17. Coordinate-Free Projective Geometry for Computer Vision

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1 17. Coordnate-Free Projectve Geometry for Computer Vson Hongbo L and Gerald Sommer Insttute of Computer Scence and Appled Mathematcs, Chrstan-Albrechts-Unversty of Kel 17.1 Introducton How to represent an mage pont algebracally? Gven a Cartesan coordnate system of the retna plane, an mage pont can be represented by ts coordnates (u, v. If the mage s taen by a pnhole camera, then snce a pnhole camera can be taen as a system that performs the perspectve projecton from three-dmensonal projectve space to two-dmensonal one wth respect to the optcal center [77], t s convenent to descrbe a space pont by ts homogeneous coordnates (x, y, z, 1 and to descrbe an mage pont by ts homogeneous coordnates (u, v, 1. In other words, the space of mage ponts can be represented by the space of 3 1 matrces. Ths s the coordnate representaton of mage ponts. There are other representatons whch are coordnate-free. The use of algebras of geometrc nvarants n the coordnate-free representatons can lead to remarable smplfcatons n geometrc computng. Kanatan [128] uses the three-dmensonal affne space for space ponts, and the space of dsplacements of the affne space for mage ponts. In other words, he uses vectors fxed at the orgn of R 3 to represent space ponts, and uses free vectors to Ths wor has been supported by Alexander von Humboldt Foundaton (H.L. and by DFG Grants So and So (G.S.

2 416 Hongbo L, Gerald Sommer represent mage ponts. Then he can use vector algebra to carry out geometrc computng. Ths algebrac representaton s convenent for two-dmensonal projectve geometry, but not for three-dmensonal one. The space representng mage ponts depends nether on the retna plane nor on the optcal center. Bayro-Corrochano, Lasenby and Sommer use R 4 for both two-dmensonal and three-dmensonal projectve geometres[19, 17, 222]. They use a coordnate system {e 1, e 2, e 3, C} of R 4 to descrbe a pnhole camera, where the e s are ponts on the retna plane and C s the optcal center. Both space ponts and mage ponts are represented by vectors fxed at the orgn of R 4, the only dfference s that an mage pont s n the space spanned by vectors e 1, e 2, e 3. Ths algebrac representaton s convenent for projectve geometrc computatons usng the ncdence algebra formulated n Clfford algebra. However, t always needs a coordnate system for the camera. The space representng mage ponts depends only on the retna plane. We notced that none of these algebrac representatons of mage ponts s related to the optcal center. By ntuton, t s better to represent mage ponts by vectors fxed at the optcal center. The above-mentoned coordnatefree representatons do not have ths property. Hestenes [113] proposed a technque called space-tme splt to realze the Clfford algebra of the Eucldean space n the Clfford algebra of the Mnows space. The technque s later generalzed to projectve splt by Hestenes and Zegler [118] for projectve geometry. We fnd that a verson of ths technque offers us exactly what we need: three-dmensonal lnear spaces mbedded n a four-dmensonal one, whose orgns do not concur wth that of the four-dmensonal space but whose Clfford algebras are realzed n that of the four-dmensonal space. Let C be a vector n R 4. It represents ether a space pont or a pont at nfnty of the space. Let M be another vector n R 4. The mage of the space pont or pont at nfnty M by a pnhole camera wth optcal center C can be descrbed by C M. The mage ponts can be represented by the threedmensonal space C R 4 = {C X X R 4 }. The Clfford algebra of the space C R 4 can be realzed n the Clfford algebra of R 4 by the theorem of projectve splt proposed later n ths chapter. The space representng mage ponts depends only on the optcal center. The representaton s completely projectve and completely coordnate-free. Usng ths new representaton and the verson of Grassmann-Cayley algebra formulated by Hestenes and Zegler [118] wthn Clfford algebra, we have reformulated camera modelng and calbraton, eppolar and trfocal geometres, relatons among eppoles, eppolar tensors and trfocal tensors. Remarable smplfcatons and generalzatons are obtaned through the reformulaton, both n concepton and n applcaton. In partcular, we are to derve and generalze all nown constrants on eppolar and trfocal tensors [76, 80, 81, 83] n a systematc way.

3 17. Coordnate-Free Projectve Geometry for Computer Vson 417 Ths chapter s arranged as follows: n secton 17.2 we collect some necesssary mathematcal technques, n partcular the theorem of projectve splt n Grassmann-Cayley algebra. In sectons 17.3 and 17.4 we reformulate camera modelng and calbraton, and eppolar and trfocal geometres. In secton 17.5 we derve and generalze the constrants on eppolar and trfocal tensors systematcally Preparatory Mathematcs Dual Bases Accordng to Hestenes and Sobczy [117], let {e 1,..., e n } be a bass of R n and {e 1,..., e n } be the correspondng dual (or recprocal bass, then e = ( 1 1 (e 1 ě e n, e = ( 1 1 (e 1 ě e n, (17.1 for 1 n. Here s the dual operator n G n wth respect to e 1 e n. The bass {e 1,..., e n } nduces a bass {e j1 e js 1 <... < j s n} for the s-vector subspace G s n of the Clfford algebra G n of R n. We have (e j1 e js = e j s e = ( 1 j1+ +js+s(s+1/2 (e 1 ě j1 ě js e n. Let x Gn, s then x = x (e j1 e js = 1 <...<j s n 1 <...<j s n e j1 e js ( 1 j1+ +js+s(s+1/2 e j1 e js (e 1 ě j1 ě js e n x. (17.2 (17.3 Let an nvertble transformaton T of R n maps {e 1,..., e n } to a bass {e 1,..., e n}. Let T = (T T 1. Then T maps the dual bass {e 1,..., e n} to the dual bass {e 1,..., e n }. Any lnear mappng T : R n R m has a tensor representaton n R n R m. Then n T = e e. (17.4 =1 For example, let Π n be the dentty transformaton of R n, then n tensor representaton, Π n = n e e for any bass {e 1,..., e n }. =1

