TOTAL LEAST SQUARES ALGORITHMS FOR FITTING 3D STRAIGHT LINES
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1 IJMML 6: (07) Mrch 07 ISSN: vll t DOI: OL LES SQURES LGORIHMS FOR FIING 3D SRIGH LINES Cupg Guo Juhu Pg d Chuto L School of Scc Ch Uvrst of Gosccs (Bjg) Bjg P R Ch School of Ld Scc d cholog Ch Uvrst of Gosccs (Bjg) Bjg P R Ch strct o ddrss th prolm of fttg 3D strght l th LS mthod sd o th Lgrg fucto s usd to solv t h umr of prmtr to stmtd s dcrsd from s to four chgg th stdrd quto of th strght l to th projctv quto of t h prolm of fttg 3D strght l s covrtd to th prolm of fttg two D strght ls wth rrors oth coordts d th th totl lst squr (LS) d lst squr (LS) mthod r mplod to ft th two D strght l smultd mpl s crrd out to dmostrt th ffctvss d pplclt of proposd lgorthms Kwords: 3D strght l fttg D strght l fttg lst squr totl lst squrs * Corrspodg uthor E-ml ddrss: gcp@cugduc (Cupg Guo) Coprght 07 Sctfc dvcs Pulshrs 00 Mthmtcs Sujct Clssfcto: 6F0 Sumttd Jqg Go Rcvd Frur 0 07
2 36 Cupg Guo t l / IJMML 6: (07) Itroducto 3D l fttg s commo prolm m mtrologcl d msurmt sstms Nowds s most of th strumts provd 3D coordts grs d sctsts hv to work wth 3D coordts frm For ll th coordts cot rrors th 3D l fttg prolm c dscussd totl lst squr (LS) frmwork Sc Prso [9] solvd th prolm of fttg D ls to dt wth rrors oth coordts qut umr of totl lst squrs (LS) mthods wr dvlopd to dl wth th D l fttg Now thr r m rsrchs out th totl lst squrs lgorthms such s th sgulr vlu dcomposto (SVD) lgorthm (Golu d V Lo []) d th lgorthm sd o th Lgrg fucto (Schffr d Wsr []) For mor formto out th mthodolog of LS o c rfr to Huffl t l [3 4] York [5] gv dtld dscusso of th clculto of th st strght l th mthod of lst squrs (LS) Rd [0] rtrtd York s soluto d dctd sr w to solv for th slop of th st-ft l Nr t l [8] solvd th l rgrsso prolm wth strghtforwrd ltcl pproch tht uss th mmto of th shortst dstc tw ch prmtl pot d th thortcl l Wog [4] dscussd d comprd th lklhood-sd mthods for otg ppromt cofdc trvls for th slop smpl lr rgrsso wh oth vrls wr msurd wth rrors Schffr t l [] otd LS soluto solvg o-lr orml qutos v wl dvlopd trtv ppromto lgorthms Fttg strght l to dt wth ucrtts oth coordts ws dscussd rducg to odmsol srch for mmum Krstk d to [6] d grld to th cs wh thr r corrltos Krstk d to [7] mr-smkoo t l [] prstd smpl d rll formulto for th lr rgrsso ft usg th wghtd totl lst squrs (WLS) prolm wh oth vrls r sujctd to dffrt d possl corrltd os Howvr ths mthods cot rdl tdd to solv th prolm of fttg sptl l to pots wth os coordts thr dmsos
