OP = OO' + Ut + Vn + Wb. Material We Will Cover Today. Computer Vision Lecture 3. Multi-view Geometry I. Amnon Shashua
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1 Comuer Vson 27 Lecure 3 Mul-vew Geomer I Amnon Shashua Maeral We Wll Cover oa he srucure of 3D->2D rojecon mar omograh Marces A rmer on rojecve geomer of he lane Eolar Geomer an Funamenal Mar ebrew Unvers of Jerusalem Israel Com Vs 7 Lecure 3: mulvew-i Com Vs 7 Lecure 3: mulvew-i 2 he srucure of 3D->2D rojecon mar Com Vs 7 Lecure 3: mulvew-i 3 O he Srucure of a rojecon Mar W U w = V W b n O U X V O = OO U Vn Wb R = [ n b] X U = R V W Com Vs 7 Lecure 3: mulvew-i 4
2 he Srucure of a rojecon Mar he Srucure of a rojecon Mar,, f ), ) = f X = f X = X = f X = f U ) f, X. f. * -. = R V w ) = [R;] W M 3 4 Com Vs 7 Lecure 3: mulvew-i 5 Com Vs 7 Lecure 3: mulvew-i 6 Generall, he Srucure of a rojecon Mar cos f f sn f = sn, ) s calle he rncle on s = f cos sn s calle he skew Com Vs 7 Lecure 3: mulvew-i 7 f f s asec rao he Camera Cener M 3 4 M has rank=3, hus c Mc = Wh s c he camera cener? Conser he ocal ra MQ ) = M M All ons along he lne Q ) are mae o he same on Q) such ha Q ) = ) c c Com Vs 7 Lecure 3: mulvew-i 8 s a ra hrough he camera cener 2
3 he Eolar ons Choce of Canoncal Frame M = MWW c e = M c e = Mc c Com Vs 7 Lecure 3: mulvew-i 9 M = M WW Com Vs 7 Lecure 3: mulvew-i W s he new worl coornae frame We have 5 egrees of freeom 6 uo scale) Choose W such ha MW = [I;] Le Choce of Canoncal Frame M = [M ;m] MW = [M;m] M M m)n /)M m ) n / [ ] = I mn mn ; /)m /)m = [ I; ] We are lef wh 4 egrees of freeom uo scale): n,) Com Vs 7 Lecure 3: mulvew-i [I;] [;e] Choce of Canoncal Frame = = µ [I;] I I n ) / *n ) * -, / I n ) =, /, /, / µ. Com Vs 7 Lecure 3: mulvew-i 2 3
4 Choce of Canoncal Frame [;e] * = [;e] I -, µ n / /). µ [;e] I n = en ; /)e / [ ] [ en ; e ] [ en ; e ] ) µ where n,) are free varables Com Vs 7 Lecure 3: mulvew-i 3 rojecon Marces j Le be he mage of on where n,) j are known [ I;] j a frame number j [ j e j n ; e j ] are free varables Com Vs 7 Lecure 3: mulvew-i 4 Faml of omograh Marces j [ j e j n ; e j ] = e n Sans for he faml of 2D rojecve ransformaons beween wo fe mages nuce b a lane n sace Com Vs 7 Lecure 3: mulvew-i 5 Recall, U V [ R; ] W Faml of omograh Marces 3D->2D from Euclean worl frame o mage X U = R V worl frame o frs camera frame W Le, be he nernal arameers of camera,2 an choose canoncal frame n whch R=I an = for frs camera. X [ ; ] = [ R ] Com Vs 7 Lecure 3: mulvew-i 6 4
5 5 Com Vs 7 Lecure 3: mulvew-i 7 Faml of omograh Marces ] ; [ R ] [ = X = X Recall ha 3 r row of s ),, R R ] [ = R Com Vs 7 Lecure 3: mulvew-i 8 R Faml of omograh Marces Assume X are on a lanar surface n = ) X n = n R Com Vs 7 Lecure 3: mulvew-i 9 Faml of omograh Marces n R n R ) Le, e = an = n n = R n e ) Image-o-mage mang where he machng ar, are nuce b a lanar surface. Com Vs 7 Lecure 3: mulvew-i 2 n e Faml of omograh Marces n frs camera frame when R R s he homograh mar nuce b he lane a nfn
