A New Free-form Deformation Through the Control of Parametric Surfaces

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1 A New Fee-fom Defomation Though the Contol of Paametic Sufaces Jieqing Feng Lizhuang Ma and Qunsheng Peng State Key Lab. of CAD&CG, Zhejiang Univesity Hangzhou , P. R. of CHINA ABSTRACT A new fee-fom defomation method is esented in this ae. The defomation of an object is achieved by attaching it to two aametic faces, namely the shae face S u v (, ) and the height face H ( u, v). A contol oint o vetex of object is ojected onto the shae face along its nomal and a coesondence between the oint and its ojection on the shae face is established. The oint is then embedded into the aametic sace defined by the shae face. By egading the height face as a dislacement function, the diected distance fom the samle oint to its ojection on the shae face can be futhe adjusted. The oosed method is indeendent of the eesentation of undelying object. Exeimental elts show that the method is intuitive, easy to contol and un fast. Keywod: fee-fom defomation, shae face, height face, B-sline face 1. INTRODUCTION Defomation methods have been widely used in the fields of both geometic modeling and comute animation. In fact defomation can be teated as a maing fom R 3 to itself excet Bechmann woks in R n [3]. All the defomation methods indeendent of the eesentation of undelying objects can be divided into two classes accoding to whethe it equies a defomation tools o not [2]. Ba s global and local defomation [1] belongs to the class which does not equie tools. In his method, defomations ch as scaling, otating, taeing, twisting, bending o the combination of them ae eesented as the matix function of location of the sace oints. This method was develoed by Watt late [14] to include intuitive inteactive tools in contolling defomation and the concet of facto cuve was intoduced. Anothe kind of defomation method fee of tools was sace defomation intoduced by Boel and Bechmann [3]. The defomation of object is simly secified by the dislacements of abitay selected oints called constaints. The size and the bounday of a bounding box centeed aound each constained oint allows contol of the extent of the defomation. Deending on this extent, the whole object can be included (global defomation) o only a limited aea aound the constaint oint (local defomation). A lage ange of defomation shaes ch as abitay shaed bums can be designed using this technique. Boel esented a simlified sace defomation method late [4]. The othe class of defomation methods emloy some defomation tools. Reesentative methods include AxDf ( Axial Defomation ) and FFD ( Fee-Fom Defomation ). The AxDf [12] accomlishes the defomation by adoting a aametic cuve as the axis. The object is then attached to the axis and defoms accodingly when the shae of the cuve is changed. This method is vey effective. Nevetheless as the defomation feedom offeed by this method is limited, an object can only be 1

2 defomed along the axis. It would be difficult to geneate an abitay shaed bum on an object by AxDf. The FFD, oiginally oosed by Sedebeg and Pay [13], has been widely used and vaious genealized foms of FFD have been develoed to meet the alication demands. The oiginal lattice of the FFD consists of contol oints along thee mutually eendicula axes, which defines a sace of tivaiate Bezie volume. Giessmai[9] adoted a tivaiate B-sline volume as encomassing sace to efom defomation. In the EFFD (Extended Fee-fom Defomation) [6], the aalleleied lattice is extended to lattice of abitay shae. Late, the AFFD ( Animated Fee-fom Defomation) [7] was develoed to geneate the shae animation of objects. Chadwick [5] used the FFD to contol the defomation of muscles. The RFFD ( Rational Fee-fom Defomation ) was intoduced by Kala et.al. [10] to simulate the movement of facial muscles. In RFFD, a ational aametic volume is defined oviding use with additional contol means. The NFFD ( NURBS-Based Fee-Fom Defomation ) was ggested to simulate the movement of legs [11]. The initial lattice of NFFD is a NURBS volume. The unique featue of FFD is that the object to be defomed is fist embedded in a sace which is defined by a 3D aametic volume. Any shae modification of the volume will oduce defomation of the object. The imlementation of FFD can be decomosed into fou stes: (1) select the contol oints of the initial lattice accoding to the defomation egion of the object (2) embed the object into the lattice sace, i.e., fo each samled oint o contol oint P( x, y, z) on the object, find the coesonding local coodinates L ( u, v, w) in the lattice sace (3) defom the lattice sace by adjusting the ositions of the elevant contol oints in the intial lattice (4) ass defomation to object: comute the new osition of P in the defomed lattice sace at the same local coodinates L ( u, v, w ) Although both AxDf and FFD establish a maing between the global object sace and a local aametic sace detemined by the defomation tool, fom the use oint of view, the AxDf elates a oint on the object to a oint on an axial cuve and FFD elates that to a oint within a 3D volume and it is moe intuitive to contol the defomation of the object by modifying the shae of a cuve athe than modifying a 3D volume. On the othe hand, FFD ovides much feedom fo global defomation of an object. Seveal FFD blocks of lattice may be equied fo a single object if a numbe of local defomation effects ae desied, though. In natue, thee ae also a class of defomation which is affected by the shae of a face, examles include a snake cawing on the gound, a soft object laced on a ock, etc. In this ae we esent a new defomation aoach which adots two aametic faces, namely a shae face and a height face, as defomation tools. The defomation is quite simila to that of both AxDf and FFD. The shae face and height face jointly define a 3D aametic sace. A maing between the object sace and the local aametic sace is set u to attach the object to be defomed to the two faces. While the shae of the two faces ae modified, the defomation will be assed to the object automatically. Ou new aoach takes the advantage of both AxDf and FFD : it ovides intuitive means fo contolling the defomation and it is caable of geneating the defomation effects on an object both globally and locally. It also includes Ba defomation as secial cases. The est of the ae is oganized as follows. Section 2 ovides the definition of two aametic faces. Section 3 discibes how to attach an object to the aametic faces. Section 4 gives the details of how to 2

