2D Shape Deformation Based on Rigid Square Matching

Transcription

1 D Shape Deformaton Based on Rgd Square Matchng A. Author Author s Afflaton Author s E-Mal Author s Webste B. Author Author s Afflaton Author s E-Mal Author s Webste C. Author Author s Afflaton Author s E-Mal Author s Webste Abstract In ths paper, we propose a fast and stable method for D shape deformaton based on rgd square matchng. Our method utlzes unform quadrangular control meshes for D shapes and tres to mantan the rgdty of each square n the control mesh durng user manpulaton. A rgd shape matchng method s performed to fnd an optmal pure rotatonal transformaton for each square n the control mesh. An teratve solver s proposed to compute the fnal deformaton result for the entre control mesh by mnmzng the dfference between the deformed vertces and ther counterparts n the neghborng rgd square. The deformaton result on the D shape s as rgd as possble and the detals of the shape are preserved well. As extensons, we present a shape-aware splttng method to mprove the deformaton effect for coarse meshes and a smple sketch-based clusterng method for skeletal deformaton. Experments wth varous D shapes show that our method s effcent and easy to use, and can provde physcally plausble result for shapes of objects n real world. Therefore, our shape deformaton method s especally sutable for applcatons n cartoon character anmaton. Keywords: shape deformaton, character anmaton, shape matchng, rgd transformaton, skeletal deformaton Fgure 1: Horsemanshp actons obtaned by our method. The orgnal D shape wth ts control mesh s on the top left. Note the natural skeletal behavor n deformng the horse legs. 1. Introducton D shape deformaton (or manpulaton) ams at deformng objects represented n the form of D mages by usng graphcal method. It s a very useful tool for applcatons such as character anmaton [1], real-tme lve performance [] and enrchng graphcal nterface [3], and therefore, has receved a lot of attentons n recent years. In such typcal applcatons, the D shape n the mage to be deformed often represents an object n real world, and more often a character or an anmal. As a result, the deformed shape must exhbt deformaton effects that are physcally plausble. Effcency

2 s another mportant feature to be consdered of the deformaton method, because the method must provde the user nteractve tools to manpulate the deformaton result. The nteracton metaphor for the user should also be ntutve and easy to use. Many methods have been proposed to satsfy these requrements. Shape deformaton methods usng spacebased technques deform the shapes by manpulate the space n whch they are embedded. Free-form deformaton (FFD) methods [4, 5, 6] and skeleton-based technques [7, 8, 9] are of ths category. They are very effcent n computaton and easy to be mplemented. However, they don t provde convenent or meanngful nteracton tools for the user. For FFD methods, the control s not ntutve, and for skeletonbased technques, the weght tunng for rggng s a panful process for users. Physcally-based smulaton ncludng fnte element method [1] and mass-sprng system [11] can provde precse deformaton result. But they are often too expensve n computaton or too slow to converge for nteractve applcatons. Moreover, precson of the deformaton result s not very mportant n such applcatons as compared to effcency and usablty. Therefore, almost all the shape deformaton methods proposed n recent years utlze geometrcally-based approach nstead of physcally-based smulaton. Recently, many nonlnear mesh deformaton methods [1, 13, 14, 15] are proposed for 3D models. They try to mnmze a nonlnear energy functonal representng local propertes of the surface. Weng et al. [1] use a varant of these methods for D shape deformaton. They use a hybrd mesh generated by usng the shape boundary as the deformable model. Ther method not only preserves the Laplacan coordnates lke the deformaton methods for 3D meshes do, but also preserves local area of the shape nteror. All these terms add up nto a non-quadratc energy functon whch s mnmzed by usng an teratve Gauss-Newton method. 