SEGMENTATION AND CLASSIFICATION OF MOVING VIDEO OBJECTS

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1 Ths s rlmr vrso of rtcl ublshd I B. Furht d O. Mrqus (dtors, Hdbook of Vdo Dtbss, Vol. 8, gs CRC Prss, Boc Rto, FL, USA, Stmbr b Drk Fr, Thoms Hslm, Sth Kof, Grld Küh d Wolfgg Efflsbrg 5 SEGMENTATION AND CLASSIFICATION OF MOVING VIDEO OBJECTS Drk Fr, Thoms Hslm, Sth Kof, Grld Küh, Wolfgg Efflsbrg Prktsch Iformtk IV Uvrst of Mhm L5, 6, 683 Mhm, Grm fflsbrg@formtk.u-mhm.d. INTRODUCTION Dst vr otmstc rdctos th rl ds of Artfcl Itllgc rsrch, comutr vso sstm tht trrts mg squcs cqurd from rbtrr rl-world scs stll rms out of rch. Nvrthlss, thr hs b grt rogrss th fld sc th d umbr of lctos mrgd wth dffrt rs. Of rtculr trst for svrl lctos r cblts for objct sgmtto d objct rcogto. Algorthms from th formr ctgor suort th sgmtto of th obsrvd world to smtc tts, thus llow trsto from sgl rocssg towrds objct-ortd vw. Objct rcogto rochs llow th clssfcto of objcts to ctgors d bl for coctul rrsttos of stll mgs or vdos. Th gol of ths chtr s th dvlomt of clssfcto sstm for objcts tht r vdos. Ths formto c b usd to d or ctgorz vdos d t thus suorts objct-bsd vdo rtrvl. I ordr to k th subjct mgbl, th sstm s mbddd to st of costrts: Th sgmtto modul rls o moto formto, thus t c ol dtct movg objcts. Furthrmor, th clssfcto modul ol cosdrs th two-dmsol sh of th sgmtd objcts. Thrfor, just cors clssfcto of th objcts to grc clsss (.g., crs, ol s ossbl. Th rmdr of th chtr s orgzd s follows: Frst of ll, summrzg our roch, Scto srvs s gudl through th subsqut sctos. I Scto 3, cmr modls d th stmto of thr rmtrs r dscrbd. Nt, w dscuss our roch to objct sgmtto Scto 4. Scto 5 troducs th vdo objct

2 Chtr 5 clssfcto sstm d Scto 6 cocluds th chtr wth rmtl rsults.. SYSTEM ARCHITECTURE Our sstm for vdo objct clssfcto cossts of two comots, ml sgmtto modul d clssfcto modul (cf. Fgur. Bsd o moto cus th cmr moto wth th sc s dtrmd (moto stmto d bckgroud mg for th tr squc s costructd (bckgroud mosc. Durg th costructo rocss, rts blogg to forgroud objcts r rmovd b tmorl fltrg. Th, objct sgmtto s rformd b vlutg dffrcs btw th currt frm d th rcostructd bckgroud mosc (sgmtto. Th objct msks dtrmd b th sgmtto lgorthm r fd forwrd to th clssfcto modul. For ch msk, ffct sh-bsd rrstto s clcultd (cotour dscrto. Th, ths dscrto s mtchd to r-clcultd objct dscrtos stord dtbs (mtchg. Th fl clssfcto of th objct s chvd b tgrtg th mtchg rsults for umbr of succssv frms. Ths dds rlblt to th roch sc urcogzbl sgl objct vws occurrg th vdo r sgfct wth rsct to th whol squc. Morovr, t llows utomtc dscrto of objct bhvor. mg squc Objct Sgmtto moto stmto bckgroud mosc sgmtto Objct Clssfcto cotour dscrto mtchg rocss dtbs wth objct rotots clssfd vdo objct Fgur : Archtctur of th vdo objct clssfcto sstm.

3 Sgmtto d Clssfcto of Movg Vdo Objcts 3 3. CAMERA MOTION COMPENSATION If vdos r rcordd wth movg cmr, ot ol th forgroud objcts r movg, but lso th bckgroud. Th frst st of our sgmtto lgorthm dtrms th moto du to chgs th cmr rmtrs. Ths llows to stblz th bckgroud such tht ol th forgroud objcts r movg rltv to th coordt sstm of th rcostructd bckgroud. It s usull ssumd tht th bckgroud moto s th domt moto th squc,.., ts r of suort s much lrgr th th forgroud objcts. I ordr to dffrtt btw forgroud d bckgroud moto, o hs to troduc rgulrzto modl for th moto fld. Ths modl should b grl ough to dscrb ll ts of moto tht c occur for sgl objct, but o th othr hd, t should b suffctl rstrctv tht two motos tht w cosdr dffrt c ot b dscrbd b th sm modl. Th moto modl lso llows to dtrm moto rs whch th ttur cott s ot suffct to stmt th corrct moto. 3. CAMERA MOTION MODELS W us world modl whch th mg bckgroud s lr d odformbl. Ths ssumto, whch s vld for most rctcl squcs, llows us to us much smlr moto modl s would b dd for th grl cs of full thr-dmsol structur. Usg homogous coordts, th rojcto of 3D sc to mg l c b formultd th most grl cs b ( w T P ( z T, whr P s 3 4 mtr (s [3,6,7]. As w r ol trstd th trsformto of o rojctd mg to othr rojctd mg t dffrt cmr osto (c.f. Fg., w c rbtrrl chg th world coordt sstm such tht th bckgroud l s loctd t z. I ths cs, th rojcto quto rducs to ( w Th 3 3 mtr o th rght dots l-to-l mg (homogrh. Lt H b th homogrh to rojct th bckgroud l oto th mg l of frm. Th, w c dtrm th trsformto from mg l to j s h h h3 H j H j H h h h3. ( h3 h3 h33

4 4 Chtr 5 bckgroud l H z Hj j Fgur : Projcto of bckgroud l world coordts to mg coordt sstms of mgs d j. Sc homogous coordts r sclg vrt, w c st h h / h d gt wth rmg of mtr lmts j j 33 h h h 3 t H j h h h 3 t. (3 h 3 h 3 h 33 Hc, th trsformto btw mg frms c b wrtt s t, t. (4 Ths modl s clld th rsctv cmr moto modl. I ths formulto, t s s to s tht th j corrsod to ff trsformto, t, t r th trsltorl comots, d, r th rsctv rmtrs. A dsdvtg of th rsctv moto modl, tht wll bcom rt th t scto, s tht th modl s olr. If th vwg drcto dos ot chg much btw succssv frms, th rsctv rmtrs, c b glctd d c b st to zro. Ths rsults th ff cmr moto modl t t. (5 Sc th ff modl s lr, th rmtrs c b stmtd sl. Th slcto of th rort cmr modl dds o th lcto r. Whl t s ossbl to us th most grl modl ll stutos, t m b dvtgous to rstrct to smlr modl. A sml moto modl s ot ol sr to mlmt, but th stmto lso covrgs fstr d s mor robust th modl wth mor rmtrs. I som lctos, t m v b ossbl to rstrct th ff modl furthr to th trsltorl