4 418 Hongbo L, Gerald Sommer Projectve and Affne Spaces An n-dmensonal real projectve space P n can be realzed n the space R n+1, where a projectve r-space s an (r+1-dmensonal lnear subspace. In G n+1, a projectve r-space s represented by an (r + 1-blade, and the representaton s unque up to a nonzero scale. Throughout ths chapter we use x y to denote that f x, y are scalars, they are equal up to a nonzero ndex-free scale, otherwse they are equal up to a nonzero scale. An n-dmensonal real affne space A n can be realzed n the space R n+1 as a hyperplane away from the orgn. Let e 0 be the vector from the orgn to the hyperplane and orthogonal to the hyperplane. When e 2 0 = 1, a vector x R n+1 s an affne pont f and only f x e 0 = 1. An r-dmensonal affne plane s the ntersecton of an (r + 1-dmensonal lnear subspace of R n+1 wth A n, and can be represented by an (r + 1-blade of G n+1 representng the subspace. The space of dsplacements of A n s defned as A n = {x y x, y A n }. It s an n-dmensonal lnear subspace of R n+1. Any element of t s called a drecton. When A n s taen as an (n 1-dmensonal projectve space, any element n t s called a pont at nfnty, and A n s called the space at nfnty of A n. Let I n = e 0 I n+1. Then t represents the space A n. The mappng In : x e 0 x = I n x, for x G n+1, (17.5 maps G n+1 to G( A n, called the boundary mappng. When I n s fxed, In s often wrtten as. Geometrcally, f I r+1 represents an r-dmensonal affne space, then I r represents ts space at nfnty. For example, when x, y are both affne ponts, (x y = y x s the pont at nfnty of lne xy. Let {e 1,..., e n+1 } be a bass of R n+1. If e n+1 A n, e 1,..., e n A n, the bass s called a Cartesan coordnate system of A n, wrtten as {e 1,..., e n ; e n+1 }. The affne pont e n+1 s called the orgn. Let x A n, then x = e n+1 + n λ e. (λ 1,..., λ n s called the Cartesan coordnates of x wth =1 respect to the bass. Below we lst some propertes of the three-dmensonal projectve (or affne space when descrbed n G 4. Two planes N, N are dentcal f and only f N N = 0, where N, N are 3-blades. A lne L s on a plane N f and only f L N = 0, where L s a 2-blade. Two lnes L, L are coplanar f and only f L L = 0, or equvalently, f and only f L L = 0. A pont A s on a plane N f and only f A N = 0, or equvalently, f and only f A N = 0. Here A s a vector.

5 17. Coordnate-Free Projectve Geometry for Computer Vson 419 A pont A s on a lne L f and only f A L = 0. Three planes N, N, N are concurrent f and only f N N N = 0. For two lnes L, L, L L = L L. For pont A and plane N, A N = A N Projectve Splts The followng s a modfed verson of the technque of projectve splt. Defnton Let C be a blade n G n. The projectve splt P C of G n wth respect to C s the followng transformaton: x C x, for x G n. Theorem [Theorem of projectve splt n Grassmann-Cayley algebra 1 ] Let C be an r-blade n G n. Let C G n = {C x x G n }. Defne n t two products C and C : for x, y G n, (C x C (C y = C x y, (C x C (C y = (C x (C y, (17.6 and defne (C x C = C (C x. (17.7 Then vector space C G n equpped wth C, C, C s a Grassmann- Cayley algebra somorphc to G n r, whch s taen as a Grassmann-Cayley algebra. Proof. Let C R n = {C x x R n }. It s an (n r-dmensonal vector space. By the lnear somorphsm of {λc λ R} wth R, t can be verfed that (C G n, C s somorphc to the Grassmann algebra generated by C R n. A drect computaton shows that the composton of C wth tself s the scalar multplcaton by ( 1 n(n 1/2 C 2. That C G n s a Grassmann-Cayley algebra follows from the dentty (C x C C (C y C = ((C x C (C y C, (17.8 whch can be verfed by the defntons (17.6 and ( Theorem can be generalzed to the followng one, whch s nevertheless not needed n ths chapter: [Theorem of projectve splt n Clfford algebra] Let C be a blade n G n. The space C G n equpped wth the followng outer product C and nner product C s a Clfford algebra somorphc to G(C : (C x C (C y = C x y, (x C C (C y = C 2 C ((x C (C y, for x,y G n.

6 420 Hongbo L, Gerald Sommer Let {e 1,..., e n } be a bass of R n. The projectve splt P C can be wrtten as the composton of the outer product by C and the dentty transformaton. It has the followng tensor representaton: P C = n s=0 1 <...<j s n (C e j1 e js (e j1 e js. (17.9 For example, when C s a vector and P C s restrcted to R n, then P C = n (C e e. (17.10 =1 In partcular, when {e 1,..., e n 1, C} s a bass of R n, then n 1 P C = (C e e. (17.11 =1 When P C s restrcted to Gn, 2 then P C = (C e j1 e j2 (e j1 e j2. ( < n In partcular, when {e 1,..., e n 1, C} s a bass of R n, then P C = 1 < n 1 (C e j1 e j2 (e e. (17.13 When n = 4, we use the notaton 1 2 to denote that, 1, 2 s an even permutaton of 1, 2, 3. Let then ê = e 1 e 2, ê = e 1 e 2. (17.14 P C = 3 (C ê ê. (17.15 =1 The followng theorem establshes a connecton between the projectve splt and the boundary mappng. Theorem When C s an affne pont, the boundary mappng realzes an algebrac somorphsm between the Grassmann-Cayley algebras C G n+1 and G( A n.

7 17. Coordnate-Free Projectve Geometry for Computer Vson Camera Modelng and Calbraton Pnhole Cameras Accordng to Faugeras [77], a pnhole camera can be taen as a system that performs the perspectve projecton from P 3 to P 2 wth respect to the optcal pont C P 3. To descrbe ths mappng algebracally, let {e 1, e 2, e 3 ; O} be a fxed Cartesan coordnate system of A 3, called the world coordnate system. Let {e C 1, ec 2, ec 3, C} be a bass of R4 satsfyng (e C 1 ec 2 ec 3 C = 1, called a camera projectve coordnate system. When C s an affne pont, let e C 3 be the vector from C to the orgn O C of the retna plane (or mage plane, and let e C 1, ec 2 be two vectors n the retna plane. Then {ec 1, ec 2, ec 3 ; C} s a Cartesan coordnate system of A 3, called a camera affne coordnate system. Let M be a pont or pont at nfnty of A 3, and let m C be ts mage. Then M can be represented by ts homogeneous coordnates whch s a 4 1 matrx, and m C can be represented by ts homogeneous coordnates whch s a 3 1 matrx. The perspectve projecton can then be represented by a 3 4 matrx. e3 C e2 C e1 O C C e3. C e1 O e2 M m Fg A pnhole camera. In our approach, we descrbe a pnhole camera wth optcal center C, whch s ether an affne pont or a pont at nfnty of A 3, as a system performng the projectve splt of G 4 wth respect to C R 4. To see how ths representaton wors, we frst derve the matrx of the project splt P C restrcted to R 4. We consder the case when the camera coordnate system {e C 1, ec 2, ec 3, C} s affne. Accordng to (17.11,

