Haptic Sculpting of Volumetric Implicit Functions

Transcription

1 Haptc Sculptng of Volumetrc Implct Functons Jng Hua Hong Qn Department of Computer Scence State Unversty of ew York at Stony Brook Stony Brook, Y , U.S.A. {jnghua qn}@cs.sunysb.edu Abstract Implct functons characterzed by the zero-set of polynomal-based algebrac equatons and other commonly-used analytc equatons are extremely powerful n graphcs, geometrc desgn, and vsualzaton. But the potental of mplct functons s yet to be fully realzed due to the lack of flexble and nteractve desgn technques. Ths paper presents a haptc sculptng system founded upon scalar trvarate B-splne functons. All the solds sculpted n our envronment are sem-algebrac sets of volumetrc mplct functons. We develop a large varety of sculptng toolkts equpped wth an ntutve haptc nterface to facltate the drect manpulaton of mplct functons n real-tme. To facltate multresoluton edtng and dfferent levels of detals, we employ three technques: herarchcal B-splnes, CSG-based functonal composton, and knot nserton. Our experments demonstrate that our algorthms and haptcs-based technques can greatly overcome the elng dffcultes assocated wth mplct functons. The novel elng technques and ther haptcs-based desgn prncple are extensble to the desgn of arbtrary mplct functons. Keywords: Geometrc Modelng, B-splnes, Implct Functon, Volume Sculptng, Marchng Cubes Renderng, Haptc Interface.. Introducton and Motvaton The effcency and flexblty of shape elng are vtal to the success of graphcs, geometrc desgn, and vrtual envronments. Despte the prevalence of parametrc forms n vsual computng felds, the tradtonal representaton of geometrc enttes such as commonlyused analytc shapes comes from mplct functons because of many of ther attractve propertes []. It can be shown that the set of mplct algebrac surfaces or solds s actually larger than that of ratonal parametrc surfaces or solds. Ths set s also closed under certan geometrc operatons. Every ratonal parametrc curve/surface/sold can be represented by an mplct algebrac equaton, but not vce versa [3]. In contrast wth parametrc forms, mplct functons have a number of advantages such as pont classfcaton, ntersecton computaton, unbounded geometry. Consder polynomal-based algebrac equatons for example, the smplest form for mplct functons s the power bass expresson of degree n j k a x y z = 0 () jk, j, k, + j+ k n Despte ther representaton potental, exstng technques assocated wth mplct functons have certan severe shortcomngs. Frst, effectvely dgtzng and renderng an mplct functon are oftentmes far from trval. It s extremely dffcult to control the shape of mplct solds whle re-renderng the fed regons fast enough for ther use wthn an nteractve envronment. Second, a desgner often has no ntutve understandng of the effect of alterng polynomal coeffcents or addng/deletng components. Consder (), the coeffcents provde nether drect and natural geometrc nterpretaton nor ntutve nsght nto the underlyng shape. Thrd, there are no convenent tools for the ntutve shape control of ths type of algebrac solds. Moreover, general mplct functons usually have a property of global control. In contrar a large number of technques and tools have been developed to afford global and local control of conventonal parametrc surfaces or solds. Yet, flexble and drect elng technques for mplct solds are under-explored. We propose a novel elng approach and a haptcsbased sculptng prncple that can ntegrate mplct functons wth parametrc representaton such as pecewse scalar B-splnes, whch permt nteractve and drect manpulaton of mplct solds n real-tme. Local control of the mplct solds can also be easly accomplshed. Ths wll enable desgners to beneft from the low degree and computatonal effcency of mplct functons. Ultmatel our endeavor should make t possble to acheve the full potental of mplct functons n commercal desgn systems.

