Fast Algorithm for Triangular Mesh Simplification Based on Vertex ...

Download PDF
7 downloads 0 Views 554KB Size Report
Comment
for triangular mesh reduction based on the principle of vertex decimation. Introduction. Due to the wide technological advancement in the field of computer ...
)DVW $OJRULWKP IRU 7ULDQJXODU 0HVK 6LPSOLILFDWLRQ %DVHG RQ 9HUWH[ 'HFLPDWLRQ 0DUWLQ )UDQF 9DFODY 6NDOD 'HSDUWPHQW RI &RPSXWHU 6FLHQFH DQG (QJLQHHULQJ 8QLYHUVLW\ RI :HVW %RKHPLD 8QLYHU]LWQL    3LOVHQ &]HFK 5HSXEOLF {marty|skala}@kiv.zcu.cz http://herakles.zcu.cz

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loot et al. (Eds.): ICCS 2002, LNCS 2330, pp. 42−51, 2002.  Springer-Verlag Berlin Heidelberg 2002

Fast Algorithm for Triangular Mesh Simplification

43

7KLVSDSHULVVWUXFWXUHGDVIROORZV,QVHFWLRQZH GLVFXVVRXUSUHYLRXVZRUNDQG IXQGDPHQWDO WHFKQLTXHV IRU WKH PHVK VLPSOLILFDWLRQ HVSHFLDOO\ GHFLPDWLRQ :H VKDOO DOVR PHQWLRQ WKH DVSHFW RI SDUDOOHOL]DWLRQ 7KLV VHFWLRQ LQFOXGHV DQ RYHUYLHZ RI WKH VLPSOLILFDWLRQ WHFKQLTXHV ZH UHFHQWO\ XVHG 7KH EXFNHWLQJ KDVK  IXQFWLRQ LV GHVFULEHGLQVHFWLRQDQGLWLVXVHG WRDYRLGYHUWH[ VRUWLQJLQWKH GHFLPDWLRQSURFHVV LQRUGHUWRGHFUHDVH DOJRULWKPFRPSOH[LW\,QVHFWLRQZH SUHVHQWRXUQHZ DSSURDFK LQ GHWDLO ± PDLQ K\SRWKHVLV GDWD VWUXFWXUHV DQG WKH DOJRULWKP 6HFWLRQ  SUHVHQWV UHVXOWVREWDLQHGDQGVHFWLRQSURYLGHVDFRQFOXVLRQ

2XU3UHYLRXV:RUN $VDOUHDG\PHQWLRQHGRXUDSSURDFK LVEDVHGRQWKH YHUWH[GHFLPDWLRQ:H FKRVH WKH YHUWH[GHFLPDWLRQEHFDXVH RI LWVVLPSOLFLW\VWDELOLW\DQGIDVWSURFHVVLQJ7KLVPHWKRG LVDOVRHDV\WRSDUDOOHOL]H 'HFLPDWLRQ ,QJHQHUDOGHFLPDWLRQWHFKQLTXHV FDQEH GLYLGHGLQWRWKUHH PDLQFDWHJRULHVDFFRUGLQJ WRWKH PHVK SULPLWLYHVZKLFK WKH PHWKRGLVGHDOLQJZLWK7KHUHIRUH ZH UHFRJQL]H WKH IROORZLQJFDVHV x 9HUWH[GHFLPDWLRQ x (GJHGHFLPDWLRQ FRQWUDFWLRQRUFROODSVH  x 7ULDQJOH SDWFK GHFLPDWLRQ 7KH SULQFLSOH RI DOO WKH DIRUHPHQWLRQHG PHWKRGV LV VLPLODU ,QLWLDOO\ ZH KDYH WR HYDOXDWH WKH LPSRUWDQFH RI WKH SULPLWLYH YHUWLFHVHGJHVSDWFKHV LQWKH PHVK7KHQ WKH OHDVWLPSRUWDQWRQH LVUHPRYHGDQGDIWHUUHPRYDOWKH UHVXOWLQJKROH LVWULDQJXODWHG :HWKHUHIRUHKDYHDQHZ PHVKZLWKDVPDOOHUQXPEHURIWULDQJOHV ,QRXUPHWKRGZH VWDUWHGZLWK WKH YHUWH[GHFLPDWLRQSURSRVHGE\6FKURHGHUÿ(DFK YHUWH[LVHYDOXDWHG DFFRUGLQJWRLWVGLVWDQFH IURPDQDYHUDJH SODQH RUOLQH JLYHQE\ LWVQHLJKERXULQJYHUWLFHV)LJ OHIW  9 G

