Self-intersection Avoidance for 3-D Triangular Mesh Model

Transcription

1 Self-tersecto Avodace for 3-D Tragular Mesh Model Habtamu Masse Aycheh 1) ad M Ho Kyug ) 1) Departmet of Computer Egeerg, Ajou Uversty, Korea, ) Departmet of Dgtal Meda, Ajou Uversty, Korea, 1) hab01@ajou.ac.kr ) kyug@ajou.ac.kr ABSTRACT Self-tersecto s a fudametal problem computatoal geometry. Model repar s a ecessty may applcatos lke CAD because t destroys the tegrty of a mesh surface that may leads to crashes. Geometrc tersecto occur due to operatos o mesh models havg arrow clearace. I accordace, we propose a self-tersecto avodace algorthm for 3-D tragular mesh models. The purpose of our work s to cotrol usafe 3-D modes that are lable to self-tersectos. Our approach s based o fdg a optmal perturbed values eeded to update the vertex postos of a tragular mesh model wthout chagg the basc shape of the model so that the operato o the model wll ot create self-tersecto. We deployed umercal root fdg methods wth dstace metrcs. Evaluato of our expermetato shows promsg results. Keywords: computatoal geometry, self-tersecto, tragular mesh 1. INTRODUCTION Geometrc modelg s the ma buldg block of computer graphcs. I relato to the growth of computer graphcs applcatos, 3-D geometrc models are commoly represeted by polygos (Shrley 009). The ma essece of ths ppele s due to easy of computatoal descrpto, rederg smplcty ad hardware mplemetato of graphcs algorthms. Tragular mesh s a wdely accepted ad smple polygoal represetato of 3-D modelg (Botsch 006). It s composed of a set of vertces, edges ad tragular faces that defes the shape of a object 3-D space. Accordgly, polygoal models must satsfy some correctess crtera demaded by a target applcato (Botsch 006, Ju 009). A mesh model should be closed, mafold ad free of self-tersecto. For stace, geometrc correctess s essetal egeerg ad maufacturg where umercal computatos eeded for sold objects, lke fte elemet aalyss ad for real producto lke quck prototypg. However, geometrc errors are evtable due to practcal actvtes (Ju 009). There are dfferet ways of creatg polygoal models. Curretly, most 3-D objects desged by usg teractve 3-D modelg software. Some recostructed from magg 1) Graduate Studet ) Professor

2 devces such as scattered pots 3-D scag ad from gray scale volume bomedcal magg. Other 3-D modes are from NURBS, subdvso surfaces ad costructve sold geometry that ofte eed to trasform to polygoal forms for vsualzato ad computato. Most computer graphcs applcatos use large mesh szes. Usually, 3-D polygoal mesh models are composed of huge umber of coected vertces ad assocated polygoal faces that eable detaled represetato of the model so as to crease the vsual qualty of the model. However, huge sze 3-D models requres compresso etworkg evromets (Abderrahm 013, Paya 005). The recostructo of the compressed model may create artfacts. Amog well-detfed geometrc errors, a tragular mesh self-tersecto s oe of a fudametal problem computatoal geometry (Botsch 006). I coecto to ths, we propose self-tersecto avodace for tragular mesh models. Our algorthm does ot requre collso detecto procedure. We used dstace metrcs ad optmal olear root fdg umercal methods. Evaluato of our system expermetato shows promsg results. The remag part of the paper s orgazed as follows. Secto presets prevous works. Secto 3 descrbes the geeral research problem. Secto 4 descrbes methodologes ad proposed soluto. The mplemetato of the proposed soluto ad expermetal results wll be preseted Secto 5 ad Secto 6 respectvely. The cocluso of the work s preseted secto 7.. PREVIOUS WORK I 3-D geometrc models, detectg exstece of self-tersecto ad removg them s a usual operato dfferet applcato domas such as sold modelg ad computer aded desg (CAD). Geometrc self-tersecto destroys the tegrty of a mesh surface that may leads to crashes applcatos. Based o ths, may research works have bee coducted o mesh self-tersecto (Campe 010, Ju 009, Ladlaw 1986). However, the majorty of works focused o detectg ad removg selftersectos (Jug 004, Yamakawa 009). Moreover, collso detecto ad selftersecto computato are computatoally very complex. I ths le, progressve mprovemets has bee show (Gottschalk 1998, Möller 1997, Park 004, Sabharwal 013) but stll there are challeges relato to the eed of effcet ad robust self-tersecto cotrol systems dfferet applcatos. The proposed system attempts to avod self-tersecto wthout gog through the collso detecto process. That s, the put s mesh model whch s free from selftersecto. The, operatos o the model that causes possble self-tersectos are cotrolled by keepg the model safe from mesh elemets colldg each other. 3. PROBLEM FORMULATION There should exst a specfed clearace betwee layered 3-D mesh models order to safeguard geometrc objects from possble self-tersectos. Thus, we have to keep apart a surface of a geometrc model so that they wll ot create self-tersecto after some operatos of a mesh model. We ca descrbe the problem aalytcally usg

