An Estimation of Earthenware s Surface Shape Using Quadric Surfaces

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1 芸術科学会論文誌 Vol. 3 No A Estmto of Erthere s Srfce She Usg Qrc Srfces Tstom Kosht ) (Memer) Ktstsg Mtsm ) (Memer) Koch Koo ) (Memer) ) Lttce Techolog Co.Lt. ) Grte school of Egeerg Ite U. 566@ft.com kmts@te-.c.j koo@eecs.te-.c.j Astrct I most cses erthere s roke he t s ecte from rs. Ths stto reqres restorto for ssemlg the erthere's eces. The ot clos mesre three-mesol ece re sefl to restore the erthere. Some methos to f jcet eces of erthere sg comter he ee roose. These methos m to restore roke erthere coectg jcet eces th sg gtl t. If hoeer erthere th lrge mssg ortos s restore t s mossle to estmte the jcet ece sg locl jcec. De to ths t s ecessr to estmte the hole she of erthere. It s lso ossle to cote seqetl locl estmto for estmtg hole she of erthere t the storto occrs m cses. I ths er e roose metho to estmte the hole she of erthere th qrc srfces. Sce most erthere s roce to e s close s rottol shes qrc srfces re stle to rereset rottol shes. I or metho the ee tter o the srfce s remoe the frst ste ccorg to the T's smoothg metho. Net the ot clo from hch ee tters re remoe s e to seerl gros. After tht ech ot clo s ftte to qrc srfces reresete the lgerc eqto. Fll the seqeces of ots geerte from qrc srfces re terolte the free-form srfces corresog to the srfce of the erthere re geerte.. Itrocto I most cses erthere s roke he t s ecte from rs. Geerll ll rocesses from clssfcto of erthere to sseml restorto re oe mll. Therefore t tkes mch tme to restore lrge mer of eces of roke erthere. The restorto lso reqres techcl kolege eerece restorto ork of relc ecomes he re the rcheologcl fel. Moreoer ml restorto rses the rsk for mgg erthere eces. I recet ers the reserches he ee crre ot to sort erthere restorto ork mesrg erthere th three-mesol mesremet ece coectg erthere eces sg mesre t [-]. Restorto of erthere s clssfe to to tes: oe s to restore ctl erthere the other s to crete relcs. If erthere s so recos to e esgte s mortt cltrl roertes f ther relcs re crete t ll e ossle to eht them t fferet ehto hlls t sme tme. Whe relcs re crete t s lso ecessr to terolte mssg ortos s mch s ossle to restore the correct coto of erthere. For teroltg mssg ortos of erthere sg the mesremet t metho to etrct the ortos from the erherl shes hs ee roose []. I [] hoeer t s ffclt to restore mssg ortos ecse the she se of the mssg ortos he some restrcto. I to f srfces for the mssg ortos re geerte from the erherl srfces the hole she mght get storte. I ths er e reset metho to estmte erthere s srfce she ee f the srfce t s comlete. I or metho the ot clo s mesre three-mesol mesremet ece. After ee tters re remoe from the ot clo [3] t s romte for qrc srfce the ceter s of the srfce s etrcte. After tht to geerte recse srfces the ot clo s e to seerl gros ccorg to the ceter s. The ech gro s ftte to stle qrc srfce. Fll free-form cre mesh s crete from qrc srfces free-form srfce mesh s crete from the free-form cre mesh. Ths the comlete

2 芸術科学会論文誌 Vol. 3 No erthere srfce s reresete free-form srfces. As reslt of eltg the stce etee the mesremet ot clo the geerte free-form srfce goo reslts ere ote. Sce ll rts of the she re efe s qrc srfces the she s ot storte the orml ectors or cttg fces of rt of the erthere srfces c e esl ote. Sch formto c e se ot ol for rerocto of the she t for lses o creto of mesre grms of erthere.. Relte Works. Remol of ee ortos If srfce s geerte from the olgo crete from the ot clos of erthere thot rocessg the ot clo t the qlt of the geerte cre srfce ecomes lo ecse ee tters rem. De to ths the ee tter s regre s ose of erthere or roose metho erforms the smoothg rocess to remoe the ee tter. Geerll the metho of lo-ss flter smoothg s se for ths rocessg. Ths er hoeer ses the T s