A Hierarchical Skeleton-based Implicit Model

Transcription

1 A Herarchcal Skeleton-based Imlct Model MARCELO DE GOMENSORO MALHEIROS WU, SHIN-TING Gruo de Comutação de Imagens (GCI-DCA-FEEC) Unversdade Estadual de Camnas (UNICAMP) Abstract. Imlct modelng s a recent trend n Comuter Grahcs and a varety of skeleton-based models has been roosed. In ths aer e resent a herarchcal frameork that can encomass a de range of these varants, rovdng a more flexble ay for defnng the shae of mlct obects. We also derve a ne decay functon, hch s comutatonally chea. 1 Introducton Imlct modelng s a recent trend n the feld of Comuter Grahcs that bulds geometrcal obects on the bass of mlct mathematcal functons. We are nterested n a secfc modelng technque, here the mlct obects are defned by skeletons. Informally, a skeleton s a set of rmtves that are combned to defne the shae of an obect. The skeleton-based aroach rovdes a very flexble ay to construct mlct obects, hch can be altered by smly changng ther rmtves. It can also be used n comlement th arametrc methods to enrch the modelng doman. Partcularly, skeletons are attractve to model a de range of real obects, manly organclke shaes (for examle, arts of the human body or anmals) and lquds [7]. These models can also be drectly anmated [8]. The most dely used technques based on the skeleton aroach are blobs [1], metaballs [6], and soft obects [7, 8]. Wth these technques, obects are secfed n terms of dstance measures to a set of onts. These technques are all smlar, and dffer on the dstance functon used. Several varants of these technques ere roosed n order to mrove ether the reresentatvty or the controllablty of the skeleton-based models [3, 5]. Snce each of these models offer some dstnctve features, t s desrable to have ther functonaltes ntegrated n order to facltate the modelng of comlex shaes. In ths aer e roose a frameork that can encomass a de range of skeleton-based models. A skeleton s defned herarchcally usng an n-ary tree structure, here the leaf nodes reresent any rmtve and the nternal nodes denote ntermedate results from a combnaton of ther chld nodes. Syntactcally, there s no restrcton on ether the knds of rmtves or the tyes of combnng oeratons. For our shae secfcaton, e consder the defnton of an mlct obect as the comoston of three geometrcally meanngful classes of functons shae, decay, andblendng functons. Ths aer s organzed as follos. We begn th the basc concets and notatons used to descrbe skeleton-based mlct obects, then e dscuss the related revous ork. Next e resent our roosed herarchcal model. Then e descrbe our mlementaton and some results. We conclude th a dscusson of future ork. 2 Skeleton-based Modelng Ths secton summarzes some basc concets related to skeleton-based modelng. For further detals, e suggest the reference [5]. In the follong dscusson, e denote onts by boldface loer case letters, transformatons by boldface catals and scalars by talcs. Subscrts denote members of a set of obects, hle suerscrts denote artcular levels n a herarchy, beng 0 the tomost one. For examle, Tx corresonds to the transformaton T of the ont x, andf (n) s the -th scalar functon belongng to the n-th deth level. Gven a scalar functon F : R m! R, hch mas m-dmensonal onts to real values, and a real value c, the set of onts x 2 R m that satsfy F (x) =c s sad to be an mlct obect. In fact, ths ont set corresonds to the level set of F assocated to c. Generally, skeleton-based models are bult from a set of scalar functons f (x), combned through elldefned oeratons knon as blends. The resultng functon F defnes a man scalar feld n R m. Once defned the value c, the set of scalar functons s called the skeleton of the mlct obect. For ractcal uroses, the functons used to defne the man scalar feld should satsfy some smoothness