5 Sgmtto d Clssfcto of Movg Vdo Objcts 5 modl. Hr, ( j quls th dtt mtr d ol th trsltorl comots t, t rm: T T T ( ( ( t t. (6 ( trslto (b sclg (c rotto (d shr ( rsctv Fgur 3: Dffrt l trsformtos. Whl trsformtos (-(d r ff, rsctv dformtos ( c ol b modld b th rsctv moto modl. 3. MODEL PARAMETER ESTIMATION I rmtr stmto, w srch for th cmr modl rmtrs tht bst dscrb th msurd locl moto. Algorthms for cmr modl rmtr stmto c b corsl dvdd to two clsss: ftur-bsd stmto [] d drct (or grdt-bsd stmto [8]. Th d of modl stmto bsd o ftur corrsodcs s to dtf st of ostos th mg tht c b trckd through th squc. Th cmr modl s th clcultd s th bst ft modl to ths corrsodcs. I drct mtchg, th bst modl rmtrs r dfd s thos rsultg th dffrc frm wth mmum rg. Ths roch s usull solvd b grdt dsct lgorthm. Hc, t s mortt to hv good tl stmt of th cmr modl to rvt gttg trd locl mmum. As th robblt for rug to locl mmum crss wth lrg dslcmts, rmd roch s oft usd. Th mg s scld dow svrl lvls d th stmto bgs t th lowst rsoluto lvl. Aftr covrgc, th stmto cotus t th t hghr rsoluto lvl utl th rmtrs for th orgl rsoluto r foud. Sc drct mthods rovd hghr stmto ccurc th fturbsd rochs, but rqur good tlzto to ssur covrgc, w r usg two-st rocss. Frst, ftur-bsd stmto whch c co wth lrg dslcmts s usd to obt tl stmt. Bsd o ths modl, drct mthod s usd to crs th ccurc. 3.3 FEATURE-BASED ESTIMATION Ftur-bsd stmto s bsd o st of fturs th mg tht c b trckd rlbl through th squc. If fturs c b wll loclzd, mg moto c b stmtd wth hgh cofdc. O th othr hd, for ls sd uforml colord rgo, w c ot dtrm th corrct objct moto. Ev for ls tht l o objct dgs, ol th moto comot rdculr to th dg c b dtrmd (s Fgur 4. To b bl to trck ftur rlbl, t s rqurd tht th ghborhood of th ftur shows structur tht s trul two-dmsol. Ths s th cs t corrs of rgos, or ots whr svrl rgos ovrl (s Fgur 4b-d.

6 6 Chtr 5? ( (b (c (d Fgur 4: Ftur ots o dgs ( cot b trckd rlbl bcus hgh ucrtt bout th osto log th dg rms. Ftur ots t corrs (b, crossgs (c, or ots whr svrl rgos ovrl (d c b trckd vr rlbl Ftur Pot Slcto For th slcto of ftur ots, w mlo th Hrrs (or Plss corr dtctor [5] whch s dscrbd th followg. Lt P { } b th st of ls mg wth ssoctd brghtss fucto I(. To lz th structur t l ( I, smll ghborhood N( I roud s cosdrd. W dot th mg grdt t s ( g ( g ( T I (. Lt us m how th dstrbuto of grdts hs to look lk for ftur ot cddts. Fgur 5 dcts scttr-lots of th grdt vctor comots for ll ls sd th ghborhood of som slctd mg ostos. W c s Fgur 5c tht for ghborhoods tht ol hbt o-dmsol structur, th grdts r ml ortd to th sm drcto. Cosqutl, th vrc s lrg rdculr to th dg d vr smll log th dg. Ths smll vrc dcts tht th ftur cot b wll loclzd. Fvorbl, fturs should os ghborhood whr th grdt comots r wll scttrd ovr th l d, thus, th vrc both drctos s hgh (cf. Fgur 5d,. Aromtg bvrt Guss dstrbuto, w dtrm th rcl s of th dstrbuto b usg rcl comot dcomosto of th corrlto mtr g ( g ( g ( g ( N ( N ( C. (7 g ( g ( g ( g ( N ( N ( λ of C. Th lgth of th rcl s corrsods to th gvlus Bsd o ths gvlus, w c troduc clssfcto of th l. W dffrtt btw th clsss flt for low λ,λ, dg for λ << λ, or corr, tturd for lrg λ,λ. Sc th comutto of gvlus s comuttoll sv (ot tht th comutto hs to b rformd for vr l th mg, Hrrs d Sths roosd to st th clssfcto boudrs such tht lct comutto of th λ

7 Sgmtto d Clssfcto of Movg Vdo Objcts 7 4 g 3 ( orgl mg 4 g 3 (b dtctd ftur ots 4 g g gvlus s ot rqurd. Elotg th fct tht λ λ Tr( d λ λ Dt( C, th dfd corr rsos vlu s λ k( λ Dt( C k Tr( C C r λ λ (8 whr k s usull st to.6. Th clss boudrs r chos s show Fgur 6. Aftr r(, hs b comutd for ch l, ftur ots r obtd from th locl mm of r(, whr λ λ Tr ( C > tlow (.., th l s ot clssfd s flt l. To mrov th loclzto of th ftur ots, Equto 7 s modfd to wghtd corrlto mtr whr th grdts r wghtd wth Guss krl w( s -3 g g (c wdow (d wdow ( wdow 3 Fgur 5: Scttr lots of grdt comots for slctd st of wdows. w( g ( g ( w( g ( g ( N ( N ( C. (9 w( g ( g ( w( g ( g ( N ( N ( Ths crss th wght of ctrl ls d th ftur ot s movd to th osto of mmum grdt vrc. Wthout ths wghtg, th bst osto to lc th ftur ot s ot uqu. Th dtctor rsos s qul s log s th corr s comltl cotd th ghborhood wdow. A sml rsult of utomtc ftur ot dtcto s show Fgur 5b. -3