metho to crete smooth shercl she sce the T's metho se o the Llce trsform forml [3] c e eecte short tme.. Qrc form ellse fttg As the metho of fttg qrc form to ot clo Ftgo et l. [4] resete the metho to ft ellse to to-mesol ot seqece. To e more secfc qrc form s reresete to-mesol cre tht mmes the stce etee the ot seqece the cre sg the Lgrge mltler metho. Ths metho s sole lg the elltcl costrt to the Lgrge mltler metho. Uforttel the rose of Ftgo et l. [4] s to romte ll ots sgle ellse. If the Ftgo s metho s le to or metho comle shes sch s erthere mst e reresete th oe qrc srfce. De to ths e jge tht t s ffclt to l the Ftgo s metho rectl to three-mesol eces of erthere..3 Qrc srfce fttg to ot clo Doros resete the metho [5] hch etes the metho of Ftgo et l [4] to three-meso fts qrc srfces to three-mesol ot clos. The Doros metho efes the qrc form s geerl oe the sme mer s of the Ftgo s metho ot clos re ftte to qrc srfce locll sg the Lgrge mltler metho. To e more secfc qrc srfce s ftte to orto of ot clo. The Doros metho rojects ots oto qrc srfce clcltes the crtre of the rojecto osto s the crtre of the ot. The Doros metho oes ot meto ho to se crtre. He ls to se crtre orer to re 3D mtchg lgorthm. The Doros metho geertes lot of smll qrc srfces oes ot coser the cott of srfces. Moreoer hs metho cot hle the she tht hs mssg rts ecse t s ot ecessr to ko elemets ecet crtre. Therefore hs metho cot e le rectl for fttg srfces from the erthere hose ortos re mssg..4 Clssfcto of srfces Ol fe CAD softre lctos c sort ll ks of qrc srfces s the she reresetto. De to ths qrc srfces m e romte free-form oes he the qrc srfce er. Whe qrc srfces re coerte to free-form oes seerl rocessg re reqre: qrc srfce so creto of for-se regos or cqsto of tget ectors or orml ectors. These rocessg re reqre to e e eeg o the te of qrc srfces. I orer to clssf qrc srfces t s ecessr to coert the form of Eqto () of qrc srfce reresete geerl qrc to the str qrc form sch s the oe sho Tle.As the coerso metho rcl-s trsformto escre secto.5 s troce. F( X A) AX c q r A c q r [ ] T [ ] T X Whe geerl qrc srfces ecet rtclr shes sch s mgr srfces les or strght les re rtte the str form the re clssfe s sho Tle. Whe the srfce she of geerl erthere s cosere t s lle to romte t coe ellso oe-sheet herolo to-sheet herolo herolc rolo elltc cler. Tle. Str forms of qrc srfces Nme Coe Str form c Ellso c Oe-sheet herolo c To-sheet herolo c Elltc rolo Herolc rolo Elltc cler Herolc cler Prolc cler () Qrc srfces c e rmetrcll eresse sg to gle rmeters for the coorte of ot o cre srfce. The rmetrc eresso res eeg o the clssfcto of qrc srfces. Eqtos () (3) sho rmetrc eressos of qrc srfces.

3 芸術科学会論文誌 Vol. 3 No Ellso: Acosθ sφ θ π B sθ sφ φ π C sφ Oe-sheet herolo: Acosθ coshφ θ π B sθ shφ π φ π C shφ () (3) Tle. Qrc srfces ther rks Nme Rk Coe 3 Ellso 3 Oe-sheet herolo 3 To-sheet herolo 3 Elltc rolo Herolc rolo Elltc cler Herolc cler Prolc cler.5 Prcl-s trsformto Ths secto escres rcl-s trsformto hch geerl form of qrc srfces sho Eqto () s trsforme to the str form orer to clssf qrc srfces escre secto.4. Eqto () s trsforme to lger-stle eqto (4) sg the follog mtr α β : F α j β X 3 j T T F X α X β (4) Sce mtr α s efe s rel smmetrc mtr sg orthogol mtr P eles golto lke Eqto (5). A orthogol se s se the Lgrge mltler metho s ell s the rcl-s trsformto. I ths er orthogol se s clclte sg the Grm-Schmt orthogolto. λ (5) P T αp P αp λ λ 3.6 Locl srfce fttg th octree Ohtke resete metho [6] tht fts locl re of ot clo sg mlct fcto. A orgl ot clo s segmete octree metho. If srfce cot e ote th the secfe tolerce the cell s se seqetll. Sce srfce fttg s erforme locll from the ot set th the cell the hole she c e reresete leg of locl srfces. Therefore t s cler hether rcl s c e ere so tht the hole srfce she oes ot coser srfce of reolto. I to ther metho reqres the orml