9þ G 9

9

L

9

M

9

9ÿ ÿDþýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýÿEþý

)LJ  9HUWH[ LPSRUWDQFH FRPSXWDWLRQ OHIW  HGJH FROODSVH ULJKW

7KH VHFRQG VWHS LV WR UHPRYH WKH OHDVW LPSRUWDQFH YHUWH[ DQG PDNH D QHZ WULDQJXODWLRQ 6LQFH ZH DUH LQ ' WKH WULDQJXODWLRQ LV TXLWH FRPSOH[ DQG WLPH FRQVXPLQJ &RQVLGHULQJ WKLV ZH GHFLGHG WR FRQWLQXH ZLWK WKH HGJH GHFLPDWLRQ LQVWHDG :H HYDOXDWHG DOO RI WKH HGJHV JRLQJ RXW RI WKH YHUWH[ DQG WKH OHDVW LPSRUWDQW HGJH LV FRQWUDFWHG WR ]HUR OHQJWK 7KLV PHDQV WKDW WKH YHUWH[ LV PRYHG WR WKH HQGSRLQW

44

M. Franc and V. Skala

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ÿþýüû WRFDUU\RXWVXFK D FRQGLWLRQ7ZRYHUWLFHVFDQEH LQWKH LQGHSHQGHQWVHWRI YHUWLFHVLI WKH\GRQ¶WVKDUHDQHGJHDVVKRZQLQ)LJ

9ÿ

9þ

9ý

)LJ  ,QGHSHQGHQW VHW RI YHUWLFHV

:H DVVLJQ DQ LPSRUWDQFH YDOXH WR HDFK YHUWH[ DQG WKHQ VHOHFW DQ LQGHSHQGHQW VHW WR UHPRYH YHUWLFHV ZLWK WKH ORZHVW LPSRUWDQFHú 7R FRQVWUXFW DQ LQGHSHQGHQW VHW IURP WKH DVVLJQHG LPSRUWDQFH YDOXHV ZH SURFHVV DOO YHUWLFHV LQ RUGHU RI WKHLU LPSRUWDQFH DQGVHOHFWDYHUWH[LIQRQHRILWVQHLJKERXUVKDYHEHHQWDNHQ 'XH WR WKH VSHFLDO GDWD VWUXFWXUH ZKLFK ZH XVHG DQG LQ RUGHU WR DYRLG FULWLFDO VHFWLRQV LQ SDUDOOHO FRGH ZH KDYH KDG WR XVH D VWULFWHU UXOH :H KDYH XVHG D VXSHU LQGHSHQGHQW VHW RI YHUWLFHV ZKHUH QRQH RI WZR WULDQJOHV VKDUH DQ HGJH YHUWLFHV Y DQGYDUHLQWKH VXSHULQGHSHQGHQWRQ)LJ 

+DVK)XQFWLRQ 8VLQJ LQGHSHQGHQW RU VXSHU LQGHSHQGHQW VHWV RI YHUWLFHV ZH QHHG WR VRUW YHUWLFHV DFFRUGLQJWRWKHLULPSRUWDQFH DQGDOVRWRFUHDWH WKH LQGHSHQGHQWVHWRI YHUWLFHV7KLV DSSHDUHGWREH DFULWLFDOSDUWRI WKH SUHYLRXVDSSURDFKHV6LQFH ZH GLGQRWZDQWWRXVH VRUWLQJ DOJRULWKPVù EHFDXVH RI WKHLU WLPH FRPSOH[LW\ ZH XVHG D VSHFLDO IXQFWLRQ WR WKUHVKROG YHUWLFHV DQG OHW RQO\ WKH OHDVW IHZ LPSRUWDQW YHUWLFHV EH FRQVLGHUHG DV FDQGLGDWHVIRUWKH UHGXFWLRQ 7KH LQLWLDO LGHD ZDV WR GLYLGH GDWD VHW YHUWLFHV  DFFRUGLQJ WR WKH QXPEHU RI IUHH SURFHVVRUV DQG UXQ GHFLPDWLRQ DV VHYHUDO LQGHSHQGHQW SDUWV $V ZH DOUHDG\