8 8 Chtr 5 λ dg r r(, l grd Δ v( Δ corr flt dg Fgur 6: Pl clssfcto bsd o Hrrs corr dtctor rsos fucto. Th dshd ls r th sols of r. λ Fgur 7: Sub-l ftur ot loclzto b fttg qudrtc fucto through th ftur ot t 3.3. Rfmt to Sub-Pl Accurc Th corr dtctor dscrbd so fr locts ftur ots ol u to tgr osto ccurc. If th tru ftur ot s loctd r th mddl btw ls, jttr m occur. Ths c b rducd b stmtg th subl osto of th ftur ot. Th rfmt s comutd ddtl for th d coordt. I th followg, w coctrt o th drcto. Th drcto s hdld smlrl. For ch ftur ot, w mtch rbol v ( Δ ( Δ b Δ c through th Hrrs rsos surfc r (,, ctrd t th cosdrd ftur ot. Th fttd rbol s dfd b th vlus of r t th ftur ot osto d ts two ghbors. B sttg Δ, whr s th ftur ot osto (cf. Fgur 7, w gt v ( r, b c ( v ( r, c ( ( v ( r, b c ( Aftr sttg d v / dδ, ths lds to r(, r(, Δ. ( r(, r(, r(,

9 Sgmtto d Clssfcto of Movg Vdo Objcts 9 Sc r, s locl mmum, t s gurtd tht Δ <. Th w ( ftur ot osto s st to th mmum of v,.., Δ Dtrmg Ftur Corrsodcs Aftr rort ftur ots hv b dtfd, w hv to stblsh corrsodcs btw ftur ots succssv frms. Thr r two m roblms stblshg th corrsodcs. Frst, ot vr ftur ot hs corrsodg ftur ot th othr frm. Bcus of mg os or objct moto, w ftur ots m r or dsr. Fortutl, th Hrrs corr dtctor s vr stbl so tht most ftur ots o frm wll lso r th t [9]. Th scod roblm s tht th mtchg c b mbguous f thr r svrl ftur ots surroudd b comrbl ttur. Ths m h,.g., wh thr r objcts wth rgulr ttur or svrl dtcl objcts th mg. Lt F, F b th st of ftur ots of two succssv mgs I, I. Our ftur mtchg lgorthm works s follows:. For ch r of fturs F, j F t ostos ( ;,( ; j j, clcult th mtchg rror d Δ Δ Δ Δ, j I 8 Δ < 8 8 Δ < 8 (, (, I j j. If th Eucld dstc btw th ftur ots cds thrshold t d m, whch s st to bout /3 of th mg wdth, d, j s st to ft. Th rtol for ths thrshold wll b gv shortl.. Sort ll mtchg rrors obtd th lst st scdg ordr. 3. Dscrd ll mtchs whos mtchg rror cds thrshold t m. 4. Itrt through ll rs of ftur ots wth crsg mtchg rror. If thr of th two ftur ots hs b ssgd t, stblsh corrsodc btw th two. Cosqutl, th mtchg rocss s grd lgorthm, whr bst fts r ssgd frst. If thr r sgl fturs wthout coutrrt, th robblt tht th wll b ssgd rroousl s low sc ll fturs tht hv corrct corrsodcs hv b ssgd bfor d, thus, r ot vlbl for ssgmt mor. Morovr, th mtchg rror wll b hgh so tht t wll usull cd t. d m Thr s o scl cs tht justfs th troducto of t m. Cosdr cmr. M ftur ots wll dsr t o sd of th mg d w ftur ots wll r t th oost sd. Aftr ll ftur ots tht r both frms r ssgd, ol thos fturs t th mg bordr rm. Thus, f th mtchg rror s low, corrsodcs wll b stblshd btw just to dsr d just rd fturs cross th comlt mg, whch s obvousl ot corrct. As w kow tht thr wll lws b lrg ovrl btw succssv frms, w lso kow tht th mmum moto c ot b fstr th, s, /3 of th mg wdth

10 Chtr 5 btw frms. Hc, w c crcumvt th roblm b troducg th mmum dstc lmt m t Modl Prmtr Estmto b Lst Squrs Rgrsso Lt, b th msurd osto of ftur, whch hd osto, th lst frm. Th bst rmtr st should mmz th squrd Eucld dstc btw th msurd ftur locto d th osto ccordg to th cmr modl ( (. Hc, w mmz th sum of rrors E ( (. ( Usg th ff moto modl wth ( t t, w obt th soluto b sttg th rtl drvtvs E / to zro, whch lds to th followg lr quto sstm: t t (3 Clrl, ths quto sstm c b solvd ffctl b slttg t u to two ddt 3 3 sstms. Whl ths drct roch works for th ff moto modl, t s ot lcbl to th rsctv modl bcus th rsctv modl s olr. O soluto to ths roblm s tht std of usg Eucld dstcs ( ( t t (4 to us lgbrc dstc mmzto, whr w tr to mmz th rsduls r ( ( ( ( (

11 Sgmtto d Clssfcto of Movg Vdo Objcts ( t (5 ( t Not tht bcus of th multlcto wth (, mmzto of r s bsd,.., ftur ots wth lrgr, hv grtr fluc o th stmto. Ths udsrbl ffct c b slghtl rducd b shftg th org of th coordt sstm to th ctr of th mg. Th sum of rsduls r c b mmzd b lst squrs (LS soluto of th ovrdtrmd quto sstm t t. (6 A umbr of ffct umrcl lgorthms (.g., bsd o sgulr vlu dcomosto c b foud for ths stdrd roblm [5] Robust Estmto Th m drwbck of th LS mthod s dscrbd bov s tht outlrs,.., obsrvtos tht dvt strogl from th ctd modl, c totll offst th stmto rsult. Fgur 8 llustrts ths roblm. For ths uros, w clcultd moto stmts from two frms of rl-world squc rcordd b g cmr. Th moto vctors du to th r clrl vsbl th bckgroud. I ddto, th wlkg rso th forgroud cuss moto stmts tht dvt from thos ducd b cmr moto. Fgur 8b dsls th globl moto fld stmtd b th LS mthod. Obvousl, th mtur of bckgroud d objct moto dd ot ld to stsfctor rsults. Istd, th cmr rmtrs wr otmzd to romt both, bckgroud d objct moto. I ordr to stmt cmr rmtrs rlbl from mtur of bckgroud d objct moto, o hs to dstgush btw obsrvtos blogg to th globl moto modl (lrs d thos rsultg from objct moto (outlrs. Fgur 8c dsls th sm moto stmts s bfor. Ths tm, howvr, ol subst of vctors (drw blck s usd for th LS stmto. Th obtd cmr moto modl dctd Fgur 8d s stmtd clusvl from lrs d, thus, clrl romts th g orto.

