Spiraling Edge: Fast Surface Reconstruction from Partially Organized Sample Points

Transcription

1 Sialing Edge: Fa Suface Reconucion fom Paially Oganized Samle Poin ABSTRACT Paicia Cono Sandia Naional Laboaoie Many alicaion oduce hee-dimenional oin ha mu be fuhe oceed o geneae a uface. Suface econucion algoihm ha a wih a e of unoganized oin ae exemely ime-conuming. Someime, howeve, oin ae geneaed uch ha hee i addiional infomaion available o he econucion algoihm. We een Sialing Edge, a ecialized algoihm fo uface econucion ha i hee ode of magniude fae han algoihm fo he geneal cae. In addiion o amle oin locaion, ou algoihm a wih nomal infomaion and knowl of each oin. Ou algoihm oduce a localized aoximaion o he uface by ceaing a a-haed iangulaion beween a oin and a ube of i neae. Thi uface ach i exended by locally iangulaing each of he oin along he of he ach. A each oin i iangulaed, i i emoved fom he and new oin along he ach ae ineed in i lace. The udaed ial ou ove he uface unil he encoune a uface bounday and o gowing in ha diecion, o unil he educe o a mall hole ha i filled by he final iangle. CR Caegoie: I.3.5 [Comuing Mehodologie]: Comue Gahic - Comuaional Geomey and Objec Modeling. Keywod: Suface econucion, advancing fon, iangulaion. 1 INTRODUCTION In ecen yea, he econucion of uface fom e of unoganized amle oin in R 3 ha been an acive aea of eeach [1][3][5][6]. The ublihed eul of hi eeach focu on olving he geneal cae whee no addiional infomaion beyond he locaion of he uface oin i known. Time eoed fo conucing uface wih weny houand oin ange beween 10 and 20 minue of CPU ime, deending on he mehod ued [1][3][5][6]. Alhough he mehod develoed fo he geneal cae can be ued o conuc uface fom amle oin whee ome oganizaion aleady exi, he ime euied o conuc a uface of ignifican ize i unacceably long. We wee able o ue he nomal and infomaion ha we aleady had fo each oin o ignificanly eedu he oce. We have called hi he Sialing Edge (SE) algoihm. The SE algoihm wa develoed o geneae a uface fom aicle oiion afe he aicle have finihed diibuing hemelve ove an iouface in a volume daa e [2]. In addiion o he aicle o oin locaion, he algoihm ue a uface nomal a each oin, a oin ye deignaion, and fo each oin a li of neighbo oin odeed by diance. Given hee inu, he SE algoihm geneae a iangulaion ha aoximae he uface much moe aidly han eviou geneal cae mehod. Ou algoihm euie only 2.3 econd o econuc a uface fom a e of ove foy houand oin. In addiion, he SE Algoihm doe no aume ha he oin fom a ingle conneced uface. 2 RELATED WORK Edwad Angel Univeiy of New Mexico In 1992, Hoe e al. eened a uface econucion algoihm ha a wih a e of unoganized oin, and geneae a iangulaion ha aoximae he unknown undelying uface [5]. The algoihm doe no exloi any ouide infomaion abou he naue of he uface. Timing on a 20 MIPS wokaion fo daa e anging fom 1000 o 21,740 oin wee beween 19 and 2,135 econd. Edelbunne and Mücke ublihed hei VKDSHV wok in 1994 [3]. Fo a oin e PÃLQÃWKUHHGLPHQVLRQDOÃVSDFHÃ VKDSHV ae a family of hae ha ae a genealizaion of he convex hull. 6HWWLQJÃ WKHÃ YDOXHÃ RIÃ Ã EHWZHHQÃ LQILQLW\Ã DQGÃ ]HURÃ JHQHUDWHV GLIIHUHQWÃ VKDSHVÃ Ã :KHQÃ Ã LVÃ VHWÃ WRÃ LQILQLW\Ã WKHÃ VKDSHÃ LVÃ WKH convex hull of PÃ Ã $VÃ Ã GHFUHDVHVÃ WKHÃ VKDSHÃ VKULQNVÃ FDYLWLHV DQGÃ KROHVÃ PD\Ã DSSHDUÃ 7KHÃ DOJRULWKPÃ IRUÃ FRQVWUXFWLQJÃ VKDSHV euie finding he hee-dimenional Delaunay iangulaion of he oin e. Edelbunne and Mücke genealized a wodimenional -fliing algoihm o hee dimenion. In hee dimenion, he Delaunay iangulaion i acually a eahedalizaion of he convex hull of he oin. The