1.1 Related Work D shape deformaton s closely related to a more general technque named mage deformaton, whch focuses on reasonably warpng the entre space of the mage. Image deformaton has a longer hstory and s typcally used for D morphng [16] and medcal magng [17]. Schaefer et al. [18] proposed an mage deformaton method whch can also be used for D shape deformaton. They used movng least squares to mantan the rgdty of the entre mage space. Though t s very effcent, the method doesn t take the nformaton of the shape nto account. Therefore, the detals of the shape cannot be preserved well for large scale deformaton. We nstead utlze the explct D rotaton expresson n [18] to mantan the local rgdty of the shape. Müller et al. [19] proposed a meshless deformaton method based on shape matchng. Ther method s fast and stable, and can produce physcally plausble deformaton results. Because of the smple modes of deformaton (only lnear and quadratc) used n shape matchng, the method s only sutable for modest deformaton complexty. A practcal enhancement s proposed n [19] usng overlappng domans for many shape matchng clusters to enrch the deformaton effect, but the result stll have blendng artfacts. Rvers and James [] proposed a fast lattce shape matchng (FastLSM) method to address these problems. Our method also utlzes regular lattces and shape matchng technque, but there are also many sgnfcant dfferences. Our method s appled for D shape deformaton amng at applcatons n character anmaton and lve performance. Ther method overlaps many cells and therefore need a specal fast algorthm to get the fnal result whle our method only perform shape matchng for each square ndependently. FastLSM uses regonbased convoluton to get the fnal result, whle our method uses an teratve solver. Igarash et al. proposed the concept of asrgd-as-possble manpulaton n [1]. In ther work, the D shape boundary s frstly represented by a D smple polygon and s then trangulated to form a trangular mesh. Durng manpulaton, the user drags some vertces of the trangular mesh as handles, and the system computes the poston of other free vertces by mnmzng the deformaton dstorton of every trangle. 1. Our Contrbutons In ths paper, we propose a fast and stable method for D shape edtng. It s motvated by the shape matchng method for dynamc deformaton n [19] and s very smple to be mplemented. Because our shape edtng me-

3 thod s manly desgned for applcatons n character anmaton, we also adopt the concept of as-rgd-as-possble manpulaton n [1] to gve physcally plausble results. Our contrbutons can be summarzed as follows: (1) We propose a rgd square matchng method for D shape deformaton. Ths method s especally sutable for applcatons n character anmaton, because t tres to mantan the local rgdty of the D shape and can produce physcally plausble de-formaton effects for shapes of objects n the real world. To solve for the entre control mesh, we propose an teratve solver whch utlzes the rgd transformatons acqured by rgd square matchng. The method s uncondtonally stable, easy-mplemented and effcent for nteractve D shape edtng. () As a practcal enhancement, we propose a shape-aware splttng method for the unform quadrangular lattce. It explots the topologcal nformaton of the D shape to mprove the deformaton effect under a relatvely coarser mesh; (3) We mplement skeletal deformatons by usng a user defned clusterng method. The sketch-based metaphor s very convenent to use and the method s easy to mplement.. Overvew In ths secton, we gve an overvew of our method whch s shown n Fgure. Our D shape deformaton method takes an mage as the nput. We suppose that the D shape to be consdered has salent dfferences n llumnaton or color from the background n the mage. We also suppose that all the mages are stored n the btmap format for convenence of dscusson, and the method proposed n ths paper can be generalzed to other mage format easly. Frstly, the background whch we are not nterested n s removed manually. Dfferent from some mesh-based methods [1, 1], our method doesn t need the shape boundary nformaton. Therefore, the next step sn t the boundary extracton