12 Chtr 5 ( (b (c (d Fgur 8: Estmto of cmr rmtrs. ( moto stmts, (b stmto b lst squrs, (c moto stmts (wht vctors r cludd from th stmto, (d stmto b lst-trmmd squrs. rgrsso. To lmt outlrs from th stmto rocss, o c l robust rgrsso mthods. Wdl usd robust rgrsso mthods comrs rdom sml cocsus (RANSAC [4], lst md of squrs (LMdS, d lst-trmmd squrs (LTS rgrsso [7]. Thos mthods r bl to clcult th rmtrs of our rgrsso roblm v wh lrg frcto of th dt cossts of outlrs. Th bsc sts r smlr for ll of thos mthods. Frst of ll, rdom subst of th dt st s drw d th modl rmtrs r clcultd from ths subst. Th subst sz quls th umbr of rmtrs to b stmtd. For stc, clcultg th 8 rmtrs of th rsctv modl rqurs 4 ftur corrsodcs, ch troducg two costrts. A rdoml drw subst cotg outlrs rsults oor rmtr stmto. As rmd, N substs r drw d th rmtrs r clcultd for ch of thm. B choosg N suffctl hgh, o c ssur u to crt robblt tht t lst o good subst,.., sml wthout outlrs, ws drw. Th robblt for drwg t lst o subst N wthout outlrs s gv b P ( ε,, N : ( ( ε whr ε dots th frcto of outlrs. Covrsl, th rqurd umbr of smls to sur hgh cofdc c b dtrmd from ths quto.

13 Sgmtto d Clssfcto of Movg Vdo Objcts 3 For ch rmtr st clcultd from th substs, th modl rror s msurd. Fll, th modl whch fts th gv obsrvto bst s rtd. Th m dffrc btw th thr mthods RANSAC, LMdS, d LTS s th msur for th modl rror. W hv chos to us th LTS stmtor bcus t dos ot rqur slcto thrshold s RANSAC d t s comuttoll mor ffct th LmdS [8]. Th LTS stmtor c b wrtt s m h ( r : (7 whr s th ut dt sz d ( r : ( r : dot th squrd rsduls gv scdg ordr. Smlr to th LS mthod, th sum of squrd rsduls s mmzd. Howvr, ol th h smllst squrd rsduls r tk to ccout. Sttg h romtl to / lmts hlf of th css from th summto, thus, th mthod c co wth bout 5% outlrs. Sc th comutto of th LTS rgrsso coffcts s ot strghtforwrd, th bsc sts r summrzd th followg. For dtld trtmt d fst mlmtto clld FAST-LTS, w rfr to [7,8]. To vlut Equto 7, tl stmt of th rgrsso rmtrs s rqurd. For ths uros, subst S of sz s drw rdoml from th dt st. Solvg lr sstm crtd b S lds tl stmt of th rmtrs dotd b. Usg w c clcult th rsduls r,,, for ll css th dt st. Th stmto ccurc s crsd b lg svrl comcto sts. Sortg r b bsolut vlu lds subst H cotg th h css wth th lst bsolut rsduls. Furthrmor, frst qult of ft msur c b clcultd from ths subst s Q : ( r H :. Th, bsd o H, lst squrs ft s clcultd ldg w rmtr stmt. Ag, th rsduls r r clcultd d sortd b bsolut vlu scdg ordr. Ths lds subst H whch cots th h css tht ossss th lst bsolut rsduls wth rsct to th rmtr st. Th qult of ft for H s gv b Q : ( r H :. Du to th rorts of LS rgrsso, t c b ssurd tht Q Q. Thus, b trtg th bov rocdur utl Q k Q k, th otml rmtr st c b dtrmd for gv tl subst.

14 4 Chtr DIRECT METHODS FOR MOTION ESTIMATION Although moto stmto bsd o ftur corrsodcs s robust d c hdl lrg motos, t s ot ccurt ough to chv sub-l rgstrto. Prtculrl th rocss of buldg th bckgroud mosc, smll rrors quckl sum u d rsult clr mslgmts. Gv good tl moto stmt from th ftur mtchg roch, vr ccurt rgstrto c b clcultd usg drct mthods. Ths tchqus mmz th moto comstd mg dffrc gv s m E m (, γ ( m (, γ ( I (, I (,. (8 I(, dots th mg brghtss t osto (, d I (, dots th brghtss t th corrsodg l (ccordg to th slctd moto modl th othr mg. For th momt, w ssum tht γ (,.., w us th sum of squrd dffrcs s dffrc msur. Ltr, w wll rlc ths b robust M-stmtor. Mmzto of E wth rsct to th moto modl rmtr vctor s dffcult roblm d c ol b tckld wth grdt dsct tchqus. W r usg Lvbrg-Mrqurdt [9,,5] mmzr bcus of ts stblt d sd of covrgc. Th lgorthm s combto of ur grdt dsct d mult-dmsol Nwto lgorthm Lvbrg-Mrqurdt-Mmzto ( Strtg wth stmto, th grdt dsct rocss dtrms ( th t stmto b tkg smll st dow th grdt ( ( α E (, (9 whr α s smll costt, dtrmg th st sz. O roblm of ur grdt dsct s th choc of good α sc smll vlus rsult slow covrgc rt, whl lrg vlus m ld fr w from th mmum. Th scod mthod s th Nwto lgorthm. Ths lgorthm ssums tht f ( s r mmum, th fucto to b mmzd c oft b romtd b qudrtc form Q s E Q E ( E ( E ( whr E dots th Hss mtr of E. If th Hss mtr s ostv dft, (m (m T, ( rg m Q ff Q. ( Hc, bcus of Q ( E ( E ( ( ( ( (

15 Sgmtto d Clssfcto of Movg Vdo Objcts 5 w c drctl jum to th mmum of Q b sttg ( ( E ( E (. (3 Not th smlr structur of ths quto comrd to Equto 9. Istd ( of tkg th vrs of th Hss, th st δ ( ( c lso b comutd b solvg th lr quto sstm ( E δ E. (4 Th Lvbrg-Mrqurdt lgorthm solvs two roblms t oc. Frst, th fctor α Equto 9 s chos utomtcll, d scod, th lgorthm combs stst dsct d Nwto mmzto to ufd frmwork. ( If w r comrg th uts of δ wth thos th Hss, w c s tht ol th dgol trs of th Hss rovds som formto bout scl. So, w st th st sz (ddtl for ch comot k s k λ ( E α. (5 kk λ s w sclg fctor whch s cotrolld b th lgorthm. To comb stst dsct wth th Nwto lgorthm, [9,] roos to df w mtr D wth d jk kk ( E for j k d ( E ( λ jk kk (lmts o th dgol Rlcg th Hss of Equto 4 wth D, w gt ( D δ E. (6 Not tht for λ Equto 6 rducs to Equto 4 (.., th Nwto lgorthm, whl for lrg λ th mtr D bcoms dgol domt d th lgorthm thus bhvs lk stst dsct lgorthm. Th Lvbrg- Mrqurdt mmzto lgorthm uss λ to cotrol th mmzto rocss d works s follows:. Choos tl λ (.g., λ.. ( δ.. Solv Equto 6 to gt th rmtr udt vctor 3. If E ( ( E (, th udt dos ot mrov th soluto. Hc, δ w crs λ b fctor of (to rduc th st sz d go bck to St. E E 4. If ( ( (, th udt mrovs th soluto. Hc, w st < δ ( ( ( δ d dcrs λ b fctor of.