addiional FRQVWUDLQWÃ RIÃ WKHÃ VKDSHÃ VSKHUHÃ WKHQÃ UHPRYHVÃ DÃ VXEVHWÃ RIÃ WKH eahedal face, o iangle, fom he Delaunay iangulaion. The iming of hei incemenal-fli algoihm on a 50 MHz MIPS R4000 anged beween 8.97 and 2, econd fo oin e anging in ize fom 318 o 15,000 oin. 7KHUHÃDUHÃVHYHUDOÃSUREOHPVÃZLWKÃXVLQJÃ econucion. VKDSHVÃWRÃGRÃVXUIDFH In objec wih hole, i euie ome H[SHULPHQWDWLRQÃWRÃILQGÃDQÃ YDOXHÃWKDWÃSURGXFHVÃWKHÃ DSSURSULDWH VXUIDFHÃ Ã,QÃ VRPHÃ FDVHVÃ WKHUHÃ LVÃ QRÃ YDOXHÃ WKDWÃ SURGXFHVÃ WKH deied uface. Eihe he uface i webbed in aea whee hee hould no be a uface, o hole ae beginning o aea in wha hould be a olid uface. Teichmann and Ca ovecame hee OLPLWDWLRQVÃ RIÃ WKHÃ VKDSHVÃ DOJRULWKPÃ LQÃ Ã ZLWKÃ DQLVRWURSLF deniy-caling [6]. Howeve, hei algoihm alo euie inenive calculaion o comue he hee-dimenional Delaunay iangulaion of he oin e. In 1998, Amena, Ben, and Kamvyeli eened a uface econucion algoihm baed on he hee-dimenional Voonoi diagam and Delaunay iangulaion [1]. The algoihm inu unoganized amle oin and ouu a e of iangle, which hey call he cu of he amle oin. The veice of he cu iangle ae all amle oin, o he algoihm ineolae he amle oin inead of aoximaing hem a Hoe algoihm doe. The cu connec only adjacen amle oin on he uface, auming ha he uface i mooh and ufficienly amled. Amena, Ben, and Kamvyeli ued an SGI Onyx wih 512 megabye of memoy o e he unning ime of hei algoihm again eveal daa e. The daa e anged beween 939 and 35,947 oin, while he unning ime wen fom 2 o 23 minue. The amoun of ime euied o do a uface econucion i in he 15-minue ange fo an examle wih 20,021 oin. Thei ae ae ha he unning ime i dominaed by he ime needed o comue he Delaunay iangulaion, which hey ue o geneae he Voonoi veice of he oin e.

2 3 OVERVIEW The aing oin fo he SE algoihm i a e of oin geneaed by an iouface algoihm. A a of he oce of obaining hee oin, infomaion in addiion o hei locaion alo i geneaed. Fo he aicle yem aoach o iouface, hi exa infomaion coni of an eimaed nomal a each oin, a li of each oin, and a ye claificaion fo each oin. A oin ae found by examining a heedimenional hee of influence aound he oin. Only hoe oin whoe nomal diffe fom he nomal of he cenal oin by an angle of le han ixy degee ae included he li. The ize of he hee of influence i baed on he local uface cuvaue a ha oin. Thi wok ha aleady been done by he aicle yem. Tye i baed on a oin oiion elaive o he boundaie of he volume daa e fom which he iouface i aken. Ineio oin ae no ouching he volume boundaie. Bounday oin lie on he volume boundaie. And cone oin lie along he ineecion beween mulile volume boundaie. The ye claificaion enable he algoihm o geneae uface whee he daa e boundaie cu he uface. Thi emi he ceaion of oen uface. We wok locally o ceae a nealy lana iangulaion beween a oin and i neae. Uing a oin a he cene of a of i, uch a oin 1 in Figue 1, we ak he following ueion. Which fom a aound he cenal oin uch ha he euao of hee aken hough each ai of adjacen on he and he cenal oin ae emy? Thoe ha fail he emy cicumhee e, uch a 8, 9, 10, 11, 14 and 18, ae emoved fom he. The emaining ae iangulaed along wih he cene oin. The iangulaion coni of lacing beween he cene oin and each of he emaining of, a well a lacing beween each of he adjacen on he. Thi oce ceae a wheel-like iangulaion wih he cene oin a he ivo and he of along he im Figue 1: Cenal oin wih of neighbo oin. On a moe global cale, ou aoach i o iniialize an wih a oin and i neae. Afe he algoihm iangulae a of oin aound he fi oin, i emove he encicled oin fom he and add he new oin ued in he iangulaion o he in