operaton. Instead, the D shape s drectly embedded nto a regular quadrangular lattce. Then, by elmnatng the lattce vertces outsde the shape, we can obtan a regular quadrangular lattce tghtly boundng the D shape. A unform quadrangular mesh M = ( V, S) s generated wth no dffculty from the lattce and s then used as the deformaton control mesh, where V s the set of n vertces n the mesh and S s the set of m square cells (because we try to preserve ther orgnal shape durng the deformaton, we always call them squares n ths paper despte they are not real squares any more after beng deformed). Our method performs some pre-processng for ths control mesh beforehand to decrease the real-tme computaton. Then the user can drag the mesh vertces freely to deform the D shape. Durng the deformaton process, our method tres to mantan the local rgdty of each square to reserve the local detals of the D shape. After the control mesh s deformed accordng to our method, the fnal D shape s rendered by usng smple lnear texture mappng technque for each square. Orgnal shape Deformed shape Control mesh generaton Deformed shape renderng Control mesh deformaton Fgure : Overvew of our D shape deformaton method. The remander of ths paper s organzed as follows. Secton 3 descrbes the core part of our method n detals, ncludng the rgd square matchng algorthm and the teratve solver to calculate the deformaton result of the entre mesh. In Secton 4, two extensons for our method are proposed to mprove the deformaton result and enrch the deformaton style. Expermental results are shown n Secton 5, followed by conclusons n Secton Shape Deformaton Based on Rgd Square Matchng After the control mesh generaton phase, a unform quadrangular mesh M = ( V, S) s generated. Then the user can drag any vertces n ths mesh to deform the D shape. As depcted n Fgure, our method frstly deforms the control mesh accordng to the as-

4 rgd-as-possble prncple. The rgd square matchng method s proposed to mantan the rgdty of each square n the control mesh. We use a smple teratve solver to compute the deformaton result for the entre control mesh. 3.1 Rgd Square Matchng The progress of the rgd square matchng algorthm s llustrated n Fgure 3. It tres to fnd a rgd square whch fts the current deformed square best. Frstly, we consder an ndvdual square s S n the control mesh. The orgnal postons of ts four vertces are x. After the user s manpulaton, the new postons of these vertces are x R R (1 4). Accordng to the shape matchng technque used n [19], the optmal rgd transformaton, ncludng an optmal D rotaton R R and a translaton vector t R, are defned to mnmze the dfference between transformed x and x 4 = ω 1 Rx + t x (1) where ω (1 4) are weghts of ndvdual vertces. Whle Müller et al. [19] use the mass of each pont as ts weght because they am at real-tme dynamc deformaton, we use the weghtng scheme to mplement poston constrant for handles. In all the examples presented n ths paper, we fnd that assgnng n for constraned handles and 1 for other free vertces s a good choce. We can actually remove the translaton vector t n Equaton 1 to smplfy the mnmzaton problem. Settng the partal dervatves wth respect to t n Equaton 1 to zero yelds the optmal translaton vector t = xc Rx c () where x and are weghted centrods of the c xc square before and after deformaton respectvely: ω x xc = ω (3) ω x xc = ω Then we can substtute Equaton nto Equaton 1 to get a smpler formula wth only R as the unknown 4 ω ˆ ˆ = 1 Rx x (4) where x = x x and x = x x. ˆ c ˆ c Snce the rotatonal transformaton n 3D space s nonlnear, Müller et al. [19] frstly relax the problem to fnd an optmal lnear transformaton and then extract the rotatonal part from t. We don t need to do ths because of the lnearty of D rotaton, so we can drectly solve for R. As a D rotaton matrx, T R s an orthogonal matrx,.e. RR= I. If R s represented n the form of a block matrx R = ( R1 R ) where R, then, and 1, R R RR 1 = T T RR 1 1 = RR = 1. By usng ths property and mnmzng Equaton 4, the optmal rotaton matrx can be gven as T 1 4 xˆ T R = ω 1 ( ˆ ˆ = T x x ) (5) μ xˆ where 4 T 4 T ( ˆ ˆ 1 ) ( ˆ ˆ = = 1 ) μ = ω xx + ω xx (6) and s an operator on D vectors such that ( x, y) = ( y, x). Then, as the result of the rgd square matchng method, the postons of the four vertces of the ftted rgd square are gven by usng ths optmal rotaton and translaton x = Rxˆ + x (1 4) (7) x 1 x 4 x c x x 3 c x x x 1 x 1 x x 3 x x 3 c c x 4 x 4 Fgure 3: Progress of rgd square matchng algorthm. 