3 defom an object to attain the desie shae. Section 5 shows some examles obtained by this method while section 6 makes the conclusions and ggests futue eseach diections. 2. Initial shae face and height face We define the shae face and the height face as Bezie-tye B-sline faces. A Bezie-tye B-sline face is ch that the oeties at the bounday of the face ae just like that of a Bezie face. Theefoe the ange of face can be detemined by its contol net intuitively; the nomals at the fou cones of the face can be deived easily fom the contol olygon, etc. Let S ( u, v) and H( u, v) denote the shae face and the height face esectively and ms ns S( u, v) = Sij Bi, k ( u) B j, k ( v) u u, u, v v, v i= 0 j= 0 whose knots sequences { u i } and { } ui = 0, i = 0, 1,, k i k ui =, i = k + 1,, ms ms k + 1 ui = 1, i = ms + 1,, ms + k + 1, [ k ms + 1] [ k ns + 1 ] v i ae defined in the following way: vi = 0, i = 0, 1,, k i k vi =, i = k + 1,, n ms k + 1 vi = 1, i = ns + 1,, ns + k + 1 whee k, k and m s +1, n s +1 ae the face degee, the numbe of contol oints in u, v diection esectively. The height face can be defined similaly : mh nh H( u, v) = ( H, H, H ) = H B ( u) B ( v) Initially, the contol oints { S ij } and { } x y z ij i, khu j, khv i= 0 j= 0 each foming a ectangula gid. Note that S ( u, v) and H ij lie esectively on the XZ-lane and the lane Y = 1, H( u, v) may be of diffeent degee and with diffeent numbe of contol oints in thei coesonding diections, that is to say it is ossible that : k k k k m m n n 3. Attaching the object to the shae face hu hv Let P = ( x, y, z) be a samled oint o a contol oint of the object to be defomed and P = ( x, y, z ) be the ojection of P on S ( u, v). The ojection diection is along the nomal of the shae face at P, Let N s be the unit face nomal at P, then h = PP N. Hee h is the diected distance fom P to S ( u, v) and " " means dot oduct. If P lies above the S ( u, v), h is ositive, else it is negative. A numeical method is emloyed hee to solve out ( u, v ) ch that S( u, v ) = P (shown by Fig. 1). We then set u a coesondence of between P and P as follows P = S ( u, v ) + h N s ( u, v ) The face S ( u, v) can be modified by taditional inteactive means, ch as ulling the contol oints selectively etc. Let S new ( u, v ) be the modified shae of S ( u, v). S new ( u, v ) then ovides a base fo object defomation. To achieve the local defomation effect, we include an addtional aamete ch that the diected distance fom P to S new ( u, v ) can also be modified. We adot one comonent of H( u, v) to efom this function. In the cuent imlementation, the y comonent of H( u, v) is selected. Let H( u, v) be modified as H new ( u, v ) whee H new = ( H x, H newy, H z ). Then the new osition of P can be detemined by P = S ( u, v ) + H ( u, v ) h n ( u, v ) new new newy new s h s h s s 3