8 422 Hongbo L, Gerald Sommer P C = 3 =1 (C e C ec. (17.1 In the camera coordnate system, let the coordnates of e, = 1, 2, 3, and O, be (e 1, e 2, e 3, 0 = (e T, 0, and (O 1, O 2, O 3, 1 = ( c T, 1, respectvely. Here e and c represent 3 1 matrces. The followng matrx changes {e C 1, ec 2, ec 3, C} to {e 1, e 2, e 3, O}: e T 1 0 e T 2 0 e T. ( c T 1 Its transpose changes {e 1, e 2, e 3, O } to {e C 1, ec 2, ec 3, C }. Substtutng e C, = 1, 2, 3 expressed by e 1, e 2, e 3, O nto (17.1, we get the matrx of P C : P C = (e 1 e 2 e 3 c. (17.3 When C = O, e C 1 = e 1, e C 2 = e 2 and e C 3 = fe 3, where f s the focal length of the camera, P C = , ( /f 0 whch s the standard perspectve projecton matrx. Ths justfes the representaton of the perspectve projecton by P C and the representaton of mage ponts by vectors n C R 4. In the case when the camera coordnate system s projectve, let the 4 1 matrces e C, = 1, 2, 3 represent the coordnates of e C wth respect to {e 1, e 2, e 3, O }. By (17.1, P C = (e C 1 e C 2 e C 3 T. (17.5 Below we derve the matrx of P C restrcted to G 2 4. Let ê C = e C 1 e C 2, (17.6 where 1 2. It represents the coordnates of ê C wth respect to the bass of G4 2 nduced by {e 1, e 2, e 3, O }. Accordng to (17.15, the matrx of P C s P C = (ê C 1 ê C 2 ê C 3 T. (17.7

9 17. Coordnate-Free Projectve Geometry for Computer Vson Camera Constrants It s clear that as long as det(e 1 e 2 e 3 0, the matrx P C = (e 1 e 2 e 3 c represents a perspectve projecton. When there s further nformaton on the pnhole camera, for example vectors e C 1, ec 2 of the camera affne coordnate system are perpendcular, then P C needs to satsfy addtonal equalty constrants n order to represent the perspectve projecton carred out by such a camera. Let 3 represent the dual n G( A 3. Let the dual bases of {e 1, e 2, e 3 } and {e C 1, e C 2, e C 3 } n A 3 be {e 3 1, e 3 2, e 3 3 } and {ec 3 1, e C 3 2, e C 3 3 }, respectvely. Then e C 1 = (ec 3 2 e C = e C 3 2 e C 3 3, e C 2 = (e C 3 3 e C = e C 3 3 e C 3 1, (17.8 where s the cross product n vector algebra. The perpendcularty constrant can be represented by e C 1 ec 2 = (ec 3 2 e C 3 3 (e C 3 3 e C 3 1 = 0. (17.9 Let the 3 1 matrx e C 3 to {e 3 1, e 3 2, e 3 bass, e C 3 represent the coordnates of e C 3 wth respect 3 }. Under the assumpton that {e 1, e 2, e 3 } s an orthonormal e C 3 j = e C 3 e C 3 j for any 1, j 3. Then (17.9 s changed to (e C 3 2 e C 3 3 (e C 3 3 e C 3 1 = 0, (17.10 whch s a constrant on P C because (e C 3 1 e C 3 2 e C 3 3 = (e 1 e 2 e 3 T. ( Camera Calbraton Let M be a space pont or pont at nfnty, m C be ts mage n the retna plane. Assume that m C s a pont, and has homogeneous coordnates (u, v, 1 n the Cartesan coordnate system of the retna plane. Let the 4 1 matrx M represent the homogeneous coordnates of M n the world coordnate system. Then (u v 1 T P C M = (e C 1 M e C 2 M e C 3 M T, (17.12 whch can be wrtten as two scalar equatons: (e C 1 ue C 3 M = 0, (e C 2 ve C 3 M = 0. (17.13 The matrx P C = (e C 1 e C 2 e C 3 T can be taen as a vector n the space R 4 R 4 R 4 equpped wth the nduced nner product from R 4. By ths nner product, (17.13 can be wrtten as

10 424 Hongbo L, Gerald Sommer (M 0 um T P C = 0, (0 M vm T P C = 0. (17.14 Gven M and (u, v for = 1,..., 6, there are 12 equatons of the forms n ( If there s no camera constrant, then snce a 3 4 matrx representng a perspectve projecton has 11 free parameters, P C can be solved from the 12 equatons f and only f the determnant of the coeffcent matrx A of these equatons s zero,. e., Λ 6 =1 (M 0 u M Λ 6 =1 (0 M v M = 0, (17.15 where the outer products are n the Clfford algebra generated by R 4 R 4 R 4. Expandng the left-hand sde of (17.15, and changng outer products nto determnants, we get ɛ(σɛ(τu σ(1 u σ(2 v τ(1 v τ(2 det(m σ(1 M σ(2 M τ(1 M τ(2 σ,τ det(m σ( =3..6 det(m τ(j j=3..6 = 0, (17.16 where σ, τ are any permutatons of 1,..., 6 by movng two elements to the front of the sequence, and ɛ(σ, ɛ(τ are the sgns of permutaton. For expermental data, (17.16 s not necessarly satsfed because of errors n measurements Eppolar and Trfocal Geometres Eppolar Geometry There s no much dfference between our algebrac descrpton of the pnhole camera and others f there s only one fxed camera nvolved, because the underlyng Grassmann-Cayley algebras are somorphc. Let us reformulate the eppolar geometry of two cameras wth optcal centers C, C respectvely. The mage of C n camera C s E CC = C C, called the eppole of C n camera C. Smlarly, the mage of C n camera C s E C C = C C, called the eppole of C n camera C. An mage lne passng through the eppole n camera C (or C s called an eppolar lne wth respect to C (or C. Algebracally, an eppolar lne s a vector n C C R 4 = (C G 2 4 (C G 2 4. (17.1 An eppolar lne C C M corresponds to a unque eppolar lne C C M, and vce versa. Let there be two camera projectve coordnate systems n the two cameras respectvely: {e C 1, e C 2, e C 3, C} and {e C 1, e C 2, e C 3, C }. Usng the relatons and (C e C (C ê C = C, for 1 3, (17.2 (C ê C 1 (C ê C 2 = C ê C, for 1 2, (17.3