3 the elemets of 3-D tragular mesh prmtves such as vertces, edges ad tragular faces as follows. Assume that we have the followg parameters from a gve 3-D mesh model. N = umber of tragles, N M; M = umber of tragle pars; K = umber of vertces, K 3N ad = A varable vector formed from a set of vertex postos;.e. V, V, V, V, V, V,, V, V, V? 1x 1y 1z x y z Kx Ky Kz Mmum update value of vertex postos;.e.,,,,,,,,, 1 1Y 1Z Y Z K KY KZ Where, (x, y, z) s a vertex posto of the model a 3-D space. Accordgly, we ca uderstad that has a dmeso of 3K. Moreover, we ca form at most M dstace equatos, say f as show eq. (1) where d s the dstace betwee a par of 3-D tragles. f f f (x, x,..., x ) 1 1 K 1 (x, x,..., x ) 1 K (x, x,..., x ) d d d M 1 K M (1) Based o eq. (1), let we specfed a threshold value,, whch s a mmum dstace betwee the two layers of mesh model that keeps the model safe or free of possble self-tersecto. The, whe d, we have to deduce a algorthm that gves a optmal updatg values of say vector ; Therefore, our major goal s to fd a mmum x such that 4. PROPOSED SOLUTION f for all 1,,, M. The aalytcal descrpto of our research problem show secto 3 ecompasses two core parts. The prmary part s 3-D dstace computato betwee tragles par. The secod s updatg vertex postos of a model by small accordace wth a predefed threshold value,. I relato to these, our proposed algorthm s descrbed as follows. Iput: 3-D tragular mesh free of self-tersecto; 1. Defe a mmum threshold value betwee tragles par: ; f x, x,.,? x f ( ) ;. Compute the dstaces betwee 3-D tragles par: 1 K 3. Idetfy tragle pars havg dstace value less tha the threshold value: f ;

4 4. Update vertex postos,, for each f by small vector value ; 5. Repeat steps -4 utl all f. The detal descrpto of our proposed algorthm wll be preseted the followg subsectos. 4.1 Shortest Dstace betwee Two 3-D Tragles I order to detfy tragles pars havg a dstace below the specfed threshold value, a 3-D tragle-to-tragle dstace computato s requred. So, our prelmary step s computg dstaces betwee tragles. As descrbed (Eberly 006), the shortest dstace betwee two 3-D tragles falls at most four cases. They are: 1) The mmum dstace betwee vertces of the two tragles; ) The mmum dstace betwee a vertex of a tragle to the edge of other tragle; 3) The mmum dstace betwee edges of the tragles ad 4) The mmum dstace from a vertex of a tragle to the face of other tragle. However, case 1 ad ca be obtaed ether of case 3 or case 4. I case 3, by treatg each edge ad vertex pars as a le segmet ad eumeratg the dstace betwee the two les segmets; case 4, by a projecto of a vertex to a plae, f t les o a permeter.e. o edge or a vertex, case 1 or case ca be obtaed Edge to Edge Dstace The edge-to-edge dstace of tragle par ca be computed relato to the cocept of shortest dstace betwee two le segmets. As llustrated Fg, assume that we have two le segmets L1 ad L as gve eq. (). L : p s u 1 0 L : q tv 0 () Where, 0 s 1, 0 t 1, p0 s a pot o L1 ad u s ormal drecto of L 1 ; q0 s a pot o L ad v s ormal drecto of L ; Fg. 1 A orthogoal Le to Two Les