3. Iteratve Solver for Control Mesh Deformaton By usng the rgd square matchng method proposed n Secton 3.1, we can fnd a ftted rgd square for each deformed square. However, for the entre mesh, these rgd squares are not necessary to conform to ther neghbors due to the arbtrary manpulaton of the user. Thus, we propose an teratve solver to compute the new postons of the vertces n the entre control mesh by mnmzng the dfference between the resultng vertces n the deformed control mesh and ther counterparts n the neghborng ftted rgd squares. For th (1 n) vertex n the control mesh, ts x 1 x 4 x x c x 3

5 current poston s x and we suppose the set of ts neghborng squares to be N() S. Then, the error functon for th vertex s defned by ω x ( Rsx + ts) (8) s N() where R and s t are the optmal rotaton s matrx and translaton vector computed for the neghborng square s, respectvely. Then the global error functon over the entre control mesh s gven by ω x ( Rsx + t s) (9) s N() We need to mnmze ths error functon to fnd the new postons for each vertex n the deformed control mesh. However, R s and t s are dependent n the postons of other vertces n the control mesh, so Equaton 9 s nonquadratc and consequently, t cannot be solved drectly. Therefore, we propose a smple teratve solver to lnearze t. R s and t are s supposed to be nvarant for each teraton and they can be vewed as constant n Equaton 9. Then ths quadratc functon has a unque mnmzer, whch yelds the teratve solver for the fnal postons of all the vertces n the control mesh 1 x ( R sx + t ) (1) s N () s N() For each teraton step, the ftted rgd squares for all the deformed squares are computed ndependently and then the correspondng rotatons and translatons are used to update the postons of all the vertces by usng Equaton 1. The teratve solver repeats untl the total error functon of the entre mesh expressed by Equaton 9 vares less than a gven threshold n several successve teratons. The expermental results show that the total error functon becomes stable n tens of teratons after user manpulaton (Fgure 4). Snce all the computaton performed n each teraton s ndependent for each square or vertex and only deal wth D matrx and vectors, the tme complexty for one teraton s Om ( ). As a result, our deformaton method can be very effcent even for control meshes wth very fne resoluton. Moreover, we make a lttle modfcaton for the teratve solver n practce to further mprove ts nteractve performance. When the user drags the handles, we perform the teraton for a fxed number of tmes (we use 3 tmes n all the examples) to guarantee the real-tme response, because the ntermedate results durng user nteracton need not to be precse. When the user releases the handles and fnshes the nteracton, our solver proceeds to fnd the precse result at convergence. Fgure 4: Error/teraton curve. The peak of the curve occurs when the user drags and releases the handle. 4. Extensons 4.1 Shape-aware Mesh Splttng In order to smplfy the control mesh generaton, we use unform quadrangular control meshes for shape deformaton nstead of adaptve meshes. As depcted n Fgure 5, for some partcular shapes wth two geodescally remote parts located near n space, coarse meshes wll ntroduce ncorrect connectons between the two parts. Ths problem s especally obvous for shapes lke anmals and humans whch have obvous elongated parts such as lmbs. One drectly approach to solve ths problem s to use a fner lattce for the shape to produce a fner control mesh, but ths wll ncrease the computaton cost of the deformaton algorthm. Instead, we propose a shapeaware mesh splttng method to allevate ths problem for relatvely coarser mesh. The man dea of the shape-aware mesh splttng method s to splt parts of the mesh by duplcatng vertces n order to break the ncorrect connecton n the control mesh. To acheve ths, we further explot the nformaton of the shape, defnng the boundary vertces to be the mesh vertces located outsde the shape. Ideally, all the boundary vertces of a shape wll form one or more closed loops,.e. each boundary vertex has exactly two neghborng boundary vertces. But for ncorrect connectons n the control mesh, one boundary vertex (whch s called the nvald boundary vertex ) may have more than two neghborng boundary vertces as depcted n Fgure 5. For