11 17. Coordnate-Free Projectve Geometry for Computer Vson 425 E CC C m.. C. C M E C C. C m Fg Eppolar geometry. we get the coordnates of eppole E CC : E CC = ((C ê C C =1..3 = ((C ê C C =1..3 = ((e C e C =1..3. (17.4 The followng tensor n (C R 4 (C R 4 s called the eppolar tensor decde by C, C : F CC (m C, m C = m C m C. (17.5 Let m C C R 4, m C C R 4. They are mages of the same space pont or pont at nfnty f and only f F CC (m C, m C = 0. Ths equalty s called the eppolar constrant between m C and m C. In matrx form, wth respect to the bases {C e C 1, C ec 2, C ec 3 } and {C e C 1, C e C 2, C e C 3 }, F CC can be represented by F CC = ((C e C (C e C j,j=1..3 = ((C e C (C e C j,j=1..3 = (ê C ê C j,j=1..3. (17.6 (17.6 s called the fundamental matrx. The eppolar tensor nduces a lnear mappng F C;C from C R 4 to (C R 4 = C G4, 2 called the eppolar transformaton from camera C to camera C :

12 426 Hongbo L, Gerald Sommer F C;C (m C = C m C. (17.7 Smlarly, t nduces an eppolar transformaton from camera C to camera C as follows: F C ;C (m C = C m C. (17.8 Both transformatons are just projectve splts. The ernel of F C;C s the one-dmensonal subspace of C R 4 represented by C C, the range of F C;C s the two-dmensonal space C C R 4. In geometrc language, F C;C maps the eppole of C to zero, and maps any other pont n camera C to an eppolar lne wth respect to C. Furthermore, we have the followng concluson: Proposton Let L C be an eppolar lne n camera C. If ts dual s mapped to eppolar lne L C n camera C by F C;C, then the dual of L C s mapped bac to L C by F C ;C. The proof follows from the dentty that for any vector M R 4, C (C (C C M C C C C M. ( Trfocal Geometry Let there be three cameras wth optcal centers C, C, C respectvely. Let M be a space pont or pont at nfnty. Its mages C M, C M and C M n the three cameras must satsfy parwse eppolar constrants. Let us consder the nverse problem: If there are three mage ponts m C, m C, m C n the three cameras respectvely, they satsfy the parwse eppolar constrants, s t true that they are mages of the same space pont or pont at nfnty?. C M. C m C m C. C m C Fg Pont correspondence n three cameras.

13 17. Coordnate-Free Projectve Geometry for Computer Vson 427 A smple counter-example shows that the eppolar constrants are not enough. When the 2-blades m C, m C, m C belong to G(C C C, the eppolar constrants are always satsfed, but the blades do not necessarly share a common vector. Assume that the eppolar constrant between m C and m C s satsfed. Let M be the ntersecton of the two lnes m C and m C n P 3. Then m C represents the mage of M n camera C f and only f m C M = 0, or equvalently, m C (M x = 0, for any x R 4. (17.10 When C, C, M are not collnear, snce M R 4 = (C M R 4 (C M R 4, (17.11 (17.10 can be wrtten as m C (m C C m C 0 (m C C m C 0 = 0, (17.12 for any mage ponts m C 0, mc 0 n cameras C, C respectvely. When C, C, M are collnear, snce m C m C, (17.12 s equvalent to the eppolar constrant between m C and m C. So the constrant (17.12 must be satsfed for m C, m C, m C to be mages of the same space pont or pont at nfnty. Defnton The followng tensor n (C R 4 (C G 2 4 (C G 2 4 s called the trfocal tensor [105, 106, 214] of camera C wth respect to cameras C, C : T(m C, L C, L C = m C L C L C, (17.13 where m C C R 4, L C C G 2 4, LC C G 2 4. Two other trfocal tensors can be defned by nterchangng C wth C, C respectvely: T (m C, L C, L C = m C L C L C, T (m C, L C, L C = m C L C L C. (17.14 In ths secton we dscuss T only. Let {e C 1, e C 2, e C 3, C}, {e C 1, e C 2, e C 3, C }, {e C 1, ec 2, ec 3, C } be camera projectve coordnate systems of the three cameras respectvely. Then T has the followng component representaton: T = ((C e C (C ê C j (C ê C = ((C e C ((C ê C j (C ê C = (ê C = ( (ê C (e C j,j,=1..3 j e C,j,=1..3.,j,=1..3,j,=1..3 The trfocal tensor T nduces three trfocal transformatons: (17.15

14 428 Hongbo L, Gerald Sommer 1. The mappng T C : (C G 2 4 (C G 2 4 (C R 4 = C G 2 4 s defned as T C (L C, L C = C (L C L C. (17.16 When L C C G4 2: s fxed, T C nduces a lnear mappng T CC L C : C G 2 4 T CC L C (LC = C (L C L C. (17.17 If L C s an eppolar lne wth respect to C, the ernel of T CC s all L eppolar lnes wth respect to C, the range s the eppolar lne represented C by L C ; else f L C s an eppolar lne wth respect to C, the ernel s the eppolar lne represented by L C, the range s all eppolar lnes wth respect to C. For other cases, the ernel s zero. Geometrcally, when T C (L C, L C 0, then L C L C represents a lne or lne at nfnty L of A 3, both L C and L C are mages of L. T C (L C, L C s just the mage of L n camera C. 2. The mappng T C : (C R 4 (C G4 2 (C G4 2 = C R 4 s defned as T C (m C, L C = C (m C L C. (17.18 When m C s fxed, T C C nduces a lnear mappng T C C m : C G 2 C 4 C R 4 : T C C m C (LC = C (m C L C. (17.19 If m C s the eppole of C, the ernel of T C C m s all eppolar lnes wth C respect to C, the range s the eppole of C. For other cases, the ernel s the eppolar lne C m C, the range s the two-dmensonal subspace of C R 4 represented by C m C. Geometrcally, when T C (m C, L C 0, then m C L C represents a pont or pont at nfnty M of A 3, m C s ts mage n camera C, and L C s the mage of a space lne or lne at nfnty passng through M. T C (m C, L C s just the mage of M n camera C. 3. The mappng T C : (C R 4 (C G4 2 (C G4 2 = C R 4 s defned as T C (m C, L C = C (m C L C. (17.20 We prove below two propostons n [81, 83] usng the above reformulaton of trfocal tensors. Proposton Let L C be an eppolar lne n camera C wth respect to C and L C be the correspondng eppolar lne n camera C. Then for any lne L C n camera C whch s not the eppolar lne wth respect to C, T C (L C, L C L C.

