Background. 3D Object recognition. Modeling polyhedral objects. Modeling polyhedral objects. Object and world coordinate systems

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1 3D Object ecognton Backgound Backgound n-pont pespectve algothms Geometc hashng Vew-based ecognton Recognton as pose estmaton Object pose defnes t embeddng n thee dmensonal space 3 degees of postonal feedom 3 degees of otatonal feedom Undelyng mathematcal methods fnd applcatons n othe aeas of mage analyss camea calbaton mage egstaton Modelng polyhedal objects How can we epesent a polyhedal object so that we could ecognze t fom an abtay mage? Collecton of postons of thee dmensonal ponts coespondng to the vetces of the polyheda Collecton of edges that meet at those vetces Colos of the plana facets that ae bounded by edges Shapes and colos of suface makngs on the plana facets Modelng polyhedal objects (1,1,1) (0,1,0) (1,1,0) (0,1,1) (0,0,0) (1,0,0) Collecton of 3-D ponts coodnates of ponts ae epessed n object coodnate system when we see an mage of the object ths means that thee s an nstance of the object n the wold so, we can thnk of the object model as beng tansfomed to the wold coodnate system thnk of the wold coodnate system as a coodnate system used to descbe locatons of ponts n a wokspace fo a obot Object and wold coodnate systems What s the object to wold coodnate system tansfomaton? t s a gd body tansfomaton tanslaton of the object otaton of the object t s called a gd body tansfomaton because tanslatons and otatons do not change the dstances between ponts -.e., the set of ponts n the object and wold coodnate systems ae conguent. y z y z Object and wold coodnate systems Let p o = (,y,z) T be the coodnates of a pont n the object coodnate system we fst otate p usng otaton mat R that detemne the oentaton of the object n the wold coodnate system: p R = Rp o we then tanslate p R by the tanslaton vecto, t, to detemne ts poston n the wold coodnate system: p w = Rp o + t

2 Wold and camea coodnates Coodnate fames The camea coodnate system s anothe 3-D coodnate system n whch the -y plane s the mage plane, the z as s othogonal to the mage plane, and the mage plane s dstance f fom the cente of pojecton, whch s gven coodnates (0,0,0) Ths s geneally NOT the same coodnate system as the wold coodnate system we can place ou camea anywhee n ou wokspace n patcula, t may be at the end of a obot am that moves though the wokspace But, we wll assume these systems ae algned y w z w y c z c c y o z o o obot cayng camea w Rp o + t y o z o o Choosng the ponts Eample mages Gven a 3-D object, how do we decde whch ponts fom ts suface to choose as ts model? choose ponts that wll gve se to detectable featues n mages fo polyheda, the mages of ts vetces wll be ponts n the mages whee two o moe long lnes meet these can be detected by edge detecton methods ponts on the nteos of egons, o along staght lnes ae not easly dentfed n mages. Choosng the ponts Eample: why not choose the mdponts of the edges of a polyheda as featues mdponts of pojectons of lne segments ae not the pojectons of the mdponts of lne segments f the ente lne segment n the mage s not dentfed, then we ntoduce eo n locatng mdpont Modelng polyhedal objects - collecton of edges Repesent each edge by ts end ponts encode whch edges meet at vetces Same gd body tansfomaton can be used to model the object to wold coodnate tansfomaton lne segment tansfoms to lne segment defned by tansfomaton of ts end ponts But mage analyss s now much hade - must fnd long edges n mage (1,1,1) (0,1,0) (1,1,0) (0,1,1) (0,0,0) (1,0,0)