6 such cases, the nvald boundary vertex s duplcated and assgned to the neghborng squares accordng to the D shape nformaton by usng a seres of rules whch s depcted n Table 1. In ths table, the shaded part n each square represents the nner regon of the shape. The red ponts are nvald boundary vertces to be consdered and the blue ponts are duplcated boundary vertces whch become vald. The duplcated vertces are offset a lttle for clarty. It should be notced that new nvald boundary vertces may be ntroduced when one nvald boundary vertex s resolved accordng to the rules. Actually, the rules are appled to the control mesh repeatedly untl there s no nvald boundary vertex. As a result, the ncorrect connectons of the control mesh are splt and more natural deformaton effect can be produced even for a relatvely coarser mesh (Fgure 7). Fgure 5: An example of ncorrect connectons. The zoomed depcton of the part enclosed by the black rectangle s shown on the rght. All the colored ponts represent the boundary vertces. Red and blue ponts represent ncorrect connectons to be splt. Intally, red ponts are recognzed as nvald boundary vertces by our mesh splttng method. Number of neghborng squares 3 4 Table 1: Mesh splttng rules. Duplcaton and assgnng rules 4. Skeletal Deformaton Usng Sketch-based Clusterng Skeletal deformaton s very mportant for applcatons n character anmaton, because almost all cartoon characters are artculated anmals. Therefore, we propose a skeletal deformaton approach to our basc D shape deformaton method. Ths approach s mplemented by usng clusterng technque and provdes a smple sketchng metaphor to the user. We vew each ndvdual square n the control mesh as a deformaton cluster and generate larger clusters whch are composed of squares accordng to the user nteracton to represent the skeletal structure of the shape. The entre progress of ths approach s shown n Fgure 6. The user draws lnes on the shape to desgnate the skeletons. Then, the squares whch ntersect the same lne are assembled nto one cluster. When the user deforms the skeletal shape, each newly generated cluster s treated as a square n the above-mentoned rgd square matchng algorthm and replaces the squares whch make up t. Because our D shape deformaton method tres to mantan the rgdty of every cluster, the skeletal deformaton effect can be presented by usng ths smple sketch-based clusterng approach. Fgure 6: Skeletal deformaton usng sketchbased clusterng. 5. Expermental Results We have mplemented the descrbed D shape deformaton method on a workstaton wth a.33ghz Intel Core TM Duo CPU and GB memory. Table shows the statstcs for the examples used n ths paper and tmngs for our shape deformaton method. In Table, Lattce grd means the resoluton of the space lattce n whch the shapes are embedded to generate the control meshes; Soluton tme 1 means tme need for our method to perform 3 teratons when the user drags the handles as descrbed n Secton 3.; and Soluton tme means tme need to exactly perform the teratve solver untl t satsfes the stop condton to

