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1 Computng Multpl Imag Motons S. S. Bauchmn, K. Danlds and R. Bajcsy GRASP Laboratory Dpartmnt of Computr and Informaton Scnc Unvrsty of Pnnsylvana Phladlpha PA USA Abstract Th computaton of mag moton for th purposs of dtrmnng gomoton s a challngng task as mag moton ncluds dscontnuts and multpl valus mostly du to scn gomtry, surfac translucncy and varous photomtrc cts such as surfac rctanc. W prsnt algorthms for computng multpl mag motons arsng from occluson and translucncy whch ar capabl of xtractng th nformaton-contnt of occluson boundars and dstngush btwn thos and addtv translucncy phnomna. Sts of xprmntal rsults obtand on synthtc mags ar prsntd. Ths algorthms ar basd on rcnt thortcal rsults on occluson and translucncy n Fourr spac. 1 Introducton Th mportanc of moton n mag procssng cannot b undrstatd: n partcular, approxmatons to mag moton may b usd to stmat 3D scn proprts and moton paramtrs from a movng vsual snsor, to prform moton sgmntaton, to comput th focus of xpanson and tm-to-collson, to prform motoncompnsatd mag ncodng, to comput stro dsparty and to masur bologcal paramtrs n mdcal magry [1] Basd on rcnt thortcal dvlopmnts n dscontnuous moton, w dvs multpl moton algorthms. W consdr 1D and D sgnals, adopt a constant modl of vlocty and us a robust statstcal procdur to xtract multpl motons from local frquncy spctra. Th moton nformaton provdd by th algorthms ncluds sngl vlocty, multpl () vlocts, assssmnt of transparncy vrsus occluson, and upon occluson vnts, th orntaton of th occluson boundary and th dntcaton of th occludng sgnal. 1.1 Ltratur Survy Many phnomna may caus multpl mag motons. Occlusons, translucncs and varous photomtrc ffcts such as spcularts ar among probabl causs. In addton, occlusons contan valuabl nformaton concrnng th gomtry of th scn and may b usd to dcoupl optcal ow lds nto thr rotatonal and translatonal componnts, dntfy dpth dscontnuts, sgmnt th scn wth rspct to moton and so on. Computng multpl motons s a complx and rarly undrtakn task. Indd, most of th xstng optcal ow mthods that hav appard n th ltratur mak an xplct us of th optcal ow constrant quaton ri T v? I t ; (1.1) whr ri (I x ; I y ) T s th spatal ntnsty gradnt and v (u; v) T s th mag vlocty. At moton dscontnuts, whr th nformaton contnt of a sgnal mostly rsds, th us of (1.1) bcoms problmatc as th sngl moton hypothss s volatd. Ara-basd and fatur-basd corrlaton tchnqus ar qually snstv to occluson as local mag structurs and faturs appar and dsappar from on mag to th nxt. To furthr complcat mattrs, rgularzaton tchnqus whch mpos a dgr of contnuty to optcal ow ar also clarly nadquat ovr occluson boundars. Howvr, n th mor rcnt rsarch n optcal ow, th non-lnar, dscontnuous and multplvalud natur of mag moton n th coordnats of th mag plan has bn rcognzd [1]. In ordr to allow multpl moton vnts n optcal ow stmaton procsss, a numbr of stratgs hav bn dvsd, such as strong ntnsty gradnts actng as nhbtors of ow cohrnc [8] and robust stmators dsgnd to captur domnant motons [3]. Othr tchnqus such as clustrng [9], suprposd moton layrs and dstrbutons [1], paramtrc modls of moton wth dscontnuous functons [4] and mxturs of probablty dnsts [7] hav appard. 1