5 The shortest dstace betwee L 1 ad L s W whch s a orthogoal le both to L1 ad L. Thus, the shortest dstace s the magtude of vector W as gve eq. (3). W :? w su? tv (3) Where, w0 p0 q0 ; s ad t ca be foud as gve eq. (4): 0 s be cd ac b ( ) ( ) t ae bd ac b ( ) ( ) (4) a u. u, b u.? v c v? v, d u.? w ad e v? w Where, 0 0 I coecto to ths, the sged dstace betwee two le segmets L 1 ad L ca be computed by usg vector projecto approach. As llustrated Fg., the procedure of ths scearo ca be summarzed as follows. 1. Take a vector parallel to L 1 ad pot p 0 o L 1 ;. Take a vector parallel to L ad pot q 0 o L ; 3. Create a vector u v, so that s perpedcular to both les 4. Create two parallel plaes: oe plae from p 0 ad ; the other plae from q 0 ad. Fg. Parallel Plaes from Two Les Thus, the shortest dstace betwee the two le segmets becomes the same as the dstace betwee the two parallel plaes. Therefore, t ca be computed as a pot to plae dstace as gve eq. (5). d w. 0 (5) Note that the codto eq. () must hold so that the projected pot should le o the le segmets. I geeral, edge-to-edge shortest dstace computato, there are e possble edge-to-edge combatos.

6 4.1. Vertex to tragle Dstace As llustrated Fg. 3, gve, a ormal drecto of the tragle V 0 V 1 V such that u v, the sged dstace betwee a tragle ad a pot, P, whch s the vertex of the other tragle, ca be computed by usg the scalar projecto of (P-V 0 ) o to as gve eq. (6). Where, w P V0 w. d (6) Fg. 3 Dstace from a Pot to a Tragle We have to ote that the projected vector should le sde the tragle. Barycetrc coordate system ca be used to verfy whether a pot les sde a tragle or ot ' P x, y, z, a tragle s descrbed (Skala 013). A barycetrc coordate pot, eq. (7). Where, s t ad s t ' P 1s t V0 sv1 tv (7) 0, 0 1. s ad t ca obtaed from barycetrc coordate computato as show eq. (8). s u v w v v v w u u v u u v v (. )(. ) (. )(. ) (. ) (. )(. ) t u v w u u u w v u v u u v v (. )(. ) (. )(. ) (. ) (. )(. ) (8) Where, u = V 1 - V 0, v = V -V 0, ad w = P 0. -VNote that the pot ' P les o a edge ' f s=0, t=0 or (s+t) =1. P les o vertex f V 0 (0,0),V 1 (0,1) or V (1,0) where the umbers the brace dcates the value of (s, t). I vertex to tragle dstace computato, there are sx possble vertex to tragle combatos. 4. Multvarable No-lear Root Fdg The secod major compoet of our proposed algorthm s updatg vertex postos by small vector values so that the predefed mmum clearace dstace,,