ector for ech ot. Therefore t s ecessr to estmte the orml ector o ll ots. Ths mes tht the clclto cost creses. As the reslt ther metho cot e le rectl to ft srfces from the erthere s rottol shes. 3. Proose Metho 3. Remol of ee ortos o erthere The smoothg techqe [] se to the T's metho s le to the olgo moel crete from the ot clo. To e more effcet some rmeters the trsfer fcto s chge. I Fg. the olgo moel the left s crete rectl from the ot clo the olgo moel the rght s smoothe lg the smoothg metho for tmes. It s oos tht most of the ee ortos re remoe. I the follog escrto olgo ot clos cte those fter smoothg less otherse secfcll ote. here λ λ re egeles of mtr α. λ 3 Whe the solto of αx β s X X s ector tht moes o qrc srfce P s mtr tht rottes o qrc srfce. If the rk of mtr α s 3 qrc srfces re cetrl qrcs. If the rk s or qrc srfces re o-cetrl qrcs. If the ceter ests the qrc srfce shol e rotte fter t s moe. If the ceter oes ot est the qrc srfce shol e moe fter t s rotte. Ths eles trsformto of Eqto () to the str form. Tle shos the clssfcto of qrc srfces ther rks. Fg. Polgo crete from ot t smoothe olgo 3. Qrc srfce fttg We tteto to the fct tht erthere srfces re close to the rottol shes. If the re rottol shes ther cross sectos ct le orthogol to the rcl s ll e ellses. A 3

4 芸術科学会論文誌 Vol. 3 No qrc srfce s ell-ko s she hose cross secto s elltc. If rt of qrc srfce c e eterme ts rottol she c e efe. Ths mes tht ee f some ortos of ot clo re mssg the c e terolte esl from the rest of the ot clo. Wth these resos for estmto of the srfce she of erthere qrc srfce fttg s rorte the techqe s eme. Geerl roch of the qrc srfce fttg s to l the lest sqres metho to the Ecle stce etee the ot clo the qrc srfce. I the lest sqres metho tht mmes the Ecle stce etee the ot clo the qrc srfce the elto fcto s o-ler. Therefore to ot solto t s ecessr to se the Leeerg-Mrqrt l or the Neto's metho. I these methos the tl les re mortt fctors. It s ffclt hoeer to set sch tl les of qrc srfces from erthere hose mfctrg recso s lo. Therefore these methos re ffclt to l to estmte the srfce she of erthere. To eterme the stce etee the ot the qrc srfce lgercll the Lgrge mltler metho s effecte. I orer to l the metho to erthere t s ecessr to coser herolos rolos or elltc clrcl srfces s ell s ellsos. Therefore or metho the etee metho Doros [5] s se for qrc srfce fttg. I to g e costrts qrc srfce fttg s erforme sg to tes of ccrc. For comtg the coeffcets of Eqto () the Lgrge mltler metho s se. The etls of the solto re roe Ae Prolem of the Lgrge mltler metho. I orer to sole the Lgrge mltler metho t s ecessr to cte the costrt eresso. Ths er ses Eqto (6) of the Doros metho [5] s the frst costrt for Eqto (): (6) Doros et l. le Eqto (6) s costrt ecse the solto cot e ote th the secfe tolerce lthogh the costrt of Eqto (8) ges the mthemtcll otml soltos. For the sme reso e se ths costrt of Eqto (6) to geerte qrc srfces. Eqto (6) s eresse the form of Eqto (7) sg the mtr C : [ ] r q c C r q c A T CA Whe Eqto (7) s set s the costrt mtr C s clclte s follos: C Whe the Lgrge mltler metho s se Eqto (8) ges the mthemtcll otml solto. Therefore Eqto (8) s le to the seco costrt. Doros et l. ot l ths costrt ecse the solto cot e ote th the secfe tolerce lthogh the costrt ges the mthemtcll otml soltos. I or metho Eqto (8) s troce to clclte the rtl rcl s. The rtl rcl s s escre the et secto. To eterme rtl rcl s t s ecessr to e ot clo to gros. Sce Eqto (8) s troce s the costrt ol for the estmto of the rcl s of the qrc srfces t s ossle to set lrger tolerce. r q c (8) Mtr C s clclte s follos reresete eresso smlr to Eqto (7). C I ths er s Eqtos (6) (8) re set s costrts the Lgrge mltler metho s se to sole the geerl forml of qrc srfces (). Usg the metho escre secto.4 Eqto () s trsforme to the str qrc form s sho Tle. The the qrc srfces re clssfe s escre secto.5 to ot the qrc srfce formto. 3.3 Determto of rcl s I or metho ot clo s e to gros erthere srfce s efe s seerl eces of qrc srfces. Whe ot clo s e the so log s mst e eterme. The rcl s s stle f ll of the ot clos c (7)