i lace. The aveal oceed in couneclockwie ode aound he nomal of he fi oin, hu oviding an oienaion fo he iangulaion ha i conien aco he uface. The algoihm hen ial aound he, iangulaing each of he oin along he in un. The algoihm aume ha exce a oin wih bounday ye deignaion he uface i cloed and hee ae no nonmanifold ucue. The algoihm alo aume ha hee i only one oin a any aicula locaion. The algoihm gow a uface fom a ingle oin unil one of wo hing haen. Eihe he uface encoune oin ha ae uface boundaie and o gowing in ha diecion o he uface fill unil he educe o a mall hole ha fill ielf in. Coneuenly, bounday oin euie ecial eamen. Each ime he become emy, a diconneced uface comonen ha finihed iangulaion. If all of he oin have been ued, iangulaion i comlee. Ohewie, he algoihm einiialize he wih a new oin and i neae and coninue. Hee i a eudo-code deciion of he main e in he SE algoihm: Clea All Poin Sau Flag o UNUSED Reea unil All Poin ae USED Iniialize Edge Ring While Edge Ring No Emy If Poin on Bounday Tiangulae Bounday Poin Ele Tiangulae Ineio Poin 4 INITIALIZATION The algoihm exec oin o be claified a ineio, bounday o cone ye. The bounday oin ae fuhe ubdivided ino one (o moe) of ix diffeen ubye, oiive x, oiive y, oiive z, negaive x, negaive y, and negaive z, ome examle of which ae hown in Figue 17. We eeen each of he bounday ye by eing a uniue bi wihin he oin ye field. Any oin ha fall ino wo o moe of hee ubye alo e he cone bi o eed u ye checking oeaion. Ineio oin ae hoe wih no bi e. Each oin ha a au flag ha ha hee ae: UNUSED, USED, and DONE. Befoe iangulaion he flag ae all cleaed o UNUSED. Once a oin ha been ued in a iangulaion, i au flag i e o USED, which coeond o a oin being added o he. Becaue hee ae cicumance whee he can include a oin moe han once (moe will be aid abou hi cae lae), he au flag alo ac a an inance coune o ha any value of he flag geae han 0 ignal he USED ae. When a oin i emoved fom he, i au flag i decemened. If he au flag i eual o 1 io o a decemen, eeening a ingle eny in he, he au flag i e o DONE inead of being decemened (he value of he DONE flag i 1). A DONE au indicae ha a oin ha been eniely uounded by a of iangle. Coneuenly, even if a DONE oin ha link o oin ha ae no DONE, i canno be a candidae in hei iangulaion. Thi iuaion can occu when link connec oin ha do no hae a Voonoi face o. A he flag ae being cleaed a au, a oine i iniialized o he fi oin. Then each ime he i einiialized afe a uface ecion ha been comleed, hi oine i udaed o oin o he cuen aing oin. In ha way, lae eache fo a new aing oin will no have o eeaedly examine DONE oin. Once he oine advance a he la oin, he algoihm know ha all oin mu be DONE becaue i call hi ouine only when he i emy. The i acually a doubly linked of elemen conaining oine o he oin. The ae eeened imlicily by he link beween he oin in he ; coneuenly he need o have a lea wo oin. To

3 iniialize he, he algoihm ine he fi oin ha i find ha i no e o DONE, and he oin neae ha i no DONE. The algoihm begin he aveal wih he fi oin. Someime he algoihm encoune oin ha eihe do no have any o whoe ae all DONE. In he fi cae, hee oin eeen uch a iny uface aea ha only a ingle oin will fi in ha ace. We view hee oin a noie and eliminae hem. In he econd cae, he algoihm ha ejeced he oin in he iangulaion of all i and i ha been lef anded. Alhough we efe o ue all of he oin in eeening he uface, he co of locaing he einen meh egion and e-iangulaing ha ecion o include he oin ouweigh he benefi of including i. So in boh of hee cae, he algoihm mak hee oin a DONE and i coninue he each fo a oin o iniialize he. 