2 Commonly used graphcs systems often rely upon 2D mouse-based nterfaces for 3D nteracton. Drect operatons on vrtual objects wth a 2D mouse are not as natural and ntutve as nteracton va a 3D nterface. To amelorate ths, we offer users a haptc nterface for the ntutve and natural sculptng of volumetrc mplct functons. Haptcs provdes users a hand-based mechansm for ntutve, manual nteractons wth vrtual envronments towards realstc tactle exploraton and manpulaton. Haptcs-based nteracton has emerged as a crtcal metaphor n the felds of medcne, educaton, ndustr entertanment, and computer arts. Our objectve s to allow users to reach toward an object, feel the physcal presence of ts shape and manpulate t. Wth a standard haptc devce, our approach permts users to nteractvely sculpt vrtual materals havng realstc propertes and feel the physcally realstc presence wth force feedback throughout the desgn process. Usng haptcs n a vrtual envronment, desgners are able to feel and sculpt real objects n a natural 3D settng, rather than beng restrcted to depend on 2D projectons for nput and output. Force feedback provdes addtonal sensory cues to desgners. Ths tactle exploraton can afford desgners to gan a rcher understandng of the 3D nature. The use of haptcs n a vrtual desgn envronment promses to ncrease the bandwdth of nformaton between desgners and the synthetc elng world. Pror research s prmarly focused on haptc renderng (.e. the feelng of rgd surfaces/solds). In contrast, our haptc sculptng system allows desgners to nteractvely sculpt mplct solds n real-tme. Our volumetrc mplct functons are well suted for ntegraton wth a haptc approach because of the propertes of mplct functons. In partcular, the elng of mplct functons smplfes the complcated computaton of collson detecton and depth penetraton between mplct solds and ponts n 3D. Throughout our system, the sculpted object s evaluated as a level set of a volumetrc mplct functon defned over a three-dmensonal workng space. Although we employ unform or non-unform B-splnes as the underlyng functon, consttuent functons may be of arbtrary type wth or wthout the local control property. We further enhance scalar B-splne functons wth addtonal features such as herarchcal decomposton and CSG-based operaton. Through the knot nserton, our system takes advantage of both unform and non-unform B-splne functons so that the knot dstrbuton wll nfluence the local shape. Rather than ndrectly fyng the coeffcents assocated wth the volumetrc mplct functon as exhbted n pror work, our sculptng tools support the drect manpulaton of mplct functons' scalar values. Our algorthms can automatcally determne all of the unknown control coeffcents and effectvely reconstruct a new volumetrc mplct functon after the local/global fcaton. Our system offers a wde array of ntutve sculptng tools responsble for the effectve constructon of varous complcated geometrc shapes wth dverse topologes. Ths allows desgners to nteractvely and drectly sculpt mplct solds wth ease. The use of pecewse B-splnes facltates the rapd fcaton on arbtrar localzed regons. It may be noted that the ntegraton of a haptc nterface and volumetrc mplct elng should be of nterest to much broader communtes. 2. Background Revew 2. Volume Sculptng Galyean and Hughes [4] frst ntroduced the concept of volume sculptng and developed a system wth smple tools n 99. Later, Wang and Kaufman [9] presented a smlar sculptng system wth sculptng tools of carvng and sawng. In order to acheve real-tme nteracton, the system reduced the complex operatons between the 3D tool volume and the 3D object to prmtve voxel-by-voxel operatons. Barentzen [] proposed to use octree-based volume sculptng. The possblty to support multresoluton sculptng and ts advantages were dscussed at length. In a nutshell, the aforementoned sculptng systems were all dependent on the smple, voxelbased operaton. The sculpted objects and the sculptng tools are represented usng a dscrete characterstc functon. Unfortunatel only C 0 contnuty could be acheved. In order to avod the object spatal alasng, the sculpted objects and sculptng tools need to undergo an approprate flterng operaton. Recentl Ravv and Elber [5] presented a 3D nteractve sculptng paradgm that employed a set of scalar unform trvarate B-splne functons as underlyng representaton. The sculpted object was represented as the zero set of the trvarate functons. Users can ndrectly sculpt objects to a desrable shape by drectly fyng relevant scalar control coeffcents of the underlyng functons wth tools. Ths work poneers the use of a contnuous characterstc functon n 3D sculptng. 2.2 Implct Functons Blnn [3] demonstrated that mplct functons are well suted for both scentfc vsualzaton and the elng tasks n computer graphcs. Typcal technques nclude forcng an algebrac surface to nterpolate a set of (regular or scattered) ponts or a network of spatal curves, and usng pecewse algebrac patches to form a complex shape satsfyng certan contnuty requrements across patch boundares. Sederberg [4][5] dscussed the elng technques for cubc algebrac surfaces.