2 condtons, lke beng contnuously dfferentable. Moreover, these functons tycally have the follong roerty: ther values decrease as the dstance to a gven reference ont ncreases. These functons defne felds th concentrc level sets. Therefore, e may say that each of them contrbutes th shercal-shaed felds to the man scalar feld. Because of the ay the functons ere defned, ther nfluence on the resultng feld s mostly concentrated n the neghborhood of. The usual blendng oeraton emloyed to combne those functons s a eghted sum. Note that f the functons are carefully chosen, then the resultng obect s bounded. 3 Prevous Work Skeleton-based mlct modelng as ntroduced by Blnn [1], motvated by the need to dslay electron densty mas of molecular structures. He used a ont set as skeleton, summng the contrbutons due to a Gaussan dstance functon for each ont. A varaton of the revous model as later develoed n [6], usng a ecese quadratc olynomal nstead of an exonental functon. Dfferently from Blnn s model, the dstance functon as defned n such a ay that the contrbuton of a gven ont droed to zero after a certan dstance. Ths alloed more effcent comutaton of the scalar feld because only the contrbutons of the onts thn a redefned dstance ere needed. Wyvll et al. [7] roosed a smlar model, th a cubc olynomal over the Eucldean dstance. There as an exlct arameter that controlled the radus of nfluence of each ont. These orks ere lmted to ont sets and lnear blends. Later, Bloomenthal and Shoemake [3] roosed curve segments as references for the dstance functon (generalzed cylnders), and a convoluton blend to combne them. An extensve survey of skeleton rmtves and blendng technques can be found n [5]. Bloomenthal and Wyvll [2] dscussed technques to mrove the effcency of dslay methods for mlct surfaces, hch together th control flexblty are essental for an nteractve desgn envronment. 4 Herarchcal Skeletons In ths secton our frameork s resented. Although e restrcted the dscusson to mlct surfaces n R 3, t can be easly generalzed to any dmenson. 4.1 Motvaton Careful analyss of revous skeleton-based models led us to conclude that e can get a large set of varants ether by modfyng the rmtves or by alterng the blendng functon. For the rmtves e can also dstngush to terms: the Eucldean dstance from x 2 R 3 to a reference ont, curve, or surface, hch accounts for the level set shae (that s, a shae functon); and the decay functon, hch forces the contrbuton of each reference entty to dro to an nsgnfcant value at a certan dstance from t. From the dscusson made n [2], e can say that there are bascally three searate ays to nteractvely control an mlctly defned obect: defnton and manulaton of the shae functon; defnton and adustment of the decay functon; and defnton and manulaton of the blendng functon. These statements are the bass of our frameork. 4.2 Structure The scalar feld defned by F s the result of a blendng oeraton on a set of scalar felds. Each one of these felds may be, n ts turn, comosed of other felds, and so on. Ths naturally leads to a tree structure, hose leaf nodes corresond to rmtve felds. Each nternal node at the n-th deth level reresents a scalar feld resultng from the combnaton of the felds belo t (Fg. 1). In the rest of ths aer, e ll refer to these leaf nodes as rmtve nodes and to the nternal nodes as blend nodes. f 1 T f 1 (1) f 2 T F f 1 (3) T f (1) 2 T T T f 3 T T f 2 (3) Fgure 1: Skeleton structure. f 4

3 A tree structure makes ossble the selectve combnaton of the rmtves, and s not lmted to ust one tye of blendng oeraton as n the usual technques. Furthermore, t smlfes the nteractve manulaton of comlex obects, because the user can searate the model nto subtrees and ork th only one at a tme. The man scalar feld, hch defnes the mlct obect, s gven by F (x) =f (0) 1 (x): We may dro the subscrt 1 because F s the unque feld at the tomost level, hch s the tree root node. The ntermedate scalar felds assocated to the other nodes are smly called comonent felds. We defne each feld n ts artcular doman. Ths allos the choce of a convenent reference system for each feld. There s a transformaton T (n) that relates a feld doman and ts arent doman. When combnng to or more scalar felds, e have to take ths transformaton nto account n order to assure that the combnaton occurs under the same reference system [5]. Gven a ont x n the arent coordnate system, the assocated feld value s comuted by the combnaton of ts chld feld values. After the arorate transformaton T (n) of x nto each -th chld feld doman, the chld feld values are obtaned by alyng f (n) on these transformed onts. Note that T (n) s not lmted to be a rgd moton: e also allo scalng and shearng transformatons. Wth ths, e can alter the resultng shae contrbuton of a artcular scalar feld. For examle, to ellsodal-shaed felds can be obtaned from to shercal ones by a scalng transformaton (Fg. 2). Fgure 2: Oeratng on the doman of the rmtves. 4.3 Blendng Oeratons The lnear blend s the most commonly used blendng oeraton for skeleton models, here the contrbutons of a set of functons are combned through a eghted sum. We generalze ths dea by assocatng a eght to each comonent feld f, ndeendently of the blendng functon. That s, a blendng oeraton combnes the values f (n) nstead of f (n). The eght can gve a flexble control over the nfluence of each comonent feld. For examle, a lnear blendng oeraton at the ont x may be exressed by f (n) (x) =X f (n1) ([T (n1) ],1 x); (1) here stands for ts -th comonent functon. In ths case, may assume zero and even negatve values, so that the corresondng feld has resectvely no and subtractve effects. In Fg. 3, the three rmtves n the to ro have eght 1. Belo, the eght of the left rmtve s set to 2, hle the rght one has a -2 value. Fgure 3: Influence of the eght. Although not strctly a blend, another useful combnng oeraton s the unon. Ths oeraton takes the maxmum of the feld contrbutons n a gven locaton, that s, f (n) (x) = max f f (n1) ([T (n1) ],1 x)g: Note that ths oeraton may ntroduce frst order dscontnutes on the surface, allong us to create shar corners on t. 4.4 Prmtve Felds Dfferng from blend nodes, a rmtve node reresents a feld defned by f (n) = d s ;