2 Our approach manats from rcnt thortcal rsults [] dscrbng th Fourr structur of occluson and translucncy phnomna for constant and lnar modls of optcal ow. 1. Modls of Optcal Flow Th optcal ow functon may b xprssd as an ordr n functon of th mag coordnats. Gnrally, w may wrt th Taylor srs xpanson for a th vlocty as: v (x; t) px qx rx j k j+k+l v j!k!l!@x l xj y k t l ; (1.) whr p + q + r n. For nstanc, th rst-ordr xpanson s wrttn as v (1) (x; t) J x + a t, whr J a1 a b 1 b s th Jacob matrx and a T?(a 3 ; b 3 ) s translaton 1. W adopt n what follows a constant modl of optcal ow. Structur of Occluson W procd to dscrb th structur of occluson vnts n th frquncy doman for 1D and D sgnals composd of an arbtrary numbr of dstnct frquncs. Lt I 1 (x) and I (x) b 1D functons satsfyng Drchlt condtons such that thy may b xprssd as complx xponntal srs xpansons: I 1 (x) I (x) n n c 1n nk1x c n nkx ; (.3) whr n s ntgr, c 1n and c n ar complx cocnts and k 1 and k ar th fundamntal frquncs of both sgnals. Lt I 1 (x; t) I 1 (v () (x; t)) and I 1 (x; t) I (v () (x; t)). Th frquncy spctrum of th occluson s: ^I(k;!) c 1n (k? nk 1 ;! + nk 1 a 1 ) n + (1? ) n c n (k? nk ;! + nk a ) 1 W us a ngatv translatonal rat wthout loss of gnralty and for mr mathmatcal convnnc. Th constant modl may b smply dnd as a lnar modl wth J I, yldng v () (x; t) x? a t. + n? c 1n(ka 1 +!) (k? nk 1 ) cn (ka 1 +!? nk a) (k? nk ) ; (.4) whr a a 1? a. In th 1D cas, quaton (.4) rvals that th frquncy spctra of both sgnals ar prsrvd to wthn scalng factors. In addton, th Drac dlta functons (ka 1 +!) and (ka 1 +! k a) consttut lnar spctra, ntrsctng th frquncs of both th occludng and occludd sgnals, and ar orntd n th drcton of th constrant ln prtanng to th occludng sgnal. Fgur.1 shows a typcal xampl wth 1D translatng snusods n an occluson scn. In th D cas, quaton (.7) shows smlarts wth (.4). Th frquncy spctra ar planar and prsrvd to wthn scalng factors undr occluson and th dstortons cast by th occluson boundary t orntd plans paralll to th plan contanng th spctrum of th occludng sgnal. Lt I 1 (x) and I (x) b D functons satsfyng Drchlt condtons such that thy may b xprssd as complx xponntal srs xpansons: I 1 (x) I (x) ~ ~ n? ~1 n? ~1 c 1n xt Nk 1 c n xt Nk ; (.5) whr n (n x ; n y ) T and N n T I ar ntgrs, x ar spatal coordnats, k 1 (k 1x ; k 1y ) T and k (k x ; k y ) T ar fundamntal frquncs and c 1n and c n ar complx cocnts. Lt I 1 (x; t) I 1 (v () 1 (x; t)), I (x; t) I (v () (x; t)) and th occludng boundary b locally rprsntd by: U(x) 1 f x T n 1 othrws, (.6) whr n 1 s a vctor normal to th occludng boundary at x. Th frquncy spctrum of th occluson s: ^I(k;!) ~ n? ~1 c 1n (k? Nk 1 ;! + a T 1 Nk 1) ~ +(1? ) c n (k? Nk ;! + a T Nk ) n? ~1 ~1; X c1n ((k? Nk 1 ) T n? 1? ; kt a 1 +!) (k? Nk 1 ) T n 1 n? ~1

