3D MODEL DEFORMATION ALONG A PARAMETRIC SURFACE

Transcription

1 3D MODEL DEFORMATION ALONG A PARAMETRIC URFACE Bng-Yu Chen * Yutaka Ono * Henry Johan * Masaak Ish * Tomoyuk Nshta * Jeqng Feng * The Unversty of Tokyo 7-3-1, Hongo, Bunkyo-ku, Tokyo, Japan Zheang Unversty Hangzhou, , Chna Abstract Free-Form Deformaton (FFD) s an effcent technque for edtng the shapes of 3D models, and wdely used n Computer-Aded Desgn (CAD), computer anmaton, computer graphcs entertanment etc. The basc dea of some famous prevous FFD approaches deforms a target 3D model by adustng some control ponts of a 3D lattce surroundng the model. It s a tedous work, especally when the lattce contans too many control ponts. Moreover, how to place the control ponts of the lattce to make them cover the regon of the target model s also a problem, and t s also dffcult to keep the geometrc measurement for the deformed model. Therefore, n ths paper, a novel FFD method wthout any lattcelke structure s proposed by borrowng the dea of nondstorton texture-mappng for free-form surfaces. By usng ths method, the shape of the deformed model due to a gven parametrc surface can be predcted easly, so that the user can get hs or her desrable results more ntutvely and has no necessary to place the control ponts of a proper lattce. Moreover, snce ths method s smple, effcent, and has a real-tme response, t s more sutable for dong the anmaton wth thn obects. Key Words: 3D model deformaton, non-dstorton deformaton, non-dstorton mappng, flattened surface, anmaton 1. Introducton Recently, computer graphcs technologes have been appled to several felds, such as move ndustry, vdeo game, CAD, and scentfc vsualzaton. Obvously, the expresson of 3D models s also an mportant part of the computer graphcs technologes. The 3D models are usually represented by some prmtves such as ponts, lnes, and polygons. Although complcated and beautful models can be rendered now wth a lot of such prmtves by usng graphcs hardware, how to deform such models to be people s desrable shapes generally, ntutvely, and effcently s stll a problem. FFD s an effcent technque n computer graphcs for provdng a general modfcaton of a 3D model such as twstng, bendng, and stretchng. ederberg and Parry frst ntroduced ths method from the practcal vewpont n 1986 [1]. By usng ths method, the users frst generate some control ponts called a lattce to contan a 3D model whch s a target obect for deformaton, and then adust the control ponts to deform the parametrc space nsde. Fnally, the deformed parametrc space s mapped onto the target model automatcally. A lot of related approaches are based on ths method, and t has become a useful and well-known technque for 3D model deformaton. everal methods of FFD are mostly based on the above methods, although there are some dfferences about the defntons of the lattces. However there are some problems whle dong the FFD processes. Frst, the relatonshp between the lattce and the target model s unclear, so t s hard to grasp ntutvely that how desrable deformaton can be obtaned f the control ponts of the lattce are adusted. Then, how to place the control ponts of the lattce to proper postons to make the users to control the shape well s very dffcult. Furthermore, t s also dffcult to keep the geometrc shape of the model after deformaton, and possble to have a dstorton problem of the deformed shape. Therefore, we propose a novel FFD method n ths paper wthout any lattce-lke structure to prevent the users to setup the control ponts and adust them. The basc dea of our approach s to utlze the technology of texture-mappng [2], snce texture-mappng can make a mappng relatonshp between a 2D texture mage wth a 3D free-form surface as to wrap the mage to ft the shape of the surface. Hence, we can also fnd a mappng relatonshp between a 3D model and a parametrc surface by ust replacng the texture mage n the texturemappng to be a 3D model. Unfortunately, the man problem of the texture-mappng tself s the dstorton problem f we ust use the lnear mappng. o, we borrow the technology of makng non-dstorton texturemappng [3] to prevent ths problem. Then, the user can deform a 3D surface as they wsh by ust makng a desrable parametrc surface whch s easer than adustng the control ponts of a lattce. 2. Related Work Barr ntroduced an effcent method wth some restrctons to deform a 3D model by usng the combnatons of four operatons [4]. Then, a deformaton technque s proposed by ederbarg and Parry [1], whch s wdely avalable and almost all subsequent methods are based on t. Coqullart proposed an extenson of t, called Extended Free-Form Deformaton (EFFD), whch uses several low resoluton lattces, called chunk, for deformaton [5]. Thereby, a complcated lattce can be

