the shape wth the large global stffness, makng t more sutable for the nteractve edtng. Furthermore, the proposed method can be easly extended to manpulate the 2D curves and stck fgures, whch s not trval for the straghtforward regular or adaptve grd based methods such as References [4 6]. Related Work As a useful tool, the shape manpulaton or deformaton methods have been well studed for many years. Many algorthms have been ntroduced and we dscuss those that are most relevant to our method. The prevalent FFD, 8 skeleton-drven 9 and other spacewarp methods 10 deform an ntermedate space where the underlyng shape s embedded. These methods are purely geometrc approaches and conduct the deformaton on ntermedate space, thus the generated deformaton may be dfferent from the real object behavors or appearances. Physcal-based approaches such as fnte element methods 11 can smulate the small-scale object deformaton behavors n a physcal accuracy and correctness way. In practcal applcaton, the shape manpulatons go well beyond the small-scale dsplacement, however, physcal smulaton of large-scale deformaton s stll a challenge problem. The dfferental doman methods are also geometrybased approaches. 12,13 They are not confned to smallscale deformatons and can produce physcally plausble results. In these settngs, surface detals are encoded as the dfferental coordnates and the deformaton s formulated as a local rgd transformaton optmzaton problem to preserve the dfferental coordnates. Smlarly, shape matchng based methods also preserve the local shape detals, where the local deformatons are confned to be rgd transformatons. 3 5 All of them are formulated as a nonlnear optmzaton problem snce nether a 2D nor a 3D rotaton transformaton can be expressed as a lnear functon of poston coordnates. In Reference [1], ths nonlnear optmzaton was decomposed nto two closed-form lnear sub-systems. Meanwhle, other lnearzaton methods are also proposed to approxmate the rotaton transformaton. 12,14 However, these lnear approxmatons may lead to sub-optmal results whch correspond to deformaton artfacts. The proposed method n ths paper falls nto the class of nonlnear methods, whch can avod the artfact problem and acheve the hgh-qualty deformatons through an teratve optmzaton. 3,5 Algorthm The method takes as nput a 2D shape whch s represented as a planar polygon. Then the 2D shape s dscretzed usng a regular grd 15 as shown n Fgure 1(b); meanwhle t s trangulated va a Delaunay trangulaton subject to boundary and area constrants, as shown n Fgure 1(d). The regular grd s then trmmed by the shape boundary, and only the cells n the shape fatty regons are preserved, called nteror grd, as shown n Fgure 1(e). Note that the nteror grd cells may be dsjoned snce some shape sknny regons may contan no grd cells. Meanwhle, the shape boundary curve s refned by nsertng ponts evenly on those edges whose lengthes are larger than the grd cell sze, such that the average edge length of the new shape boundary curve approaches to the grd cell sze. User manpulates the shape by placng handles on the vertces of the nteror grd or(and) the boundary curve and movng them to the desred postons. Durng the manpulatons, the nteror grd and the boundary curve are deformed wth the least dstortons subject to the user s constrants,.e., n an as-rgd-as-possble way. Meanwhle, the deformatons of the nteror grd and the boundary curve are coordnated va a juncton energy defned on a juncton trangular mesh regon. An example of the deformatons s shown n Fgure 1(f). Fnally, the deformatons of the nteror grd and the boundary curve are transferred to the shape trangulaton, and then to the embedded 2D shape, as shown n Fgure 1(g).. Copyrght 2009 John Wley & Sons, Ltd. 177 Comp. Anm. Vrtual Worlds 2009; 20:
4 W. YANG AND J. FENG Local Rgd Regons The neghborng cells of the nteror grd are grouped as local regons. Durng deformaton, these local regons tend to be as rgd as possble,.e., ther dstortons are mnmzed, thus they are called nteror rgd regons. Smlarly, the neghborng edges of the refned boundary curve are grouped as the boundary rgd regons. For descrptve convenence, the vertex n the nteror grd or the boundary curve s called node throughout the rest of the paper whlst the vertex n the trangular mesh remans called vertex. Let L n be the set of the nteror grd nodes. The onerng neghborhood of an nteror grd node conssts of those nodes that share wth t at least one nteror grd cell. Then for each node L n, an nteror rgd regon R n wth half-wdth w s defned as the set of cells that composes of nodes reachable by traversng no more than w-rng neghborhood of the node. Thus for a fully connected nteror grd, an nteror rgd regon R n s a square n general and conssts of (2w) 2 cells (or (2w + 1) 2 nodes). Obvously, Wth the ncrease of the half-wdth w, the neghborng rgd regons wll be more tghtly ted, such that they are more dffcult to be stretched or bent. Thus the half-wdth w can be regarded as a reasonable global stffness control parameter,.e., large value for stff object whlst small value for soft object. The boundary rgd regon s defned on the boundary curve smlarly. Let L b be the set of the boundary curve nodes. For each node L b, a boundary rgd regon R b wth half-wdth w s defned as the consecutve 2 w edges, whch conssts of the followng 2 w + 1 nodes: w,...,,...,+ w. Unlke the half-wdth w of an nteror regon, w s determned accordng to the w. To acheve the consstent deformatons of the nteror and boundary rgd regons, the length of each boundary rgd regon should approach to the wdth of the nteror rgd regon. Thus the half-wdth w s determned by satsfyng the equaton: 2 wl b 2wl n, where l b s the average edge length of the boundary curve and l n the nteror grd cell sze. As l b approaches to l n n our settng, w approaches to w n general. Deformaton Energy Rgdty. Ideally, the deformatons of nteror and boundary rgd regons should be rgd transformatons. To meet the user s constrants, the rgd regons can only be deformed n an as-rgd-as-possble way. To acheve ths goal, t s mportant to defne dstorton metrcs for the two types of rgd regons. Let R be a rgd regon at an nteror grd node or a boundary curve node. The ntal and deformed postons of a node j are denoted as p j and q j, respectvely. Then we defne the dstorton metrc for the rgd regon as E(R ) = q j q c A (p j p c ) 2 (1) j R where A s the optmal rotaton whch transforms the rgd regon from the ntal poston to the deformed poston, and p c (q c ) are the ntal(deformed) rotaton centers. The A can be found by mnmzng Equaton (1) and there s an analytcal soluton for 2D case. 16 Snce a rgd regon s assumed to be translated and rotated as a whole (.e., as rgd as possble), t s straghtforward to choose the ntal and deformed rotaton centers p c and q c as follows: { p c = p ; q c = q p c = p s ; q c = q s no node j R s a handle s R s a handle To acheve as-rgd-as-possble deformaton, the total rgd regons dstortons should be mnmzed,.e., mnmzng the followng energy functon, E r = E rn + E rb = L n w (2) E(R n ) #(R n ) + E(R b w ) #(R b L b ) (3) where #(R n ) and #(R b ) denote the number of nodes n the nteror rgd regon R n and the boundary rgd regon R b, respectvely. Snce the nteror grd cells are n the same sze and the boundary curve nodes are dstrbuted nearly unformly on the boundary curve, the energy normalzaton n Equaton (3) dsperses the energy on the nteror or boundary rgd regons evenly. The weght w s penalty factor for the average dstorton on an nteror or boundary rgd regon, and hence could be regarded as a local stffness control parameter. Juncton. Mnmzng the energy functon Equaton (3) wll deform the nteror grd and the boundary curve ndependently n an as-rgd-as-possble way. Because there s no explct connecton between them, thus t wll lead to the deformatons of the nteror grd and the boundary curve unmatched, as llustrated n Fgure 1(h). Ths problem can be addressed by addng edge connectons between the nteror grd and the boundary curve as n Reference [2], but the subsequent relaxng process n the numercal soluton wll not only. Copyrght 2009 John Wley & Sons, Ltd. 178 Comp. Anm. Vrtual Worlds 2009; 20:
5 2D SHAPE MANIPULATION be computatonally expensve but also dffcult to be mplemented. Furthermore, the regular neghborhood structure of the nteror grd and the boundary curve wll be destroyed, and hence the fast summaton, 7 an effcent technque for acceleratng the energy functon optmzaton, could not be employed. We address ths problem by ntroducng an addtonal juncton energy to coordnate the nteror grd deformaton and the boundary curve deformaton. The juncton energy s defned on the gaps between the nteror grd and the boundary curve. Durng deformaton, the juncton energy tes the nteror grd and the boundary curve tghtly by preservng the local areas of the gaps, such that ther deformatons can be coordnated, as shown n Fgure 1(). Here the key ssues are on how to delmt the gap regon and how to defne the juncton energy. As shown n Fgure 2, the gaps between the nteror grd and the boundary curve can be fully covered by two types of trangles n the trangular mesh. The frst type s composed of those that are outsde of the nteror grd cells (the sold trangles) and the second type s composed of those that ntersect the boundary cells of the nteror grd (the hatchng trangles). For convenence, the two types of trangles are called as juncton trangles, and we defne the juncton energy on the juncton trangles. To preserve the local areas of the juncton trangles, the juncton energy measures the dstortons of ther local propertes: the mean value barycentrc coordnates of the vertces (.e., the relatve postons among vertces) and the edge lengths, of the juncton trangles. Let V be the set of all nteror vertces n the juncton trangles, where an nteror vertex n the juncton trangles s the one whose one-rng neghborng vertces are all n the juncton trangles. An example of the V s shown n Fgure 2, where the nteror vertces n the juncton trangles are marked as crcle dots. Furthermore, let E be the set of all trangle edges {v v j } n the juncton trangles, and then the juncton energy s defned as Fgure 2. The gaps between the nteror grd and the boundary curve. follows: E s = v w,j v j 2 + (v v j ) e(v, v j ) 2 v V v j v v j E where s one-rng neghborng vertces of v, w,j s the mean value barycentrc coordnate of v wth respect to the neghbor v j, and e(v, v j ) = ( l,j /l,j )(v v j ), where l,j, l,j are the orgnal and deformed edge lengths of v v j, respectvely. Total Energy. To coordnate the nteror grd and boundary curve deformatons va the juncton energy (see Fgure 1), t should dsperse the juncton energy to the nteror grd and the boundary curve such that the local areas of the gaps between them can be preserved durng the deformaton. To acheve ths goal, the juncton energy should be reformulated n terms of the nteror grd and boundary curve nodes. For ths purpose the trangular mesh vertces are classfed nto three sets: V 1 the vertces on the mesh boundary edges, V 2 the vertces that are nsde the nteror grd cells, and V 3 the others. Obvously, each mesh boundary vertex n V 1 can be represented as a lnear combnaton of two boundary curve nodes, and each mesh vertex n V 2 can be expressed n the blnear combnaton of four nodes n the nteror grd. Therefore, for each mesh vertex v V 1,2,3, ts deformaton poston can be expressed as v = W Q, where W s a row vector composed of lnear combnaton coeffcents and Q s a column vector composed of 2D poston vectors. When v V 1, gven the deformaton postons: q 1 and q 2 of the boundary curve nodes that correspond to the end ponts of the boundary curve edge whch the mesh vertex v s on, then W = (α 1,α 2, 0, 0) and Q = (q 1, q 2, 0, 0) T, where α 