7 get the fnal result when the user releases the handles. Soluton tme 1 s approxmately lnear n the scale of the control mesh, ndcatng that our D shape deformaton method s very effcent for nteractve shape edtng. Soluton tme s affected not only by the scale of the mesh, but also the scale of the deformaton and the topology of the mesh, so the numbers lsted n the last row of Table 1 are average tme under varous condtons for each shape. In all the experments presented n ths paper, the teratve solver converges wthn.6 second. Ths cost s acceptable because t only occurs when the user stops nteractng wth the D shape. Fgure 7 compares the deformaton results wth and wthout shape-aware mesh splttng for the same shape under the same resoluton of control mesh. It s shown that the shapeaware mesh splttng method can elmnate the ncorrect connectons n the control mesh. By usng the splt control mesh, our D deformaton method can provde more natural effect. In Fgure 8, we present the effect of skeletal deformaton by usng our sketch-base clusterng approach. It s also compared wth the deformaton result wthout skeletons. Our skeletal deformaton approach provdes an easy-to-use metaphor for users and produces physcally plausble effect especally for artculated cartoon characters. Fgure 9 shows that our method can provde more realstc result than the as-rgd-as-possble shape manpulaton method n [1] does. To llustrate the versatlty of our D shape deformaton method, more examples are gven n Fgures 1 and 1. The tmngs n Table and examples n the fgures show that our method can provde as good deformaton result as the nonlnear optmzaton method n [1] at less computaton cost. Fgures 11 and 1 show that our method can also be appled to mage deformaton to provde smlar results as [18] wth no extra modfcaton except treatng the entre mage as a shape. 6. Concluson In ths paper, we propose a D shape deformaton method based on rgd square matchng. The method uses unform quadrangular meshes as control meshes whch s much easer to buld than trangular meshes used n [1] and hybrd meshes used n [1]. We adopt the concept of as-rgd-as-possble deformaton and use the shape matchng technque to mantan the local rgdty of the shape. The transformatons acqured by shape matchng technque are constraned explctly to be pure rotatons. Therefore, the detals of the D shapes can be preserved well durng deformaton. Moreover, we drectly update every vertces of one square by usng pure rotatonal transformaton wthout dynamc ssues n [19], and consequently there wll be no nverted square and our system s uncondtonally stable. We also propose a smple teratve solver to compute the fnal deformaton result of the entre mesh. The deformaton mode for each square s smply the rgd moton, but the fnal results of the entre shape exhbt very complex deformaton effects as shown n Fgures 1 and 1, because the connectons between the squares n the control mesh provde the system a very hgh degree-of-freedom. Essentally, our shape deformaton method s space-based, but by generatng the control mesh accordng to the D shape t avod the problems presented n the space warpng approaches. We also propose a practcal enhancement whch further utlzes the nformaton of the shape to mprove the deformaton effect for coarse control meshes. By usng a smple clusterng method, our method can provde skeletal deformaton effect for artculated shapes lke cartoon characters and anmals. The skeleton desgnatng metaphor s sketch-based and s very easy to use. Our method s very effcent and can provde physcally plausble deformaton effect for shapes of objects n real world. However, t stll has some lmtatons. Frst, because we use pure rotatonal transformaton for each square n the control mesh, the global area cannot be preserved by the current algorthm. Second, t s too rgd to deform shapes of soft and rubber-lke objects, such as sponges and jelles. These two problems can be addressed by usng more complex transformatons n the square matchng process. Moreover, snce we are plannng to generalze our method to 3D shape edtng, a solver more effcent than the teratve solver proposed n Secton 3. s to be consdered n the future work.