3 Hoffmann [6] systematcally revewed the mplct functon technques ncludng the mplctzaton, parameterzaton, and the parametrc/mplct converson n CAGD. Bajaj and Ihm [7] presented an effcent algorthm to mplement Hermte nterpolaton of lowdegree algebrac surfaces wth C or G contnuty. ote that, nether pont nor curve nterpolaton s an attractve mechansm for defnng an mplct surface because t s dffcult for desgners to predct the surface behavor beyond nterpolatng curves and ponts. Implct functons can also be used to represent a sold. Commonly-used, yet smple solds such as spheres, cubes, cylnders, and tor are oftentmes used as prmtves. To create more nterestng shapes, prmtve solds can be collected nto a herarchcal organzaton wth the help of Boolean operatons. More complcated operatons through the use of functonal composton are also possble to generate more nterestng shapes. The common feature essental to all mplct sold elng methods [8][9] s the creaton of an orented three-dmensonal boundary surface whch parttons the entre 3-space nto two dstnct regons, namely the one occuped by the sold nteror and the one outsde of the defned sold. 2.3 Haptc Renderng Haptc renderng s the process of applyng forces through the use of force-feedback devces and augmentng a vrtual envronment wth a haptc nteracton. Haptc renderng requres: () sensng the poston of the user's fnger; (2) locatng the contact pont; and (3) approprately generatng a force to be appled to the fnger. Thompson et al. [6] derved effcent ntersecton technques that can be appled to nearly any type of haptc nterface. Dachlle et al. [20] developed a haptc nterface to permt the drect manpulaton of dynamc surfaces. McDonnell et al. [7] employed haptc toolkts to explore the dynamc subdvson solds. Avla et al. [8] presented a haptc nteracton that s sutable for both volume vsualzaton and elng applcaton. Despte the wdespread applcaton of haptcs n vsual computng areas, haptcsbased nteracton was manly appled to parametrc representatons for shape sculptng. We ntegrate the prncple of haptc elng wth the drect manpulaton of mplct solds. 3. Volumetrc Implct Functons 3. Tensor-Product Scalar B-splnes Throughout ths paper, we utlze scalar trvarate B- splne functons as the underlyng shape prmtves for object representaton. The use of mplct B-splne functons for sold elng s strongly nspred by ther attractve propertes ncludng smplct generalt local control, etc. The generc B-splne functons are of the followng form: l m n s ( u, v, w) = p B ( u) C ( v) D ( w (2) = 0 j= 0 k= 0 jk, r j, s k, t ) where s ( u, v, w) represents the scalar value at poston (u, v, w) n parametrc doman. u, v, w change from 0 to U, V, W, whch represent the sze of samplng ponts along three dmensons of parametrc doman. p jk are the scalar control coeffcents wth the doman of I, J, K that are [0, l-], [0, m-], [0, n-], respectvely. In addton, B, r ( u), C j, s ( v) and D k, t ( w) are the bass functons correspondng to p jk, evaluated at (u, v, w). The degrees of the three bass functons are r-, s-, and l-, respectvely. To smplfy the mathematcal notaton, (2) can also be expressed as the followng matrx form: s = ( B C D) p (3) where denotes Kronecker Product, and T s = [, s, ] ( [0, U ], j [0, V ], k [0, W ]) jk p = [, p, T jk ] ( [0, l ], j [0, m ], k [0, n ]) B, C, and D are matrces composed of the samplng of bass functons. They are of the followng forms: B0, u0) B, u0 ) m BI, u0) C0, s( v0 ) C, s( v0 ) m CJ, s( v0) = B0, u ) B, u ) m BI, u) B C0, s( v ) C, s( v) m CJ, s( v ) C= o o m o o o m o B0, uu ) B, uu ) m BI, uu ) C 0, s( vv ) C, s( vv ) m CJ, s( vv ) D0, w0 ) D, w0 ) m DK, w0 ) = D0, w ) D, w ) m DK, w ) D o o m o D0, ww ) D, ww ) m DK, ww ) ( B C D) could be precomputed n order to save runtme computaton and mprove real-tme performance. 3.2 Implct Solds The mplct functon can be generally characterzed as: {( x, F( = 0} (4) The boundng surface defned by an mplct functon s a level-set w = F( w = w0 By collectng all the level sets whose return values are greater (or smaller) than a gven threshold, we could defne a mplct sold. w = F( w > w0 The advantages of mplct forms have been brefly documented n Secton. In prncple, the elng schemes founded upon mplct forms are much more (5) (6)

4 powerful than that of parametrc-drven geometrc elng. However, elng technques based on mplct functons are not yet wdespreadly explored due to the lack of drect manpulaton mechansm. Our elng system can brdge the large gap towards the full realzaton of all the elng power of mplct functons. 3.3 Volumetrc Implct Functons We shall collect dfferent B-splne patches defned over the 3D workng space to form a volumetrc mplct functon that can be collectvely used to represent objects of complcated geometry and arbtrary topology. ote that, sgnfcantly dfferent from commonly-used parametrc B-splnes, mplct B-splne functons formulate the scalar value dstrbuton n 3D where mplct solds are unquely defned as sem-algebrac pont sets. Ravv and Elber [5] used a smlar representaton to mplement a freeform sculptng system. In our system, we further enhance the B-splne representaton power by ncorporatng the elng advantages from herarchcal splnes, generalzed CSGbased Boolean operatons, and non-unform knot nserton Herarchcal Organzaton Let us assume users have defned B-splne patches over the sculptng workng space, whch are located at any locaton and wth any orentaton. In general, these patches may be formulated by dfferent number of control coeffcents n order to acheve the goal of multresoluton analyss and level-of-detals control. Then the scalar value at the locaton ( can be computed as F( = s ( T ( ) (7) = where T s an affne transformaton from the Eucldan space to the parametrc doman of patch s. Snce the trvarate B-splne has the affne nvarance propert ths transformaton can be easly mplemented. For each dfferent patch s, there s a correspondng transformaton T. ow F ( becomes a new volumetrc