3 OL LES SQURES LGORIHMS FOR / IJMML 6: (07) Up to ow thr r lttl ltrturs focusd o 3D l fttg Kh [5] prsts smpl o-trtv lr procdurs for fdg th lst squrs l thr dmsos mmg th sum of th squrd prpdculr dstcs tw th dt pots d th fttd l Sow d Schffr [3] solvd th prolm of fttg ls 3D spc usg w lgorthm for th LS soluto udr olr Guss-Hlmrt modl I Sow s ppr ol four prmtrs wr stmtd thr vodg ovr-prmtrto I ths ppr projctg th 3D strght l oto two coordt pl ol four prmtrs r to stmtd 3D l fttg s vstgtd wth th ojctv of mmg th totl sum of ll squrd rdom rrors th 3D vrls h structur of ths ppr s s follows Scto troducs 3D l fttg wth rrors coordt d coordt Scto 3 shows 3D l fttg wh ll th thr coordts cot rrors smulto stud s cludd Scto 4 d t s cocludd Scto 5 3D L Fttg udr LS Crtro Suppos tht 3D strght l B gos through th pot ( ) 0 0 d hs drcto vctor ( p p p ) d stdrd quto of l B c prssd s 0 0 p 0 p 0 p h projctv qutos of th strght l s drvd s c d () whr p p p p 0 0 c d 0 0 () p p p p
4 Cupg Guo t l / IJMML 6: (07) h strght l c rgrdd s th trscto of th two pls rprstd th two qutos d ol four prmtrs d to stmtd W c ft th d coordts of th dt to Equto () to gt th stmtd vlu of d d ft th d coordts of th dt to Equto () to gt th stmtd vlu of c d d W ssum th vrls d hv qul vrcs σ Usg osrvtos ( )( ) d th ccompg fd vlus ( ) w c gt th LS stmt of c d d ( ) ( ) Y d c X β β (3) whr Y X For th orml vctors of th pl prssd () d () w c formult rspctvl s ( ) ( ) 0 0 c d gt drcto vctor of th l B ( ) c l Lt Bcus udr th LS crtro th strght l fttd th dt gos through th ctr of th dt th plc prssd () gos through ( )( ) d th plc prssd () gos through ( )( )
5 OL LES SQURES LGORIHMS FOR / IJMML 6: (07) hrfor th stmtd l B gos through th ctr ( ) of th dt h stmtd l B c prssd s c (4) 3 3D L Fttg udr LS Crtro ssumg th vrl s cotmtd gross rrors sds vrl d th vrl d hv vrc σ Usg osrvtos ( )( ) w c form th followg qutos ccordg to () d (): (5) d c (6) Lttg Z Y X (5) d (6) c prssd mtr oto ( ) Z X (7) ( ) d Z c Y (8)
6 Cupg Guo t l / IJMML 6: (07) hrfor th prolm of 3D l fttg s trsformd to two D l fttg prolms wth rrors oth coordts (7) d (8) hv th sm form d th sm coffct mtr so th soluto of (7) d (8) hv smlr rprstto For ths rso ol th soluto of (7) s dscussd low h totl lst squrs prcpl s to mm th ojctv fucto S Q Q B mplog th quvlt trgt fucto ccordc wth Lgrg mthod w hv ( ) ( ) ( ) Z X Φ Q Q d th cssr Eulr-Lgrg codtos r drvd ml 0 Φ Q (9) 0 Φ Q (0) 0 Φ Z X () ( ) 0 Φ Z () 0 Φ (3) whr tlds dct prdctd vctors d hts dct stmtd os
7 OL LES SQURES LGORIHMS FOR / IJMML 6: (07) Now d c prssd trms of usg (9) d (0) hs lds to tht d ftr srtg ths to () w hv tht Q (4) Q (5) [ Q ] ( X Z ) Q (6) Lt th Q s vrtl W rdl ot Isrtg (8) to (3) w gt Lt Q Q Q (7) Q ( X Z ) (8) ( ) Q ( X ) Z Q (9) ( Q ) Q Q Q ( I Q ) Q ( Z Q Z ) Q ZQ Z Q Q ( X Z ) (0) Usg (0) w c gt ( Q 4 Z Q3 X ) () Isrtg () to (9) th closd-form prsso of th stmtd prmtr vctor d c drvd s Q Q ( Z Q 3X ) ( I ) X ( Q Q ) Z ) ()