7 s mataed. I relato to ths, we computed small updatg vertex values by usg multvarable o-lear root fdg method. Mostly, solvg system of olear equatos s complex. It eeds sght vestgato to uderstad the costrats of the target problem. There s o stadard way of solvg olear systems. The good way of solvg olear equatos s by usg umercal methods based o teratve methods. Iteratve computato requres tal value approxmato ad covergece whch makes challegg a usablty of exstg umercal algorthms for a hgh dmesoal root fdg problems. The most practcal ad robust multvarate olear root fdg method s Newto-Raphso method (Burde 010, Ortega 1970, Press 007). It s fast ad powerful method wth quadratc covergece. However, covergece s affected by tal value estmato that eeds to be hadled. F( ) 0 (9) For a vector fucto gve eq. (9), assume F s a m x dmesoal vector valued fucto. I other words, we wll have m o-lear equatos wth ukows. Thus, a umercal estmate of Newto Raphso terato s gve eq. (10) 1 ew= old + J F( old ) (10) Where, F() (f 1( x1, x,..., x ),..., ( 1,,..., )) T fm x x x. The Newto Raphso method procedure requres the computato of the verse of the Jacoba matrx. For a bgger dmeso of a Jacoba matrx, the verse computato s ot feasble due to the assocated computatoal complexty or the ature of matrces practcal stuatos such as rectagular or sgular matrces. Hece, we eed to use other mechasms lke a lear algebrac equato solvg method stead of a drect verse computato. However, the selecto of a proper lear solver depeds o the ature of the problem. I our case, mostly we have a uder costraed equato whch the umber of equatos are less tha umber of ukows. Our Jacoba matrx s a rectagular sparse matrx. Hece, we used a Sgular Value Decomposto (SVD) method as descrbed (Press 007) though t has a performace trade-off. A lear algebrac estmate verso of Newto method s gve eq. (11). F( ) J( ) x (11) Where,? 1 ad J s a Jacoba matrx. After estmatg the value of x by lear solver, we ca complete the terato of Newto step as eq. (1). ew old x (1) As depcted (Burde 010), the multvarable Newto Raphso root fdg method goes through the followg steps. 1. Ital vector value estmato, 0, ad settg covergece crtera, tolerace. Jacoba matrx ad resdual fucto values computato,.e. J ad F ). (

8 3. Applyg lear solver o: 4. Compute the curret terato value, 5. Based o covergece crtera, f t coverges STOP 6. If ot coverged, set ew old ad loop from step. 4.3 Covergece of Newto Method The covergece of Newto Raphso method s affected by the tal value approxmato because of ts local quadratc covergece costrat. I some practcal stuatos, the tal value approxmato s challegg. Hece, relato to a problem doma sght aalyss s eeded to devse a globally covergg Newto Rapsho method. As descrbed (Press 007), oe of the best approaches s based o le searches ad backtrackg. The key pot s a decso whe to accept or reject the value of x a gve Newto step show eq. (13). x; 0 1 (13) ew old I the ewto step of eq. (13), f( ew) f( old ) should be satsfed for the acceptace of the step so that each dstace fucto values are progressvely creasg towards the defed threshold value. 5. IMPLEMENTATION The mplemetato of our proposed algorthm has two ma parts. The frst s a 3- D tragle-to-tragle dstace computato ad the secod oe s fdg roots of multvarable o-lear system of equatos. The o-lear equatos are formed from dstace fuctos. As descrbed secto 3, f the dstace d s less tha the predefed threshold value,, the vertex postos should be updated by small vertex values so that f. That s f f, the f ( + ) = (14) Based o eq. (14), -order to solve for the ukow vector, we setup system of o-lear equatos formed from dstace fuctos. The, we apply Newto Raphso method to fd the roots of these o-lear fuctos. Let our mesh model has K vertces ad the umber of tragle pars havg dstace less tha s m. Thus we ca have m dstace fuctos wth ukows 3K. The system of o-lear equatos has the form dcated eq. (1). The Jacoba matrx s computed as the frst order partal dervatves of vector valued dstace fuctos as dcated eq. (15).

9 f1 f f ( x) ( x) ( x) x1 x x f f f ( x) ( x) ( x) J( ) x1 x x fm f fm ( x) ( x) ( x) x1 x x (15) The actual aalytcal descrpto of vertex postos update brefly descrbed the followg subsectos. 5.1 Edge to Edge dstace f, we eed to update the vertex postos by small Recall that whe so that vector value f. I le wth ths, the updated edge-to-edge dstace betwee two 3-D tragles s gve eq. (16). d ( w ). 0 ew ew ew (16) Where, (w ) = (u - v ) + ( - ) 0 ew 0 0 u0 v0 ( ) = (u - u ) + ( - ) 1 ew 1 0 u1 u0 ( ) = (v - v ) + ( - ) ew 1 0 v1 v0 = ( ) ( ) ew 1 ew ew I eq. (16), there are four ukow vectors,,,,, that updates the u0 u1 v0 v1 vector postos based o the edge-to -edge dstace codto as defed eq. (17). f ( w ). (17) 0 ew ew ( u0, v0, u1, v1) Accordgly, the Jacoba matrx of eq. (17) s show eq. 18 eq. 1 wth respect to the four ukow vertex postos updatg varables. f (.) 1 w0. ( (w 0 )) 3 ( ) u0 f (.) 1 w0. ( w0 ) 3 ( ) u1 f (.) 1 w0. ( ( 1 w0 )) 3 ( 1 ) v0 (18) (19) (0)