2 defned wthout rasng the resoluton of the chunks, snce t can be a combnaton of several chunks, but t s dffcult to mantan the contnuty of the form n the case of connecton of the chunks. Coqullart and Jancéne used EFFD method to buld anmatons called Anmated Free-Form Deformaton (AFFD) [6]. Gressmar and Purgathofer proposed a new FFD method based on B-plne wth three valuables, and optmzed the mesh dvson after deformaton [7]. By usng the general FFD method, the relatonshp between a lattce and a target model s obscure, so that to adust the control ponts of the lattce s dffcult to understand ntutvely. Hsu et al. amed to solve ths problem and proposed a method to control the target model drectly [8]. By usng ths method, a desrable shape could be generated by adustng arbtrary vertces drectly of the model, and then the correspondng control ponts of the lattce are calculated backward to acqure the shape. Therefore, desrable results can be acqured even for the local deformaton. Kalra et al. thought up a Ratonal Free-Form Deformaton (RFFD) method to use a ratonal parametrc volume to smulate the movement of facal muscles [9]. The method proposed by Lamousn and Waggenspack, Jr. allows the users to do the local deformatons by usng a B-plne or controllng the shape by weghtng the control ponts of the lattce, called NURB-based Free- Form Deformaton (NFFD), and succeeded n rasng the flexblty of FFD [10]. Lke EFFD method, t s also possble to combne two or more lattces, and the shape could stll be smoothly connected only by plng up a segment n the connecton part of the lattces. Moreover, Feng and Peng also proposed a fast accurate B-plne FFD method [11]. In ths method, the target model s surrounded by a B-plne volume, but the problems of placng and controllng the control ponts are the same as the prevous methods. Obvously, to adust the control ponts of a lattce s not ntutve; to make the 3D model deformaton to be easer, a more ntutve method s needed. Lazarus et al. used an axs nstead of usng a lattce to try to provde an effcent and ntutve deformaton method called Axal Deformatons (AxDf) [12]. Chang and Rockwood used a Bézer curve to defne the desrable skeleton of the deformed obect [13]. ngh and Fume used wres for deformaton [14]. Although these methods provde smpler control methods than prevous ones, the user also needs to place a proper axs or some sutable wres for deformaton. Feng et al. provded another FFD method by usng two parametrc surfaces called shape surface and heght surface [15]. Ths method gves users more flexblty for deformaton and has an ntutve controllng, but the deformed results have a dstorton problem. Therefore, we propose a novel method to deform the surface of the target 3D model by a gven parametrc surface, so that the user can mage what wll be generated after the deformaton. By utlzng the technology of non-dstorton texture-mappng, the dstorton effect of the deformed obect s decreased. Moreover, for dong the anmaton of the 3D model deformaton, a realtme system s desred by the users. nce the proposal method s effcent and has a real-tme response, t s sutable for dong the anmaton task. 3. Deformaton by a Gven Parametrc urface To use a gven parametrc surface to do the 3D model deformaton, the user frst selects a well-defned surface, such as the surface of a cone or a cylnder, or generates a parametrc one by hmself or herself. The selected well-defned surface or generated parametrc one s the desred shape after the target model s deformed,.e. the deformed result wll be lke the shape along the gven surface. The mappng relatonshp between the gven parametrc surface and the target 3D model s lke the mappng between a free-form surface and a 2D texture mage, because n texture-mappng, the texture mage s mapped onto the curved surface, and n our approach, the target model s mapped onto the gven parametrc surface, f we thnk of the texture mage as a plane n a three dmensonal space. Unfortunately, although texture-mappng s a wellknown technology n computer graphcs, and all graphcs lbrares support t, such as OpenGL, there s a wellknown problem that the texture mage s dstorted because the texture data on the flat mage plane s mapped onto the curved surface. Hence, f we ust use the tradtonal texture-mappng method to map the target model to the gven parametrc surface, the dstorton s also a problem for us. Therefore, how to perform the mappng wth less dstorton s what we have to solve frst. To solve the dstorton problem of the texture-mappng, some methods are proposed. Peachey frst faced to mnmze ths dstorton problem [16]. Ber and loan, Jr. separated the texture-mappng process nto two passes [17]: the 2D texture mage s frst mapped onto a smple ntermedate surface, and then ths surface s proected onto the target 3D model. Lévy and Mallet comprsed numercal computatons of physcal propertes stored n fne grds wthn texture space, and then generated the grds whch are sutable for fnte element analyss [18]. Ma and Ln also proposed an approxmate non-dstorton texture-mappng method to solve ths problem [3]. 3.1 Flattened urface Generaton In Ma and Ln s method, the curved surface s flattened to a 2D plane. Except for some knds of 3D curved surfaces such as the surface of a cylnder or a corn whch s called developable surface, other general surfaces lke the Bézer surface cannot be completely flattened to 2D planes wthout any dstorton, so they could only have approxmate flattened surfaces as shown n Fgure 1.