15 17. Coordnate-Free Projectve Geometry for Computer Vson 429 Proof. The hypotheses are L C L C, C L C 0. Usng the formula that for any C R 4, A 3, B 3 G 3 4, we get C (A 3 B 3 = (C B 3 A 3 (C A 3 B 3, (17.21 T C (L C, L C C (L C L C = (C L C L C (C L C L C L C. Proposton Let m C, m C be mages of the pont or pont at nfnty M n cameras C, C respectvely. Let L C be an mage lne passng through m C but not through E C C. Let L C be an mage lne passng through m C but not through E C C. Then the ntersecton of T C (L C, L C wth the eppolar lne C m C s the mage of M n camera C. Proof. The hypotheses are M L C = M L C = 0, C L C 0, C L C 0. So T C (L C, L C (C m C = (C (L C L C (C C M = ((C C L C L C (C M ((C M L C L C (C C = (C L C (C L C (C M C M Relatons among Eppoles, Eppolar Tensors, and Trfocal Tensors of Three Cameras Consder the followng 9 vectors of R 4 : ES = {e C, e C j, e C 1, j, 3}. (17.1 Accordng to (17.4, (17.6 and (17.15, by nterchangng among C, C, C any of the eppoles, eppolar tensors and trfocal tensors of the three cameras has ts components represented as a determnant of 4 vectors n ES. For example, E CC F CC j T j = (e C = (ê C = (ê C ; ê C j ; j. (17.2 Conversely, any determnant of 4 vectors n ES equals a component of one of the eppoles, eppolar tensors and trfocal tensors up to an ndex-free

16 430 Hongbo L, Gerald Sommer scale. Snce the only constrant on the 9 vectors s that they are all n R 4, theoretcally all relatons among the eppoles, eppolar tensors and trfocal tensors can be establshed by manpulatng n the algebra of determnants of vectors n ES usng the followng Cramer s rule [76, 80]: (x 2 x 3 x 4 x 5 x 1 = (x 1 x 3 x 4 x 5 x 2 (x 1 x 2 x 4 x 5 x 3 +(x 1 x 2 x 3 x 5 x 4 (x 1 x 2 x 3 x 4 x 5, (17.3 where the x s are vectors n R 4. In practce, however, we can only select a few expressons from the algebra of determnants and mae manpulatons, and t s dffcult to mae the selecton. In ths secton we propose a dfferent approach. Instead of consderng the algebra of determnants drectly, we consder the set of meets of dfferent blades, each blade beng an outer product of vectors n ES. Snce the meet operator s assocatve and ant-commutatve n the sense that A r B s = ( 1 rs B s A r, (17.4 for A r G4 r and B s G4 s, for the same expresson of meets we can have a varety of expansons. Then we can obtan varous equaltes on determnants of vectors n ES, whch may be changed nto equaltes, or equaltes up to an ndex-free constant, on components of the eppoles, eppolar tensors and trfocal tensors. It appears that we need only 7 expressons of meets to derve and further generalze all the nown constrants on eppolar and trfocal tensors. It should be remnded that n ths chapter we always use the notaton 1 2 to denote that, 1, 2 s an even permutaton of 1, 2, Relatons on Eppolar Tensors Consder the followng expresson: Fexp = (e C (e C 1 e C 2 e C 3 (e C (17.5 It s the dual of the blade C C C. Expandng Fexp from left to rght, we get Fexp = 3,=1 = 3,=1 ((e C e C (ê C E CC F CC e C. Expandng Fexp from rght to left, we get ê C e C

17 17. Coordnate-Free Projectve Geometry for Computer Vson 431 Fexp = 3 =1 ( ((e C e C ((e C 1 e C 2 e C 3 ec ((e C e C ((e C 1 e C 2 e C 3 ec = 3 (E C C E C C E C C E C C e C, =1 where. So for any 1 3, 3 =1 where K C CC e C E CC F CC K C CC, (17.6 = E C C E C C E C C E C C. (17.6 s a fundamental relaton on the eppolar tensor F CC and the eppoles. In matrx form, t can be wrtten as (F CC T E CC E C C E C C ; (17.7 n Grassmann-Cayley algebra, t can be wrtten as C (C C (C C C (C C. (17.8 Geometrcally, t means that the eppolar lne n camera C wth respect to both C and C s the mage lne connectng the two eppoles E C C and E C C. One should notce the obvous advantage of Grassmann-Cayley algebrac representaton n geometrc nterpretaton. Snce E C C E C C s orthogonal to E C C, an mmedate corollary s (E CC T F CC E C C = 0, (17.9 whch s equvalent to (C C (C C = 0. Geometrcally, t means that the two eppoles E C C and E C C satsfy the eppolar constrant Relatons on Trfocal Tensors I The frst dea to derve relatons on trfocal tensors s very smple: f the tensor (T j,j,=1..3 s gven, then expandng (ê C j (e C (17.10 gves a 2-vector of the e C s whose coeffcents are nown. Smlarly, expandng Texp 1 = (ê C (ê C (e C (17.11 from rght to left gves a vector of the e C s whose coeffcents are nown. Expandng Texp 1 from left to rght, we get a vector of the e C s whose

18 432 Hongbo L, Gerald Sommer coeffcents depend on eppolar tensors. By comparng the coeffcents of the e C s, we get a relaton on T and eppolar tensors. Assume that j. Expandng Texp 1 from left to rght, we get Texp 1 = (ê C = 3 =1 F CC j F CC e C. 3 (ê C =1 Expandng Texp 1 from rght to left, we get Texp 1 = 3 ( ((ê C e C ((ê C =1 ((ê C e C ((ê C = 3 (T j1 T j2 T j1 T j2 e C =1 where. So where ê C e C e C e C, e C Fj CC F CC = t C j, (17.12 t C j = T T j2 T j1 T j2. (17.13 Proposton For any 1, j, 3, Fj CC F CC t C j. (17.14 Corollary Let 1,,,, 3, then F CC F CC F CC F CC = tc t C, for any 1 3; (17.15 = tc j t C j, for any 1 j 3; (17.16 t C t C = tc t C. (17.17 Notce that (17.17 s a constrant of degree 4 on T. To understand relaton (17.14 geometrcally, we frst express t n terms of Grassmann-Cayley algebra. When C e C s fxed, T nduces a lnear mappng T C C : C G 2 4 C R 4 by T C C (L C = C ((C e C L C. (17.18 The matrx of T C C s ( T j T j,=1..3.