3 + c n((k? Nk ) T n?; 1 kt a 1 +!? a T Nk ) (k? Nk ) T n 1 (.7) whr a a 1? a. Equaton (.7) s a gnralzaton of quaton (.4) from 1D to D sgnals and ts gomtrc ntrprtaton s smlar. For nstanc, frquncs (Nk 1 ;?a T 1 Nk 1 ) and (Nk ;?a T Nk ) t th constrant plans of th occludng and occludd sgnals, dnd as k T a 1 1 +! and k T a +!. In th dstorton trm, th Drac functon wth argumnts (k? Nk ) T n? 1 and k T a 1 +!? a T Nk rprsnt a st of lns paralll to th constrant plan of th occludng sgnal k T a 1 +! and, for vry dscrt frquncy Nk 1 and Nk xhbtd by both sgnals, thr s a frquncy spctrum ttng th lns gvn by th ntrscton of plans k T a 1 +!? a T Nk and (k? Nk 1 ) T n? 1. Th magntuds of ths spctra ar dtrmnd by thr corrspondng scalng functons c 1n [(k? Nk 1 ) T n 1 ] and c n [(k? Nk ) T n 1 ]. 3 Estmaton of Multpl Imag Moton Equatons (.4) and (.7) provd a modl of th Fourr spctrum at an occluson boundary. W dvs svral algorthms opratng on local Fourr transforms whch ar capabl of xtractng multpl vlocty masurmnts along wth th nformaton-contnt of occluson boundars D Algorthm Gvn a frquncy masurmnt (^k j ; ^! j ), ts corrspondng vlocty stmat s gvn by ^v (?^! j ^k j ; 1) T. In optmal condtons, th non-zro spctrum of a purly translatng mag sgnal should b ntrly consstnt wth ts vlocty. That s to say, vry frquncy masurmnt (^k j ; ^! j ) should b consstnt wth v, th tru sgnal vlocty. Howvr, owng to multpl factors such as acquston nos, sgnal dformatons and dvatons from th locally constant vlocty modl, t may b that som varabllty n th dgr of agrmnt btwn masurmnts and th tru vlocty xsts. In lght of ths, an rror mtrc, corrspondng to th angular dvaton btwn a masurmnt ^m j (^k j ; ^! j ) and an stmat of th th vlocty ^v may b dnd as [7]! ^m T ( ^m j ; ^v ) sn j ^v : (3.8) k ^m j k k^v k v Fgur 3.: Th gomtry of th angular rror masur. Th ln dnd by ^k j (^k j ; ^! j ) T should b prpndcular to th ln paralll to v (?! j k j ; 1) T, as dpctd by vctor ^k j (k j ;! j ) T. In addton, t s mathmatcally convnnt to smplfy th rror mtrc and us th sn of th angl as th amount of dvaton: ( ^m j ; ^v ) kj θ ^ k j ^m T j ^v k ^m j k k^v k : (3.9) Undr th assumpton that th angular rror s normally dstrbutd, w dn a mxtur modl of normal dstrbutons to account for multpl motons. Consdr G to b th st of masurmnts ^m j, j 1; : : : ; n. Th probablty dnsty functon for ^m j G s rprsntd by th mxtur of g normal dstrbutons: f( ^m j ; ) gx 1 f ( ^m j ; ); (3.1) whr f ( ^m j ; ) s a normal probablty dnsty functon, ( 1 ; : : : ; g ; v 1 ; : : : ; v g ) T s th vctor of th mxtur paramtrs and (v 1 ; : : : ; v g ) T s th vctor of normal dstrbuton paramtrs. s th s th probablty of ^m j to b from normal dstrbuton f. Th ar mxtur probablts and thus must satsfy gx 1 1 (3.11) In addton, th mxtur paramtrs must satsfy th lklhood quaton gx nx 1 j1 whch ylds th constrants ln f ( ^m j ; (3.1) P P n j n j ^m j1 and v n? j j1 n ; (3.13) whr j s th postror probablty that ^m j blongs to f and ^m? j (?^! j^kj ; 1) T [7]. In ths mxtur modl,