Fast Algorithm for Triangular Mesh Simplification

45

PHQWLRQHG PRVW RI GDWD ZDV SURGXFHG E\ ' VFDQQHUV RU LVRVXUIDFH H[WUDFWLRQ PHWKRGVVXFKDV 0DUFKLQJ&XEHVÿ&RQVLGHULQJWKH SULQFLSOH RI ERWK WHFKQLTXHVZH FDQ VXSSRVH WKDW VXFK WULDQJXODU PHVK LV D VHTXHQFH RI VWULSV ZKHUH QHLJKERXULQJ YHUWLFHVDUH DOVRYHU\FORVH WRHDFK RWKHU LQGDWDILOH RUPHPRU\,QRWKHU ZRUGVLI ZH GLYLGH D GDWD VHW LQWR OHW¶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¶WILWWRWKH DVVXPSWLRQ DERXWVWULSV DQGWKH DOJRULWKP LV TXLWH LQHIIHFWLYH LQ WKLV FDVH 7KLV H[SHULHQFH OHG XV WR XVH D KDVK IXQFWLRQWKDWSURYLGHVYHUWH[EXFNHWLQJ7KLVDSSURDFKHQDEOHGXVWRDYRLGVRUWLQJ úýþ úÿþ

FH ûýþ Q HU U ûÿþ X FF ÿR W üýþ Q HF UH S üÿþ ýþ ÿþ ü

û

ú

ù

ý

ø

÷

ö

õ

üÿ

üü

üû

üú

üù

üý

üø

ü÷

üö

UHODWLYHÿLPSRUWDQFHÿ>þýüû@

)LJ  9HUWH[ LPSRUWDQFH KLVWRJUDP 1HZ $SSURDFK ,QVWHDG RI YHUWH[ VRUWLQJ RU WKUHVKROGLQJ ZH FDQ XVH D KDVK IXQFWLRQ DQG ZLWK HQRXJK PHPRU\ ZH FDQ ³VRUW´ YHUWLFHV LQ 2 1 WLPH FRPSOH[LW\ ZKHUH 1 LV D QXPEHU RI YHUWLFHV ,W LV QHFHVVDU\ WR QRWLFH WKDW ZH GR QRW XVH PRGXOR RSHUDWLRQ LQ WKH KDVK IXQFWLRQ IRU WKH DGGUHVV FRPSXWDWLRQ WR WKH KDVK WDEOH $ KLVWRJUDP RI YHUWH[ LPSRUWDQFH IRU WHVWHG GDWD VHWV LV VKRZQ LQ )LJ  ,W LV REYLRXV WKDW RYHU  RI DOO YHUWLFHV KDYH WKH LPSRUWDQFH EHORZ  RI PD[LPXP LPSRUWDQFH YDOXH :H XVHG D VLPSOH KDVK IXQFWLRQ VKRZQ LQ )LJ  OHIW 

\

[ [ N



46

M. Franc and V. Skala

&

ÿþü÷

ÿ

þ ýüû ýüú ýüù ýüø ý ÿýü÷ ÿýüø ý ÿýüù ÿýüú ÿýüû ÿþ

ÿþ

D ýü÷

þ

þü÷

Nÿþýü Nÿþýû Nÿü

E

D

ÿþýýýýýÿüþýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýýÿüüþ

)LJ  *UDSK RI WKH IXQFWLRQ XVHG OHIW  GHILQLWLRQ RI WKH KDVK IXQFWLRQ ULJKW

7KLV KDVK IXQFWLRQ HQDEOHV XV WR PDS WKH LQWHUYDO  f WR   QRQOLQHDUO\ %HFDXVH ZH QHHG WR PDS RQO\  RI LPSRUWDQW YHUWLFHV WKH KDVK IXQFWLRQ  PXVW EH PRGLILHG DFFRUGLQJO\ VHH )LJXUH  ULJKW  DQG VFDOLQJ FRHIILFLHQW & PXVW EH LQWURGXFHG \

&

[ [N

 