3 Gven: N- pont pose ecovey a 3-D model oganzed as collecton of ponts Image of a scene suspected to nclude an nstance of the object, segmented nto featue ponts Goal Hypothesze the pose of the object n the scene by matchng (collectons of) n model ponts aganst n featue ponts, enablng us to solve fo the gd body tansfomaton fom the object to wold coodnate systems, and Vefy that hypothess by pojectng the emande of the model nto the mage and matchng look fo edges connectng pedcted vete locatons suface makngs ? 4 - pont pespectve soluton unque soluton fo 6 pose paametes computatonal complety of n 4 m pont pespectve soluton geneally two solutons pe tangle pa, but sometmes moe educed complety of n 3 m ? 1 - pont pespectve! Object estng on a known gound place Reduces poblem to only 3 degees of feedom Solved by matchng a sngle oented edge pont fom the mage to an oented model edge. Complety of O(mn) Reducng the combnatocs of pose estmaton Bg poblem: we ae lookng fo an object n an mage but the mage does not contan the object then we would only dscove ths afte compang all n 4 quaduples of mage featues aganst all m 4 quaduples of object featues. How can we educe the numbe of matches? consde only quaduples of object featues that ae smultaneously vsble - etensve pepocessng damete 2 subgaphs of the object gaph but n some mages no such subgaphs mght be vsble Reducng the comnbatocs of pose estmaton 3 pont tue pespectve M 2 Reducng the numbe of matches consde only quaduples of mage featues that ae connected by edges ae close to one anothe but not too close o the nevtable eos n estmatng the poston of an mage vete wll lead to lage eos n pose estmaton fom damete 2 subgaphs of the mage vete gaph moe geneally, ty to goup the mage featues nto sets that ae pobably fom a sngle object, and then only constuct quaduples fom wthn a sngle goup C n 2 α θ 2 k D 2 θ H 1 D 2 1 j M 0 H 1 m R 2 φ m m 1 0 γ 2 γ 1 O Eact Pespectve n 1 M 1

4 Tangle pose estmaton algothms Thee ae two basc appoaches to solvng poblems lke pose estmaton Analytcal methods based on constuctng systems of equatons and eplctly solvng fo unknown pose paametes fo tangle pose estmaton ths nvolves solvng a quadatc equaton n one pose angle, and then usng the solutons to the quadatc equaton to solve fo emanng pose paametes poblem: eos n estmatng locaton of mage featues can lead to ethe lage pose eos o falue to solve the quadatc equaton Appomate numecal algothms fnds solutons when eact methods fal due to mage measuement eo moe computaton A α C A B δ λ C β Numecal method B If dstance, R, to C s known, then possble locatons of A (and B) can be computed they le on the ntesectons of the lne of sght though A and the sphee of adus A-C centeed at C Once A and B ae located, the dstance can be computed and compaed aganst the actual dstance A-B A α C A B δ λ C β Numecal Method B Not pactcal to seach on R snce t s unbounded Instead, seach on one angula pose paamete, α. R a = R c cos δ ± A-C sn α R b = R c cos λ ± B-C sn β R c = A-C cos α / sn δ Ths esults n fou possble lengths fo sde A-B Geometc hashng Consde the followng smple 2-D ecognton poblem. We ae gven a set of object models, M each model s epesented by a set of ponts n the plane M = {P,1,..., P,n } We want to ecognze nstances of these pont pattens n mages fom whch pont featues (junctons, etc.) have been dentfed So, ou nput mage, B, s a bnay mage whee the 1 s ae the featue ponts We only allow the poston of the nstances of the M n B to vay - oentaton s fed. We want ou appoach to wok even f some ponts fom the model ae not detected n B. Geometc hashng We have aleady studed solutons to ths poblem Coelaton - match each of the M aganst B, fndng locatons at whch the coelaton s hgh Hough tansfom - to speed up the basc coelaton algothm. Geometc hashng - we tade off pepocessng tme fo seach tme dung ecognton we wll seach fo all possble models smultaneously coelaton must seach fo them one at a tme Geometc hashng Consde two models M 1 = {(0,0), (10,0), (10,10), (0,10)} M 2 = {(0,0), (4,4), (4,-4), (-4,-4), (-4,4)} We wll buld a table contanng, fo each model, all of the elatve coodnates of these ponts gven that one of the model ponts s chosen as the ogn of ts coodnate system ths pont s called a bass, because choosng t completely detemnes the coodnates of the emanng ponts. eamples fo M 1 choose (0,0) as bass obtan (10,0), (10,10), (0,10) choose (10,0) as bass obtan (-10,0), (0,10), -10,10))