1, α 2 are the lnear combnaton coeffcents. Note that α 1 or α 2 may be 0 when the mesh vertex v concdes wth a boundary curve node. When v V 2, we have v = α 1 q 1 + α 2 q 2 + are the deformed postons α 3 q 3 + α 4 q 4, where {q j } 4 j=1 of the four nodes of the correspondng nteror grd cell n whch the mesh vertex v les, and {α} 4 j=1 are the blnear combnaton coeffcents. Thus W = (α 1,α 2,α 3,α 4 ) and Q = (q 1, q 2, q 3, q 4 ) T. At last, for the v V 3, t s just expressed as tself,.e., W = (1, 0, 0, 0) and Q = (v, 0, 0, 0) T. It should be noted that snce the vertex v V 3 does not le ether n the nteror grd cell or on the boundary curve edge, we can draw that the vertex wll belong to the juncton trangles.. Copyrght 2009 John Wley & Sons, Ltd. 179 Comp. Anm. Vrtual Worlds 2009; 20: (4)
6 W. YANG AND J. FENG By substtutng the expresson v = W Q nto the juncton energy (4), the energy functons E r and E s wll be descrbed by the common varables,.e., the deformaton postons of all the nteror grd and boundary curve nodes: q n and q b, and the deformaton postons of the vertces v V 3. Therefore, a unform energy functon can be defned and expressed n terms of the common varables as follows: {q n mn },{q b },{v V 3 } w r (E rn + E rb ) + w s E s (5) In ths way the juncton energy on the juncton trangles s dspersed to the nteror grd and the boundary curve. The number of varables n Equaton (5) equals to the total numbers of the nteror grd nodes and the boundary curve nodes as well as the vertces n V 3. In general, the number of vertces n V 3 s small, e.g., 0 10 n all our examples. In our experments, we have tested dfferent values for the weghts w r and w s, and found that (w r,w s ) = (6, 1) could produce well-coordnated deformatons of the nteror grd and the boundary curve, as shown n Fgure 1(). Of course, users can tune the two weghts, but should be n a reasonable range. For example, when the value w s s too small the gaps between the nteror grd and the boundary curve could not be well preserved (Fgure 1j); when the value w s becomes too large, the gaps may be preserved strongly so that the nteror grd or the boundary curve mght be dstorted (Fgure 1k). Optmzaton The optmal problem (5) s nonlnear as the terms {A } and {e(v, v j )} cannot be expressed as lnear combnatons of ther varables. Thus we can optmze the energy functon (5) can be optmzed by usng an teratve scheme. The optmzaton s subjected to the poston constrants, whch are the handles specfed nteractvely on the nteror grd or boundary curve nodes. The teraton process works as follows wth an ntal guess of the deformaton postons of the nteror grd and boundary curve nodes as well as the trangular mesh vertces,.e., {q n }, {q b}, and {v V 3 }. In our mplementaton, the ntal guess s taken as the value at the last manpulaton and t works very well for the nteractve applcatons. (1) Starts wth the ntal guess {q n } (0), {q b }(0), and {v V 3 } (0). (2) At the kth teraton, compute {A } (k) for each nteror and boundary rgd regon by mnmzng the energy functon E(R ) n Equaton (1) usng {q n } (k 1) or {q b}(k 1) where {A } (k) has the analytcal soluton; 16 compute each {e(v, v j )} (k) and {w j } (k) n Equaton (4). (3) Substtute above {A } (k), {e(v, v j )} (k), and {w j } (k) nto the Equaton (5), and then mnmze the energy functon (5) by solvng a lnear system to determne {q n } (k), {q b}(k), and {v V 3 } (k). (4) If a local mnmum of energy functon (5) s acheved, stop; otherwse go to step 2. At each teraton, after substtutng the nonlnear terms, {A } (k), {e(v, v j )} (k) and {w j } (k), the nonlnear optmal problem (5) wll become lnear and can be solved by usng a standard lnear least squares mnmzaton. By calculatng the dervatves of energy functon (5) wth respect to all varables and settng them to be zero, a lnear system s obtaned: Mx = b. The lnear system s sparse and the matrx M only depends on the ntal confguratons of the