4 Fgur.1: (from lft to rght): a) Gaussan-wndowd 1D sgnal wth snusodals actng as occludng and occludd surfacs. Th occludng sgnal has spatal frquncy k 1 and vlocty v 16 1 (1; 1). Th occludd sgnal has frquncy k and vlocty v 8 (; 1). b) Fourr spctrum of a). c) Gaussan-wndowd D sgnal wth snusodals actng as occludng and occludd surfacs. Th occludng sgnal has spatal frquncy k 1 ( ; ) and vlocty v 1 (1; 1; 1). Th occludd sgnal has frquncy k ( ; ) and vlocty v 8 8 (; ; 1). d) Fourr spctrum of c). w hypothsz homoscdastcty, that s to say, th normal dstrbutons wthn th mxtur shar th sam standard dvaton, whch w consdr as a constant. W also us an outlr dtcton mchansm basd on Jpson and Black's modl. W rst pos th hypothss that outlyng masurmnts ar unformly dstrbutd ovr th paramtr spac of th mxtur, and thus w us a constant masur for th outlr probablty of a masurmnt. W only updat a mxtur proporton for thos. Constrants ^m j at a prdtrmnd dstanc from othr dstrbutons should b consdrd as nosy masurmnts and not ntr th vlocty stmaton procss. Th constant probablty of obsrvng a nosy masurmnt can b xprssd as? 1 p n ; (3.14) v from whch w not that masurmnts at standard dvatons from th mans of th normal dstrbutons ar consdrd as corruptd by nos. Furthr, th magntud of masurmnts ^m j ar rlvant as th frquncs composng th dstorton trms ar typcally smallr n magntud than th frquncs of th sgnals from whch thy orgnat. In lght of ths, w ncorporat th magntud as th strngth of masurmnts P by rplacng n, th numbr of masurmnts n by ( ^m j j), whr s a masur of th magntud of th local Fourr transform at ^m j. Wth th hypothss of homoscdastcty, constant standard dvaton and unform dstrbuton of nosy masurmnts, w stablsh th tratv quatons for th Expctaton-Maxmzaton algorthm. Th xpctaton stp s th computaton of postror probablts for th normal dstrbutons, whch w wrt as j P g t1 ^(k) t ( ^m j;^v ) v ( ^m j;^v t) (k) v + ^? n (3.15) for 1; : : : ; g and j 1; : : : ; n and, for th unform dstrbuton of nosy masurmnts, w wrt j P g t1 ^(k) t? v ( ^m j;^v t) (k)? v + ^ n (3.16) for j 1; : : : ; n. Th quatons for th maxmzaton stp, n whch th paramtrs of th dstrbutons ar updatd, ar wrttn as follows for th mans ^v (k+1) P n (k) j1 ^ j ( ^m j) ^m? j P n ( ^m j1 j) (3.17) for 1; : : : ; g, and th mxtur proportons ar updatd as ^ (k+1) P n j1 ( ^m j) j P n j1 ( ^m j) (3.18) for ; : : : ; g. Fgur 3.3a shows an xampl of obsrvatons randomly chosn from a suprpopulaton composd of angular normals and a unform dstrbuton. Th mxtur paramtrs ar ( 1 :4; :4; v 1 1; v ) T for th normals and ( :) for th unform dstrbuton. An EM algorthm wth angular rror masur (3.9) was appld to ths st of obsrvatons. Aftr 15 tratons th algorthm convrgd to :18, 1 :411, :46, v 1 1:17 and v :. Fgurs 3.3b and c show obsrvatons for whch th nal postror probablts j ar abov.95

5 Fgur 3.3: a (lft): Mxtur of two angular normals randomly gnratd wth :, 1 :4, :4, v 1 (1; 1), v (; 1) and v :75 radans. b) (cntr): EM rsults for obsrvatons wth 1j > :95 and c) (lft): wth j > :95. Th algorthm convrgd to :18, 1 :411, :46, v 1 (1:17; 1) and v (:; 1) n 15 tratons. Masurmnts at a dstanc :5 standard dvatons ar consdrd as outlrs. for 1;. Thrsholdng on th postror probablts allows to assocat th obsrvatons wth th varous probablty dnsty functons composng th mxtur. In ordr to dntfy th spctra assocatd wth occludng boundars, w rst nd pak frquncy masurmnts for both sgnals. That s to say, w nd for sgnal t, th frquncy ^m t such that tk > k for t 6 and ( ^m t ) s maxmal and dtrmn th strngth