5 M 1,1 M 2,2 Hashng table M 2,2 M 1,1 Hash table ceaton M 2,2 5 0 M 2,2 M 2,1 M 2,1 M 2,1 M 1,1 M 2,1 M 1,2 M 1,2 M 1,2 How many entes do we need to make n the hash table. Mode M has n pont each has to be chosen as the bass pont coodnates of emanng n -1 ponts computed wth espect to bass pont enty (M, bass) enteed nto hash table fo each of those coodnates And ths has to be done fo each of the m models. So complety s mn 2 to buld the table But the table s bult only once, and then used many tmes. Usng the table dung ecognton Pck a featue pont fom the mage as the bass. the algothm may have to consde all possble ponts fom the mage as the bass Compute the coodnates of all of the emanng mage featue ponts wth espect to that bass. Use each of these coodnates as an nde nto the hash table at that locaton of the hash table we wll fnd a lst of (M, p j ) pas - model bass pas that esult n some pont fom M gettng these coodnates when the j th pont fom M s chosen as the bass keep tack of the scoe fo each (M, p ) encounteed models that obtan hgh scoes fo some bases ae ecoded as possble detectons Some obsevatons If the mage contans n ponts fom some model, M, then we wll detect t n tmes each of the n ponts can seve as a bass fo each choce, the emanng n-1 ponts wll esult n table ndces that contan (M, bass) If the mage contans s featue ponts, then what s the complety of the ecognton component of the geometc hashng algothm? fo each of the s ponts we compute the new coodnates of the emanng s-1 ponts and we keep tack of the (model, bass) pas eteved fom the table based on those coodnates so, the algothm has complety O(s 2 ), and s ndependent of the numbe of models n the database 1 On to 3-D Consde the case whee the pont pattens can undego not only tanslaton, but also otaton now one pont s not a suffcent bass to compute the coodnates of the emanng ponts but two ponts ae suffcent 2 y y 4 3 Table constucton Revsed geometc hashng need to consde all pas of ponts fom each model fo each pa, constuct the coodnates of the emanng n-2 ponts usng those two as a bass add an enty fo (model, bass-pa) n the hash table complety s now mn 3 Recognton pck a pa of ponts fom the mage (cyclng though all pas) compute coodnates of emanng pont usng ths pa as a bass look up (model, bass-pa) n table and tally votes

6 t(p 2 -P 1 ) P 1 P 2 P 1 + t(p 2 -P 1 ) Let P 1 and P 2 be ponts Affne combnatons of ponts Consde the epesson P = P 1 + t(p 2 - P 1 ) P epesents a pont on the lne jonng P 1 and P 2. f 0 <= t <= 1 then P les between P 1 and P 2. We can ewte the epesson as P = (1-t)P 1 + tp 2 Defne an affne combnaton of two ponts to be a 1 P 1 + a 2 P 2 whee a 1 + a 2 = 1 P = (1-t)P 1 + tp 2 s an affne combnaton wth a 2 = t. Geneally, P 1 P 3 P P 2 Affne combnatons f P 1,...P n s a set of ponts, and a 1 + a n = 1, then a 1 P 1 + a n P n s the pont P 1 + a 2 (P 2 -P 1 ) + a n (P n -P 1 ) Let s look at affne combnatons of thee ponts. These ae ponts P = a 1 P 1 + a 2 P 2 + a 3 P 3 = P 1 + a 2 (P 2 -P 1 ) + a 3 (P 3 -P 1 ) whee a 1 + a 2 + a 3 = 1 f 0 <= a 1, a 2, a 3, <=1 then P falls n the tangle, othewse outsde (a 2, a 3 ) ae the affne coodnates of P homogeneous epesentaton s (1,a 2,a 3 ) P 1, P 2,, P 3 s called the affne bass Affne combnatons Affne tansfomatons Gven any two ponts, P = (a 1, a 2 ) and Q = (a 1, a 2 ) Q-P s a vecto ts affne coodnates ae (a 1 -a 1, a 2 -a 2 ) Note that the affne coodnates of a pont sum to 1, whle the affne coodnates of a vecto sum to 0. Rgd tansfomatons ae of the fom [,y ]= a b b a y + t ty whee a 2 + b 2 =1 An affne tansfomaton s of the fom [,y ]= a b c d y + t ty fo abtay a,b,c,d and s detemned by the tansfomaton of thee ponts Repesentaton n homogeneous coodnates In homogeneous coodnates, ths tansfomaton s epesented as: a b t [,y,1]= c d ty y Affne coodnates and affne tansfomatons The affne coodnates of a pont ae unchanged f the pont and the affne bass ae subjected to the same affne tansfomaton Based on smple popetes of affne tansfomatons Let T be an affne tansfomaton T(P 1 - P 2 ) = TP 1 - TP 2 T(aP) = atp, fo any scala a.