[t

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ÿþ &

7KHQ 6ROYLQJWKDWZH JHW

E  E ÿ N z ÿ EÿN

& þ EÿN E

Nþ

IRU E 

D þ & D  D ÿ N z ÿ

DÿN

DQG

D þ EÿN E

D DÿN

& þ E ÿ N þÿÿ N E E

DEýþ ÿ D ÿ DE ÿ D

N þ Düÿ ý D ý þ D ýD

& þ ÿÿ N

$FFRUGLQJ WR RXU H[SHULPHQWV RQ ODUJH GDWD VHWV ZH KDYH IRXQG WKDW WKH RSWLPDO YDOXHV RI FRHIILFLHQWVDUH IROORZLQJD D 7KLVPHDQVWKDWSHUFHQWVRI WKH OHDVW LPSRUWDQWYHUWLFHVZLOOEH PDSSHGRQWRRI WKH ZKROH KDVK WDEOH7KH XVHU KDV WR VHW YDOXHV RI ERWK SDUDPHWHUV 7KH SDUDPHWHU D KDV D GLUHFW LQIOXHQFH WR WKH FOXVWHU OHQJWK LQWKH KDVK GDWDVWUXFWXUHZKHUH WKH FOXVWHU OHQJWK LVHTXDOWRQXPEHU RI YHUWLFHVLQWKH VDPHEXFNHW

Fast Algorithm for Triangular Mesh Simplification

47

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

5HVXOWV ,Q WKLV VHFWLRQ ZH SUHVHQW UHVXOWV RI RXU H[SHULPHQWV WKDW FRPSDUH DFKLHYHG WLPH TXDOLW\ RI DSSUR[LPDWLRQ DQG DOVR VKRZ VRPH H[DPSOHV RI UHGXFHG PRGHOV :H KDYH XVHG VHYHUDO ODUJH GDWD VHWV EXW ZH PHQWLRQ H[SHULPHQWDO UHVXOWV RQO\ ZLWK  GLIIHUHQW GDWD VHWV VHH 7DEOH  7DEOH  'DWD VHWV XVHG 0RGHOQDPH 7HHWK %XQQ\ +RUVH %RQH +DQG 'UDJRQ +DSS\%XGGKD 7XUELQHEODGH

1R RIWULDQJOHV        

1RRI YHUWLFHV        

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

48

M. Franc and V. Skala

ÿýþþþ

@ F H V >ÿ H P L W

+DVKÿIXQFWLRQ

7KUHVKROGLQJ

4þ6RUW

ÿþþ

ÿþ

ÿ

7HHWK

%XQQ\

+RUVH

%RQH

+DQG

'UDJRQ

+DSS\ %XGGKD

7XUELQH EODGH

PRGHOÿQDPH

)LJ  $FKLHYHG WLPH FRPSDULVRQ IRU WKUHH PHQWLRQHG DSSURDFKHV

8QIRUWXQDWHO\ RXU UHVXOWV DUH QRW GLUHFWO\ FRPSDUDEOH ZLWK RWKHU NQRZQ DOJRULWKPV GXH WR WKH GLIIHUHQW SODWIRUPV 7R PDNH WKH UHVXOWV URXJKO\ FRPSDUDEOH DW OHDVW ZH XVH WKH RIILFLDO EHQFKPDUNV SUHVHQWHG E\ 63(& DV VKRZV 7DEOH  ZKHUH K SUHVHQWV WKH VXSHULRULW\ RI '(// FRPSXWHU DJDLQVW WKH 6*, 7DEOH  SUHVHQWV RXU UHVXOWV DFFRUGLQJ WR UHVXOWV REWDLQHG UHFHQWO\ WDNLQJ WKH UDWLR K LQWR WKH FRQVLGHUDWLRQ 7DEOH  %HQFKPDUN WHVW SUHVHQWHG E\ 6WDQGDUG 3HUIRUPDQFH (YDOXDWLRQ &RUSRUDWLRQ

%HQFKPDUN WHVW  PDFKLQH 63(&IS 63(&LQW

6*, 5 '(//  3UHFLVLRQ K '(//6*,      

Table 3. Rough comparison of running times of reduction of the Bunny model.