19 17. Coordnate-Free Projectve Geometry for Computer Vson 433 Defne a lnear mappng t C C : C R 4 C G 2 m C C R 4 and m C = L C 4 as follows: let 1 L C 2, where L C 1, L C 2 C G4, 2 then t C C (m C = T C C (L C 1 C T C C (L C 2. (17.19 We need to prove that ths mappng s well-defned. Usng the formula that for any 2-blade C 2 G 2 4 and 3-blades A 3, B 3 G 3 4, we get (C 2 A 3 (C 2 B 3 = (A 3 B 3 C 2 C 2, (17.20 t C C (m C = C ((C e C LC 1 ((C e C LC 2 = L C 1 LC 2 (C ec C C e C = m C (C e C C C e C = 3 m C (C e C (C C e C e C C ê C. =1 (17.21 So t C C s well-defned. Let j and, then snce t C C (C e C j = T C C (C e C j e C C T C C (C e C j e C = ( 3 =1 T j2 C e C C ( 3 =1 = 3 (T j1 T j2 T j1 T j2 C ê C =1 the matrx of t C C s (t j T j,=1..3. So (17.14 s equvalent to T j1 C e C T C C (L C 1 C T C C (L C 2 = m C (C e C C C e C. (17.22 Geometrcally, T C C maps an mage lne n camera C to an mage pont on the eppolar lne C C e C n camera C. (17.22 says that the mage lne connectng the two mage ponts T C C (L C 1 and T C C (L C 2 n camera C s just the eppolar lne C C e C. Ths s the geometrc nterpretaton of (17.14., Relatons on Trfocal Tensors II Now we let the two ê C s n Texp 1 be dfferent, and let the two e C be the same,. e., we consder the expresson

20 434 Hongbo L, Gerald Sommer Texp 2 = (e C 1 e C j (e C 2 e C j (e C (17.23 Assume that 1 2. Expandng Texp 2 from left to rght, we get ( 3 Texp 2 = (e C e C 1 e C 2 j ((e C j ê C e C = 3 =1 E C C j T j ec. Expandng Texp 2 from rght to left, we get =1 Texp 2 = 3 ( ((ê C 2 j e C ((ê C 1 j e C =1 ((ê C 2 j e C ((ê C 1 j e C = 3 (T 1j T 2j T 1j T 2j e C =1 where. So where, e C E C C j T j = tc j, (17.24 t C j = T 1j T 2j T 1j T 2j. (17.25 Proposton For any 1, j, 3, E C C j T j t C j. (17.26 Corollary For any 1, 1, 2, j,,, 3, T 1j T 2j = tc 1j t C 2j ; T j T j = tc j t C j. (17.27 Same as before, to understand relaton (17.26 geometrcally, we frst express t n terms of Grassmann-Cayley algebra. When C ê C j s fxed, T nduces a lnear mappng Tj CC : C G4 2 C G2 4 by T CC j (L C = C ((C ê C j LC, (17.28 whose matrx s ( T j,=1..3. T also nduces a lnear mappng T CC j : C R 4 C R 4 by j (m C = C (m C (C ê C j, (17.29 T CC

21 17. Coordnate-Free Projectve Geometry for Computer Vson 435 whose matrx s ( T j T,=1..3. Defne a lnear mappng tcc j : C R 4 C R 4 as follows: let m C C R 4 and m C = L C 1 LC 2, where LC 1, LC 2 C G4, 2 then t CC j (m C = Tj CC (L C 1 T j CC (L C 2. (17.30 We need to prove that ths mappng s well-defned. Usng the formula that for any C R 4 and A 3, B 3 G 3 4, (C (A 3 B 3 B 3 = (B 3 C(A 3 B 3, (17.31 we get ( t CC j (m C = C ((C ê C j LC 1 ( = C C ( ( (C ê C j LC 1 C ((C ê C j LC 2 (C ê C j LC 2 = (C ê C j C C ((C ê C j LC 1 L C 2 = (C ê C j C C ((C ê C j mc = E C C j T CC j (m C. (17.32 So t CC j s well-defned. Usng (17.30, t can be verfed that the matrx of t CC j s (t C j,=1..3. Thus, (17.26 s equvalent to Tj CC (L C 1 Tj CC (L C 2 = (C ê C j C C ((C ê C j L C 1 L C 2. (17.33 Geometrcally, Tj CC maps an mage lne L C n camera C to the mage lne n camera C, whch s the mage of the space lne on both planes C ê C j and L C. (17.33 says that the ntersecton of the two mage lnes Tj CC (L C 1 and Tj CC (L C 2 s just the mage of the ntersecton of the plane C ê C j wth the lne L C 1 L C 2 n the space. Ths s the geometrc nterpretaton of ( Relatons on Trfocal Tensors III Consder the followng expresson obtaned by changng one of the e C s n Texp 1 to an e C : Texp 3 = (ê C j (ê C Expandng Texp 3 from left to rght, we get (e C (17.34

22 436 Hongbo L, Gerald Sommer Texp 3 = (ê C = 3 l=1 j e C T j Fl CC e C l. ( 3 (ê C l=1 ê C l e C l Expandng Texp 3 from rght to left, we get ( Texp 3 = (ê C j (ê C e C = (T j1 F CC where. T j F CC 1 e C (ê C + T j2 F CC 2 e C T j F CC 2 e C, Proposton For any 1, j 3, 3 =1 e C T j F CC = 0. (17.35 By (17.16, F CC = F1 CC t C j /tc j1. So (17.35 s equvalent to 3 T j t C j = det(t j j,=1..3 = 0, (17.36 =1 for any 1, j 3. (17.36 can also be obtaned drectly by expandng the followng expresson: (ê C 1 (ê C 2 (ê C 3 (e C (17.37 Expandng from left to rght, (17.37 gves zero; expandng from rght to left, t gves det(t j j,=1..3. To understand (17.36 geometrcally, we chec the dual form of (17.37, whch s C ((C e C (C ê C 1 ((C e C (C ê C 2 ((C e C (C ê C 3. (17.38 (17.38 equals zero because the ntersectons of a lne wth three planes are always collnear. Interchangng C wth C, we get det(t j,j=1..3 = 0 for any 1 3. By (17.26, we have