16 Chtr Wh λ cds hgh thrshold, v th lst smll sts dd ot mrov th soluto d w sto. Adtg ths tchqu to our moto stmto roblm, w must dtrm th Hss mtr d grdt vctor for gv rmtr stmt. I th followg, w ssum th rsctv moto modl from Equto 4. Th grdt vctor c b dtrmd sl from Z I Z I 3 t Z I (7 7 I I Z wth th shortcut Z. Hc, wth ( γ th grdt vctor s sml E 8. (8 W smlf th comutto of th Hss b gorg th scod ordr drvtv trms: k j k j k j k j. (9 Hc, w gt k j jk E (. ( Alg M-stmtor Th lgorthm dscrbd bov ssums tht th whol mg movs ccordg to th stmtd moto modl. Howvr, our lcto, ths s ot th cs sc forgroud objcts grll mov dffrtl. Ths troducs lrg mtchg rrors rs of th forgroud objcts. Th cosquc s tht ths msmtch dstorts th stmto bcus th lgorthm lso trs to mmz th mtchg rror th forgroud rgo. As th tru objct osto s ot kow t, t s ot ossbl to clud th forgroud rgos from th stmto rocss. Ths roblm c b

17 Sgmtto d Clssfcto of Movg Vdo Objcts 7 llvtd b usg lmtd rror fucto for γ ( std of th squrd rror. For smlct, w us for < t γ ( t ls. Itroducg ths rror fucto to Equto 8 d comutg th grdt d Hss s rtculrl sml s γ ( f < t ls. Fgur 9 shows two dffrc frms, th frst usg squrd rror s mtchg fucto d th scod usg th robust dstc fucto. It c b s tht th rgstrto rror th bckgroud rgo s smllr for th robust dstc fucto. (3 (3 ( (b Fgur 9: Dffrc frm ftr Lvbrg-Mrqurdt mmzto. ( shows rsduls usg squrd dffrcs s rror fucto, (b shows rsduls wth sturtd squrd dffrcs. It s vsbl tht th robust stmto chvs bttr comsto. Escll ot th tt th rght rt of th mg. 4. DETERMINING OBJECT MASKS Th rcl of our sgmtto lgorthm s to comut th dffrc of th currt frm to sc bckgroud mg whch dos ot cot forgroud objcts. Th bckgroud mg s utomtcll costructd from th squc such tht th bckgroud dts tslf to chgs or vrg llumto. Ev f th bckgroud s vr vsbl wthout forgroud objcts, th lgorthm s cbl to rtfcll rcrt t.

18 8 Chtr 5 Sc th dffrc btw ut mg d bckgroud cots much rror du to cmr os or smll rts of th objct hvg th sm color s th bckgroud, rgulrzto of th objct sh s ld to th dffrc frm. Fgur : Rcostructo of bckgroud bsd o comsto of cmr moto btw vdo frms. Th orgl vdo frms r dctd wth bordrs. 4. BACKGROUND RECONSTRUCTION Th moto stmto st rovds th moto modl j, j btw coscutv frms j d j. B cosdrg th trstv closur s th coctto of moto trsformtos, w c df ll j, k btw rbtrr frms j, k. If w f th frst frm s th rfrc coordt sstm for th bckgroud rcostructo, w c dd frm j to th bckgroud b lg th trsformto, j. To rvt th drft from slght rrors th moto stmto st, th drct stmto st s ot ld to succssv frms but to th ut frm wth rsct to th currt bckgroud mosc. Fgur shows how ut frms r ssmbld to combd mosc. I grl, th ut vdo wll cot forgroud objcts most of th frms. Howvr, t s mortt tht th rcostructd bckgroud dos ot cot ths objcts. As t s ot -ror clr whch rts r forgroud d whch r bckgroud, w df vrthg s bckgroud tht s stbl for t lst b frms. Th rcostructo lgorthm stors th lst b bckgroud moscs obtd so fr. Th rcostructd bckgroud mg s th dtrmd b lg tmorl md fltr [,] ovr ths cturs (cf. Fgur. Clrl, f t lst b cturs hv rl th sm color t l, ths vlus wll b st th bckgroud rcostructo.

19 Sgmtto d Clssfcto of Movg Vdo Objcts 9 md fltrg tm rcostructd bckgroud Fgur : Algd ut frms r stckd d l-ws md fltr s ld th tmorl drcto to rmov forgroud objcts. Ths roch works wll f th objcts r movg th sc. If th st too log t th sm osto, th wll vtull bcom bckgroud. A sml rcostructd bckgroud from th stf squc c b s Fgur. 4. CHANGE DETECTION MASKS Th rcl of our sgmtto lgorthm s to clcult th chg dtcto msk (CDM btw th bckgroud mg d th ut frms. I th r whr th forgroud objct s loctd, th dffrc btw bckgroud d ut frm wll b hgh. Not tht th roch of tkg th dffrc to rcostructd bckgroud hs svrl dvtgs ovr tkg dffrcs btw succssv frms:. Th sgmtto boudrs r mor ct. If dffrcs r comutd btw succssv frms, ot ol th w osto of objct wll hv lrg dffrcs, but lso th ucovrd bckgroud rs. Ths rsults og rtfcts bcus fst movg objcts r vsbl twc.. Objcts tht do ot mov for som tm or tht r ol movg slowl c ot b sgmtd. Morovr, slowl movg rgo wth lmost uform color would ol show dffrcs t th dgs succssv frms. 3. Th rcostructd bckgroud c b usd for objct-bsd vdo codg lgorthms lk MPEG-4 whr th bckgroud c b trsmttd ddtl (s so clld bckgroud-srt, whch rducs th rqurd bt-rt s ol th forgroud objcts hv to b trsmttd. ( frm (b frm (c dffrc Fgur : Comutg th dffrc btw succssv frms rsults uwtd rtfcts. Th frst two cturs show two ut frms wth forgroud objcts. Th rght ctur show th dffrc. Two kds of rtfcts c b obsrvd. Frst, th crcl rs twc sc th lgorthm cot dstgush btw rg d dsrg. Scod, rt of th r r of th olgo s ot flld bcus th ls ths r do ot chg thr brghtss.