mplct functon defned over the 3D workng space. Wthout loss of generalt we make use of cubc B-splnes wth nonperodc knot vectors. In order to make the boundares of dfferent trvarate patches acheve C contnut the frst and last 4 layer control coeffcents along three prncpal drectons of the parametrc doman should be set to zero CSG-based Operatons Users may ntend to sculpt mplct solds to form sharp features over ther boundares or change the contnuty requrements across ther smooth boundares. Featurebased sculptng tools can sgnfcantly mprove the system performance. In lght of ths demand from users, our system provdes CSG-based operatons on any userdefned trvarate patche n order to facltate the rapd constructon of complcated els satsfyng many feature-orented requrements. Therefore, complcated geometry s readly avalable n our system through the use of F( = s ( T ( ) (8) Ω= where Ω s a Boolean operaton such as Unon, Intersecton, or Dfference. In addton, T and s have the same geometrc meanng as those appeared n (7). In our system, the Boolean operaton nformaton wll be stored n a tree structure n order to speedup the data query on-unform Knot Dstrbuton The use of unform knots to el sophstcated objects may result n a extremely large number of knots and control ponts. Ths wll lead to nformaton redundancy and deformaton dffculty as many degrees of freedom must be employed n any localzed small regon. However, the use of non-unform knot sequences affords addtonal shape control and the elng of a much larger class of shapes than what the unform knot vectors can offer. Furthermore, the use of non-unform B-splnes can overcome certan elng dffcultes assocated wth unform B-splnes. For example, t s almost mpossble for B-splnes wth unform knot vectors to nterpolate hghly unevenly spaced data ponts wthout the unwanted scenaro of oscllatons or loops [2]. Our system allows users to specfy a non-unform vector durng the ntalzaton phase of the object desgn sesson. In addton, users could nsert more knots anywhere nto current knot vector at any tme after the sculptng manpulaton s underway. When new knots are nserted, the system wll generate correspondng control coeffcents and the sculpted object wll be reevaluated upon the refned knot vector. In our system, the knot spacng s proportonal to the dstances of the data ponts: = + x + x where represents the spatal dfference between the (+)th knot and th knot, x represents the spatal dfference between the (+)th data pont and th data pont. Thus, the underlyng el represented by volumetrc mplct functons s essentally a non-unform B-splne. Through the dfferent combnaton of these three technques, our system could offer users a large array of elng operatons and enhance the already-powerful shape varaton of mplct B-splnes wth the addtonal flexblty n a herarchcal fashon.

5 4. System Descrpton The sculpted object of an mplct B-splne functon s dscretzed nto a voxel raster n our system for renderng purpose. Every voxel contans a scalar value, called densty value, sampled at a grd pont. The volumetrc mplct functon descrbed n Secton 3 s employed to assgn the densty value to the sample ponts to ndcate f the locaton has materal. The functon wll be used to formulate the densty dstrbuton over the 3D workng space and represent the sculpted object by a gven level set. Fg. (see the color page) shows voxel maps n 2D space and 3D space, respectvely. Ths voxelmap defnes a functon, where the sold partcles (colored n red) denote locatons n whch materal exsts and the empty partcles (colored n gray) denote locatons n whch there s no materal. Although we use bnary materal dstrbuton n Fg. to llustrate the concept, however, n our system the characterstc functon s not a bnary functon, rather t s a contnuous functon. When a sculptng tool s used to sculpt the object, the densty values of the workng space nsde the tool volume wll be fed correspondngly. Then the system wll reconstruct the volumetrc mplct functon to represent the new, fed object undergong deformaton. By usng local Marchng Cubes technque [2][2], the sosurface of the object could be dsplayed nteractvely. The haptc nterface of our system allows users to reach toward an object, feel the physcal presence of ts shape, and sculpt t wth force feedback. Through the use of many haptc tools avalable n our system, users can obtan both ntutve feelng and better understandng of the vrtual sculptng. The feedback forces are computed drectly based on the object representaton. 4. Octree-based Data Structure Snce the sculpted object s dscretzed n a voxel raster, usually there are many homogeneously empty regons outsde the object of nterest. If those regons could be quckly separated from the sculptng regon, t wll sgnfcantly reduce the memory consumpton and speed up the volume renderng and elng tasks. Therefore, an octree-based data structure s employed n our system, smlar to the scheme used n []. The workng space s recursvely subdvded untl ether the subdvded volume s empt or the subdvson has reached a pre-defned maxmal subdvson depth. In the frst case, the subdvded volume s an empty leaf node, whle the second stuaton means that the current locaton s not empty and the materal property at that locaton should be recorded. Every tme the sculpted object s fed by a sculptng tool, the octree data structure could locate where the fcaton s performed and only needs to locally update the volumetrc mplct functon for effcency purpose. Our system uses Marchng Cubes technque to render the sosurface of the sculpted object. Ths local update property can speed up the Marchng Cubes renderng by only conductng the reevaluaton task of the fed parts. 