7 Let P 1 = (,y,1) and P 2 = (u,v,1) Note that P 1 - P 2 s a vecto TP 1 = (a + by + t, c + dy + t y, 1) TP 2 = (au + bv + t, cu + dv + t y,1) TP 1 - TP 2 = (a(-u) + b(y-v), c(-u) + d(y-v), 0) P 1 - P 2 = (-u, y-v,0) T(P 1 -P 2 ) = (a(-u) + b(y-v), c(-u)+d(y-v), 0) Poof Geometc hashng Let P 1, P 2, P 3 be an odeed affne bass tplet n the plane. Then the affne coodnates (α,β) of a pont P ae: P = α(p 2 - P 1 ) + β (P 3 - P 1 ) + P 1 Applyng any affne tansfomaton T wll tansfom t to TP = α(tp 2 - TP 1 ) + β (TP 3 - T P 1 ) + TP 1 So, TP has the same coodnates (α,β) n the bass tplet as t dd ognally. What do affne tansfomatons have to do wth 3-D ecognton Suppose ou pont patten s a plana patten -.e., all of the ponts le on the same plane. we constuct out hash table usng these coodnates, choosng thee at a tme as a bass We poston and oent ths plana pont patten n space fa fom the camea and take ts mage. So, the t z component of the model to wold gd tansfomaton s lage - I.e., the Z coodnates of the 3D object ponts ae lage. the tansfomaton of the model to the mage s an affne tansfomaton so, the affne coodnates of the ponts n any gven bass ae the same n the ognal 3-D plana model as they ae n the mage. Why s ths tue? If ou model, M, s a plana pont patten, then n the object coodnate system the ponts ae epesented as P = (X, Y, 0) Just ceate the model n the Z=0 plane M s then placed nto the camea 3-D coodnate system by some gd tansfomaton wth ts otaton mat and tanslaton vecto. Let p = (u,v ) be the mage of the gdly tansfomed P. Then 11X + 12Y + t u = f v 31X + 32Y + tz = f X X + 22Y + t y + Y + t 32 z P = RP + T w o [ X = w, Yw, Zw] X + t w = 11X o + 12Yo + 13Z0 13 X o t + 23 Yo t y Z 33 o tz And the mage coodnates of (X w, Y w, Z w ) ae 11X + 12Yo + 13Zo + t u = fx w / Z w = f X + Y + Z + t 31 o o 32 o 33 o z Remembe Why s ths tue? Placng M fa fom the camea means that n the denomnato of these epessons, t z domnates. So we ewte them as: u = v = y + t f = [ f11 / tz] + [ f12 / tz]y + t / tz tz a b t y + ty f tz Ths s an affne tansfomaton = [ f 21 / tz] + [ f 22 / tz]y + ty / tz c d t 2