nteror grd and the boundary curve as well as the trangular mesh. Thus a drect solver wth a pre-factorzaton of the matrx M could be employed here, n whch the matrx M needs to be factored only once. 17 In ths way, the lnear system could be solved effcently. In fact, after pre-factorzaton, at each teraton only one smple back-substtuton s executed for solvng the unknown varables,.e., x, and the major runtme cost s the computaton of the rght sde of the lnear system,.e., the vector b. In our expermental examples, about 94 99% computatonal cost of the vector b s spent on the summatons of the nteror or boundary rgd regons. The summatons have the form j R C j, where R s an nteror or boundary rgd regon and C j s the lnear combnaton of the data attached on each node j of the regon R, e.g., j R w A p j or j R w j A j p etc. Obvously, the computatonal cost for the summatons wll ncrease quadratcally for nteror rgd regon (lnearly for boundary rgd regon) wth the ncrease of the rgd regon half-wdth w. Thus the nave calculatons wll be neffcent for nteractvely manpulatng a stff object whch has a large half-wdth value w. Thanks to the regulartes of the nteror grd and the boundary curve, the above problem could be addressed by usng the fast summaton technque. 7 The key dea of the fast summaton here s to fully reuse the redundant summatons on the nteror or boundary rgd regons by explotng ther regulartes, and eventually reduce. Copyrght 2009 John Wley & Sons, Ltd. 180 Comp. Anm. Vrtual Worlds 2009; 20:
7 2D SHAPE MANIPULATION the computatonal cost sum of all j R C j to be nearly lnear n the number of the nteror grd or boundary curve nodes,.e., ndependent of the regon half-wdth w. The detals of fast summaton algorthm can be referred to Reference [7]. Fnally, the nonlnear optmzaton could be accelerated so as to facltate the nteractve manpulaton of both soft and stff objects. Expermental Results and Dscusson The method has been mplemented n a sngle thread way on a PC wth 2.4 GHz Intel Core 2 Duo CPU and 2 GB RAM. The numercal optmzaton for all our expermental examples are convergent after 6 9 teratons n general. Table 1 lsts some performance statstcs data. The runtme costs of the nteror grd and boundary curve generatons as well as the constraned trangulaton are not ncluded as they are all less than 3 mllsecond for all examples n ths paper, and thus could be neglgble. From Table 1, we can see that wth the ncrease of the regon half-wdth w, the number of nonzero elements n the sparse lnear system wll ncrease accordngly, and the performance of the sparse lnear system solver drops ether. 17 However by employng the fast summaton technque the optmzaton runtme can stll meet the requrement of nteractvely manpulatng a stff model n the moderate sze whlst t s prohbtve wthout employng the fast summaton technque. Fgure 1(g) shows the deformaton results usng our method. Obvously, our method effectvely avods the topology-unaware results, as shown n Fgure 1(c), whch s generated by the straghtforward regular grd based method. 4 Fgure 1(h ) shows the mportance of the juncton energy n Equaton (4) whch coordnates the nteror grd and boundary curve deformatons. Even for the nteror grd n the dsjoned shape fatty regons, as shown n Fgure 1(e), the juncton energy can stll coordnate the deformatons between each part of the nteror grd and the boundary curve well (see Fgure 1f and g). The proposed method can generate natural and physcally plausble results wth a few constrants due to ts nonlnear nherence. Fgure 3(a c) shows some natural deformatons of a non-artculated 2D snake by manpulatng only two constrant handles. In Fgure 3(d g), a 2D bottle s edted by manpulatng only four handles. These deformaton effects are comparable to those generated by the two-step lnear approxmaton method, 1 as well as the nonlnear method. 