20 Chtr IMPROVED CHANGE DETECTION BASED ON THE SSD-3 MEASURE Wth stdrd chg dtcto bsd o squrd or bsolut dffrcs, tcl rtfct c b obsrvd. If th mgs cot shr dgs or f ttur, ths structurs usull c ot b cclld comltl bcus of mror fltrg d lsg th mg cqusto rocss. Hc, f ttur d shr dgs r oft ccdtll dtctd s movg objct. O tchqu to rduc ths ffct s to us th sum of stl dstcs (SSD-3 msur [] to comut th dffrc frm. Th rcl of ths msur s to clcult th dstc tht l hs to b movd to rch l of smlr brghtss th rfrc frm. I th o dmsol cs, t s dfd s wth d SSD 3 m( d, d, d (33 I( I ( ( I ( ( ( d I. (34 For th two-dmsol cs, ths msur s comutd ddtl for th horzotl d vrtcl drcto d th mmum s tk. For dth lto of ths msur, s []. Th dffrc frms obtd wth ths msur comrd to stdrd squrd rror s dctd Fg. 3. ( squrd rror (b SSD-3 Fgur 3: Dffrc frms usg squrd rror d SSD-3. Not tht SSD-3 shows cosdrbl lss rrors t dgs cusd b lsg th sub-smlg rocss.

21 Sgmtto d Clssfcto of Movg Vdo Objcts ( ghborhood (b strght clqus (c dgol clqus Fgur 4: Dfto of l ghborhood. Pctur ( shows th two clsss of l ghbors; strght ( d dgol (. Ths two clsss r usd to df th scod ordr clqus. Strght clqus (b d dgol clqus (c. 4.4 SHAPE REGULARIZATION USING MARKOV RANDOM FIELDS If th grto of br objct msk from th dffrc frm s do l b l wth fd thrshold, w hv to fc lot of wrogl clssfd ls (cf. Fgur 5. Sc most rl objcts hv smooth boudrs, w mrov th sgmtto b sh rgulrzto, whch s do usg Mrkov rdom fld (MRF modl. Th forml dfto of our sgmtto roblm s tht w wt to ssg lbl out of th lbl st L{ bckgroud, forgroud } to ch l osto. For ch l osto thr s rdom vrbl F wth vlus f L. Th robblt tht l s ssgd lbl f s dotd s P ( F f. A rdom fld s Mrkov f th robblt of lbl ssgmt for l s ol ddt o th ghborhood of ths l: P F f F P( F f F, (35 ( I { } N ( whr FI { } dots th lbl cofgurto of th whol mg ct l d F N ( th cofgurto ghborhood of l. W df th ghborhood of s th 8-ghborhood, whl dffrttg btw strght d dgol ghbors (s Fgur 4. P F f F r usull hrd to df, Mrkov Sc th robblts ( N ( rdom flds r oft modlld s Gbbs rdom flds (GRF. It c b show tht both dscrtos r quvlt []. A GRF s dfd through th totl lbl cofgurto robblt P ( f s P( f U ( f T (36 Z whr Z s ormlzto costt to sur tht f P( f. I th followg, w wll lws st th tmrtur rmtr T. U ( f s th rg fucto, whch s dfd s U ( f V c ( f, (37 c C

22 Chtr 5 whch th sum s ovr ll clqus th mg d V c ( f s clqu ottl. Hghr clqu ottls rsult lowr robblts for ths clqu cofgurto. Clqus r substs of rltd l ostos th mg. I our lcto, w r usg Auto-Logstc modl, whch ol uss clqus of sgl ordr (th ls thmslvs d of scod ordr (cf. Fgur 4. Thus, th rg fucto c b wrtt s U ( f V f V f f. (38 ( (, N ( Frst ordr clqu ottls V ( r st ccordg to th dffrc frm formto,.., how robbl l blogs to forgroud objcts gv ts dffrc frm vlu d (. V f d ( β β (. ( d ( for for f f forgroud, d bckgroud. (39 Th scod ordr clqu ottls r st such tht smooth rgos r rfrrd. I.., clqus whch cot dffrt lbls r ssgd mor rg. Mor scfcll, w us V ( f, f μ μ f f f f f f. (4 Th rmtr μ s st dffrtl for th two ts of clqus wth lowr vlus for dgol clqus s th corrsodg ls r frthr w. Th lbl cofgurto tht mmzs th totl fld robblt (Eq. 36 s obtd through trtv Gbbs smlg lgorthm []. Fgur 5b shows th sgmtto msk obtd wth our MRF modl comrd to l bsd clssfcto. Alg MRF-bsd clssfcto to ch dffrc frm lds br objct msks for th tr vdo. ( r-l clssfcto (b MRF-bsd clssfcto Fgur 5: Sgmtto rsults for r-l dcso btw forgroud d bckgroud objct d MRF bsd sgmtto

23 Sgmtto d Clssfcto of Movg Vdo Objcts 3 5. VIDEO OBJECT CLASSIFICATION Th utomtc sgmtto rocdur s dscrbd bov rovds objct msks for ch frm of th vdo. Bsd o ths msks, furthr hgh-lvl rocssg sts r ossbl. To bl smtc sc lss, t s rqurd to ssg objct clsss (.g., rsos, mls, crs to th dffrt msks. Furthrmor, objct bhvor (.g., rso s sttg, stds u, wlks w c b dscrbd b obsrvg th objct ovr tm. I our roch, objct clssfcto s bsd o comrg slhoutts of utomtcll sgmtd objcts to rototcl objcts stord dtbs. Ech rl-world objct s rrstd b collcto of two-dmsol rojctos (or objct vws. Th slhoutt of ch rojcto s lzd b th curvtur scl sc (CSS tchqu. Ths tchqu rovds comct rrstto tht s wll sutd for dg d rtrvl. Objct bhvor s drvd b obsrvg th trstos btw objct clsss ovr tm d slctg th most robbl trsto squc. Sc frqut chgs of th objct clss r ulkl, occsol fls clssfctos rsultg from rrors whch occurrd rvous rocssg sts r rmovd. 5. REPRESENTING SILHOUETTES USING CSS Th curvtur scl sc tchqu [,3,4] s bsd o th d of curv voluto,.., bscll th dformto of curv ovr tm. A CSS mg rovds mult-scl rrstto of th curvtur zro crossgs of closd lr cotour. Cosdr closd lr curv Γ(u rrstg objct vw, Γ( u {( ( u, ( u u [,] }, (4 wth th ormlzd rc lgth rmtr u. Th curv s smoothd b o-dmsol Guss krl g( u, σ of wdth σ. Th dformto of th closd lr curv s rrstd b Γ( u, σ {( X ( u, σ, Y ( u, σ u [,] }, (4 X u d Y u dot th comots (u d (u ftr g. Vrg σ s quvlt to choosg fd σ whr (, σ (, σ covoluto wth ( u, σ d lg th covoluto trtvl. Th curvtur (, σ drvtvs (, σ (, σ (, σ κ u of volvd curv c b comutd usg th X u u, X uu u, Y u u, d Y uu ( u, σ s X u ( u, σ Yuu ( u, σ X uu ( u, σ Yu ( u, σ κ ( u, σ. (4 3/ ( X ( u, σ Y ( u, σ A CSS mg I ( u, σ s dfd b {( u, σ κ ( u, } u u I ( u, σ σ. (43