5 TRAVERSING THE EDGE RING Figue 2: Sialing iangulaion of oin on he. Uing Figue 2, we will e hough eveal ieaion of how he i maniulaed when i coni excluively of ineio oin. The nomal of he oin ae aumed o be oining ou of he age. The i iniially ju oin 1 and i neae, oin 2, wih oin 1 a he fi oin o be iangulaed. Afe he algoihm iangulae he of oin aound oin 1, i emove oin 1 fom he and add oin 3 hough 7 o he. The algoihm hen advance o he nex oin on he, oin 2, and iangulae he ecion of oin 2 of ha ha no ye been iangulaed a a of oin 1 iangulaion. Thi ub- i hown in dake gay in he figue. The algoihm elace oin 2 on he wih oin 8, 9, and 10. Advancing o oin 3, he algoihm iangulae oin 3 (whoe iangulaion i hown in whie) and elace oin 3 on he wih oin 11. Coninuing in hi manne, he algoihm iangulae oin 4 and 5, leaving an coniing of oin 6 hough 16, wih oin 6 a he nex oin o be iangulaed. Each ucceive iangulaion e i hown a a diffeen colo. Noe ha ohe han he oiginal of iangle aound oin 1, all of he ucceeding iangulaion oduce a fan of iangle ha begin and end wih veice on he. 6 TRIANGULATING INTERIOR POINTS The SE algoihm a wih hee oin: he cene o ivo oin, he oin ha ecede he ivo oin on he (efeed o a he fi oin), and he oin ha ucceed he ivo oin on he (efeed o a he la oin). When he only coni of wo oin, a i doe iniially, he fi and la oin ae he ame Ceaing he Neighbo Ring The nomalized veco beween he ivo oin,, and he fi oin i a efeence veco whee he angle i zeo. We efe o hi efeence veco a z. The algoihm o each of he neighbo oin by angle ode aound. If wo neighbo oin ae found o have he ame angle, wihin ome eilon, he oin cloe o i included in he and he ohe oin i eliminaed fom conideaion. To calculae he angle, he algoihm doe he following. Fi, i ake he do oduc of he veco z and he nomalized veco beween he oin and a neighbo oin, which we will call v, o obain he coine of he angle beween hem. Becaue he coine i ymmeic abou, he algoihm need ome way o diffeeniae which ide of he coine lie in. Hence, we conuc a lane hough he ivo oin and he fi oin by aking he co oduc of z and nomal, oducing he nomal of he lane licing he a 180 degee. Thi veco i called m. The do oduc of m and v i comaed o zeo. If m v 0 ÃWKHQà ÃLVÃVHWÃWRÃWKHÃDUFFRVLQHÃRIà z v à RWKHUZLVHà ÃLV e o 2 minu he accoine of z v. The algoihm ue he angle beween z and he veco beween and he la oin, la, a he maximum allowable angle. When he fi and la oin ae he ame, he ending angle i 2. The ending angle even including neighbo oin in he ha would euie he iangulaion o ovela egion ha ae aleady iangulaed. In Figue 3, oin and ae linked a, bu becaue he angle fomed by he veco fom o and z i geae han he angle beween la and z, i no included in he. la fi z Figue 3: Uing max angle o emove oin fom. end la new end a fi z new a Figue 4: Changing he a and end oin of he. Thee ae iuaion ha aie ha euie he algoihm o change he aing and/o ending angle ha i eablihed uing fi and la. An examle whee boh mu be changed i eened in Figue 4. A in he eviou dicuion, he fo i iniialized o conain only he wo oin fi and la, which ae he wo adjacen oin on he. The algoihm kee oine o he oin whoe angle eeen he aing and ending angle wihin he. We call hee