4 P new ( u, v) S( u, v) P P H ( u, v ) h n ( u, v ) newy new S new ( u, v ) ( u, v ) Fig.1 Attach P onto shae face S( u, v) Fig.2 Comute the defomed oint P new whee n new ( u, v ) is the unit nomal of S ( new u, v ) at ( u, v ) (shown as Fig. 2). Fig.3, 4, 5, 6 illustate the defomation effects by modifying S ( u, v) and H ( u, v). H( u, v) H( u, v) S( u, v) object S ( new u, v ) defomed object Fig.3 The object and the two faces Fig.4 The defomation effect by modifying the shae face H new ( u, v ) S new ( u, v ) H new ( u, v ) S( u, v) defomed object defomed object Fig.5 The defomation effect by modifying the height face only Fig.6 Joint defomation effect by modifying both shae face and height face 4. Object Defomation Let B 0 be the ojected aea of the bounding box of the object on the XZ-lane. Use fist defines a quadate S 0 as the bounday of S( u, v) in the XZ-lane. Then the contol net of initial shae face is detemined inteactively. The height face is defined similaly. 4

5 If B 0 lies inside S 0 totally, the defomation is global. Some elementay defomations ch as bending, taeing, twisting can be accomlished by global defomation. They can be defined as stadad oeations of ou aoach. (1). Bending: The object will be bent if the shae face S ( u, v) is bent. Fig.7 and Fig.8 show an examle whee S ( u, v) is defined by a 2 1 degee B-sline face with contol oints distibuted unifomly. The height face just emains unchanged. H ( u, v ) H ( u, v ) S( u, v) object S ( new u, v ) defomed object Fig.7 the oiginal object Fig.8 the bent object (2). Taeing: When the shae face S ( u, v) is changed into a tiangle and the height face H( u, v) is defined by a lane with sloe, the defomed object will be taeed. The simlest way is to set S ( u, v ) and H ( u, v ) as an 1 1 degee B-sline face. Then the two contol oints on u=0 of S ( u, v ) ae meged into its middle oint, the two contol oints on u=0 of down to the Y=0. This is illustated by Fig.9 and Fig.10. H( u, v) H new ( u, v ) H( u, v) ae dagged S( u, v) object defomed object S ( new u, v ) Fig.9 the oiginal object Fig.10 the taeed object (3). Twist: When the shae face S u v Fig.11 and Fig.12 ) H( u, v) (, ) is twisted, so does the defomed object. (shown as H( u, v) object S( u, v) defomed object Snew ( u, v) Fig.11 the oiginal object Fig.12 the twisted object 5

6 If B 0 lies atially inide S 0, the defomation of object is local. We adot a olygon cliing algoithm to find the intesection of B 0 and S 0. Let BS denote the intesection. If continuity of face of the defomed object is desied, the ositions of some contol oints in the BS which ae adjancent to the bounday will emain unchanged in the ocess of defomation. Fig. 13 shows an examle. object S( u, v) Fig.13 The contol oints labeled with small squaes will emain unchanged. In actice the defomation attained is only an aoximation [8]. The defomation could not be icise if the object to be defomed is samled infficiently. So thee is a comomise between the equested shae and the comutation cost. This is justified by uses. How to avoid the self-intesection duing object defomation is a common oblem fo both AxDf and FFD. Ou oosed method faces the same oblem. Fo examle when the shae face has a tight concave bent, the defomed object may be self-intesected. In ou cuent imlementation this oblem is solved with use s inteaction. 5. Exeimental elt We have imlemented the oosed algoithm on HP720 gahics wokstation. The test object is a teaot which is comosed of fou B-sline faces. In all of the following wie fame figues, S ( u, v) is maked with geen, H ( u, v) is in yellow colo. Red gid eesents the lane Y=0. The defomation is assed to the contol oints of the object. Fig.1 shows the oiginal teaot. Fig.2 and Fig.3 show the taeed teaot. In Fig.4 and Fig.5, H ( u, v) is defomed while S( u, v) emains unchanged. Fig.6 and Fig.7 show the teaot with wave shae. In Fig.8 and Fig.9, the teaot undegoes tae and wave-shaed defomations. Fig.10 and Fig.11 show the abitay shaed defomation of the teaot. 6. Conclusion A new kind of fee-fom defomation method has been esented. The defomtion of an object is contolled by two aametic faces which ae temed esectively the shae face and the height face. The defomation can be eithe global o local. Thee is one degee of feedom in the AxDf and thee degees of feedom in the FFD. Ou aoach offes two and a half degees of feedom fo object defomation. The alication domain of the esented defomation is much wide than the AxDf and it can simulate most effects ceated by 6