3 two vertces of the target model could be kept on the deformed model. Fgure 1: a Bézer surface and ts flattened surface. The algorthm for flattenng the parametrc surface s descrbed n Appendx. When generatng a flattened surface, the range of P n Appendx s a crtcal pont to be concern about. Assume the range s set to be r, where r s a postve nteger. If r s enlarged, the precson of the flattened surface s better, and snce there were few steps to convergence n practce, t could also get a good performance. flatten performance (ms) number of grds Fgure 3: performance testng for flattened surface generaton due to the number of grds and r. r=1 r=2 r=3 Fgure 4: samplng ponts for r = 1and r = D Model Deformaton Fgure 2: another flattened result of Fgure 1 when settng r =1. As the Bézer surface shown n Fgure 1, f we set r =1, the flattened result wll be unusable as shown n Fgure 2. To get a proper flattened surface, a larger r s needed. The flattened surface shown n Fgure 1 s the result of settng r = 3. How to set a proper r for a good flattened surface s an expermental problem and depends on the number of grds of the Bézer surface. In Fgure 3, the relatonshp of the number of grds of the Bézer surface s shown n Fgure 1 and the flattened surface generaton performance wth dfferent r s measured. When the number of grds s ncreased, the performance becomes worse, and the result wth a small r becomes unusable lke Fgure 2. Moreover, to enhance the calculatng performance, the samplng ponts for the range r s set to be the dstance of the grd of the parametrc surface as shown n Fgure 4. Therefore, the correspondng ponts for mappng are computed by comparng coordnate system of the flattened surface wth that of the texture, so that the texture-mappng wth less dstorton whch geometrcal dstances are mantaned s acheved. In our approach, before dong the 3D model deformaton, the gven parametrc surface s flattened to a 2D plane by usng ths method. That means the geometrcal dstance between By usng a gven parametrc surface to do the 3D model deformaton, we frst flatten the surface, and then put the model onto the flattened surface, so that the model can get a proper correspondng mappng wth the parametrc surface. Moreover, unlke the other conventonal FFD methods, the proposed method s done wthout generatng any lattce-lke structure, and the deformaton task s also easer than before. To get a smooth deformed obect, we have to subdvde the target 3D model, f the sze of the polygon of the model s larger than the sze of the grd of the flattened surface. We frst proect all the vertces, edges, and faces of the target model to the flattened surface, and then compute the followng three types of ponts wth the grd of the flattened surface of the gven parametrc surface: The vertex ponts whch are defned as the orgnal proected vertces of the target model. The edge ponts whch are generated by ntersected the orgnal proected edges of the target model wth the grd of the flattened surface. (c) The face ponts whch are generated by nserted the cross ponts of the grd of the flattened surface nsde the orgnal proected faces of the target model.