5 芸術科学会論文誌 Vol. 3 No e ftte to oe qrc srfce. Qrc srfces cot t lest oe rcl s thot the resece of the srfce cetrl es. Ths mes tht ee f ot clo s ftte to cre srfce t s ossle to secf the rcl s. I to ee f qrc srfce stt from the ot clo s clclte the crete qrc srfce s rottol she therefore ts rcl s ll ot e fferet so mch. As the costrt for qrc srfce fttg or metho troces Eqto (8) to eterme the rcl s tht c ss throgh the rcl rottol s. Ths s ecse Eqto (6) s sffcet coto to ere the stle rcl s. Coseqetl s the costrt coto for rogh rcl s oth Eqtos (6) (8) re erfe to e rorte. Fg. shos the frot ck es of the olgo moels of Fg. hose rcl es sle th re les re ote sg Eqtos (6) (8). The rcl s clclte sg Eqto (6) oes ot le t the ceter of the olgos. () Prcl s (re le) ote Eqto (6) (frot e) fttg I orer to e the ot clo to seerl gros the ot clo s ftte to qrc srfce th the secfe tolerce sg the rcl s clclte secto 3.3. Frst the terl of so s eterme log the rcl s. Net sg the coorte of the ot clo efe from the terl qrc srfce s ftte sg the metho escre secto 3.. The stce etee the qrc srfce the ot clo s elte qrc srfce s efe for ech ot clo th the tolerce. I qrc srfce fttg escre ths secto Eqto (6) s se s the costrt. Ths s ecse the solto mght ot e ote he the Eqto (8) s se. Ee f the solto s ote the mer of gros the ot clo ll e lrge lrge mer of qrc srfces mst e efe for erthere. Therefore s for the resole mer of so Eqto (6) s se s the costrt. The oere of the rocess s escre elo:. Dece the legth of the rcl s Sce the rcl s clclte secto 3.3 s fte the rcl s s efe ol for the terl here ot clo ests. All ot clos re rojecte oto the rcl s to ots tht form the mmm stce o the rcl s re ote. The ote ots re set s the strt e ots of the s. Fg. 3 shos the ot clo the re le shos the rcl s. For eser e the est of the ot clo s ecrese to % of the ctl oe. () Prcl s (re le) ote Eqto (6) (ck e) Fg. 3 Pot clo rcl s (c) Prcl s (re le) ote Eqto (8) (frot e) () Prcl s (re le) ote Eqto (8) (ck e) Fg. Dfferece of the rcl es eeg o costrts 3.4 Dso of ot clo for qrc srfce. De the ot clo ccorg to the les Sce there re te coeffcets for Eqto () t lest te ots re reqre for qrc srfce fttg. Ples hose ormls re the rcl es ste re cosere. O the rcl s t s ossle to retree ot clos cle rego sche the to les. I to the ot clo s e so tht ech of the e gros cots te or more ots. For emle f the rcl s s e to terls gros of ot clos re geerte. Ths ech gro ll cot ot 5 ots cse of the ot clo sho Fg.. Accorgl / of the legth of the rcl s s set s the t of moemet. The crete qrc srfce s elte th orgl ot clo sg mmm erge stce. The legth ote g the rcl s to terls s set to the strt ot of the rcl s s set to s the e ot s 5

6 芸術科学会論文誌 Vol. 3 No set to e. Ple s efe s le tht sses throgh oe of the ots s or e cots the orml ector o the rcl s. Ple s efe s le trslte from le stce. Pot clos elmte le le re ote. Fg. 4 shos the ot clos etee to les here s mltle seerl tmes from the ctl tlto les. I Fg.4 the re le s the rcl s sho Fg.3. Ple Ple Fg. 4 Pot clo elmte to les 3. Qrc srfce fttg elto A qrc srfce s ftte to the ot clo smle ste. If qrc srfces re crete from ot clos e rge s mch s ossle the mer of qrc srfces to e crete c e rece. Accorgl th the secfe tolerce (erge stce mmm stce) the stce etee the ot clos qrc srfces s elte. The erge stce etee the ot clo the qrc srfce s ere from the erge ote rojectg ech ot rego etee the to les. The mmm stce s clclte the sme mer s the erge stce. If the stce s eql to or smller th the tolerce trslte le log the s to crese the terls elmte les. The reet the fttg. If the stce s greter th the tolerce qrc srfces ll e efe for the ot clos the