8 Pepocessng Suppose we have a model contanng m featue ponts Fo each odeed noncollnea tplet of model ponts compute the affne coodnates of the emanng m-3 ponts usng that tple as a bass each such coodnate s used as an enty nto a hash table whee we ecod the (base tplet,model) at whch ths coodnate was obtaned. Complety s m 4 pe model. p1, p2, p3,m p1, p2, p3,m p1, p2, p3,m p1, p2, p3,m Recognton Scene wth n nteest ponts Choose an odeed tplet fom the scene compute the affne coodnates of emanng n-3 ponts n ths bass fo each coodnate, check the hash table and fo each enty found, tally a vote fo the (bass tplet, model) f the tplet scoes hgh enough, we vefy the match If the mage does not contan an nstance of any model, then we wll only dscove ths afte lookng at all n 3 tples. Vew Based Recognton Recognze a 3-D object by compang an mage to typcal pctues of the object wll need many pctues of each 3-D object that cove dffeent poses and scales each such pctue needs to be compaed aganst evey mage wndow to detemne f any match suffcently well coelaton-lke measues ae used to compute smlaty of a specfc vew to an mage wndow Vew Based Recognton Canoncal poblem: Face ecognton We ae povded wth a galley of fontal mages of faces we want to ecognze mages n galley have been nomalzed to fed sze mages have been nomalzed so that the centes of the left and ght eyes ae n standad postons thee s no backgound tetue aganst whch the faces ae vewed. Now, gven an mage of an unknown face nomalze t by fndng the eyes and scalng/otatng the unknown mage so that they ae n standad postons compae aganst each face n the galley Repesentng the galley Typcal galley contans O(10,000) faces ths makes mage coelaton mpactcal Fnd a low dmensonal epesentaton fo mages Chaacte ecognton - educed a 5050 chaacte (2500 bts of nfomaton) to 7-8 featues, each of whch equed ~32 bts of nfomaton (10:1 educton) Featues wee chosen n an ad hoc manne - no way to judge the qualty othe than to epement wth classfcaton Image codng model An mage epesentaton s good f t can be used to econstuct a close appomaton to the mage t codes Gven two epesentatons that econstuct an mage wth the same accuacy, the one equng fewe bts s pefeable Codng-based epesentatons - Foue tansfoms Foue s theoem: Gven any (well-behaved) one dmensonal functon, f(), t s possble to epesent the functon as a weghted sum of sne and cosne tems of nceasng fequency. The functon, F(u), s the Foue tansfom and descbes the weghts. F(u) = f ()e 2πu d e 2πu = cos(2πu) sn(2πu)

9 Foue tansfoms Gven F(u), t s possble to econstuct f() usng the fomula: f () = F(u)e 2πu du Foue tansfoms Typcally, F(u) deceases wth u small u coespond to snes/cosnes wth low fequences - they gossly encode lage objects n the sgnal lage u coespond to hgh fequency snes/cosnes - they encode fne detal n the sgnal Fo dgtal functons, the ntegals ae eplaced wth summatons, and only dscete values of u ae used an appomaton to f can be computed by tuncatng the econstucton - usng only a fnte ange of u Foue tansfoms of 2-D mages Sne and cosne functons ae eplaced by snusodal gatngs. Each gatng has spatal fequency oentaton Image s then epesented as a weghted sum of these bass functons By tuncatng the summaton, we get an appomaton of the ognal mage Poblems and solutons The nvese Foue tansfom conveges slowly to the ognal functon ths means we need a lot of coeffcents to obtan a good epesentaton of the mage Thee s nothng magcal about the use of snes and cosnes as a bass fo mage epesentaton any othonomal and complete set of functons wll do natual bass of the m-dmensonal mages So, maybe we can fnd a set of bass functons that ae good fo epesentng a patcula class of mages poblem s solved by pncpal component analyss Foue tansfoms of 2-D mages Pncpal components analyss We ae gven: a set of n objects (mages) each s epesented, ntally, by a set of m featues (512 2 ) pels Ths data s oganzed as a (vey lage!) n m mat Let s look at a small eample y p 1 ponts ae 2-D ponts we fnd the as that most closely passes though these ponts f the as passed eactly though these ponts, then we would need only one coodnate to epesent each pont.