2 However, as the authors descrbed, the two-step lnear approxmaton method 1 may produce physcally mplausble results, as shown n Fgure 4(b). Our nonlnear method effectvely avods the mplausble effects and tends to generate the physcally plausble ones as llustrate n Fgure 4(c). Furthermore, our method sometmes can generate better results than the nonlnear method, 2 because the energy based on the shape matchng,.e., the Equaton (1) s more sutable to preserve the shape nteror rgdty (.e., nternal resstance to deformaton) than the energy based on the local shape area preservaton n Reference [2], as shown n Fgure 4(d f). The proposed method can also provde user an ntutve and convnent way to tune the shape global and local stffnesses. The examples n Fgures 3(d g) and 4(c) and show the deformaton results of the shapes wth dfferent global stffnesses. Curve and Stck Fgure Edtng Besdes 2D shapes, the proposed method can also be appled to other 2D objects, such as planar curves or Fgure VER NOD w PRE OPT1 OPT2 3(a c) (d g) ;6;9 37.1;175.6; ;96.5; ;256.1; (c) ;3;6 7.3;11.5; ;10.7; ;18.6;36.5 4(f) Table 1. Performance statstcs The runtme cost s measured n mllsecond. VER: number of trangular mesh vertces; NOD: number of the nteror grd and boundary curve nodes; w: regon half-wdth; PRE: runtme of pre-computaton; OPT1/OPT2: runtme of teratons wth/wthout fast summaton technque.. Copyrght 2009 John Wley & Sons, Ltd. 181 Comp. Anm. Vrtual Worlds 2009; 20:
8 W. YANG AND J. FENG Fgure 3. (a c) Manpulaton of a non-artculated 2D snake wth two handles: (a) the rest shape. (d g) Deformatons of a 2D bottle wth the ncreasng global stffness from left to rght, where w = 1, 6, 9, respectvely: (d) the rest shape. Fgure 4. Comparson between (a) the rest shape, (b) the two-step lnear approxmaton method 1, and (c) our method. The upper row of (c) shows the shape deformatons wth the dfferent global stffness parameters,.e., w = 1, 3, 6, respectvely. Comparson between (d) the rest shape (e) the nonlnear method 2, and (f) our approach. In (e) the shrnkage occurs at the frog rght arm. stck fgures etc. A planar curve could be regarded as a 2D shape wthout nteror part. Therefore, the proposed method appled to the planar curve edtng could be smplfed by only optmzng the energy functon E rb n Equaton (3). Fgure 5(a b) show an example of the sprng-lke curve deformatons wth dfferent global stffnesses, n whch the curve detals are well preserved and the sprng deformaton results mmc well the real sprng deformatons. A stck fgure s defned as a connected planar graph, whch s composed of open and close 2D shapes as shown n Fgure 5(c). The proposed method could also be. Copyrght 2009 John Wley & Sons, Ltd. 182 Comp. Anm. Vrtual Worlds 2009; 20:
9 2D SHAPE MANIPULATION Fgure 5. (a b) Curve edtng wth dfferent global stffnesses: (a) soft (w = 1) and (b) stff (w = 6). They mmc the behavors of stretchng and squashng a soft and stff sprngs n a physcally plausble way. (c f) An example of the stck fgure manpulaton: (c) the rest fgure. appled to the stck fgure deformaton straghtforwardly. Whlst the boundary rgd regons are defned on the refned edges of both open and close shapes, the nteror rgd regons are only defned on the nteror grds of the close shapes. The energy functons E rb, E rn, and E s could be defned accordngly, and a total energy functon for the stck fgure manpulaton s as follows: E f = n w r E r b + =1 m w r E r n + w s E s (6) where n s the number of both the open and close shapes, and m s the number of the close ones n the stck fgure. Fgure 5(c f) show an example of the stck fgure manpulaton. =1 Concluson and Future Work Ths paper presents a two-dmensonal shape deformaton method va drect manpulaton. It adopts a hybrd shape representaton structure whch composes of an nteror regular grd and a trangular mesh. The approach constructs a global and local stffness-tunable deformaton model based on the nteror regular grd and elmnates the unnatural results by consderng the topologcal structure