8 D shapes Lattce grd n Control mesh m statstcs Soluton tme 1 Soluton tme Bee (Fg. 8, 1) ms.39s Horse (Fg. 1) ms.56s Gecko (Fg. 6) ms.14s Character (Fg., 1) ms.19s Flower (Fg. 1) ms.59s Mona Lsa (Fg. 11) ms.3s Leanng Tower (Fg. 1) ms.51s Table : Statstcs and tmngs. Fgure 7: Deformaton of a gecko wth (rght) and wthout (mddle) shape-aware mesh splttng. The orgnal shape of the gecko wth the orgnal control mesh s shown on the left. Boundary vertces are marked as green ponts and blue ponts wth blue ones ndcatng ncorrect connectons before mesh splttng. Fgure 8: Deformaton of a bee wth (rght) and wthout (mddle) skeleton desgnaton by sketch-based clusterng. The orgnal shape s shown on the left and the correspondng control meshes are shown besde the shapes. Clustered squares are panted n red for the control mesh of the deformed shape on the rght. Red ponts represent the handles manpulated by the user. Fgure 9: Comparson between the method n [1] and our method. Acknowledgements References [1] Y. Weng, W. Xu, Y. Wu, K. Zhou, and B. Guo. D shape deformaton usng nonlnear least squares optmzaton. The Vsual Computer, (9-11):653-66, 6 [] T. Ngo, D. Cutrell, J. Dana, B. Donald, L. Loeb, and S. Zhu. Accessble anmaton and customzable graphcs va smplcal confguraton modelng. In Proceedngs of ACM SIGGRAPH, pages 43-41, [3] H. T. Bruce and P. Calder. Anmatng drect manpulaton nterfaces. In Proceedngs of UIST 95, pages 3-1, 1995 [4] R. MacCracken and K. Joy. Free-form deformatons wth lattces of arbtrary

9 topology. In Proceedngs of ACM SIGGRAPH 96, pages , 1996 [5] T. Mllron, R. Jensen, R. Barzel, and A. Fnkelsten. A framework for geometrc warps and deformatons. ACM Transactons on Graphcs, 1(1):-51, [6] T. Sederberg and S. Parry. Free-form deformaton of sold geometrc models. In Proceedngs of ACM SIGGRAPH 86, (4):151-16, 1986 [7] J. P. Lews, M. Cordner, and N. Fong. Pose space deformaton: a unfed approach to shape nterpolaton and skeleton-drven deformaton. In Proceedngs of ACM SIGGRAPH, pages , [8] H. -B. Yan, S. -M. Hu, R. R. Martn, and Y. -L. Yang. Shape deformaton usng a skeleton to drve smplex transformatons. IEEE Transactons on Vsualzaton and Computer Graphcs, 14(3):693-76, 8 [9] S. Forstmann, J. Ohya, A. Krohn- Grmberghe, and R. McDougall. Deformaton styles for splne-based skeletal anmaton. In Proceedngs of Eurographcs/ACM SIGGRAPH Symposum on Computer Anmaton, pages , 7 [1] G. Celnker and D. Gossard. Deformable curve and surface fnte-elements for freeform shape desgn. In Proceedngs of ACM SIGGRAPH 91, pages 57-66, 1991 [11] S. F. F. Gbson and B. Mrtch. A survey of deformable modelng n computer graphcs. Techncal report TR-97-19, Mtsubsh Electrc Research Laboratores, 1997 [1] O. K. C. Au, C. L. Ta, L. Lu, and H. Fu. Mesh edtng wth curvature flow laplacan operator. Techncal report, Computer Scence Techncal Report, HKUST-CS5-1, 5 [13] J. Huang, X. Sh, X. Lu, K. Zhou, L. We, S. Teng, H. Bao, B. Guo, and H. Y. Shum. Subspace gradent doman mesh deformaton. In Proceedngs of ACM SIGGRAPH 6, pages , 6 [14] A. Sheffer and V. Kraevoy. Pyramd coordnates for morphng and deformaton. In Proceedngs of 3DPVT, pages 68-75, 4 [15] M. Botsch, M. Pauly, M. Wcke, M. Gross. Adaptve space deformaton based on rgd cells. In Proceedngs of Eurographcs 7, 6(3): , 7 [16] D. Smythe. A two-pass mesh warpng algorthm for object transformaton and mage nterpolaton. Tech. Rep. 13, ILM Computer Graphcs Department, Lucasflm, San Rafael, Calf, 199 [17] T. Ju, J. Warren, G. Echele, C. Thaller, W. Chu, and J. Carson. A geometrc database for gene expresson data. In SGP 3: Proceedngs of the 3 Eurographcs/ ACM SIGGRAPH Symposum on Geometry Processng, pages , 3 [18] S. Schaefer, T. McPhal, and J. Warren. Image deformaton usng movng least squares. In Proceedngs of ACM SIGGRAPH 6, 5(3):533-54, 6 [19] M. Müller, B. Hedelberger, M. Teschner, M. Gross. Meshless deformatons based on shape matchng. In Proceedngs of ACM SIGGRAPH 5, 4(3): , 5 [] A. R. Rvers and D. L. James. FastLSM: Fast Lattce Shape Matchng for Robust Real-Tme Deformaton. In Proceedngs of ACM SIGGRAPH 7, 6(3):8, 7 [1] T. Igarash, T. Moscovch, and J. F. Hughes. As-rgd-as-possble shape manpulaton. In Proceedngs of ACM SIGGRAPH 5, 4(3):

10 Fgure 1: More results of our shape deformaton method. In each row, the orgnal shape s shown one the leftmost mage and others are the deformaton results. Fgure 11: Image deformaton for Mona Lsa by usng our method. After deformaton, her face s thnner and she s n a sad mood. Fgure 1: Image deformaton for the Leanng Tower of Psa by usng our method. We mplement deformaton wth lne segment handles [18] by usng the skeletal deformaton method proposed n Secton 4..