4 here the functons s and d are called shae and decay, resectvely Shae Functon As already exlaned, the shae functon defnes a scalar feld n R 3 and accounts for the form of the rmtve. Because each feld s defned n ts on doman, the oston, the sze, and the orentaton of any rmtve can be arbtrary. Hence, e may alays choose a convenent reference system to defne t. For examle, a shercal rmtve, centered at the orgn and th nfluence radus gven by r, can be exressed by kxk s (x) = ; (3) r here x 2 R 3. The scalar r s a user-defned arameter of ths rmtve 1. Note that ths feld has value 1 at onts belongng to the surface of a shere th radus r and centered at the orgn Decay Functon Usually, the feld gven by a shae functon ncreases as e move aay from the rmtve. Thus e need to aly the decay functon, to force that the feld contrbuton goes toard zero th the augment of dstance. In ths ay, local (or seudo-local) control of the model s ensured. Wyvll et al. [7] derved a decay functon smlarly shaed to the one roosed by Blnn [1], th the advantage of drong to zero outsde ts range of nfluence. Ther functon requred three addtons and fve multlcatons. Based on the same rncle aled by Wyvll et al., e derved another smlar functon, also th lmted nfluence. The addtonal advantage reles on the fact that our functon s smler, requrng feer oeratons. We obtaned ths functon by mosng that d (0) = d (1) = 0, nstead of requrng that d (0:5) = 0:5. Thus, our decay functon s gven by d (t) = 8< : 1; f t 0,(t, 1) 3 ; f 0 <t<1 0; otherse : (4) Observe that the result of the comoston d s, gven resectvely by equatons (4) and (3), s a ecese monotonc and non-ncreasng functon, hose value vanshes after a dstance greater than or equal to r from the orgn (see Fg. 4). 1 In fact, ths arameter s redundant, because the same effect could be obtaned by alyng a unform scalng transformaton on the doman of a shercal rmtve th untary radus. Ths s left n the rototye as a convenence. 1 Value 0 0 r Dstance Fgure 4: Shercal rmtve feld functon. 5 Imlementaton Snce the mlct obect s recursvely defned from rmtve felds, an adequate data structure for ts reresentaton s an n-ary tree. In ths tree, the leaf nodes denote the rmtve feld functons and the nternal nodes reresent the blendng functons, hch are used to combne the functon values of ts chld nodes. Each node contans a homogeneous transformaton matrx T that relates ts doman coordnate system to the one from ts arent. It also contans a eght that affects the contrbuton of the node to the arent scalar feld. We develoed a C class hch mlements ths data structure. The methods rovded by ths class ere desgned amng nteractve manulaton on the mlct obects, as lsted n the follong: Methods to create and destroy rmtve nodes. There are also methods to ndvdually query and modfy the arameters of ther shae and decay functons. Methods to create and destroy blend nodes. Ther arameters can be quered and modfed (ncludng the eght ), and the blendng oeratons can also be changed. Methods to modfy the transformaton matrx T beteen a node and ts arent. Methods to manulate the tree structure (e.g., get root node, query arent, query chld, remove subtree, etc). A method to query and modfy the current level set value c of the man scalar feld. A method to evaluate the man scalar feld at a gven ont, returnng both the scalar value and the assocated gradent vector. We used abstract classes for defnng rmtve felds and blendng oeratons. Hence, ne rmtves and ne