4.2 Volume Sculptng 4.2. Tool Modelng Tools are represented by any 3D mplct functon w 0 = G(. It s easy to determne whether a locaton s nsde the tool volume by smply evaluatng the functon. In order to prevent object spatal alasng, a flterng operaton must be used nsde the tool volume. The flterng algorthm used n our system s smlar to the one n [4][9]. Gven a locaton (, the shortest dstance from ( to the boundary of the tools s computed usng the evaluaton functon. Then ths shortest dstance s used to flter the densty values at the locaton (. Here we use a lnear flter. The mnmal densty value s assgned to the boundary and the maxmal one s assgned to the center of the tool. The densty values at the ntermedate locatons are lnearly nterpolated. So the densty value at ( s proportonal to the shortest dstance to the boundary. Later we wll explan how to further generalze ths concept n haptc nterface to obtan realstc force feedback Tool-Object Interacton When users assgn a sculptng tool to a new locaton, the tool s mapped to the coordnate system, whch contans the sculpted object. The boundary box of the tools s then computed. And the densty values at the locatons nsde the tool volume are fed as descrbed n Secton If the tool s to add materal, those densty values should be greater than the object so-value. If the tool s to remove materal, those densty values should be less than the so-value. After ths ntal fcaton on materal dstrbuton, we have to reconstruct the volumetrc mplct functon of B-splnes accordng to the new densty dstrbuton. Currentl our system only allows sngle operaton to fy exactly one patch at any tme durng the sesson of volume sculptng. So only control coeffcents that belong to one B-splne patch need to be fed at every tme of sculptng. Realtme performance wth realstc haptc feedback can be easly acheved. The mathematcs of B-splne manpulaton s formulated as follows: ( B C D) p new = s new (9) where s new represents the new densty dstrbuton over the sculpted patch regon and p new are new control coeffcents, B, C, D are samplng matrces of bass functons as shown n Secton 3.. Because of the local support property of B-splnes, only a very small subset of

6 the control coeffcents needs to be fed. Hence, we only need to solve ths system of lnear equatons wthn the tool sculptng regon. Therefore, (9) can be further smplfed nto: B C D p = (0) ( ' ' ') s where s and p only come from the fed regon. B', C' and D' are small sets of the orgnal bass matrces, whch are correspondng to the local fed regon. ow, n essence the problem of volumetrc sculptng s equvalent to a typcal data fttng applcaton: Gven a set of ponts ( x, y j, zk ) n the parametrc doman, and the densty value d jk at every pont, fnd the best possble soluton that fts the data set ether through nterpolaton (when one unque soluton exsts) or approxmaton (when the system becomes over-constraned). Usually the number of control coeffcents s less than the hardwarepermtted resoluton of 3D workng space. So we employ the Mean Square Error for data approxmaton: MSE = LM L M = j= k= ( d jk s( x, y, z )) j k 2 () where LM represents the total number of data ponts that have been fed usng certan sculptng tools. Therefore, t s necessary to seek a functon s(x,y, that mnmzes the mean square error. It s convenent to use matrx algebra to symbolcally formulate the soluton to the precedng problem. Usng the matrx forms, the Mean Square Error can be wrtten as: = LM [ d ( B' C' D') p ] 2 MSE (2) where d s a vector of densty values whose elements are d jk. Dfferentatng wth respect to the elements of p, and settng the dervatve to zero leads to the soluton: T T p = [( B' C' D' ) ( B' C' D' )] ( B' C' D' ) d (3) whch s equvalent to the least-square fttng. After the new control coeffcents are generated, the system uses the Local Marchng Cubes algorthm to render the fed part to generate the new sosurface of the deformed object. 4.3 Haptc Feedback In order to enhance the realsm of the vrtual sculptng, our system offers haptc nteractons, whch can gve users a realstc feel of the vrtual objects. Thus, users can gan a rcher understandng of ther sculpted el. Our work sgnfcantly extends the noton of smply touchng complant objects (.e., haptc renderng) to nteractvely and drectly sculptng of vrtual solds (.e., haptc elng). From the standpont of volume sculptng, the followng problems must be addressed n order to provde meanngful force feedback for haptc nteracton: Force computatonal rate: the computatonal rate must be hgh and latency must be low. Inapproprate values can cause an mproper feel of the vrtual envronment. Generaton of contactng forces: ths creates the "feel" of the object. Contactng forces can represent the stffness of the object, dampng, frcton, surface texture, etc. Fast data fcaton and renderng: ths could make the sculptng operaton consstent wth the haptc force feedback. Snce the sculptng could only be performed wthn a small regon at every tme step, t s natural to only allow the computaton of haptc nteractons to occur wthn a localzed regon, n order to meet the hgh frequency requrement set by the haptc devce and make the sculptng consstent wth the force feedback. In addton, oftentmes users' meanngful sculptng operatons would not exceed the regon lmtaton wthn a very tny tme step along tme axs (e.g., only ~2 ms n our system). So ths assumpton s reasonable