mpled n the trangular mesh. Meanwhle, by explotng the nteror grd and boundary curve regulartes, the method can be speeded up usng the fast summaton technque so as to manpulate both the soft and stff objects nteractvely. Furthermore, the proposed method can be extended to manpulate the 2D curves and stck fgures. As the future work, the proposed method wll be extended to 3D case. However, t s not a trval work. Compared wth the 2D trangular mesh, the 3D tetrahedralzaton of a mesh s more dffcult to be mplemented and computatonally expensve, as well as s not robust. Therefore, the key step n 3D case s to construct a sutable tessellaton structure that s able to capture the 3D surface topology. ACKNOWLEDGEMENTS The work was jontly supported by the NSF of Chna ( ), the 973 Program of Chna (2009CB320801), the Natonal Key Technology R&D Program (2007BAH11B02), and the NSF of Zhejang Provnce (R106449).. Copyrght 2009 John Wley & Sons, Ltd. 183 Comp. Anm. Vrtual Worlds 2009; 20:
10 W. YANG AND J. FENG References 1. Igarash T, Moscovch T, Hughes JF. As-rgd-as-possble shape manpulaton. ACM Transactons on Graphcs 2005; 24(3): Weng Y, Xu W, Wu Y, Zhou K, Guo B. 2d shape deformaton usng nonlnear least squares optmzaton. The Vsual Computer 2006; 22(9): Sorkne O, Alexa M. As-rgd-as-possble surface modelng. In SGP 07: Proceedngs of the 5th Eurographcs Symposum on Geometry Processng, July 2007; Yang W, Feng J, Jn X. Shape deformaton wth tunable stffness. The Vsual Computer 2008; 24(7 9): Botsch M, Pauly M, Wcke M, Gross M. Adaptve space deformatons based on rgd cells. Computer Graphcs Forum 2007; 26(3): Stenemann D, Otaduy MA, Gross M. Fast adaptve shape matchng deformatons. In Proceedngs of the ACM SIGGRAPH/Eurographcs Symposum on Computer Anmaton, 2008; Rvers AR, James DL. Fastlsm: fast lattce shape matchng for robust real-tme deformaton. ACM Transactons on Graphcs 2007; 26(3): Sederberg TW, Parry SR. Free-form deformaton of sold geometrc models. Computer Graphcs (SIGGRAPH 1986) 1986; 20(4): Lews JP, Cordner M, Fong N. Pose space deformaton: a unfed approach to shape nterpolaton and skeleton-drven deformaton. In SIGGRAPH 00, July 2000; Botsch M, Kobbelt L. Real-tme shape edtng usng radal bass functons. Computer Graphcs Forum 2005; 24(3): Schwetz T, Georg J, Westermann R. Freeform mage. In Proceedngs of Pacfc Graphcs 2007, October 2007; Sorkne O, Cohen-Or D, Lpman Y, Alexa M, Rössl C, Sedel H-P. Laplacan surface edtng. In SGP 04: Proceedngs of the 2004 Eurographcs/ACM SIGGRAPH Symposum on Geometry Processng, July 2004; Yu Y, Zhou K, Xu D, Sh X, Bao H, Guo B, Shum H-Y. Mesh edtng wth posson-based gradent feld manpulaton. ACM Transactons on Graphcs 2004; 23(3): Lpman Y, Sorkne O, Levn D, Cohen-Or D. Lnear rotatonnvarant coordnates for meshes. ACM Transactons on Graphcs 2005; 24(3): James DL, Barbč J, Twgg CD. Squashng cubes: automatng deformable model constructon for graphcs. In SIGGRAPH 04: ACM SIGGRAPH 2004 Sketches, August 2004; Schaefer S, McPhal T, Warren J. Image deformaton usng movng least squares. ACM Transactons on Graphcs 2006; 25(3): Davs TA. UMFPACK, an unsymmetrc-pattern multfrontal method. ACM Transactons on Mathematcal Software 2004; 30(2): Authors bographes: Wenwu Yang s a doctoral student n the State Key Lab of CAD&CG, Zhejang Unversty, Peoples Republc of Chna. He receved hs MSc n computer graphcs from Zhejang Unversty n Hs research nterests nclude computer graphcs, geometrc modelng, and computer anmaton. Jeqng Feng s a professor n the State Key Lab of CAD&CG, Zhejang Unversty, Peoples Republc of Chna. He receved hs BSc n appled mathematcs from the Natonal Unversty of Defense Technology n 1992 and hs PhD n computer graphcs from Zhejang Unversty n Hs research nterests nclude geometrc modelng, renderng, and computer anmaton.. Copyrght 2009 John Wley & Sons, Ltd. 184 Comp. Anm. Vrtual Worlds 2009; 20:
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