24 4 Chtr 5 It cots th zro crossgs of th curvtur wth rsct to thr osto o th cotour d th wdth of th Guss krl (or th umbr of trtos, s Fgur 6. Durg th dformto rocss, zro crossgs vsh s trstos btw cotour sgmts of dffrt curvtur r smoothd out. Cosqutl, ftr crt umbr of trtos, flcto ots dsr d th sh of th closd curv bcoms cov. Not tht du to th ddc o curvtur zro crossgs, cov objct vws cot b dstgushd b th CSS tchqu. Sgfct cotour rorts tht st tct for lrg umbr of trtos rsult hgh ks th CSS mg. Sgmts wth rdl chgg curvturs cusd b os roduc ol smll locl mm. I m css, th ks th CSS mg rovd robust d comct rrstto of objct vws cotour. Not tht rotto of objct vw o th mg l c b ccomlshd b shftg th CSS mg lft or rght horzotl drcto. Furthrmor, mrrord objct vw c b rrstd b mrrorg th CSS mg. Ech k th CSS mg s rrstd b thr vlus, th osto d hght of th k d th wdth t th bottom of th rc-shd cotour. Th wdth scfs th ormlzd rc lgth dstc of th two curvtur zro crossgs frmg th cotour sgmt rrstd b th k th CSS mg [6]. It s suffct to trct th sgfct mm (bov crt os lvl from th CSS mg. For stc, th ml dctd Fgur 6, ol fv dt trls rm d hv to b stord ftr smll umbr of trtos. Th dtbs dscrbd th followg scto stors u to sgfct dt trls for ch slhoutt. trtos ( (b (c (d rc lgth Fgur 6: Costructo of th CSS mg. Lft: Objct vw ( d trtvl smoothd cotour (b-(d. Rght: Rsultg CSS mg.

25 Sgmtto d Clssfcto of Movg Vdo Objcts 5 5. BUILDING THE DATABASE Th ortto of objct d th osto of th cmr hv grt mct o th slhoutt of th objct. Thrfor, to bl rlbl rcogto, dffrt vws of objct hv to b stord th dtbs. Rgd objcts c b rrstd b smll umbr of dffrt vws,.g., for cr, th most rlvt vws r frotl vws, sd vws, d vws whr frotl d sd rts of th cr r vsbl. For o-rgd objcts, mor vws hv to b stord th dtbs. For stc, th cotour of wlkg rso vdo chgs sgfctl from frm to frm. Smlr vws of o t of objct r ggrgtd to o objct clss. Our dtbs stors 75 slhoutts collctd from cl rt lbrr d from rl-world vdos. Th lrgst umbr of mgs show ol (4 mgs, mls (67 mgs, d crs (48 mgs. Bsd o bhvor, th objct clss ol s subdvdd to th followg objct clsss: stdg, sttg, stdg u, sttg dow, wlkg, d turg roud. 5.3 OBJECT MATCHING Ech utomtcll sgmtd objct vw s comrd to ll objct vws th dtbs. I frst st, th sct rto, dfd s quott of objct wdth d hght, s clcultd. For two objcts vws wth sgfctl dffrt sct rtos, o mtchg s crrd out. If both sct rtos r smlr, th ks th CSS mgs of th two objct vws r comrd. Th bsc sts of th mtchg rocdur r summrzd th followg [6]: Frst, both CSS rrsttos hv to b lgd. For ths uros t mght b cssr to rott or mrror o of thm. As mtod bov, shftg th CSS mg corrsods to rotto of th orgl objct vw. To lg both rrsttos, o of th CSS mgs s shftd so tht th hghst k both CSS mgs s t th sm osto. A mtchg k s dtrmd for ch k gv CSS rrstto. Two ks mtch f thr hght, osto d wdth r wth crt rg. If mtchg k s foud, th Eucld dstc of th ks th CSS mg s clcultd d ddd to dstc msur. If o mtchg k c b dtrmd, th hght of th k s multld b lt fctor d ddd to th totl dffrc.

26 6 Chtr 5 Fgur 7: Wghts for trstos btw objct clsss. Thckr rrows rrst mor robbl trstos. 5.4 CLASSIFYING OBJECT BEHAVIOR Th mtchg tchqu dscrbd bov clcults th bst dtbs mtch for ch utomtcll sgmtd objct vw. Sc th dtbs trs r lbld wth rort clss m, th vdo objct c b clssfd ccordgl. Th objct clss ssgd to sgmtd objct msk c chg ovr tm bcus of objct dformtos or mtchg rrors. Sc th robblt of chgg from o objct clss to othr dds o th rsctv clsss, w ssg ddtoll mtchg costs for ch clss chg. Lt d k ( dot th CSS dstc btw ut objct msk t frm d objct clss k from th dtbs. Furthrmor, lt w k, l dot th trsto cost from clss k to l (cf. Fgur 7. Th, w sk th clssfcto vctor c, ssgg objct clss to ch ut objct msk, whch mmzs m c d ( w. (44 c c, c Ths otmzto roblm c b solvd b shortst th srch s dctd Fgur 8. Wth rsct to th fgur, d k ( corrsods to costs ssgd to th ods d w k, l corrsods to costs t th dgs. Th otml th c b comutd ffctl usg dmc rogrmmg lgorthm. Th objct bhvor c b trctd sl from th ods log th mmum cost th.