and does not ntroduce any lmtatons n our volume sculptng system. In our system, we use a pont contact force el [8]: F = R( V ) + S( ) (4) where V s movng speed of a contactng pont, R (V ) s a dampng force that tends to resst moton along the opposte drecton of the contactng pont's movement, S ( ) s a stffness force along the normal of the contactng pont ( ). F s the feedback force to users, whch s equal to the sum of the moton dampng force and stffness force. The force calculaton should be very fast to meet the PHAToM update rate (.e., greater than kh. Otherwse, users would have uncomfortable feelngs such as buzzng durng the haptc nteracton, hence, destroyng the purpose of usng haptcs to augment realsm. As we descrbed n Secton 4.2., the densty dstrbuton n our el s proportonal to the dstance map. So t s natural for us to use the densty feld nstead of the dstance feld to calculate the force. The moton dampng force and stffness force are calculated, respectvel as follows: R( V ) V f ( d) = (5) S( ) f s ( d ) r = (6) where d s a densty value, f r and f s are transfer functons, whch map densty values to force magntudes. The transfer functons f r and f s that we are currently usng n our envronment are as follows: ( ) = ( ) (7) f d a d d r new + b

7 f ( d) k( d d) s = new (8) where d s current densty value at (, d new represents the new densty value at ( x + y + z +, whch s the very next tme step. a, b and k are control varables whch can be nteractvely set up by users. Usng dfferent transfer functons, we could let users feel dfferent force effects, ncreasng the flexblty of our haptc nteracton. In our system, the densty dstrbuton over the 3D workng space s represented as a contnuous volumetrc mplct functon of B-splnes. Ths property can help to avod the dscontnuous force feedback, whch leads to unrealstc feelngs such as buzzng. Another advantage for ntegratng haptc nteracton wth mplct functons s that t s much easer to compute the contactng pont and determne f the contactng pont s nsde the object to be sculpted. The densty value at any locaton could be gven by smply evaluatng the volumetrc mplct functon. Therefore, the dampng force and stffness force could be computed effcently to satsfy the hgh update rate of haptc nteracton, R ( V) = V ( a( F( x + y + z + F( ) + b) (9) S( ) = k ( F( x + y + z + F( ) (20) where F s obtaned from (7) or (8). The normal at ( can be computed analytcally as F F F (,, ). x y z 5. Implementaton Our system s mplemented on a Mcrosoft Wndows T PC wth a 550MHz CPU and 52 MB RAM. A PHAToM.0 3D Haptc nput/output devce from Sensable Technologes s employed to provde natural and realstc force feedback. The entre system s wrtten n Mcrosoft Vsual C++ and the graphcs renderng component s bult upon OpenGL. Fg. 2 (see the color page) shows the system nterface. When usng haptc tools, to reduce the latency and maxmze the throughput, we resort to a parallel technque that can multthread the haptcs, graphcs, and sculptng processes wth weak synchronzaton. Ths technque leads to the possble performance mprovement and ultmatel the parallel processng of haptc sculptng gven the hghend mult-processor envronment. Therefore, our system s readly avalable n many dfferent confguratons. Fg. 3 shows the structure of the multthreads, where thck arrows represent data flow and thn arrows represent control flow. The haptc loop s mplemented n a sngle thread. It mantans the haptc refresh rate whch s no less than KHz. Ths requrement s crtcal to the realstc feedback of haptc nteracton. If the update rate s below the threshold of KHz, users would feel uncomfortably. In our system, the haptc thread has the hghest prorty. Sculptng thread Get cursor poston Object sculptng Update object Object dataset Graphcs thread Local Marchng Cubes Update dsplay Fg. 3 The structure of multthreads The object sculptng loop s mplemented n another thread. It controls the object sculptng. In order to keep up wth haptc update rate, any sculptng operaton wthn one tme step s lmted to a small regon. As we specfed n Secton 4.3, usually users' sculptng operatons would not exceed ths lmted regon wthn a tny tme step. The graphcs loop s developed to handle the renderng of volumetrc objects. The renderng task makes use of local Marchng Cubes algorthm and only update the very small regon n order to acheve nteractve speed and make graphcs dsplay consstent wth sculptng operaton and force feedback. 6. Interactve Sculptng Toolkts Haptc thread Get haptc nput Compute forces Send back forces Our system offers several haptc tools such as haptc so-surface feelng, haptcs-based probng, and haptcsbased drller. Besdes feelng the boundary surface of a volumetrc object, users can choose any so-value from the allowable range of the volumetrc mplct functon and the system generates correspondng sosurfaces quckly. The fgure on the left shows two dfferent so-surfaces wth the wreframe dsplay e n order to make two surfaces vsble at the same tme. Users can feel dfferent so-surfaces of the sculpted object by movng the cursor and navgatng over those so-surfaces. Ths tool allows users to examne the smoothness of objects' surface and the nteror structure tactlely. Through the use of the contactng force el descrbed n Secton 4.3, our system can afford users to feel the tny dfference of an object's stffness (or densty) whle users move the cursor nsde the object. When actve, the probng tool exerts a force on the user's fnger proportonal to the local densty values wthn a gven radus of the tool.