4 oine a and end and hey iniially oin o fi and la. The algoihm hen look a he wo oin along he o eihe ide of fi and la, and, eecively, o deemine if he i convex o concave a hoe wo oin. In hi cae boh oin ae concave. If a he algoihm add in, i find ha i aleady on he, i adjacen o he oin oined o by a, and he i concave a hi oin. The algoihm hen advance he a oine o oin o. The algoihm kee z a he efeence veco, bu now he beginning angle i geae han zeo. Similaly, he algoihm move he end oine when i add o he. Lae, when he algoihm conide adding and o he, hey ae excluded becaue hey fall ouide he angle limi fomed by he new a and end oin. If and had aleady been included in he, he algoihm would emove hem along wih any ohe oin in he beween a and. In oing he of oin, we have made he aumion ha hey ae in a nealy lana egion. Howeve, hi aumion i no alway ue. We ue he convex labeling of he a and end oin o eliminae oin fom he which have aed he angle e, bu which have been oed incoecly due o he hee-dimenional dioion in a highly cuved egion. a end 4 wih a USED au ae given an addiional check befoe being added o he. In he examle hown in Figue 6, he oin e i eaed a he ivo oin and he angle fomed by he eceding and ucceeding oin on he li (ela and efi) ae calculaed. Then he angle of wih eec o hee beginning and ending angle fo e i found. If could be included in e hyoheical, hen e i included in. In hi cae, could no be included, o e i eliminaed fom conideaion fo. 6.2 Filling Hole Once all of he neighbo oin have been evaluaed fo incluion in he, he algoihm mu evaluae he oin i ha choen fo oible emoval. Since a hole could have been ceaed by elf-ineecion of he du iangulaion of an ealie ivo oin, he algoihm check fo he ecial cae of filling a hole fomed by, fi, and la. Once he algoihm ha ued he e aleady decibed o exclude unwaned fom he, hee ae wo oible ye of emaining. Thee ae UNUSED oin inide he hole and UNUSED oin ouide he hole. la fi ULQJ ULQJ Figue 5: Convexiy e needed o eolve angle o failue. An examle whee he angle o fail i hown in Figue 5. The i hown a a doed line ha ovela he beween 1 and 2, and 3 and 4. The incoecly include oin 2, which ha been dawn lighly above oin 1 fo emhai, bu which i eally level wih and oin 1 and fuhe back ino he age. Poin 3 i alo fuhe back ino he age, while oin 4 i cloe o he viewe. The convexiy e eliminae oin 2 fom he becaue oin 2 i he nex oin on he li afe he oin oined o by a (oin 1) and 1 i convex. efi e ela la fi Figue 6: Ovela due o failue of angle e o exclude oin e. The convexiy e only alie o oin in he ha ae adjacen o he oin oined o by a o end ince beyond hi oin on he could be on he ohe ide of a hole being filled and could be legiimaely included. On he ohe hand, hee i he oibiliy of ovelaing iangle caued by he incluion of oin on he ha ae in a configuaion uch a ha in Figue 6. Fo oin, oin e ae he angle e fo he ange of angle beween fi and la. Howeve, if e i included in, he euling iangulaion will ovela he egion aleady iangulaed. Coneuenly, oin Figue 7: Eliminaing ouide he hole. An examle of hi configuaion i hown in Figue 7 whee he coni of fi,,, and la. The goal i o kee and emove. The algoihm conuc a lane by aking he co oduc of he veco beween fi and la and he aveage nomal of, fi and la. Only hoe oin ha ae on he ame ide of he lane a ae ke in he. The ohe oin, uch a, ae emoved fom he. Noe ha he algoihm will no ecognize a hole fomed by moe han hee oin. If he examle in Figue 7 conained an addiional oin on he li foming he hole, oin would be ke in he and oduce an ovela in he iangulaion. Alhough check fo lage hole could be made, i would be a cae of diminihing eun. Since he diance beween and any of i i limied, he likelihood of being linked o oin on he ohe ide of a iangulaed i deceae wih he ize of hole. Howeve, hee ae ill cicumance whee ovela occu. 6.3 Removing Neighbo fom he Ring The algoihm now evaluae each oin in he fo oible emoval. The algoihm calculae he cene of he cicumcicle aing hough he ivo oin and he oin on eihe ide of he oin being evaluaed uing a echniue decibed in Gahic Gem IV [4]. Figue 8 een an examle evaluaing oin fo emoval fom he. A cicumcicle i fomed uing oin,, and. The algoihm ea he cicumcicle a hough i wee a he euao of a coeonding cicumhee. The cene of he cicumcicle can be ued a he cene of he cicumhee. The algoihm find he diance d in hee-dimenion beween he cene and, and