5 blends can be easly ntegrated by dervng secalzed classes. We have already mlemented the rmtve functon shon n Fg. 4 and the oeratons gven by the equatons (1) and. Several authors [2, 4] have observed that resentng an mlct obect as a samle of onts offers greatly mroved seed relatve to olygonzaton. Moreover, use of arorate deth-cueng roves to be very effectve for vsualzng clouds of onts n sace. Snce e are nterested on the rad rototyng of the mlct obect, e oted for vsualzng the mlct obect as a set of artcles. We used the algorthm resented by Fgueredo and Gomes [4] to lace these artcles on the surface of the mlct obect. Roughly seakng, ther algorthm randomly scatters artcles n R 3 and uses a modfed gradent vector feld to force them to mgrate to the surface of the obect. For the renderng e used the OenGL grahcs lbrary. The deth cue as obtaned by usng ts fog feature. 6 Results A model of a hand as bult from a set of 17 rmtves, hose outlnes and relatve ostons are shon n Fg. 5. The resultng model s resented n Fg. 6, beng vsualzed by 3200 artcles lyng at the obect surface. f4a f3a f3b f4b f5a f5b f1b f1a f2b f2a Fgure 5: Outlnes of the rmtves. Frst the alm as bult, usng 7 rmtves, flattened from ther orgnal shercal shaes. Then each fnger as modeled searately, as a subtree th to rmtves. Next, the fngers ere ostoned relatve to the alm, as f they ere sngle rmtves. Only lnear blendng oeratons ere used (denoted by ), th untary eght for each rmtve (Fg. 7). Fg. 8 exemlfes a modfcaton of ths hand, here the fngers are slghtly moved aart from each other. Due to the herarchcal structure, to acheve ths effect e ust need to aly rotatons to the corresondng subtree root nodes: f1, f2, f3, f4,andf5. Fgure 6: Imlct model of a hand. 4 f5 6 f4 5 f3 f1b f1 1 f2a f2b 2 3 f1a F f3a f3b f4a f4b f5a f5b 7 f2 Fgure 7: Structure of the hand. The next examle demonstrates the use of to combnng oeratons lnear blend and unon n the same mlct obect. The obect shon on the to of Fg. 9 s made of 3 rmtves, th ts corresondng tree structure sketched belo t. The rmtves labelled g1a and g1b are combned through a unon oeraton (ndcated by U ), makng the subtree labelled g1. Ths subtree s then combned th the horzontal rmtve g2 through a lnear blend, thus yeldng the fnal obect. Observe the dfferent connectons beteen the rmtves. The use of a unon oeraton leads to a shar corner (A), hle the lnear blend gves a smooth ont (B or C).

6 Fgure 8: Modfed hand. 7 Conclusons We resented a frameork that not only encomasses several exstng skeleton-based models but also allos ther extenson. The roosed herarchcal structure rovdes adequate defnton and manulaton semantcs for comlex obects. Our goal has been to suort modelng of comlex mlct obects n an nteractve desgn envronment. In ths aer e addressed only the modelng roblem, but e beleve that t can be the underlyng structure for nteractve systems. Observe that the control flexblty s accomlshed by the set of arameters nvolved n the defnton of the model, hch allo a vast range of ntutve geometrc varatons. Regardng to the modelng roblem, to contrbutons are orth to be mentoned. Frst, e derved a comutatonally chea decay functon. Then, e have used geometrcal transformatons on the functon doman to mrove the modelng flexblty. As future ork, e lan to exlore the nteractvty oer of the roosed frameork. 8 Acknoledgments We sh to thank Marcos K. Agulera and the referees for ther helful suggestons on the mrovement of ths aer. We also thank CNPq for the fnancal suort. References [1] Blnn, J. F., A Generalzaton of Algebrac Surface Drang, ACM Transactons on Grahcs, 1(3), A B C g1 g2 F g1a g1b U Fgure 9: Examle of multle blends. [2] Bloomenthal, J. and Wyvll, B., Interactve Technques for Imlct Modelng, Comuter Grahcs (1990 Sym. on Interactve 3D Grahcs), 24, [3] Bloomenthal, J. and Shoemake, K., Convoluton Surfaces, Comuter Grahcs, 25(4), [4] de Fgueredo, L. H. and Gomes, J., Samlng Imlct Obects th Physcally-based Partcle Systems, Comuter & Grahcs, 20(3), [5] Gomes, J. and Velho, L., Imlct Obects n Comuter Grahcs, IMPA, [6] Nshmura, H., Hra, M., Kaa, T., Kaata, T., Shrakaa, I., and Omura, K., Obect Modelng by Dstrbuton Functon and a Method of Image Generaton, Jaan Electroncs Communcaton Conference 85, [7] Wyvll, B., McPheeters, C., and Wyvll, G., Data Structure for Soft Obects, TheVsual Comuter, 2(4), [8] Wyvll, B., McPheeters, C., and Wyvll, G., Anmatng Soft Obects, The Vsual Comuter, 2(4), 1986.