8 Usng the haptcs-based drller, user can make a drll to the object along any drecton. In ths process, users could feel the realstc force comng from the object's stffness and the moton resstance. The drller could be any knd of shape such as spheres, cubes or stars. For tool operatons, Fg. 4 (see the color page) shows a number of toolkts ncludng sphere-based carvng and addton tools, cylnder-based carvng and addton tools, rectangle-based carvng and addton tools, torus-based carvng and addton tools, chsel tools, copy and composng tools, squrt tools, nflaton and deflaton tools, movng and bendng tools, etc. ext we explan some tool confguratons and ther operatons. When usng copy and composng tools to buld up a complcated scene wth the same object, users can defne a number of trvarate patches at any locatons and along any drectons. The collecton of those patches s based on the unon operaton for CSG els. Through the use of the smple copy operaton, the patch coeffcents can be duplcated from one regon to another regon of nterest, a set of smlar objects of the same geometry could be easly created. In Fg. 4 where copy and composng operatons were undertaken, 3 3 patches were created parallel to each other n the workng space. When the char sculptng was completed nsde one patch, the coeffcents of that patch were coped and loaded n other 8 patches to create the scene. The nflaton and deflaton tools cause a localzed regon to grow or shrnk. Users can nteractvely defne a regon. When usng a deflaton tool, the coeffcents of the scalar trvarate mplct functon nsde the regon decrease n order to shrnk the specfed part of the object. For the nflaton tool, those coeffcents should be ncreased nstead. In Fg. 4 where nflaton and deflaton operatons were undertaken, the soccer player's head was nflated and one of hs arms was deflated. The movng tool moves a selected regon of the coeffcents to other locatons. The bendng tool deforms a selected regon of coeffcents to new postons. In Fg. 4 where movng and bendng operatons were undertaken, we frst moved both arms up through the use of the movng tool, and then, bended the player to get the second gesture. 7. Expermental Results We have developed a novel elng system for haptc sculptng of the volumetrc mplct functon based on nonunform B-splnes. The mplct sold can be generated wth a varable number of control coeffcents and wth varable samplng rates. We have conducted a large number of experments and recorded the runnng tme for the sculptng of volumetrc mplct functons. The experments are based on a workng space sampled at The tool sze s gven as the number of data ponts that the tool affects. The results are detaled n Table. Table : Run tme of Tool-Object nteracton Control coeffcent Update tme Tool sze resoluton (ms) Wthn our mplct functon elng framework and wthout usng any other external resource, we have created several nterestng objects and scenes from scratch. Fg. 5 (see the color page) shows a seres of actons of a soccer player. They were sculpted wth unform knots and control coeffcents. Fg. 5(a) shows a standng one. We shall use ths example to explan how to sculpt an object usng our system. We began wth a cubc block. By carvng and haptcally drllng the cubc block several tmes, a rectangular body was created. The neck was sculpted usng cylnder-based addton. The head was placed on top of the neck usng a sphere-based addtve tool. By way of a rectangular tool, the shoulder part was sculpted. The two arms were created usng cylnder-based addton. The two legs were obtaned n a smlar fashon. The feet were sculpted by addng rectangular materals and sculptng wth sphere-based tools and the haptcsbased drller. Other motons of the soccer player are all based on the ntalzed el. Through the use of movng and bendng operatons, the anmated sequences of els were subsequently created. Fg. 6 (see the color page) shows several characters mounted on a rectangular stone. The workng space contans a sngle patch wth non-unform knots and control coeffcents. The regon that contans all characters has much more knots and control coeffcents than the flat regon where no deformaton s undertaken. Fg. 7, Fg. 8 and Fg. 9 (see the color page) show three scenes, whch were sculpted entrely usng our system (wthout resortng to any other external resource) and rendered usng the commercal software of POV-Ray. For example, Fg. 7 shows a cartoon tran runnng n a desert envronment. The tran was sculpted wth nne patches. One patch was for sculptng the body of the tran. And other eght patches were used for sculptng the eght wheels. The ralroad was sculpted n a patch. Every cactus was sculpted n a patch. The collecton of the patches n the workng space was based on herarchcal organzaton and unon operaton.