5 comae hi o he adiu of he cicumcicle. If fall inide he cicumhee o i emoval would ceae an angle diffeence beween he veco fom o and o of moe han 120 degee, mu be ke. Ohewie, i emoved fom he. In hi cae, i emoved and he evaluaion move on o oin. We aived a 120 degee a he cu off angle by exeimenaion. cicumcicle cene d Figue 8: Evaluaing oin fo emoval. 6.4 Tiangulaing he Ring la fi Ã!Ã degee ovelaing iangle Figue 9: Teing fo an ovelaing iangle veu a hole. Once he enie ha been evaluaed, one final check mu be made befoe iangulaing he emaining oin. A i hown in Figue 9, if he coni of only wo oin and he angle hey fom aound he ivo oin i geae han 180 degee, he euling iangle will ovela exiing iangle. In hi cae, he will be dicaded and he ivo oin will advance o he nex oin on he. Evenually, will be iangulaed indiecly by iangulaing he oin adjacen o i on he. The occuence of hi configuaion indicae ha i inufficienly linked o i. Thi iuaion can aie when he neighbo oin lie ju ouide he each ange fo. If he angle beween fi and la i le han 180 degee, a wo-oin eeen a hole ha i being filled and iangulaion may oceed. la o fi, fi o, and o, in ha ode. Aow demonae he ode of he veice in he iangle. Afe he iangle i added o he ouu li he iangulaion advance o and, hen and, and finally and la. Once a iangle i added o he ouu li i may no be emoved. 6.5 Udaing he Edge Ring Once he iangulaion i comlee, he ivo oin i emoved fom hi ecion of he. If he ivo oin only aeaed in he once (indicaed by a au of 1, ince he au double a an inance coune), he oin au i e o DONE. Ohewie, he au of he ivo oin i decemened. acive acked la fi end a Figue 11: Poin emoved fom while iangulaing. If fi and la ae no longe he a and end oin in he, hen all of he oin befoe a and afe end in he ha ae dulicaed in he need o be emoved fom hi ecion of he. If hi lace i he only one whee a dulicaed oin aea in he, a in he examle of la in Figue 11, hen he oin au field i e o DONE. Noice la i now uounded by he iangulaed uface, o i canno be ued again. Ohewie, a in he cae of fi in Figue 11, he dulicaed oin au i decemened. Once he dulicae have been emoved fom he, he emainde of he oin in he, excluding he oin oined o by a and end, ae ineed in couneclockwie ode ino he. In Figue 11, he emainde of he coni of he wo oin and. A each of hee oin i ineed ino he, i i maked a USED. If a oin i encouneed ha i aleady USED, he fi ohe inance of he oin in he i locaed. Deending on he locaion of he ohe inance, one of wo hing will haen. If he ohe inance i in he acive, he i li ino wo and one of he i uhed ono a ack. In Figue 11, du an ealie oin in he iangulaion fi wa ued fo a econd ime by a oin above fi and he wa li. fi Figue 10: Ceaing a iangle uing, fi, and. Tiangulaion coni of aveing he fom fi o la uing wo adjacen oin on he a a ime. The ae ceaed in a couneclockwie diecion wih eec o he uface nomal. In he examle in Figue 10, he uface nomal i oining ou of he age. Edge ae dawn fom Figue 12: Rejoining he. If he ohe inance i no found in he acive, he ack of i eached in o down ode o ha he mo ecenly li will be found fi. Once he ohe inance of he oin i found in a acked, he wo ae joined ino a ingle wih he haed