reos terl. To e more secfc le moe log the s ots re efe the terl elmte les. A qrc srfce s ftte to these ots the qrc srfce formto s se. The le s moe to the osto of le the rocess of stes 3 s reete. As les coce the rocessg s termte. I Fg. 5 the re ots re fe s qrc srfce les re reefe the re le s the rcl s Fg.3. Ple Ple Fg. 5 Pot clos set s qrc srfce efto of e les Fg. 6 sles qrc srfces geerte from the es of rcl ot clo gros. These re rts of qrc srfces geerte ol for the ortos here ot clos est. The tl tolerce s.5mm for the erge stce.mm for the mmm stce. If srfce fttg fls the tolerce ll e set to greter le from 5 to % of the tl oe for the et fttg. For geertg the srfces Fg. 6 the follog tolerce s set: mm or smller for the erge stce 4mm or smller for the mmm stce. Alog the rtl rcl s qrc srfces re ftte to the ot clos orer. The fe ot clos re ecle for the sseqet lcto. I other ors heeer gro of the ot clo s fe s qrc srfce the mer of the ots lle ll cote to ecle. I or metho the ellso sho Fg. 6 () s ftte fter the qrc srfces of () () (c) re fe. De to ths ot clos cot e chose so flel qrc srfces cot e ftte th sg the sme tolerce. Therefore qrc srfces re ftte sg the eflt tl tolerce. For the terls for hch the srfces cot e ftte the tolerce s loosee to.5mm for the erge stce 6mm for the mmm stce. Fg. 6 (e) shos the ll the srfces of () () (c) () r t the sme tme. () Ellso (c) Oe-sheet herolo (e) All gros () Oe-sheet herolo () Ellso Fg.6 Qrc srfces for ot clo gros 3.5 Free-form srfce geerto from qrc srfces Erthere srfces c e reresete th to or more qrc srfces sg the metho escre secto 3.4. If erthere srfces c e reresete th qrc srfces the metho c e se for e rge of lctos. Emles re s follos:. There re so m mssg rts erthere tht the cot e terolte from the erherl formto. 6

7 芸術科学会論文誌 Vol. 3 No The srfce s set s the referece for otmg the locto of erthere eces. I ths er e roose metho to reroce free-form srfces from qrc srfces. For the rerocto rocessg mssg ortos etee serte qrc srfces mst e terolte. I or metho the tolerce s set to grtee the stce eql to or smller th mm o erge (.5 mm for some ortos). Ths smle ots re geerte o qrc srfces for teroltg mssg ortos free-form cre mesh s geerte from the smle ots. A rego eclose the geerte for free-form cres s terolte srfce th goo ccrc for erthere. I to sg the for free-form cres t s ot ecessr to geerte trmme srfces hch ll mke oerto eser for the crete free-form srfces. The oere of the rocessg s escre elo:. Geerte smle ots o ech qrc srfce Ech qrc srfce hs ts rcl s. The s s e eqll to the secfe mer of terls le hose orml s the rcl s s efe to ss throgh oe of the so ots. Smle ots re geerte o the tersecto cre etee the qrc srfce the efe le. Accorg to the clssfcto escre secto.5 ech ot s reresete rmetrcll. B settg the mer of so for the rcl s tht for tersecto (ellse) the coortes of the smle ots c e esl clclte. Fg. 7 shos the smle ots geerte o qrc srfces. Fg. 7 Smle ots o qrc srfces. Geerte cre mesh The smle ots re terolte ech of rectos. The recto here ctes the recto of rcl of ech qrc srfce the recto ctes the recto of the rcl s. B geertg Beer cre tht sses throgh ot [7] oe Beer cre s ls geerte etee ertces. I ths er free-form cre ssg throgh the ot s geerte hle t ech ot the orml ector the tget ectors the rectos c e ote. It s lso ossle to geerte free-form cre th hgh ccrc sg the ots ormls tget ectors hch re troce the ocmet [7]. Fg. 8 s emle here smle ots re terolte th Beer cres. Fg. 8 Geerte cre mesh 3. Iterolte cre mesh Sce Beer cres ls ss throgh smle ots eer e ot of Beer cres coces th of the e ots of Beer cres. De to ths t s es to geerte srfce to rego eclose th for ses. I ths er Gregor srfce [8] s geerte sce t s es to geerte the srfce from for Beer cres. Fg. 9 shos Gregor srfces geerte to regos eclose th for Beer cres. Fg. 9 Reslt of srfce terolto 4. Eermetl Reslt Ths secto escres the reslts of lg or metho to erthere eces. The erthere eces se the eermet re orroe from Isek No M-k t Morok Ct. We se PC th the CPU of Itel Core Do.8GH 3.48GB memor the eermets. Fg. shos olgo moel ote smoothg olgos hose locto ostre re eterme three-mesoll from mesre ot clos sg the metho escre secto 3.. The mer of olgos s 76. The eermet s crre ot for the moel sho Fg.. The og o se of the moel Fg. s 5.6mm (th) 88.mm (heght) 8.mm (eth). Fg. Smoothe olgo t 7