10 f (.) 1 w0. (w 0 1 ) 3 ( 1 ) v1 (1) 5. Vertex to tragle Dstace Smlar to the edge-to-edge dstace codto, whe so that update the vertex postos by small vector value updated vertex -to - tragle dstace s gve eq. (). Where, d ew f, we eed to f. Thus, the ( w) ew. ew () ew (w) = (u - v ) + ( - ) ew 0 0 u0 v0 ( ) = (v - v ) + ( - ) 1 ew 1 0 v1 v0 ( ) = (v - v ) + ( - ) ew 0 v v0 ew = ( 1) ew ( ) ew Accordg to eq. (), there are also four ukow vectors, ( u0, v0, v 1, v), to update the vector postos. The Jacoba matrx of the vertex to tragle dstace fucto s derved as show eq. 3 - eq.6. f(.) v0 u0 f (.) 1 w. ( (w ) ( 1 w0 )) (( 3 ) ( 1 )) f (.) 1 w. ( w) ( 3 ) v1 f (.) 1 w. (w 1) ( 3 1) v (3) (4) (5) (6) 6. EPERIMENTS AND RESULTS We expermeted our system o Itel Core7CPU 3.07GHz ad 1GB RAM platform. We developed a C++ program for the mplemetato of our algorthm. Halfedge data structure ad Wave frot.obj fle format are used to represet mesh model. We corporated optmal best practces of Newto method ad SVD preseted (Press 007). Heurstcally, we defed the clearace threshold value betwee mesh objects as Models are created usg Bleder software. Accordgly, we evaluated our algorthm wth respect to 3-D tragular mesh compresso. We used Ope 3D Graphcs Compresso (O3DGC) tool to ecode/decode our expermetal models. The three scearos of our expermetal results are preseted as follows. The box model expermet s show Fg. 4. It has 636 vertces ad 10 tragular faces. The put model to our system s dcated Fg 4(a). As show Fg.

11 4(b), whe the orgal model compressed ad decoded, artfacts occurred due to a self-tersecto. However, after the mplemetato of our algorthm, the artfacts are clearly removed as dcated Fg. 4(c). We used dfferet colors to dfferetate the teror ad exteror faces so that we ca see easly the artfacts f ay. (a) Orgal (b) Decoded (c) Updated the decoded Fg. 4 Box Model The car-wall collso expermet s dcated Fg 5. The wall model has 88 vertces ad 484 tragles. The car model has 37,074 vertces ad 7,368 tragles. We mplemeted the 3-D compresso o the wall model so that t wll geerate artfacts due to self-tersecto. The, we smulated a collso betwee a car ad the wall usg Bullet Physcs to vestgate the uderlg results. As show Fg 5(b), after compresso ad decode of the wall, artfacts were geerated. Ths created a wrog collso result betwee the car ad wall as show Fg. 5(c) due to the selftersecto error, whch affected the ormal drecto of the uderlg faces of the wall. The mplemetato of our algorthm showed the correct smulato result of the car to wall collso as show Fg. 5(d). (a) Orgal (b) Wall Decoded (c) Car-wall collso smulato after wall decoded (d) Car-wall collso smulato after wall update the decoded Fg. 5 Smulato of Car to wall Collso