8 4 Cupg Guo t l / IJMML 6: (07) Cs Stud I ths scto th proposd LS pproch wll ppld to smultd mpl comprd wth th LS pproch 3D strght l s gv s 3 4 (3) h projctv quto of th strght l s gv s 4 4 (4) kg s umrs dstrutd uforml [ 0 0] d clcultg th vlus of d ccordg to Equto (4) prs of pots r formd h orml rdom rror s grtd d ddd to th coordts of ch pot h dsg s s follows: th LS mthod th proposd LS mthod r mplmtd for comprso purposs h LS soluto r gv s th tl vlus for trtos of th LS mthod d ε s chos s th covrgc tolrc h ms d th 0 0 root m squr rrors (RMSE) of th prmtrs β d β d th mmum dvto tw th clcultd vlu d th tru vlu r computd for 000 prmts for th two mthods h rsults r show l l Comprsos of th LS mthod d th proposd LS mthod LS LS ru vlu() 4 ru vlu() ru vlu(c) ru vlu(d) 4 M() RMSE() M() RMSE() M(c) RMSE(c) M(d) RMSE(d)
9 OL LES SQURES LGORIHMS FOR / IJMML 6: (07) s w c s from l for ll th prmtrs to stmtd th prmtr stmts otd LS r closr to th tru vlus th LS d hv smllr RMSE 5 Cocluso B chgg th stdrd quto of th 3D strght l to th projctv quto of t th umr of prmtr to stmtd s dcrsd from s to four Usg mmum prmtrto w hv solvd th 3D strght l fttg prolm covrtg t to th prolm of fttg two D strght ls wth rrors oth coordts d th th LS mthod s doptd to ft th D strght l mmg th sum of th squrd rror Morovr smultd mpl s crrd out to dmostrt th ffctvss d pplclt of th proposd lgorthm ckowldgmt h rsrch s supportd th Fudmtl Rsrch Fuds for Ctrl Uvrsts (No650593) Udrgrdut chg-rform Projct (NoJGYB059) Grdut chg-rform Projct of Ch of Uvrst of Gosccs (Bjg) NSFC ( d ) d SKLGED E Rfrcs [] R mr-smkoo F Zgh-Njd J sgr d S Jr Estmto of strght l prmtrs wth full corrltd coordts Msurmt 48() (04) [] G H Golu d C F V Lo lss of th totl lst squrs prolm SIM Jourl o Numrcl lss 7(6) (980) [3] S V Huffl d J Vdwll h otl Lst Squrs Prolm: Computtol spcts d lss Soct for Idustrl d ppld Mthmtcs Phldlph 997 [4] S V Huffl C L Chg N Mstrord C Pg d Kukush otl Lst Squrs d Errors--Vrls Modlg: lss lgorthms d pplctos Sprgr Scc d Busss Md Dordrcht 03
10 44 Cupg Guo t l / IJMML 6: (07) [5] P C Kh Smpl mthods for computg th lst squrs l thr dmsos Computrs d Chmstr 3(3) (989) 9-95 [6] M Krstk d M to wghtd totl lst-squrs lgorthm for fttg strght l Msurmt Scc d cholog 8() (007) 3438 [7] M Krstk d M to lst-squrs lgorthm for fttg dt pots wth mutull corrltd coordts to strght l Msurmt Scc & cholog (3) (0) 0350 [8] F Nr G Stt d S Choflo ccurt d strghtforwrd pproch to l rgrsso lss of rror-ffctd prmtl dt Jourl of Phscs E Sctfc Istrumts () (989) 5-7 [9] K Prso O ls d pls of closst ft to sstms of pot spc Phlosophcl Mg () (90) [0] B C Rd Lr lst-squrs fts wth rrors oth coordts mrc Jourl of Phscs 57(57) (989) [] B Schffr I L Y Flus d Y Cho otl lst-squrs (LS) for godtc strght-l d pl djustmt Bolltto D Gods E Sc ff 65(3) (006) 4-68 [] B Schffr d Wsr O wghtd totl lst-squrs djustmt for lr rgrsso Jourl of Gods 8(7) (008) 45-4 [3] K Sow d B Schffr L fttg Eucld 3D spc Stud Gophsc t Godtc 60() (06) 0-7 [4] M Y Wog Lklhood stmto of smpl lr rgrsso modl wh oth vrls hv rror Bomtrk 76() (989) 4-48 [5] D York Lst-squrs fttg of strght l Cd Jourl of Phscs 44(5) (966) g
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