8 芸術科学会論文誌 Vol. 3 No Frst qrc srfces geerte sg the metho escre secto 3.4 re elte hether the re resole. Fg. shos the olgos smoothe sg the metho escre secto 3. oerle o qrc srfces. The ortos crretl sho ello c le re re ftte to qrc srfces. A gree rt shos olgo moel ote smoothg olgos hose locto ostre re eterme three-mesoll from the mesre ot clos. I the sseqet fgres the sme rle of the colors s le. Tle 3 shos the erge stces the mmm stces etee the qrc srfces Fg. 6 the ertces o the smoothe olgos. The qrc srfces re geerte th the tolerce eql to or smller th mm for the erge stce (.5mm for some ortos) hle the stces for some ortos re the oes sho Tle 3. As the mesre mmm stces re comre to 5.6mm the mmm se of the or o of the moel the g s ot.9% (.6% for some ortos). Ths qlt s jge to e sffcet for restorto of erthere. Bt the roose metho remoes ee ortos o srfces lg the smoothg techqe. If the hole she s storte the remol t s ffclt to l ths metho. Fg. Erthere eces olgo t smoothe th the T s metho Fg. 3 shos the t from hch some olgos of the moel Fg. re remoe roml. The og o se of the moel Fg. 3 s 5.6mm (th) 88.mm (heght) 8.mm (eth). The mer of olgos s Fg. Comrso of qrc srfces olgos Tle 3. Mmm stces etee smle ots qrc srfces Qrc srfce Aerge stce Mmm stce () Ellso.57mm 3.8mm () Oe-sheet.6mm 3.93mm herolo (c) Oe-sheet.5mm 3.33mm herolo () Ellso.6mm 5.4mm Fg. 3 Polgo moel some of hose ot clos re remoe Frst qrc srfces ftte to the t of Fg. 3 the t of Fg. 6 re comre. Fg.4 shos to k of qrc srfces. Oe s from comlete olgos sho Fg.6 the other s from comlete olgos sho Fg.3. The qrc srfces geerte from the olgos Fg. 3 re sho mget. Both srfces re lmost cocet. Ths mes tht the mer of ot clos oes ot ffect qrc srfce fttg. I to sce the legth log the rcl s s ot chge gretl the mer of ot clos oes ot ffect grog of ot clos. The the geerte free-form srfces re elte hether the rereset the hole she roerl. Fg. shos the smoothe olgos oerle o the geerte free-form srfces. The ortos sho orge re free-form srfces geerte sg the metho escre secto 3.5. It s ossle to f tht the re lmost cocet. The erge serto stce s.3mm the mmm serto stce s 6.mm. () Ellso () Oe-sheet herolo (c) Oe-sheet herolo () Ellso Fg. 4 Oerl of qrc srfces 8

9 芸術科学会論文誌 Vol. 3 No I the e Fg. 5 shos the free-form srfce t geerte from the olgo t Fg. 3 oerle o those geerte from the olgo t Fg.. The free-form srfces sho mget re geerte from the t of Fg. 3 those orge re the oes sho Fg.. It s ossle to f tht the srfces re lmost cocet. Fg. 5 Oerl of free-form srfces To cofrm the effcc gst other erthere or metho s le to other erthere. I Fg. 6 () the olgo moel s crete rectl from the ot clo of erthere thot mssg ortos. The mer of olgos s The og o se of the moel s 4.4mm (th) 83.3mm (heght) 4.4mm (eth). The smoothe olgo moel s geerte lg the T's metho s sho Fg.6 (). For the olgo moel Fg. most of the ee ortos col e remoe lg the smoothg metho for tmes t the olgo moel Fg. 6 s smoothe lg the smoothg metho for tmes ecse the heght fferece t ee ortos s lrge. The the tl le of the mer of lcto tmes s set to the test s reete for tmes 3 tmes more cremets of. As reslt the mer of lcto tmes s set to hch most of the tters o srfces re remoe. Fg. 7 shos the reslts of the qrc srfce estmto the roose metho. The ot clos ere e to for segmets. For geertg the srfces Fg. 7 the follog tolerce s set:.5mm or smller for the erge stce 8.mm or smller for the mmm stce. The tolerce s set the sme metho s the oe escre secto 3.4. Fg.8 shos the olgos of Fg. 6 () oerle o qrc srfces. The gree ortos sho the olgo moels ote smoothg olgos hose locto ostre re eterme three-mesoll from the mesre ot clos. Tle 4 shos the erge stces the mmm stces etee the qrc srfces Fg. 7 the ertces o the smoothe olgos. The ellso sho Fg. 7 () hs the logest stce. As the mesre mmm stces re comre to 4.4mm the mmm se of the og o of the moel the g s ot 6.