of masurmnts ^m j along th drcton prpndcular to th hypothszd occludng vlocty at th pak frquncy of th hypothszd occludd sgnal. Thrfor, two tsts ar prformd n ordr to vrfy whch of both hypothss s th corrct on. To tst for th sgnal corrspondng to vlocty v as occludng, th procdur s to rst consdr only thos masurmnts ^m j blongng to th unform dstrbuton of th mxtur: j > j, for 1; and j 1 : : : n, as dtrmnd by th EM algorthm and th pak frquncy of th sgnal corrspondng to vlocty v t, whr t 6. W thn procd wth th computaton of th strngths of masurmnts conrmng ths hypothss. Among masurmnts blongng to th unform nos dstrbuton, w comput thr postror probablty of bng part of th dstorton spctra cast by th hypothszd occluson as: ^ j ( ^m t ; ^v ) v (( ^m j? ^m t);^v ) v : (3.19)? (( ^m j? ^m t);^v ) + n W also dtrmn th postror probablts of th masurmnts to b from th unform nos dstrbuton to th xcluson of th spctra of th occluson as: ^ j ( ^m t ; ^v ) v? n : (3.)? (( ^m j? ^m t);^v ) + Mxtur proportons may b obtand from ths postror probablts that assss th hypothss undr tst. Ths proportons ar computd as: P n j1 ^ (^v ) ( ^m j)^ j ( ^m t ; v ) P n ( ^m (3.1) j1 j) for ; 1. Thus, f vlocty ^v s occludng, thn th strngths of masurmnts conrmng ths hypothss outnumbr thos prtanng to ts contrary and thus n (^v ) (^v ) > (^v t ) (^v t ) : (3.) Hnc, varous hypothss-tstng mthods may b appld to dtrmn th mag vnts gvng rs to multpl vlocts. 3. D Algorthm Th algorthm for D sgnals s ssntally smlar to th 1D algorthm w dscrbd. Th masurmnts ^m j (k xj ; k yj ;! j ) T and vlocty stmats ^v (v x ; v y ; v t ) T ar usd n th rror mtrc (3.9) to dtrmn th postror probablts j, as s th cas wth th 1D algorthms. Howvr, th choc of vlocty stmats drs substantally. In th cas of D sgnals, th vlocty stmats at ach EM traton must maxmz

6 th numrator xponntal of (3.15). In ths cas, w follow th approach adoptd by Jpson and Black [7], and consdr th squar of th rror mtrc (3.9) as th quaton for whch th solutons yld vlocty stmats. W obsrv that ( ^m j ; ^v ) may b wrttn n matrx form as (m T j v ) v T M j v (3.3) whr M j ^m j ^m T j. By slctng th gnvctor corrspondng to th mnmum gnvalu of M j for v, w mnmz (3.3). Snc M j s ral and symmtrc, ts gnvalus ar ral and non-dgnrat and th gnvctors form an orthogonal bass n th spac of masurmnts. In lght of ths obsrvatons, w dn P n (k) j1 occluson: j j (( ^m j? ^m t);^n) v P g t1 ^(k) t (( ^m j? ^m t);^n) (k) v + ^? n P g t1 ^(k) t? v (( ^m j? ^m t);^n) (k) v + ^? n : (3.5) (3.6) whr s th rror masur (3.8). Th stmat of th spctral orntaton and th mxtur proportons ar updatd as: ^n (k+1) P n (k) j1 ^ ( ^m j j)( ^m j? ^m t ) P n ( ^m j1 j) (3.7) (k+1) j P n ( ^m j1 j) ( ^m j)m j (3.4) ^ (k+1) P n j1 (k) ^ ( ^m j j) P n ( ^m j1 j) (3.8) as th matrx from whch th vlocty stmat v (k+1) s to b obtand n th form of th gnvctor (k+1) corrspondng to th mnmum gnvalu (k+1) of. Th mnmum gnvalu holds nformaton about th vlocty stmat obtand from ts corrspondng gnvctor. A zro valu for ndcats that th vlocty masurmnt s normal, whras a non zro valu ndcats a full vlocty masurmnt [5]. To s ths, consdr a st of obsrvatons consstng of