4 For example, f we proect the polygon at the bottom of a cube onto a flattened Bézer surface, there wll be a square on the flattened surface, and the generated edge ponts and face ponts are shown n Fgure 5. The vertex ponts are the four corners of the square, the edge ponts are the blue ponts on the four edges, and the face ponts are the red ponts nsde the square. The flattened surface n Fgure 5 s the flattened result of the Bézer surface shown n Fgure 9. edge ponts are detected as shown n Fgure 6. If there s a face pont on the edge also has a curvature hgher than the threshold ε, the endponts of the other edge whch conssts of the face pont are also marked as Fgure 6. And then, check the other proected edges agan to see f there stll have un-removable edge ponts. Fnally, the target model can be subdvded due to the newly generated ponts. Moreover, the user can choose where to put the target model onto a part of the gven parametrc surface for anmaton or ust fttng the regon of the surface. To keep the contnuty of the surface of the deformed obect, the user could only put the target model onto the gven surface, and the proected vertces could only be located nsde the flattened surface. 4. Results Fgure 5: edge ponts and face ponts. Then, the edge ponts and face ponts are proected backward to the orgnal parametrc surface, and the correspondng ponts on the target model are also generated by usng b-lnear nterpolaton. Fnally, the generated correspondng ponts are dsplaced along the normal vectors of the parametrc surface. Hence, there are two correspondng ponts for each vertex pont, edge pont, and face pont: one s on the parametrc surface, and the other s on the surface of the target model. Although there are several generated edge ponts and face ponts, to make the deformed model not to contan too much newly generated vertces, the curvatures of the correspondng ponts on the parametrc surface of the edge ponts and face ponts are computed, so that only the bumpy regon due to the parametrc surface needs to be subdvded. There are two 3D models n Fgure 7: s a smple thn plate whch has only 6 polygons, and s a dolphn model. Fgures 8-9 show the deformaton results of the smple model n Fgure 7 along the shape of dfferent Bézer surfaces and a cylnder, respectvely. Fgure 7: a smple thn plate and a dolphn model. Fgure 6: mark the un-removable edge ponts and face ponts on the edges conssted by them; mark other edge ponts due to the face ponts. To reduce the numbers of the edge ponts and face ponts, we frst calculate the curvature of one proected edge wth the parametrc surface, and mark the edge ponts, where the curvatures are hgher than a gven threshold ε, to be un-removable. The edge pont at the opposte sde of the marked edge pont s also marked. Then, the face ponts on the edge lnked by the marked Fgure 8: a Bézer surface and a deformaton result along t. The performance of our approach s lsted n Table 1. The process of both flattenng the gven parametrc surface and deformng the target 3D model due to the surface are performed. The testng platform s a PC wth an Intel Xeon 2GHz CPU, 1GB memory, and a 3Dlabs Wldcat II 5110 graphcs accelerator wth OpenGL support. The range r used for flatten the surface s set to be 3. Although the tme for flattenng the surface s much larger than the tme for deformng the target model, the flattenng process s ust a pre-process and could be

5 done at off-lne and ndependent wth the complexty of the target model. polygon of the orgnal dolphn model s smaller than the grd of the flattened surface, there s no subdvson n the deformaton process. Fgure 9: another Bézer surface and a deforma- ton result along t. g ven surfaces Fgure 8 Fgure 9 Fgure 10 number of grds flatten (ms) deform (ms) Table 1: performance testng for flattenng the gven surfaces shown n Fgures 8-10 and the deformaton due to them. Fgure 12: a swmmng dolphn. Fgure 10: ( a) a Bézer surface and a dolphn model deformed along the surface. Fgure 13: a dolphn deformed along a parametrc surface. nce our method can get a real-tme response, t s sut- able for dong anmaton about deformaton. For example, the dolphn model n Fgure 7 s deformed to smulate the swmmng as shown n Fgure 12. To do the anmaton, the dolphn s put onto a parametrc surface lke Fgure 13, then the dolphn can be deformed along the surface n real-tme, although t contans 73,665 polygons. The number of grds of the gven parametrc surface affects the performance sgnfcantly, whch could be controlled by users through our system, because the complex parametrc surface has more grds to be flattened and the new ponts used for subdvdng the target model are also ncreased. However, to use a parametrc surface wth more grds for model deformaton also means to get a better qualty of the deformed result. Moreover, although the numbers of the edge ponts and face ponts are ncreased f the number of grds s ncreased, the number of vertces of the deformed model s not ncreased too much, snce the newly generated vertex s added to the target model when t s needed. 5. Concluson Fgure 11: the user nterface of our system and there s a dolphn model put on a flattened surface. Fgure 10 ( b) shows the deformaton of a complex dol- phn model as shown n Fgure 7. The deformaton s along a Bézer surface shown n Fgure 10. The graphcal user nterface (GUI) of our system s shown n Fgure 11. The desrable Bézer surface and the relatve poston of the target model on the flattened surface of the Bézer surface could be controlled through the system. In the screen of Fgure 11, the dolphn model of Fgure 7 s put onto the flattened surface of the Bézer surface of Fgure 10. Moreover, snce each We proposed a novel FFD method to realze non- dstorton deformaton whch mantans the geometrc shape even after deformaton by utlzng the technque of texture mappng wth less dstorton. Because the deformaton s performed along a gven 3D parametrc surface whch the user could generate easly, t s easy to predct the result shape after the deformaton. Moreover, usng a gven parametrc surface to do the deformaton, the adustment of the control ponts of the parametrc surface more s ntutve and ntellgble than the one of the lattce used n prevous FFD methods.