%. Ths qlt s jge to e sffcet for restorto of erthere. () Ellso () Ellso (c) Oe-sheet herolo () Oe-sheet herolo () Polgo crete from ot t (e) All gros Fg.7 Qrc srfces for ot clo gros () Polgo smoothe lg the T s metho for tmes Fg. 6 Ecte erthere Fg.8 Comrso of qrc srfces olgos 9

10 芸術科学会論文誌 Vol. 3 No Tle 4. Mmm stces etee smle ots qrc srfces Qrc srfce Aerge stce Mmm stce () Ellso.7mm 5.39mm () Ellso.mm 7.69mm (c) Oe-sheet.8mm 3.58mm herolo () Oe-sheet herolo.84mm 4.38mm Net to cofrm the effcc gst erthere th mssg ortos or metho s le to ssemle erthere ecte from rs. Fg. 9 () shos ssemle erthere ecte from rs here more th hlf of erthere s mssg. The mer of olgos s 757. The og o se of the moel s 54.mm (th) 3.mm (heght) 4.3mm (eth). Fg. 9 () shos olgo moels ote smoothg olgos sg the T's metho. sho olgo moels ote smoothg olgos hose locto ostre re eterme three-mesoll from the mesre ot clos. Tle 5 shos the erge stces the mmm stces etee the qrc srfces Fg. 9 the ertces o the smoothe olgos. As the mesre mmm stces re comre to 4.3mm the mmm se of the og o of the moel the g s ot 4.4%. Ths qlt s jge to e sffcet for restorto of erthere. It shos the effecteess for mge erthere. (Oe-sheet herolo) Fg. Qrc srfce for ot clo gros Fg. Comrso of qrc srfces olgos () Polgo crete from ot t () Polgo smoothe lg the T s metho for tmes Fg. 9 Ecte erthere Fg. shos qrc srfce geerte from the rcl s ot clos. For geertg the srfce Fg. the follog tolerce s set:.mm or smller for the erge stce 6.5mm or smller for the mmm stce. The tolerce s set the sme metho s the oe escre secto 3.4. Fg. shos the olgos of Fg. 9 () oerle o the qrc srfce. The gree ortos Tle 5. Mmm stce etee smle ots qrc srfces Qrc srfce Aerge stce Mmm stce Ellso.74mm 6.7mm 5. Coclso Ftre Isse I ths er e roose the metho of fttg qrc srfces crete the ot clos to erthere. I or metho the T s smoothg metho s le to mesre ot clos the ortos th ee tters re remoe. After tht qrc srfces re ftte to ll ot clos the rcl s s eterme for g the ot clos to gros. B reetg the so of ot clos log the rcl s fttg qrc srfces to the ot clos the ot clos re e most rortel for qrc srfce fttg. Erthere srfces c e eresse th the geerte qrc srfces. I to sg ots o qrc srfces the metho to rereset the etre srfce of erthere th free-form srfces s escre. I the ftre the metho s etee for thckeg shes or otmg the ostre of erthere eces. Prt of ths ork s sorte JSPS KAKENHI Grt Nmer

11 芸術科学会論文誌 Vol. 3 No The sc cocet of or metho hs lre ee resete NICOGRAPH Itertol 3[9] ths er s the etee oe. We re etremel grtefl for effcet ce from the er reeers. Ae As the ot s set s X coser the rolem of fttg qrc srfce to set of ots. Eqto (-) s qrc form reresetg qrc srfce. ) ( r q c AX A X F [ ] T r q c A [ ] T X For rtrr ot ) ( X s efe s follos: [ ] T X If ot ) ( les o the qrc srfce ) ( A X F s ote otherse ) ( > A X F s ote. I other ors etermg A tht mmes Eqto (-) elo Eqto (-) for the qrc srfce c e efe. { } X F ) ( (-) Ths s reefe s rolem of mtr. D If the mtr D efe th ro of X t ll ertces s efe s sho oe ths ll e the rolem to mme DA D A DA T T (-3) I to he Eqto (-3) s mme ths rolem c e relce th the oe solg the Lgrge mltler metho g the costrt to A. Ftgo et l. sets Eqto (-4) s the coto for efg Eqto (-) to e ellse. ) ( AX A X F (-4) [ ] T A [ ] T X I to the costrt coto s set Eqto (-5). 4 (-5) I ths er t s ecessr to hle geerl qrc srfces clg ellsos fferet costrts mst e set. Doros escres the costrts tht the se. The costrt s set s the rtl fferetl ector t the org s set to the t ector. If fferetl ector s efe s F r q c q r F If Eqto (-6) s rertte sg mtr the follog s ote. If C the [ ] r q c C r q c A T CA To smmre the oe the qrc srfce fttg rolem ecomes the rolem of the geerl Lgrge mltler metho sho elo here Eqto (-3) s mme er the costrts of Eqto (-7). (-6) (-7) (-) s ote hch mes