9 8. Concluson We have presented a novel haptcs-based volumetrc sculptng envronment that employs trvarate scalar nonunform B-splnes as underlyng representaton. All the volumetrc objects sculpted n our elng system are characterzed by pece-wse mplct functons. We have proposed a new approach that unfes mplct functons and parametrc representatons wthn a sngle haptcsbased sculptng system. We have developed a large varety of algorthms and toolkts that afford desgners the mechansm of nteractve and drect manpulaton of mplct solds n real-tme, augmented by a realstc and ntutve haptc nterface. To facltate multresoluton edtng and drect control on dfferent levels of detals, we have also ncorporated three popular elng technques: herarchcal B-splnes, CSG-based functonal composton, and knot nserton nto our envronment, makng our novel mplct elng technques even more powerful and flexble to handle both complcated geometry and arbtrary topologes. Our experments have demonstrated that our algorthms and drect edtng technques based on nonunform B- splne mplct functons can not only overcome the exstng dsadvantages assocated wth conventonal elng of mplct functons, but realze all the potentals exhbted n mplct functons n vsual computng felds as well. More mportantl the powerful 3D haptcs-based nterface of our system s more ntutve and natural than conventonal 2D mouse-based nterfaces, makng t possble for our mplct functon elng system to appeal to a spectrum of users rangng from hghly traned engneerng desgners, computer professonals, artsts, to even computer llterates. Our sculptng system permts desgners to create real-world, complcated els n real-tme. Fnall the novel elng technques and ther haptcs-based desgn prncple are extensble to the desgn of arbtrary mplct functons. Acknowledgements The authors wsh to thank Kevn T. McDonnell and We Hong for provdng some of the code used n mplementng ths system. Ths research was supported n part by the SF CAREER award CCR , the SF grant DMI , the SF ITR grant IIS , and research grants from Ford Motor Company and Honda Amerca Inc. References [] J. Bloomenthal, B. Wyvll. Interactve technques for mplct elng. Computer Graphcs, Vol. 24, o. 2, pp 09-6, March 990. [2] L. Pegl and W. Tller. Curve and surface constructons usng ratonal B-splnes. Computer-Aded Desgn, Vol. 9, o. 9, pp , ovember 987. [3] J. F. Blnn. Generalzaton of algebrac surface drawng. ACM Trans. On Graphcs, Vol., o. 3, pp , July 982. [4] T. A. Galyean and J. F. Hughes. Sculptng: An nteractve volumetrc elng technque. Computer Graphcs, Vol. 25, o. 4, pp , July 99. [5] A. Ravv and G. Elber. Three dmensonal freeform sculptng va zero sets of scalar trvarate functons. In Proc. of 5th ACM Symposum on Sold Modelng and Applcatons, pp , 999. [6] T. V. Thompson, D. E. Johnson and E. Cohen. Drect haptc renderng of sculptured els. In Proc. of the 997 Symposum on Interactve 3D Graphcs, pp 67-76, 997. [7] K. T. McDonnell, H. Qn and R. A. Wlodarczyk. Vrtual Clay: A real-tme sculptng system wth Haptc Toolkts. In Proc. of the 200 Symposum on Interactve 3D Graphcs, pp.79-90, 200. [8] R. S. Avla and L. M. Soberajsk. A haptc nteracton method for volume vsualzaton. In Proc. of the 7th IEEE Vsualzaton 96, pp , 996. [9] S. W. Wang and A. E. Kaufman. Volume sculptng. In Proc. of the 995 Symposum on Interactve 3D Graphcs, pp 5-56, 995. [0] D. R. Forsey and R. H. Bartels. Herarchcal B-splne refnement. Computer Graphcs, Vol. 22, o. 4, pp 205-2, August 988. [] J. Andreas Barentzen. Octree-based volume sculptng. IEEE Vsualzaton 98, Late Breakng Hot Topcs Proceedngs, pp. 9-2, 998. [2] W. E. Lorensen and H. E. Clne. Marchng Cubes: A hgh resoluton 3D surface constructon algorthm. Computer Graphcs, Vol. 2, o. 4, pp 63-69, July 987. [3] H. Qn. Physcs based geometrc desgn. Internatonal J. of Shape Modelng, Vol. 2, o. 2&3, pp 39-88, 996. [4] T. Sederberg. Technques for cubc algebrac surfaces. IEEE Computer Graphcs and Applcaton, Vol. 0, o. 4, pp 4-25, 990. [5] T. Sederberg. Technques for cubc algebrac surfaces. IEEE Computer Graphcs and Applcaton, Vol. 0, o. 5, pp 2-2, 990. [6] C. Hoffmann. Implct curves and surfaces n CAGD. IEEE Computer Graphcs and Applcatons, Vol. 3, o., pp 79-88, 993. [7] C. Bajaj and I. Ihm. Algebrac surface desgn wth Hermte nterpolaton. ACM Transactons on Graphcs, Vol., o., pp 6-9, 992. [8] J. L. Blechschmdt and D. agasuru. The use of algebrac functons as a sold elng alternatve. Advances n Desgn Automaton, B. Ravan Ed., ASME Desgn Conference, Chcago, IL, pp. 33-4, Sept [9] V. Shapro. Real functons for representaton of rgd solds. Computer Scence Tech. Report TR9-245, Cornell Unv., Ithaca, Y, 99. [20] F. Dachlle IX, H. Qn and A. E. Kaufman. A novel haptcs-based nterface and sculptng system for physcsbased geometrc desgn. Computer-Aded Desgn, Vol. 33, o. 5, pp , 200. [2] G. Wyvll, C. McPheeters and B. Wyvll. Data structure for soft objects. The Vsual Computer, Vol. 2, o. 4, pp , 986.

10 Fg. Voxelmap n 2D and 3D Fg. 2 System Interface comprsed of on-screengui, a PHAToM devce Sphere-based carvng and addton Cylnder-based carvng and addton Rectangle-based carvng and addton Torus based carvng and addton Chsel operaton Copy and composng operaton Squrt operatons Inflaton and deflaton operatons Movng and bendng operatons Fg. 4 A set of typcal toolkts and sculpted examples

11 (a) (b) (c) (d) (e) (f) (a) Model ntalzaton, (b) Startng poston, (c) Movng, (d) Kckng, (e-f) Celebratons Fg. 5 A moton seres of a soccer player and hs entre kckng actons (a, b, c, d, e, f) Fg. 6 Englsh letters mounted on stone plates Fg. 7 A Runnng Cartoon Tran Fg. 8 A corner of PG dscusson room Fg. 9 PG conference room