6 Esmang he omograh Mar Reconsrucon roblem ow man machng ons? π [ ; e ] = µ e µ h = h 3 We wsh o solve for he moon an srucure from maches Whou aonal nformaon we canno solve unquel for because s eermne u o a 4-arameer faml oson of a reference lane n sace). h = h ons make a bass for he rojecve lane Com Vs 7 Lecure 3: mulvew-i 2 Com Vs 7 Lecure 3: mulvew-i 22 rojecve Geomer of he lane a b c = Equaon of a lne n he 2D lane A rmer on rojecve geomer of he lane he lne s reresene b he vecor an l = l = a, b, c) =,,) Corresonence beween lnes an vecors are no - because a, b, c) reresens he same lne l) =, he vecor,,) oes no reresen an lne. Com Vs 7 Lecure 3: mulvew-i 23 wo vecors fferng b a scale facor are equvalen. hs equvalence class s calle homogenous vecor. An vecor a, b, c) s a reresenaon of he equvalence class. Com Vs 7 Lecure 3: mulvew-i 24 6
7 A on, ) rojecve Geomer of he lane les on he lne concen wh) whch s reresene b ff l = l = a, b, c) Bu also ) l =,, ),, 2, 3) reresens he on he vecor reresens he on, ) 2, ),,) oes no reresen an on. ons an lnes are ual o each oher onl n he 2D case). Com Vs 7 Lecure 3: mulvew-i rojecve Geomer of he lane r s r s r = r s) r = s = r s) s = noe: a b) c = e a, b, c) Lkewse, q l q Com Vs 7 Lecure 3: mulvew-i 26 l Conser lnes Lnes an ons a Infn r s r = a, b, c) s = a, b, c ) bc cb b b r ) s = ac ac = c c) a a whch reresens he on b, a ) wh nfnel large coornaes a b b All mee a he same on a Com Vs 7 Lecure 3: mulvew-i 27 on a nfn Lnes an ons a Infn he ons, 2,),, 2 le on a lne he lne l =,,),,) 2 = s calle he lne a nfn he ons, 2,),, 2 are calle eal ons. a b A lne a, b, ) mees l =,,) a b ) = *a whch s he recon of he lne) Com Vs 7 Lecure 3: mulvew-i 28 7
8 A Moel of he rojecve lane, 2,) eal on 2 A Moel of he rojecve lane l =,,) s he lane 2 3 n = {[,..., n ] [,...,] :[,..., n ] = [,..., n ], } ={lnes hrough he orgn n n R } ={-m subsaces of n R } 3, 2, ), ons are reresene as lnes ras) hrough he orgn Lnes are reresene as lanes hrough he orgn Com Vs 7 Lecure 3: mulvew-i 29 Com Vs 7 Lecure 3: mulvew-i 3 2 rojecve ransformaons n he su of roeres of he rojecve lane ha are nvaran uner a grou of ransformaons. 2 rojecve ransformaons n rojecv: 2 2 h : ha mas lnes o lnes.e. reserves colnear) An nverble 33 mar s a rojecv: Le, 2, 3 Colnear ons,.e. l = l = he ons s calle homograh, colneaon A homograh s eermne b 8 arameers. Com Vs 7 Lecure 3: mulvew-i 3 le on he lne l herefore reserves colnear. s he ual. ersecv A comoson of ersecves from a lane o oher lanes an back o s a rojecv. Ever rojecv can be reresene n hs wa. 6.o.f) Com Vs 7 Lecure 3: mulvew-i 32 8
9 2 rojecve ransformaons n Eamle, a resecv n D: Lnes ajonng machng ons are concurren l =,,) 2 rojecve ransformaons n s no nvaran uner : ons on l are 2,,) a b c a b c 2 = h 2h2 = s no necessarl arallel lnes o no reman arallel Lnes ajonng machng ons a,a ),b,b ),c,c ) are no concurren Com Vs 7 Lecure 3: mulvew-i 33 l s mae o l Com Vs 7 Lecure 3: mulvew-i 34 rojecve Bass rojecve Invarans A Smle n n R s a se of n2 ons such ha no subse Of n of hem le on a herlane lnearl eenen). Invarans are measuremens ha reman fe uner colneaons In 2 a Smle s 4 ons of neenen nvarans =.o.f of confguraon -.o.f of rans. E: D case 22 has 3.o.f A on n D s reresene b arameer. 4 ons we have: 4-3= nvaran cross rao) heorem: here s a unque colneaon beween an wo Smlees Com Vs 7 Lecure 3: mulvew-i 35 2D case: has 8.o.f, a on has 2.o.f hus 5 ons nuce 2 nvarans Com Vs 7 Lecure 3: mulvew-i 36 9