,W LV REYLRXV WKDW RXU DOJRULWKP LV TXLWH IDVW KRZHYHU ZH GR QRW NQRZ WKH DSSUR[LPDWLRQ TXDOLW\ UHDFKHG E\ WKH RWKHU DOJRULWKPV 7KH TXDOLW\ RI DQ DSSUR[LPDWLRQ FDQ EH PHDVXUHG E\ VHYHUDO DSSURDFKHV 3UREDEO\ WKH PRVW SRSXODU ZD\ LV WR FRPSXWH D JHRPHWULF HUURU XVLQJ (DYJ PHWULFû GHULYHG IURP +DXVGRUII GLVWDQFH $V RXU PHWKRG NHHSV WKH VXEVHW RI RULJLQDO YHUWLFHV ZH XVH PRUH VLPSOH IRUPXOD  

Fast Algorithm for Triangular Mesh Simplification

(DYJ 0 ÿ  0 þ

 ¦ G Yþ 0 þ  ; ÿ  3 0 ÿ Nÿ Y; ÿ

49

 

ZKHUH0ÿDQG0ÿ DUH RULJLQDODQGUHGXFHGPRGHONÿ LVRULJLQDOQXPEHURI YHUWLFHV 3 0ÿ LV D VHW RI RULJLQDO YHUWLFHV DQG GY 0þ LV WKH GLVWDQFH EHWZHHQ RULJLQDO VHW RI YHUWLFHVDQGUHGXFHGPRGHO ,I ZH FRPSDUH DOO WKUHH DERYH GHVFULEHG DSSURDFKHV ZH ZLOO ILQG WKDW WKH HUURU YDOXHV DUH DOPRVWWKH VDPH,WLVDOVRKDUGWRVD\ZKLFK PHWKRGJLYHVWKH EHVWUHVXOWV EHFDXVH IRU HDFK PRGHO ZH JHW GLIIHUHQW EHKDYLRXU RI HUURU ([DPSOHV RI UHGXFHG PRGHOVDUHSUHVHQWHGLQ)LJ

)LJ  $ EXQQ\ PRGHO FRXUWHV\ *D7HFK DW GLIIHUHQW UHVROXWLRQV WKH RULJLQDO PRGHO ZLWK  WULDQJOHV RQ WKH OHIW UHGXFHG WR DSSUR[  WULDQJOHV RQ WKH ULJKW

)LJ  $ WHHWK PRGHO FRXUWHV\ &\EHUZDUH DW GLIIHUHQW UHVROXWLRQV WKH RULJLQDO PRGHO ZLWK  WULDQJOHV RQ WKH OHIW UHGXFHG WR DSSUR[  WULDQJOHV LQ WKH PLGGOH DQG  WULDQJOHVDSSUR[LPDWLRQRQWKHULJKW

50

M. Franc and V. Skala

)LJ 7KH+DSS\EPRGHO FRXUWHV\*D7HFK DW GLIIHUHQWUHVROXWLRQVWKHRULJLQDO PRGHO ZLWK  WULDQJOHV D  UHGXFHG WR  WULDQJOHV E   WULDQJOHV F   WULDQJOHV G

&RQFOXVLRQV :H KDYH GHVFULEHG RXU VXSHULRU DOJRULWKP IRU VLPSOLILFDWLRQ RI WULDQJXODU PHVKHV ZKLFK LV FDSDEOH RI SURGXFLQJ JRRG DSSUR[LPDWLRQV RI SRO\JRQDO PRGHOV 7KH DOJRULWKP FRPELQHV 6FKURHGHU¶V GHFLPDWLRQ LWV YHUWH[ LPSRUWDQFH HYDOXDWLRQ ZLWK HGJH FRQWUDFWLRQ WR VLPSOLI\ REMHFW PRGHOV LQ D VKRUW WLPH :H KDYH LQWURGXFHG D KDVK IXQFWLRQ ZKLFK ZH XVH LQVWHDG RI H[SHQVLYH YHUWH[ VRUWLQJ 2XU DOJRULWKP KDV SURYHG LWV KLJK VSHHG DQG VLPSOLFLW\

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

Fast Algorithm for Triangular Mesh Simplification

51

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¶  $&LDPSDOLQL3&LJQRQL&0RQWDQL56FRSLJQR0XOWLUHVROXWLRQGHFLPDWLRQEDVHGRQ JOREDO HUURU 7HFKQLFDO 5HSRUW &18&( & ,VWLWXWR SHU O (ODERUD]LRQH GHOO ,QIRUPD]LRQH&RQGVLJOLR1D]LRQDOHGHOOH5LFKHUH3LVD,7$/