6 To deform the thn model by usng our system, the deformed model could get the result wth less dstorton, snce the faces of the model far from the parametrc surface have worse deformed results than others. To mnmze the dstorton, the user could move the flattened surface n the mddle of the model. Acknowledgement One of the authors, Jeqng Feng, s partally supported by Natonal Natural cence Foundaton of Chna (No ), and Henry Johan s supported n part by the Japan ocety for the Promoton of cence Research Fellowshp. References [1] T. W. ederberg and. R. Parry, Free-form deformaton of sold geometrc models, ACM Computer Graphcs (Proc. IGGRAPH 86 Conf.), 20(4), 1986, [2] E. Catmull, A subdvson algorthm for computer dsplay of curved surface (Ph.D. Thess, Report UTEC-Cc , Computer cence Dept., Unv. of Utah, 1974). [3]. D. Ma and H. Ln, Optmal texture mappng, Proc. Eurographcs 88 Conf., 1988, [4] A. H. Barr, Global and local deformatons of sold prmtves, ACM Computer Graphcs (Proc. IG- GRAPH 84 Conf.), 18(3), 1984, [5]. Coqullart, Extended free-form deformaton: a sculpturng tool for 3d geometrc modelng, ACM Computer Graphcs (Proc. IGGRAPH 90 Conf.), 24(4), 1990, [6]. Coqullart and P. Jancéne, Anmated free-form deformaton: An nteractve anmaton technque, ACM Computer Graphcs (Proc. IGGRAPH 91 Conf.) 25(4), 1991, [7] J. Gressmar and W. Purgathofer, Deformaton of solds wth trvarate B-splnes, Proc. Eurographcs 89 Conf., 1989, [8] W. M. Hsu, J. F. Hughes, and H. Kaufman, Drect manpulaton of free-form deformatons, ACM Computer Graphcs (Proc. IGGRAPH 92 Conf.), 26(2), 1992, [9] P. Kalra, A. Mangl, N. M. Thalmann, and D. Thalmann, mulaton of facal muscle actons based on ratonal free form deformaton, Computer Graphcs Forum (Proc. EUROGRAPHIC 92 Conf.), 11(3), 1992, [10] H. J. Lamousn and W. N. Waggenspack, Jr., NURB-based free-form deformatons, IEEE Computer Graphcs and Applcatons, 14(6), 1994, [11] J. Feng and Q. Peng, Acceleratng accurate B- splne free-form deformaton of polygonal obects, ACM Journal of Graphcs Tools, 5(1), 2000, 1-8. [12] F. Lazarus,. Coqullart, and P. Jancéne, Axal deformatons: an ntutve deformaton technque, Computer-Aded Desgn, 26(8), 1994, [13] Y.-K. Chang and A. P. Rockwood, A generalzed de Castelau approach to 3d free-form deformaton, Proc. ACM IGGRAPH 94 Conf., 1994, [14] K. ngh and E. L. Fume, Wres: a geometrc deformaton technque, Proc. ACM IGGRAPH 98 Conf., 1998, [15] J. Feng, L. Ma, and Q. Peng, A new free-form deformaton through the control of parametrc surfaces, Computers and Graphcs, 20(4), 1996, [16] D. R. Peachey. old texturng of complex surfaces, ACM Computer Graphcs (Proc. IGGRAPH 85 Conf.), 19(3), 1985, [17] E. A. Ber and K. R. loan, Jr., Two part texture mappng, IEEE Computer Graphcs and Applcatons, 6(9), 1986, [18] B. Lévy and J.-L. Mallet, Non-dstorted texture mappng for sheared trangulated meshes, Proc. ACM IGGRAPH 98 Conf., 1998, Appendx: Optmal Texture Mappng [3] The next equaton defnes the errors between the curved surface and ts flattened surface: C = ( d d ) P P Ω where s a set of the ponts on the curved surface, Ω s a set of the ponts whch exst contnuous wth the pont among, s the dstance between and P P d d on the curved surface, and s the dstance be- tween correspondng two ponts on the flattened surface. The flattened surface s obtaned by teratng the above equaton to mn mze C. Then, the above equaton s rewrtten n the coordnate expresson: C = P P Ω d where ( x y ) t 2, t ( X X ) ( X X ) P 2 ( t X =, and X = x, y ), and the coordnate value whch used n the next step of the teratve algorthm s obtaned accordng to the followng equaton: X k + 1 k C = X ρ X where ρ s a small postve number. The flattened surface of the target surface s obtaned by reducng the value o f C.,,