10 he cross-rao of 4 ons: ab = ac c b rojecve Invarans 24 ermuaons of he 4 ons formng 6 grous:,,,,, Com Vs 7 Lecure 3: mulvew-i 37 a b b 5 ons gves us.o.f, hus -8=2 nvarans whch reresen 2D,, z, u are he 4 bass ons smle) =< z, u, >, =< z, u,, > z u, are eermne unquel b, rojecve Invarans u on of nersecon s reserve uner rojecv eercse), unquel eermne u Com Vs 7 Lecure 3: mulvew-i 38 Eolar Geomer an Funamenal Mar Remner: [ I;] [ ; e ] [ en ; e ] = = µ = µ µ e = e n Sans for he faml of 2D rojecve ransformaons beween wo fe mages nuce b a lane n sace Com Vs 7 Lecure 3: mulvew-i 39 Com Vs 7 Lecure 3: mulvew-i 4
11 µ e lane aralla =,,, µ ) [ I,] [ e ] Remner: [ ; ] X = [ R ] π R = R e =, e R n ) wha oes µ san for? wha woul we oban afer elmnang µ Com Vs 7 Lecure 3: mulvew-i 4 Com Vs 7 Lecure 3: mulvew-i 42 R R n ) Noe ha e, are eermne each) u o a scale. Le: = e n n e = n ) e ) e Recall: X = µ e µ = Com Vs 7 Lecure 3: mulvew-i 43 Le, Be an reference on no arsng from Le µ µ e µ e be he homograh we wll use Com Vs 7 Lecure 3: mulvew-i 44
12 µ µ µ e Recall: µ = µ = µ µe lane aralla We have use 4 sace ons for a bass: 3 for he reference lane for he reference on scalng) Snce 4 ons eermne an affne bass: µ s calle relave affne srucure µ = Com Vs 7 Lecure 3: mulvew-i 45 Noe: we nee 5 ons for a rojecve bass. he 5 h on s he frs camera cener. Com Vs 7 Lecure 3: mulvew-i 46 Noe: An Affne Invaran Wha haens when camera cener s a nfn? arallel rojecon) µ e Funamenal Mar =,,, µ ) µ =, hs nvaran s neenen of boh camera osons, an s Affne. Com Vs 7 Lecure 3: mulvew-i 47 e [ e ] = 2 rank e ) = [ e] ) = F = Com Vs 7 Lecure 3: mulvew-i 48 π 2
13 Funamenal Mar Eoles from F [ e] ) = Noe: an homograh mar mas beween eoles: F = Defnes a blnear machng consran whose coeffcens een onl on he camera geomer shae was elmnae) F oes no een on he choce of he reference lane [ e] = [ e] [ e] e n ) Com Vs 7 Lecure 3: mulvew-i 49 c e e c e Com Vs 7 Lecure 3: mulvew-i 5 e Eoles from F Esmang F from machng ons Fe = [ e] e [ e] = e F e= [ e] e = F = =,..., 8 F = =,..., 7 e F) = Lnear soluon N on-lnear soluon F s he eolar lne of - he rojecon of he lne of sgh ono he secon mage. e F) = s cubc n he elemens of F, hus we shoul eec 3 soluons. Com Vs 7 Lecure 3: mulvew-i 5 Com Vs 7 Lecure 3: mulvew-i 52 3
14 Esmang F from omograhes F Inuces a omograh F s skew-smmerc.e. roves 6 consrans on F) π F = [ e] F e n ) [ e] = e] e n ) = = [ [ e] F F = F 2 homograh marces are requre for a soluon for F [ ] F s a homograh mar nuce b he lane efne b he jon of he mage lne an he camera cener Com Vs 7 Lecure 3: mulvew-i 53 Com Vs 7 Lecure 3: mulvew-i 54 rojecve Reconsrucon. Solve for F va he ssem F = 8 ons or 7 ons) 2. Solve for e va he ssem F e= 3. Selec an arbrar vecor e 4. [ I ] an [ ] e ] F [ ] F µ e are a ar of camera marces. Com Vs 7 Lecure 3: mulvew-i 55 4
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