collnar masurmnts, consstnt wth a normal vlocty. It s obsrvd that n such crcumstancs, th lns of matrx ar lnarly dpndnt, ladng to a mnmum gnvalu of valu zro. Thus, th nal gnvalus contan nformaton on th natur of th masurd vlocts that s vry rlvant n most uss of mag vlocty. Undr th hypothss of a straght-dgd occluson boundary, ts normal may b stmatd from th frquncy structur of th occluson. To prform ths stmaton, th algorthm must rcovr th orntaton of th spctrum cast by th occluson about th maxmum frquncy of th occludd sgnal, wthn a plan paralll to that of th occludng sgnal. To prform ths stmaton, t s ncssary to nclud an EM traton whch convrgs to ths lnar orntaton wthn th spcd constrant plan. W consdr only thos masurmnts whch ar consstnt wth th plan contanng th pak frquncy ^m t of th occludd sgnal and prpndcular to th occludng vlocty v, that s to say, w nd ^m j? ^m t such that k > tk, for t 6 k. W procd wth th computaton of postror probablts gvn an ntal stmat ^n () of th orntaton of th lnar spctra cast by th 4 Exprmnts W prformd numrcal xprmnts on synthtc snusodal magry composd of four 1D occluson scns and on D occluson squnc, as dscrbd by Fgur 4.4. Th mags usd n ths xprmnts ar vrtually fr from nos. Local frquncy masurmnts ar obtand for an mag locaton by computng a local Fast Fourr Transform wthn a rgon of sd sz 3. W obsrvd that 3 tratons wr sucnt for th EM algorthm to convrg. Th ntal stmats for vlocts and mxtur proportons may b chosn randomly, but w prfr to hav ntal vlocty stmats st as apart as possbl to avod convrgnc of both stmats to a sngl pak. Whn th EM tratons bgn, w st v to.618 radans, or 15 dgrs. At ach stp, w dcras v to obtan a nal valu of.1745, or 1 dgr. It s obsrvd that a largr valu for th standard dvaton durng th rst tratons brngs th ntal vlocty stmats n th nghborhood of th tru paramtrs whl a smallr valu for th last tratons mprovs th accuracy of th nal stmats. A valu of.5 for and 1. for n ar chosn for th unform dstrbutons. It was xprmntally dtrmnd that n ordr to assss th prsnc of multpl motons, th mxtur probablts must satsfy > 1 ; (4.9) whr 1 :3. In addton, to assss vlocty ^v as occludng, w rqurd that (^v ) (^v )? (^v t ) (^v t ) > ; (4.3)

7 Fgur 4.4: Synthtc magry and rsults wth k 1 and v 1 occludng. (top to bottom): 1D magry a) k 1 k, v 8 1 (1; 1) and v (; 1). b) k 1, k 16, v 8 1 (1; 1) and v (; 1). c) k 1 k v 1 (:5; 1) and v (; 1). d) k 1 k 16, v 1 (:5; 1) and v (?:75; 1). ) D magry k 1 ( 16 ; k ( 8 ; 8 ) v 1 (1; 1; 1) and v (; ; 1). (lft to rght): a) Synthtc mag. b) Optcal ow. c) Multpl vlocts. d) Occludng vlocts. ) Occludd vlocts., 16, 16 ), 16 whr 1: 1?3. Fgur 4.4 shows th rsults obtand on th occluson scns. Ths optcal ow lds ar vrtually fr from rror, du to th prfct natur of th synthtc magry. Howvr, w hav obsrvd that th dgr to whch ths algorthms ar capabl of dntfyng multpl vlocts that ar rla-