10 Pncpal components analyss PC seeks the as whch the cloud of ponts ae closest to ths s mathematcally dentcal to fndng the as on whch the vaance of the pont pojectons s geatest (that s, on whch the pojectons ae most spead out). fo hgh dmensonal objects, lke pctues, t s unlkely that thee wll be a sngle as that passes close to all of the objects. So, n ths case, afte we fnd the best as (u 1 ), we then fnd the net best one (othogonal to the fst - u 2 ), and then the thd best (u 3 ), etc. Images ae then epesented by the pojectons on these aes: v = I u. Ths s eactly analogous to the Foue tansfom, wth the u eplacng the snusods. Pncpal components analyss If we compute m pncpal aes, then we can econstuct any mage eactly fom ts pncpal components epesentaton: m I = vu =1 Ths s just anothe bass fo the m-vecto that epesents the mage the ognal bass s the natual one - (1,0,0,...0), (0,1,0,...)... the pncpal aes epesent just a otaton of the ognal hgh dmensonal coodnate system Pncpal components analyss Howeve, we don t need to use all of the pncpal aes to obtan good econstuctons of the mage. The mathematcal pocedue that detemnes the pncpal aes uses an egenvecto analyss, and assocates a scoe wth each as these scoes coespond to the amount of vaaton n the mage set that the as coesponds to and ae the egenvalues of the pocedue The scoes geneally go to zeo quckly. Fo a face database, we can geneally econstuct a face usng only pncpal aes wth vey small eo. Recognton usng pncpal component analyss Gven you galley of mages compute ts pncpal components ths s just a set of othe mages that ae used as a bass fo epesentng the mages n the galley detemne a k<<m such that the fst k pncpal components ae a good epesentaton fo the galley can be chosen based on the scoes (egenvalues) computed by the PCA epesent each mage n the galley by ts pojecton on these k pncpal components ths s just the dot poduct of the mage and the pncpal components. each mage now epesented by k numbes Recognton usng pncpal component analyss Gven an unknown mage compute ts pojecton onto the pncpal component bass ths s a set of k numbes epesentng the unknown mage compae ths k-tuple aganst each of the database mage k- tuples smple L 2 nom sometmes each component s weghted by the assocated egenvalue Challenges to appeaance-based vson Vaatons n lghtng Occluson addessed by the use of obust estmaton fo computng pojectons onto pncpal aes Nomalzaton fo sze, poston and oentaton wthn the mage Lage numbe of mages n galley fo vewpont ndependence Modelng wthn-class vaatons Rejecton ctea

11 A 2-D object to 1-D mage eample How many ponts do we need to see n the 1-D mage of a 2-D lne segment of known length to ecove the poston and oentaton of the lne segment n the plane? A 1 - clealy not enough 2 - no choose a pont, p A on ay A daw the ccle of adus centeed at p f P s not too fa away fom O, then ths ccle wll ntesect ay B at two ponts, p B1 and p B2 ab s the mage of both p A p B1 and p A p B2 a b B A 2-D object to 1-D mage eample How many ponts? 3 B = A + k 1 C = A + k 2 s a unt vecto of unknown decton defnng the pose of the lne ABC snce ABC ae collnea, the ccles of ad AB and BC centeed at B must be tangent to the ays aa and cc A B a b c C Hough tansfom fo object ecognton Hypothesze-and-test appoach to ecognton Hypotheses geneated by clusteng pose estmates fom 3-pont pespectve solutons Bute foce algothm: set up 6-D pose Hough aay Fo all n 3 m 3 pangs of tples of mage featues to tples of object featues compute possble poses usng tangle pose algothm and ncement countes n Hough aay Possble coect poses wll coespond to clustes of hgh votes n the 6-D pose Hough aay, obtaned by seachng though Hough aay and fndng 6-D neghbohoods wth hgh total counts Hough tansfoms fo pose estmaton Key subpoblems: 1) Repesentng the paamete space mpactcal to use a 6-D aay to epesent all possble poses 2) Employng geometc constants to flte clustes n ths moe comple poblem thee s no guaantee that the mage tangle/object tangle pangs that vote fo a specfc pose wll be consstent Repesentng the paamete space Impactcal to cluste dectly n a s dmensonal clusteng aay too much stoage s equed, even wth vaable esoluton technques too much computaton assocated wth clusteng clustes too spead out due to vaous souces of eo Poposed soluton - epesent only a lowe dmensonal pojecton of the 6 dmensonal space A 3-D pojecton Two paametes coespond to the lne of sght to the object centod solvng tangle pose can be used to compute ( c,y c,z c ) - the locaton of object centod n wold coodnate system ts pojecton onto the mage s then (-f c /z, -fy c /z), whee f s the focal length of the camea can egad ths as the lne of sght to the cente of the object