12 芸術科学会論文誌 Vol. 3 No [Prolem of the Lgrge mltler metho] A s ote to mme the follog: T T T L( A λ) A D DA λ( A CA ) λ > Refereces [] M.Skmoto S.Yshr M.Ko S.Kto H.Itoh A Cotor Mtg Metho sg Herrchcl Cotor Eresso ts Alcto o Erthere Recostrcto Imgg & Vsl Comtg The Jorl of the Isttte of Imge Electrocs Egeers of J Vol.34 No [] QXg Hg S.Flor N.Gelf M.Hofer H.Pottm Ressemlg Frctre Ojects Geometrc Mtchg ACM SIGGRAPH [3] B.Bro C.Toler-Frkl D.Neh M.Brs A.Vlchoolos C.Doms D.Dok S.Rske- c T.Werch A Sstem for Hgh-Volme Acqsto Mtchg of Fresco Frgmets: Ressemlg Ther Wll Ptgs ACM Trsctos o Grhcs (Proc. SIGGRAPH) 8 Vol.7 No.3 8. [4] K. Shoj K. Koo T. Koo F. Ch A Algorthm of Frctre Mtchg Bse o Mesre Pot Set of Frgmet Srfce IWAIT CD-ROM. [5] E. Alttsetseg Y. Mrk F. Ch K. Koo 3D Srfce Recostrcto of Stoe Tools Usg For-Drectol Mesremet Mche The Itertol Jorl of Vrtl Relt (IJVR) Vol. No [6] K.Hor M.Im T.Ogsr Dt Moel for Comter Recostrcto of PotshersJ Socet for Archeologcl Iformto Vol.5 No [7] K.HorI M.Im T.Ogsr Herrchcl Descrto of Cotor for Recostrcto of Broke Erthere IEICE Trs D-II Vol.J83-D-II Vol [8] M.Ko S.Kto H.Itoh A Fst Jot Detecto Metho Bse o Dffclt of Dscrmto Trsctos of Iformto Processg Socet of J Vol.4 No [9] S. Ok C.L K.Mtsm K.Koo Y.Tokm A Emto of Mtchg Algorthm Coserg Ptter Flo of Cor-Wre Stck Ptter for Erthere Restorto IWAIT CD-ROM 3. [] K.Koo N.Ae F.Ch Y.Tokm A St o Geertg Secto Les to Mke Mesre Drgs of Erthere Artfcts The jorl of the Isttte of Imge Iformto Teleso Egeers Vol.6 No [] T.Koo K.Koo Colm form etrcto shft estmto sg ot clo 3D mesremet: trl recostrcto of the soth gte oter fece of Shjo J Socet for Archeologcl Iformto Vol.3 No [] T.Kosht Y.Mrk K.Mtsm K.Koo Pece Moelg Metho ro Hole to Recostrct Erthere Vessel The Jorl of the Socet for Art Scece Vol. No [3] G.T A sgl rocessg roch to fr srfce esg I ACM SIGGRAPH Coferece Proceegs [4] A. Ftgo M. Pl R.B. Fsher Drect lest sqre fttg of ellses IEEE trs. Ptter Alss Mche Itellgece Vol. No [5] I.Doros B.Bto Three-Dmesol Srfce Crtre Estmto sg Qrc Srfce Ptches Scg Proceegs. [6] Y.Ohtke A.Bele M.Ale Greg Trk H.Seel Mlt-leel rtto of t mlcts ACM Trsctos o Grhcs Vol. No [7] H.Chokr Sol moelg th DESIGNBASE: theor mlemetto Aso-Wesle Logm Plshg Co. Ic 988. [8] H.Chokr F.Kmr Desg of sols th free-form srfce Comter Grhcs Vol [9] T.Kosht K.Mtsm K.Koo A Estmto of Erthere's Srfce She Usg Qrc Srfces NICOGRAPH Itertol 3 (th) 3. 3

13 芸術科学会論文誌 Vol. 3 No Tstom Kosht Tstom Kosht s memer of Lttce Techolog Co. Lt. He recee BS mthemtcs 993 from Scece Uerst of Toko. He recee Dr. Eg comter scece from Ite Uerst 3. Hs reserch terests cle geometrc moelg comter grhcs. He s memer of the Socet for Art Scece. Ktstsg Mtsm Ktstsg Mtsm s crretl ssstt rofessor t Ite Uerst. Hs reserch terests cle comter grhcs formto slto tercte sstems. He recee BE ME DE egrees comter scece from Ite Uerst resectel. He s reserch ssocte t Ftre Uerst-Hkote from 5 to. Koch Koo Koch Koo s rofessor of Fclt of Egeerg t Ite Uerst. He recee BS formto scece 985 from the Uerst of Tsk. He ere hs Dr. Eg. recso mcher egeerg from the Uerst of Toko 996. He joe the sol moelg roject t RICOH from 985 to 999 the XVL roject t Lttce Techolog Co. Lt.. He orke o ssocte rofessor of Fclt of Egeerg t Ite Uerst from to 9. Hs reserch terests cle rtl relt geometrc moelg 3D mesremet sstems comter grhcs. He s memer of IEEE CS. 33