8 tvly smlar n thr orntaton s not vry satsfyng. Th vlocts must b at last 15 to dgrs apart n orntaton for th algorthms to yld a postv assssmnt of multpl vlocts. Issus such as th valus of th varous standard dvatons for th mxtur and th orntatons of th ntal stmats hav a dnt nunc on ths phnomnon. On potntal soluton to obtan bttr orntatonal rsoluton would b to prform svral EM tratons n paralll wth drnt valus for thr ntal stmats and thn procd wth an analyss of th nal convrgnc valus. 5 Concluson Th natur of dscontnuous mag motons n Fourr spac has long bn unclar. Th algorthms proposd n ths contrbuton ar basd on a rm thortcal framwork whch dscrbs th cohrnt bhavor of occluson vnts n Fourr spac. Howvr, opn qustons abound: Thortcally, th structur of planar moton, quadratc n th magng plan, rmans to b stablshd n Fourr spac. In addton, th algorthms proposd hrn may srv as a rst stag nto th prcptual groupng of vlocts, allowng to dntfy th occludng and occludd sgnals not only at occludng boundars but wthn rgons xhbtng cohrnt motons, and thrfor ladng to prformng moton-basd mag sgmntaton. Furthr, Occluson dtcton oprators could also b dvlopd wthn th contxt of ths thortcal framwork and unrportd xprmnts conductd wth occluson-tund Gabor ltrs on a 1D par of translatng sgnals show ths possblty. Exprmntally, th lmtng condtons undr whch th currnt tchnqus fal must b stablshd. For nstanc, th dgr of multpl vlocty rsoluton and th factors nuncng t must b dntd. In ts currnt stat, th xprmntal valuaton only conrms that nos-fr magry undr optmal condtons yld nos-fr rsults. Howvr, t has bn clar for som tm that a numbr of vson algorthms fal to mt ths fundamntal crtron [1]. To conclud, w hav dmonstratd th fasblty of computng dscontnuous motons and othr masurmnts such as th local dntcaton of occludng vlocts and occluson boundary normals, translucncy phnomna and th dsambguaton of occludng sgnals surng from th aprtur problm. Th thortcal framwork undr whch ths algorthms hav bn dvsd consttuts a foundaton for furthr rsarch n moton analyss. Indd, w strongly blv that furthr dvlopmnts n th ld of optcal ow and moton analyss ought to b basd on rmly stablshd thortcal backgrounds rathr than ncdntal vdnc [6]. Rfrncs [1] J. L. Barron, D. J. Flt, and S. S. Bauchmn. Prformanc of optcal ow tchnqus. IJCV, 1(1):43{77, [] S. S. Bauchmn, A. Chalfour, and J. L. Barron. Dscontnuous optcal ow: Rcnt thortcal rsults. In Vson Intrfac, pags 57{64, Klowna, Canada, May [3] M. J. Black. A robust gradnt-mthod for dtrmnng optcal ow. Tchncal Rport YALEU/DCS/RR-891, Yal Unvrsty, Nw- Havn, CT, [4] M. J. Black and A. Jpson. Estmatng optcal ow n sgmntd mags usng varabl-ordr paramtrc modls wth local dformatons. Tchncal Rport SPL-94-53, Xrox Systms and Practcs Laboratory, Palo Alto, Calforna, [5] B. Jahn. Moton dtrmnaton n spac-tm mags. In Procdngs of ECCV, pags 161{173, Antbs, Franc, Aprl 199. [6] R. C. Jan and T. O. Bnford. Ignoranc, myopa and navt n computr vson systms. CVGIP:IU, 53:11{117, [7] A. D. Jpson and M. Black. Mxtur modls for optcal ow computaton. In IEEE Procdngs of CVPR, pags 76{761, Nw York, Nw York, Jun [8] H.-H. Nagl. On th stmaton of optcal ow: Rlatons btwn drnt approachs and som nw rsults. Artcal Intllgnc, 33:99{34, [9] B. G. Schunck. Imag ow sgmntaton and stmaton by constrant ln clustrng. IEEE PAMI, 11(1):11{17, [1] J. Y. A. Wang and E. H. Adlson. Layrd rprsntaton for moton analyss. In Procdngs of CVPR, 1993.