12 A 3-d pojecton Thd coodnate s the appaent sze of the object n the mage f actual sze of object s h, then ts appaant sze s fh/z h coesponds to lagest dstance between 3-D model vetces Punng false tangle matches usng geometc constants Stoe lst of matchng tangle pas at each cell of clusteng aay Constant 1 : elmnate duplcate matches of the same tangle pa occus because of nonunqueness of tangle pose Constant 2: elmnate pas n whch model tangle could not be vsble face nomal mght pont away fom mage both tangles meetng at a concave edge must be vsble fo the edge to be vsble vsblty analyss elmnates about 50% of the false tangle pas Unqueness of mage featue to object vete mappng Let T be the set of tangle pas at a pont n the clusteng aay each model vete can be matched to only one mage featue each mage featue s the mage of a unque model vete f j - the fequency of pang model vete to mage featue j. coect matches should have lage f j because mage featues odnaly belong to many tangles Punng tangles usng unqueness Fo each mage featue, choose the model vete wth hghest f j Let P be the set of esultng mage featue - model vete pangs elmnate fom T any tangle pa wth a pang nconsstent wth P elmnate fom T any tangle pa that does not contan at least one pang fom P Fnally, elmnate fom T any pa of tangles pas wth mutually nconsstent pangs Epemental esults Eample- sngle polyhedal object wth 12 vetces, nonclutteed backgound Clustes coespond to mamal 333 neghbohoods of clusteng aay Cluste Tangles Pont Pas Og Fnal Fnal Scaled othogaphc pojecton - defntons and pose estmaton M 0, M 1,..., M,...M n ae the object featue ponts M 0 s called the efeence pont object fame of efeence s M 0 u, M 0 v, M 0 w (U, V, W ) ae the known coodnates of M n the object fame of efeence. In Scaled othogaphc pojecton (SOP) we assume the ange to all object ponts s Z 0, the ange to M 0. SOP coodnates of p, the SOP mage of M ae = fx /Z 0 y = fy /Z 0

13 Notaton The pespectve mage, m, of ths pont s = fx /Z y = fy /Z The ato s = f/z 0 s called the scalng facto of the SOP = fx /Z 0 + f(x -X 0 )/Z 0 = 0 + s(x -X 0 ) y = y 0 + s(y - Y 0 ) The otaton mat R fo the object s the mat whose ows ae the coodnates of the unt vectos,j,k fom the camea coodnate system epessed n the object coodnate system (M 0 u, M 0 v, M 0 w). u v w R = ju jv jw ku kv kw z w H M 0 Z 0 C m p m 0 f k y j O N P POSE fom SOP M v u G K SOP coodnates of M ae: Iu 0 = I M 0M = [UVW] Iv Iw Ju y y0 = J M 0M =[UVW] Jv Jw I = s, J = sj, s = f/z 0 I and J can be ecoveed by solvng system of lnea equatons fom n pont coespondences Unscalng I and J gves otaton mat and dstance, Z 0, to M 0. Geometc Hashng Recognton of flat objects depth vaaton wthn object small compaed to dstance of object fom camea and focal length of camea Pespectve s then well appomated by paallel pojecton wth a scale facto Two dffeent mages of the same flat object ae n affne 2-D coespondence thee s a nonsngula 22 mat A and a 2-D tanslaton vecto b such that each pont n the fst mage s tansfomed to A + b