23 17. Coordnate-Free Projectve Geometry for Computer Vson 437 det(t C j,j=1..3 = 0. (17.39 A smlar constrant can be obtaned by nterchangng C and C. (17.37 can be generalzed to the followng one: ( ( ( 3 λ ê C 1 ( 3 λ ê C 2 =1 ( =1 ( 3 λ ê C 3 (e C =1 (17.40 where the λ s are ndetermnants. (17.40 equals zero when expanded from the left, and equals a polynomal of the λ s when expanded from the rght. The coeffcents of the polynomal are expressons of the T j s. Thus, we get 10 constrants of degree 3 on T, called the ran constrants by Faugeras and Papadopoulo [81, 83] Relatons on Trfocal Tensors IV Now, we let the two ê C s n Texp 3 be dfferent. Consder Texp 4 = (e C e C 1 j (e C e C 2 (e C (17.41 Assume that 1 2. Expandng Texp 4 from left to rght, we get Texp 4 = (e C e C 1 e C 2 j e C 1 j (e C ( 3 ((e C e C l=1 = (E C C j E CC (e C e C e C 2 ê C e C l + T 2jF CC 1 ec + T 2jF CC 1 e C + T 2jF CC 1 e C. Expandng Texp 4 from rght to left, we get Texp 4 = ((e C e C 1 (e C j + ((e C e C 1 (e C j l (ê C ( ((e C e C 1 (e C j + ((e C e C 1 (e C j 1 ê C e C (ê C = T 2jF CC 1 e C + T 2jF CC 1 e C (T 2j F CC 1 1 ê C (ê C 1 ê C + T 2j F CC 1 e C, e C (ê C 1 ê C e C

24 438 Hongbo L, Gerald Sommer where. So 3 =1 T 2jF CC 1 = E C C j E CC. (17.42 Interchangng 1, 2 n Texp 4, we obtan 3 =1 T 1jF CC 2 = E C C j E CC. (17.43 Proposton For any 1, j 3, E C C j E CC W j, (17.44 where W j = 3 =1 T 1jF CC 2. From (17.36, (17.42 and (17.43 we get Proposton For any 1 1, 2, j 3, 3 (T 1jF CC 2 =1 + T 2jF CC 1 = 0. (17.45 In fact, (17.44 can be proved by drect computaton: W j = 3 (C e C 1 (C ê C j (C ê C (C ec 2 (C e C =1 ( ( 3 = C (C e C 2 (C e C =1 êc = (C C e C 2 (C e C 1 (C ê C j = (C C e C 1 e C 2 (C C ê C j = E CC E C C j. (C e C 1 (C ê C j So (17.44 s equvalent to (C C e C 2 (C e C 1 (C ê C j = (C C e C 1 e C 2 (C C ê C j ; (17.46 (17.45 s equvalent to the ant-symmetry of C C e C 1 e C 2 wth respect to e C 1 and e C 2. Defne

25 17. Coordnate-Free Projectve Geometry for Computer Vson = 3 t C T 1 2 (17.47 =1 for 1 1, 2,, 3. By (17.12, (17.36, (17.42 and (17.43, 1 2 = 3 = F CC 1 F CC T 1 2 =1 0, f 1 = 2 ; F CC 1 E C C E CC, f 1 2 ; E CC, f 2 1. F CC 1 E C C (17.48 Two corollares can be drawn mmedately: Corollary For any 1 l, j l 3, where 1 4, = uc C 3 4j 4 = EC C 1 2j 3 3 4j 4j 3 E C C 1 2 j 3. ( Let 1 2. Then for any 1 j l 3 where 1 l 4, 1 2 C 1 2j 3j 4 = uc 1 = ECC 1j 3j 4 E CC 2. (17.50 Corollary For any 1 1, 2, 3 where 1 2,, E C C ( 1 2 j2=1..3. ( For any 1, 3, E CC ( 23 32, 23 31, T. (17.52 Now we explan (17.48 n terms of Grassmann-Cayley algebra. We have defned two mappngs T C C and t C C n (17.18 and (17.19, whose matrces are ( T j j,=1..3 and (t C j j,=1..3 respectvely. By the defnton of 1 2, 1 2 C = t C C 1 (C e C T C C 2 (C ê C. (17.53 Expandng the rght-hand sde of (17.53, we get 1 2 = U C C (C e C 1, C e C 2, C e C, C ê C, (17.54 where U C C : (C R 4 (C R 4 (C R 4 (C G 2 4 R s defned by U C C (m C 1, m C 2, m C, L C = m C 1 m C C L C C (m C 1 C m C 2. (17.55

26 440 Hongbo L, Gerald Sommer (17.54 s (17.48 n Grassmann-Cayley algebrac form. It means that the u C C s are components of the mappng U C C. Notce that (17.49 s a group of degree 6 constrants on T. It s closely related to Faugeras and Mourran s frst group of degree 6 constrants: T 1. T 1l 2. T l1l 2. T 1. T l1. T l1l 2. = T l1. T 1l 2. T l1l 2. T 1. T l1. T 1l 2., (17.56 where T 1. = (T 1 =1..3. It s dffcult to fnd the symmetry of the ndces n (17.56, so we frst express (17.56 n terms of Grassmann-Cayley algebra. Usng the fact that T 1. s the coordnates of C ((C e C (C ê C, we get T 1. T 1l 2. T l1l 2. ( ( = C (C e C (C ê C ( ( C C (C e C (C ê C l 2 ( ( C C (C e C l 1 (C ê C l ( ( s2 = C (C e C (C ê C (C e C (C ê C l 2 ( (C e C l 1 (C ê C l 2. By formula (17.20, T 1. T 1l 2. T l1l 2. = (C e C (C ê C l 2 (C ê C (C C e C (C e C l 1 (C ê C l 2 = (C e C (C ê C (C ê C l 2 C (C e C e C l 1 C (C ê C l 2. (17.57 Defne a mappng V C : (C R 4 (C R 4 (C G4 (C 2 G4 2 R as follows: V C (m C 1, m C 2, L C 1, L C 2 = (C (m C 1 C m C 2 (17.58 (C L C 2 (mc 1 LC 1 LC 2. Let v C l 1l 2 = V C (C ê C, C ê C l 1, C ê C, C ê C l 2. (17.59 By (17.57,