6 oin aea wice in he ame. The iangulaion of he in he hydogen daa e, ee Figue 21, oduced hi iuaion a he wa ejoined when he iangulaion finihed he ucue, a i hown in Figue 12. Once he i finihed being udaed, he new oin i he oin ha had been he end aicle in he. 7 TRIANGULATING BOUNDARIES Bounday oin een a oblem wih eec o he aoach eened above. A can be een in Figue 13, he fo oin will coni of wo dijoin ecion on eihe ide of. Even if he algoihm iangulae in a, i canno emove fom he. If he algoihm doe emove, he emoval would eihe fagmen he o caue nonadjacen oin o be adjacen in he (and hence o be viewed a by he algoihm which lead o ovelaing iangle). fi la bounday aicle uface bounday Figue 13: Edge ineec uface bounday a. Ou oluion ha wo a. Fi, we avoid uing bounday oin a ivo oin wih only a few exceion. Thee exceion include he beginning of a new whee boh oin ae bounday oin, and when he algoihm i iangulaing he aea aound uface bounday cone. Ohewie, he algoihm imly advance o he nex oin in he. In he examle in Figue 13, he algoihm can iangulae he o he igh of uing la a he ivo oin. A he algoihm cicle aound he back owad, i can iangulae he aea o he lef of uing fi a he ivo oin. Secondly, he algoihm leave all he bounday oin in he unil i make a comlee cicui of he wihou any new iangle being added. Then he algoihm mak he bounday oin a DONE and fee he o ha i can a a new diconneced uface comonen. 7.1 Edge Ring I Two Bounday Poin When a new coni of only wo bounday oin, he algoihm mu ue a bounday oin a a ivo oin. If we wee o ue ou aoach wihou modificaion, he algoihm would ceae eihe a zeo aea iangle o a iangle ouide he uface when i connec he wo bounday oin on eihe ide of he ivo oin. An examle of hi i hown in Figue 14, whee and fi ae he only wo oin on he. uface bounday incoec iangle = fi = la Figue 14: Unmodified oduce incoec iangle. Howeve, if he algoihm add o he and ue i in lace of fi, keeing he ohe bounday oin a la, hen he aoach can be ued wihou modificaion. So io o calling he ouine, he algoihm eache fo a hid oin ha i UNUSED and ha a bounday in common wih. Alhough ineio would alo avoid he eo, he algoihm migh elec an ineio oin ha would be culled fom he iangulaion, wheea he bounday oin would no be culled. By no eui he new oin o have a ye idenical o, he algoihm kee fom eliminaing cone oin. =la =fi =la =fi Figue 15: Ineing he hid oin in he. Once he algoihm ha found he hid oin,, i mu decide whehe o ine befoe o afe in he. The lacemen of he hid oin in he deemine whehe he hid oin i ued a fi o la in he, a i hown in Figue 15. In he examle in Figue 14, he algoihm need o ine afe o ha he ange of allowable angle will include oin inide he volume bounday. The deciion deend on which bounday i involved and whee he hid oin i locaed along he bounday elaive o. negaive x bounday Z Y ye i oiive x N oiive x bounday co oduc volume domain Figue 16: Deciding whehe o ine he hid oin befoe o afe in he. The algoihm ake he co oduc of he nomal a and a veco beween and he econd oin in he. The algoihm hen comae he diecion of he euling veco wih he ye of. The hid oin i ineed befoe in he if one of wo condiion exi. Eihe lie on a negaive bounday and he coeonding comonen of he co oduc veco i negaive, o lie on a oiive bounday and he coeonding comonen of he co oduc veco i oiive. Ohewie, he hid oin i ineed afe in he. Figue 16 illuae he ue of he co oduc fo he configuaion eviouly given in Figue Volume Bounday Cone The ohe exceion o he olicy of no uing bounday oin a ivo, i when he ha adjacen oin whoe ye indicae a aniion fom one volume bounday o anohe. In Figue 17, he oin in he aniion fom he oiive x bounday o he oiive y bounday a oin. Poin i of ye oiive x, i a oiive y oin, and oin i boh of hee ye, o i i alo a cone ye. Even if oin wee no in X