7 FFD, howeve the imlementation of the oosed method is moe intuitive and convenient than FFD. Futue eseach diections include: (1) develo an adative bdivision method o samling method to obtain the desied shae defomation ; (2) identify the self-intesection automatically duing defomation. 7. Acknowledgments The authos ae gateful to the efeees fo thei valuable comments on the manuscit of this ae. Thanks go to membes of the State Key Lab of CAD&CG at Zhejiang Univesity fo thei hel in imlementing the oosed algoithm. This oject is oted by National Natual Science Foundation of China and HUO Yingdong Foundation fo Excellent Young Teaches. Refeence [1] A.H.Ba, Global and local defomation of solid imitives, Siggah 84, ACM Comute Gahics, vol. 18, No.3, 21-34, [2] D. BECHMANN, SPACE DEFORMATION SURVEY, Comute&Gahics, Vol.18, No.4, , [3] P. Boel and D. Bechmann, Defomation of N-dimensional objects, Int. J. Comutatioal Geomety Al. Vol.1, No.4, , [4] P. Boel and A. Raoot, Simle Constained Defomations fo Geometic Modeling and Inteactive Design, ACM Tansactions on Gahics Vol. 13, No.2, , [5] J.E. Chadwich, D.R. Haumann and R.E.Paent, Layeed Constuction fo Defomable Animated Chaactes, Siggah 89, ACM Comute Gahics, Vol. 23, No. 3, , [6] S. Coquillat, Extened Fee-Fom Defomation: A Scultuing tool fo 3D Geometic Modeling, Siggah 90, ACM Comute Gahics, Vol.24, No. 4, , [7] S. Coquillat and P. Jancene, Animated Fee-fom Defomation: An Inteactive Animation Technique, Siggah 91, ACM Comute Gahics, Vol.25, No. 4, 23-26, [8] G. Fain Cuves and Sufaces fo CAGD, , Thid edition, Academic Pess, [9] J.Giessmai and W. Pugathofe, Defomation of solids with Tivaiate B-slines, EUROGRAPHICS 89, , [10] P.Kala, A.Mangili, N.Thalmann and D.Thalmann Simulation of facial muscle actions based on ational fee-fom defomation, EUROGRAPHICS 92, Com. Gah. Foum, Vol.2, No. 3 C59-C69,1992 [11] H.J. Lamousin and W.N.Waggensack, NURBS-Based Fee-Fom Defomations, IEEE tans. Comute Gahics and Alications, vol.14, No.1, [12] F. Lazaus, S. Coquillat and P. Jancene, Axial defomation: an intuitive technique, Comte-Aided Design, vol.26, No. 8, , [13] T.W.Sedebeg and R. Pay, Fee-Fom Defomation of Solid Geometic Models, SIGGRAPH 86, ACM Comute Gahics Vol.20, No.4, , [14] A. Watt, Advanced Animation and Rendeing Technique, ADDISON-WESLEY,

8 NOTE: Colos in wiefame figues ae illustated as follows: (1) The shae face (2) The height face (3) The lane Y = 0 is in ed colo. S( u, v) is in geen colo. H( u, v) is in yellow colo. (4) The defomed teaot is in white colo. Fig.1 The oiginal teaot Fig.2 The wiefame figue of the taeed teaot, height face and shae face 8

9 Fig.3 The shaded image of the taeed teaot Fig.4 The wiefame of the defomed teaot, whee the shae face emains unchanged and the height face is edited as the shae in the figue Fig.5 The shaded image of Fig.4 Fig.6 The wiefame of the defomed teaot, whee the height face is scaled with a facto 0.4 and the shae face is a wave-shaed face. 9

10 Fig.7 The shaded image of wave-shaed teaot Fig.8 A to view of teaot defomed by taeed and wave-shaed defomations,in which the height face is a sloe lane and the shae face is in a taeed and wave-shaed fom Fig.9 The shaded image of teaot with taeed and wave-shaed defomation 10

11 Fig.10 The wiefame of the defomed teaot, whee the height face is scaled with a facto 0.4 and the shae face is in an abitay fom Fig.11 The shaded image of abitay-shaed defomation teaot 11