12 The psto smulato expermet s preseted Fg. 6. The psto model has 960 vertces ad 1856 tragular faces. As dcated Fg 6(b), after the orgal psto was compressed ad recostructed, the cylder of the psto created some artfacts. I addto, the psto was ot well smulated, as the psto dd ot get to the cylder because of a small sze chage created a collso betwee the psto ad cylder. After the update of the model by our algorthm, the psto s properly smulated as show Fg. 6(c). (a) Orgal (b) Decoded (c ) Decoded the Updated Fg. 6 Psto Smulato 7. CONCLUSION We preseted a algorthm that avods possble self-tersectos of 3-D mesh models due to operatos lke compresso. Our developed system s based o umercal methods partcularly o-lear root fdg system. We have foud expected results from our expermets. The performace of the system depeds more o the umber of o-lear equatos tha the mesh sze. I other words, the more usafe modelg codtos or whe we have more umber of ( f ), the performace

13 degrades. We adopted effcet data structure ad optmzed umercal methods for our expermetato. We have tested the system o 3-D tragular mesh model. For others geometrc models, further vestgato ad aalyss may be requred. ACKNOWLEDGMENT Ths work s supported by the Basc Scece Research Program through the Natoal Research Foudato of Korea fuded by the Mstry of Scece, ICT & Future Plag (NRF-013R1A1A011733). REFERENCES Abderrahm, Z., Tech, E., ad Bouhlel, M. S. (013), Progressve compresso of 3D objects wth a adaptve quatzato, Computatoal Geometry, Vol.10 (-1), Botsch, M., Pauly, M., Rossl, C., Bscho, S., ad Kobbelt, L. (006), Geometrc modelg based o tragle meshes", Proceedgs of ACM SIGGRAPH 006, New York. Burde, R. L. ad Fars, J. D. (011), Numercal Aalyss, 9ed, Brooks/Cole, MA. Campe, M. ad Kobbelt, L. (010), Exact ad robust (self-) Itersectos for polygoal meshes, Computer Graphcs Forum, Vol. 9(), Eberly, D.H. (006), 3D Game Ege Desg, ed, CRC Press, NW. Gottschalk, S. ad L, M. C. (1998), Collso detecto betwee geometrc models: A survey", Proceedg of IMA Coferece o the Mathematcs of Surfaces. Ju, T. (009), Fxg geometrc errors o polygoal models: A survey", Joural of Computer Scece ad Techology, Vol 4(1), Jug, W., Sh, H., Cho, ad Byoug, K. (004), Self-tersecto removal tragular mesh offsettg, Computer-Aded Desg ad Applcatos, Vol. 1(1-4), Ladlaw,D. H., Trumbore, W. B., ad Hughes, J. F. (1986), Costructve sold geometry for polyhedral objects, Proceedgs of SIGGRAPH 86, New York. Möller, T. (1997), A fast tragle-tragle tersecto test, Joural of Graphcs Tools, vol., Ortega, J.M. ad Rhebolt, W.C. (1970), Iteratve Solutos of Nolear Equatos for Several Varables, Academc Press, FL. Park, S. C. (004), Tragular mesh tersecto, The Vsual Computer, Vol. 0(7), Paya, F. ad Ato, M. (005), A effcet bt allocato for compressg ormal mesh wth a error-drve quatzato, Computer Aded Geometrc Desg, Vol., Press, W.H., Teukolsky, S.A., Vetterlg, W.T. ad Flaery, B.P (007), Numercal Recpes: The Art of Scetfc Computg, 3ed, Cambrdge Uversty Press. Sabharwal, C.L., Leopold, J.L. ad McGeeha, D. (013), Tragle-tragle tersecto determato ad classfcato to support qualtatve spatal reasog, Polbts, Vol 48, 13-.

14 Shrley, P., ad Marscher, S. (009), Fudametals of Computer Graphcs, 3ed, CRC, A.K Peters, MA. Skala V. (013), Robust barycetrc coordates computato of the closest pot to a hyperplae E, Proceedg of Iteratoal Coferece o Appled Mathematcs ad Computatoal Methods Egeerg. Yamakawa, S. ad Shmada, K. (009), Removg self-tersectos of a tragular mesh by edge swappg, edge hammerg, ad face lftg, Proceedgs of the 18th Iteratoal Meshg Roudtable, Salt Lake Cty, UT.