7 he, he aniion in ye beween and would indicae he aniion beween volume boundaie. Thee ae hee cae eui ecial eamen: i ene a cone, i exiing a cone, and i on he cone. Examle of hee ae hown in Figue 18, Figue 19, and Figue 20, eecively. In each cae, due o he configuaion of bounday and cone oin, he oin would neve be included in he iangulaion due o he lack of neighbo ineio oin. Theefoe, he algoihm canno advance o he nex oin in he wihou fi iangulaing uing a he ivo oin. The algoihm ue he ame aoach a befoe, ineing anohe UNUSED bounday oin ino he eihe befoe o afe, hen iangulaing uing he egula aoach. In all of hee cae, he UNUSED oin ineed ino he i. negaive x bounday Z Y oiive y bounday oiive x bounday volume domain Figue 17: Edge aniion beween X and Y boundaie. ene cone link negaive x ye negaive y ye X negaive x & negaive y & cone ye Figue 18: The ivo oin i ene a cone. exiing cone link oiive y ye oiive y & oiive z & cone ye oiive x & oiive z & cone ye oiive x & oiive y & cone ye Figue 19: The ivo oin i exiing a cone. link oiive y ye i on a cone Figue 20: The ivo oin i on a cone. oiive y & oiive z & cone ye oiive x & oiive z & cone ye oiive x & oiive y & cone ye 8 RESULTS AND FUTURE WORK We geneaed oin e uing a aicle yem ha ioufaced vaiou volume daa e. The iming fo he SE algoihm o geneae iangulaion of hee oin e ae eened in Table 1. The hadwae lafom wa an SGI High Imac Indigo 2 unning IRIX 6.3 on an R4400 wih 192 megabye of memoy. The numbe of oin inu and he numbe of iangle geneaed ovide a meaue of he oblem ize in each cae. Rende of he iangulaed oin e ae hown in Figue Poin Se # Poin # Tiangle Time (ec) Lobe Elecon Cloud Bla Wave Hydogen Hyeboloid Table 1: Timing fo SE algoihm on vaiou oin e. In fuue wok, we would like o evaluae he addiional ime needed o ynheize vaiou inu ha he SE algoihm cuenly euie. Fo inance, Hoe geneae nomal infomaion a a of hi algoihm and comue he k-neae fo each oin. Could we find alenae way o ceae hee inu ha would mainain ou eed advanage? Alo, once we have he nomal and neae infomaion, can we deemine which oin ae ineio and which oin ae bounday oin wihou doing exenive oeaion like a Delaunay iangulaion? I would be ueful o develo echniue o fill in whicheve iece ae miing in he inu and o uanify he co of uing hem. 9 ACKNOWLEDGEMENTS The DOE Mahemaic, Infomaion, and Comue Science Office funded hi eeach. The wok wa efomed a Sandia Naional Laboaoie. Sandia i a muliogam laboaoy oeaed by Sandia Cooaion, a Lockheed Main Comany, fo he Unied Sae Deamen of Enegy unde Conac DE- AC04-94AL REFERENCES [1] Amena, N., M. Ben, and M. Kamvyeli. A New Voonoi-Baed Suface Reconucion Algoihm. In SIGGRAPH 98 Confeence Poceeding, Annual Confeence Seie, age Addion Weley, July [2] Cono, P. and E. Angel. Iouface Exacion Uing Paicle Syem. In Poceeding of Viualizaion 97, age IEEE, Ocobe [3] Edelbunne, H. and E. Mucke. Thee-Dimenional Alha Shae. ACM Tanacion on Gahic, 13 (1): 43-72, Januay, [4] Hill, F. The Pleaue of Pe Do Poduc. Gahic Gem IV, age AP Pofeional, Boon, [5] Hoe, H., T. DeRoe, T. Ducham, J. McDonald, and W. Suezle. Suface Reconucion fom Unoganized Poin. In Comue Gahic (SIGGRAPH 92 Confeence Poceeding), 26 (2): Addion Weley, July [6] Teichmann, M. and M. Ca. Suface Reconucion wih Aniooic Deniy-Scaled Alha Shae. In Poceeding of Viualizaion 98, age IEEE, Ocobe 1998.

8 Figue 23: Hyeboloid oin e. Figue 21: Hydogen oin e. Figue 24: Bla wave oin e. Figue 22: Lobe oin e. Figue 25: Elecon cloud oin e.