27 Sgmtto d Clssfcto of Movg Vdo Objcts 7 rso (wlkg frm frm frm 3 w frm 4 frm 5 lst frm, ut frms w, d (5 objct clsss rso (std u rso (stttg 3 d (5 d (5 3 cr 4 th wth mml totl cost trsto costs btw objct clsss d (5 4 d kot wth mml totl costs Fgur 8: Etrcto of objct bhvor. 6. SYSTEM IMPLEMENTATION AND RESULTS As outld bov, our objct clssfcto sstm cossts of motobsd sgmtto modul d sh-bsd clssfcto modul. W strt b dscussg rmtl rsults chvd b our sgmtto modul. For ths uros, th sgmtto lgorthm ws ld to two rl-world squcs, ml th stf squc usd throughout ths chtr d rod squc rcordd b hdhld cmr. Fgurs 9 d dct som of th rsults. I Fgur, th rcostructd bckgroud of th stf squc s dsld. W obsrv tht th sgmtto modul srts th movg objcts from th bckgroud vr wll. I th cs of th stf squc, som movg rts th udc r dtctd. I th rod squc th crs (d dstr r trctd vr rcsl. I ordr to clssf th objcts rsultg from th sgmtto rocss, th clssfcto lgorthms r ld to thr shs. Fgur dsls som sgmtto msks obtd b our utomtc sgmtto d th rsctv bst mtch clcultd from th sh dtbs. I thr css th clssfcto s succssful d lds rsobl mtch. I ddto, w obsrv o msmtch whch s du to sgmtto rror d th fct tht th dtbs dd ot cot rort rrstto of th rug ts lr. Fll, lt us cosdr th trcto of objct bhvor. I Fgur 3, th lft mg dsls trg squc usd to df objct rotots for scfc objct bhvor th dtbs. Th rght mg dcts tst squc wth th utomtcll ssgd clss lbls. Th dffrt stgs of th objct bhvor wr dtrmd from th shortst th clcultd from th CSS mtchg rsults d th objct frms wr slctd from th mddl btw clss trstos.

28 8 Chtr 5 Fgur 9: Sgmtto rsults of stf tst squc (frms 4, 8,, 6. Fgur : Rcostructd bckgroud from stf squc. Not tht th lr s ot vsbl v though thr s o ut vdo frm wthout th lr.

29 Sgmtto d Clssfcto of Movg Vdo Objcts 9 Fgur : Sgmtto rsults of rod tst squc (frms, 4, 7, 75. Fgur : CSS mtchg rsults for slctd frms of th stf -squc.

30 3 Chtr 5 wlkg stdg wlkg std u sttg stdg wlkg stdg wlkg stdg std u sttg ( (b Fgur 3: Automtcll trctd bhvor dscrto. REFERENCES [] S. Abbs d F. Mokhtr. Sh smlrt rtrvl udr ff trsform: Alcto to mult-vw objct rrstto d rcogto. Proc. Itrtol Cofrc o Comutr Vso, IEEE, 999. [] D. Fr d P. H. N. d Wth. A w smlrt msur for subl ccurt moto lss objct-bsd codg.procdgs of th 5th World Mult-Cofrc o Sstmcs, Cbrtcs d Iformtcs (SCI, , Jul. [3] O. D. Fugrs. Thr-dmsol Comutr Vso: A Gomtrc Vwot. MIT Prss, Cmbrdg, MA, 999. [4] M. Fschlr d R. Bolls. Rdom sml cocsus: A rdgm for modl fttg wth lctos to mg lss d utomtd crtogrh. Commuctos ACM, 4(6:38-395, 98. [5] C. Hrrs d M. Sths. A combd corr d dg dtctor. Proc. Alv Vso Cofrc,. 47-5, 988. [6] R. Hrtl d A. Zssrm. Multl vw gomtr comutr vso. Cmbrdg Uvrst Prss, Cmbrdg,. [7] B. K. P. Hor. Robot Vso. MIT Prss, Cmbrdg, MA, 986. [8] M. Ir, P. Ad. About Drct Mthods. Vso Algorthms: Thor d Prtc, Itrtol Worksho o Vso Algorthms, 67-77, 999.

31 Sgmtto d Clssfcto of Movg Vdo Objcts 3 [9] K. Lvbrg. A mthod for th soluto of crt roblms lst squrs. Qurt. Al. Mth., :64-68, 944. [] S. Z. L. Mrkov Rdom Fld Modlg Comutr Vso. Artfcl Itllgc. Srgr-Vrlg, Toko, 995. [] D. Mrqurdt. A lgorthm for lst-squrs stmto of olr rmtrs. SIAM J. Al. Mth., :43-44, 963. [] M. Mss d W. Bdr. Slt stlls: Procss d rctc. IBM Sstms Jourlm, 35(3/4: , 996. [3] F. Mokhtr, S. Abbs, d J. Kttlr. Effct d robust rtrvl b sh cott through curvtur scl sc. Proc. Itrtol Worksho o Img DtBss d MultMd Srch,. 35-4, 996. [4] F. Mokhtr, S. Abbs, d J. Kttlr. Robust d ffct sh dg through curvtur scl sc. Brtsh Mch Vso Cofrc, 996. [5] W. H. Prss, S. A. Tukolsk, W. T. Vttrlg, d B. P. Flr. Numrcl Rcs C : Th Art of Sctfc Comutg. Cmbrdg Uvrst Prss, Nw York, 99. [6] S. Rchtr, G. Küh, d O. Schustr. Cotour-bsd clssfcto of vdo objcts. Procdgs of SPIE, Storg d Rtrvl for Md Dtbss, volum 435, ,. [7] P. J. Roussuw d A. M. Lro. Robust Rgrsso d Outlr Dtcto. Joh Wl, Nw York, 987. [8] P. J. Roussuw d K. V Drs. Comutg LTS rgrsso for lrg dt sts. Isttut of Mthmtcl Sttstcs Bullt, 7(6, Novmbr/Dcmbr 998. [9] C. Schmd, R. Mohr, d C. Buckhg. Evluto of trst ot dtctors. Itrtol Jourl of Comutr Vso, 37(:5-7, Ju. [] R. Szlsk. Img moscg for tl-rlt lctos. Tchcl Rort 94/, Dgtl Equmt Cororto, Cmbrdg Rsrch, Ju 994. [] L. Todoso d W. Bdr. Slt vdo stlls: cott d cott rsrvd. ACM Multmd, 993. [] P. H. S. Torr, A. Zssrm. Ftur bsd mthods for structur d moto stmto. Vso Algorthms: Thor d Prtc, Itrtol Worksho o Vso Algorthms, 78-94, 999.

32 3 Chtr 5 INDEX Cmr moto 3 Prsctv moto modl 4 Aff moto modl 4 Trsltorl moto modl 4 Ftur-bsd stmto 5 Grdt-bsd stmto 5 Drct mthods 5 Hrrs corr dtctor 6 Ftur corrsodcs 9 Modl rmtr stmto Lst squrs rgrsso Robust stmto Lst-trmmd squrs rgrsso Lvbrg-Mrqurdt mmzto 4 M-stmtor 6 Bckgroud rcostructo 8 Chg dtcto msks 9 Sum of stl dstcs Mrkov rdom flds Objct clssfcto 3 Curvtur scl sc 3 Objct bhvor 6