27 17. Coordnate-Free Projectve Geometry for Computer Vson 441 T 1. T 1l 2. T l1l 2. = v C l 1l 2. (17.60 Smlarly, we can get T 1. T l1. T l1l 2. = v C l 1l 2, T l1. T 1l 2. T l1l 2. = v C l 1l 2, T 1. T l1. T 1l 2. = v C l 1l 2. (17.61 So (17.56 s equvalent to = vc l 1l 2 l 1l 2 v C v C l 1l 2 v C l 1l 2, (17.62 whch s smpler than (17.56 n appearance. By (17.58, (17.59, n Grassmann- Cayley algebra, (17.62 s just the followng dentty: C (m C 1 C m C 2 C LC 2 m C 1 LC 1 LC 2 C (m C 1 C m C 2 C LC 1 m C 1 LC 2 LC 1 = C (m C 2 C m C 1 C LC 2 m C 2 LC C (m C 2 C m C 1 C LC 1 m C 2 LC 1 LC 2 2 LC 1, (17.63 for any m C 1, m C 2 C R 4, L C 1, L C 2 C G4. 2 By (17.58, we have 0, f 1 = 2 or = ; F CC 1j E C C E CC, f 1 2 and j, v C 1 2 = or 2 1 and j ; F CC 1j E C C E CC, f 1 2 and j, or 2 1 and j. (17.64 Corollary For any 1 l,, 3 where 1 l 4, = vc v C v C 1 2 v C 3 4. ( Let 1 2. Then for any 1 j l 3 where 1 l 4, = vc 1 2j 3j 4 1 v C v C 1 2 v C 1j 3j 4. (17.66

28 442 Hongbo L, Gerald Sommer (17.56 s a specal case of (17.65 where 3 = 2, 4 = 1. Comparng U C C wth V C, we get V C (m C 1, m C 2, L C 1, L C 2 = U C C (m C 1, m C 2, L C 1 L C 2, L C 2. (17.67 It appears that we have generalzed Faugeras and Mourran s frst group of degree-sx constrants furthermore by U C C, because (17.50 s equvalent to (17.66, whle (17.65 s a specal case of (17.49 where j 4 =. An explanaton for ths phenomenon s that the varables n V C are less separated than those n U C C, so there are less constrants on T that come from V C than from U C C. Interchangng C and C n (17.56, we get Faugeras and Mourran s second group of degree 6 constrants: T 1. T 1.l 2 T l1.l 2 T 1. T l1. T l1.l 2 = T l1. T 1.l 2 T l1.l 2 T 1. T l1. T 1.l 2, (17.68 where T 1. = (T 1 =1..3. Ths group of constrants can be generalzed smlarly Relatons on Trfocal Tensors V Consder the followng expresson: Texp 5 = (e C e C 1 j (e C 2 j (e C (17.69 Assume that 1 2. Expandng Texp 5 from left to rght, we get Texp 5 = E C C j E C C j e C T 2j 3 l=1 Expandng Texp 4 from rght to left, we get Texp 5 = T 2j T 2je C T 2j T 2je C +(T 2j T 2j + T 2j T 2j e C, where. So 3 =1 T l 2j ec l. T 2j T 2j = E C C j E C C j. (17.70 Proposton For any 1, j 3, 3 =1 T j T j E C C j E C C j. (17.71

29 17. Coordnate-Free Projectve Geometry for Computer Vson 443 Usng the relaton (17.26, we get 3 =1 t C j T j = det(t j,=1..3 = (E C C j 2 E C C j. (17.72 Corollary For any 1 1, 2, 3 where 1 2, ( E C C det(t j,=1..3. (17.73 ( 1 2 j= Relatons on Trfocal Tensors VI The second dea of dervng relatons on trfocal tensors s as follows: f the tensor (T j,j,=1..3 s gven, then expandng (e C j (e C 1 e C 2 e C 3 (17.74 gves a vector of the e C s whose coeffcents are nown. Smlarly, expandng (e C j (e C j j (e C 1 e C 2 e C 3 j, (e C 1 e C 2 e C 3 (17.75 gves two 2-vectors of the e C j e C s and the e C e C s respectvely, whose coeffcents are nown. The meet of two such 2-vectors,. e., Texp 6 = ( (e C (e C 1 e C 2 e C 3 ( (e C (e C 1 e C 2 e C 3 (17.76 s an expresson of the T j s. Expandng the meets dfferently, we get a relaton on T, eppoles and eppolar tensors. Assume that j and. Expandng Texp 6 accordng to ts parentheses, we get ( 3 Texp 6 = (e C 1=1( 3 ê C 1 (e C 2=1 ê C 2 e C 1 e C 2 = 3 ( T 1 T 2 + T 2 T 1 T j1, =1 where 1 2. Usng the fact that the meet of a 4-vector wth any multvector n G 4 s a scalar multplcaton of the multvector by the dual of the 4-vector, we get

30 444 Hongbo L, Gerald Sommer So Texp 6 = (e C F C C j = F C C j E C C E C C = (e C E C C E C C. (e C 1 e C 2 e C 3 (e C 1 e C 2 e C 3 3 (T 1 T 2 T 2 T 1 T j1. (17.77 =1 Interchangng, n Texp 5, we get F C C j E C C E C C = 3 (T 1 T 2 T 2 T 1 T j2. (17.78 =1 Interchangng, n Texp 5, we get F C C j E C C E C C = 3 (T 1 T 2 T 2 T 1 T j1. (17.79 =1 Interchangng (, and (, n Texp 5, we get Then F C C j E C C E C C = 3 (T 1 T 2 T 2 T 1 T j2. (17.80 =1 When = or =, Texp 5 = 0 by expandng from left to rght. Defne v C = vj C 1 = 3 (T 1 T 2 T 2 T 1 T j1. (17.81 =1 0, f = or = ; F C C j F C C j E C C E C C, f j and, or j and ; E C C E C C, f j and, or j and. (17.82 Proposton For any 1,,, 3, v C C ; E C C vj C = EC 2 v C vj C = EC 1 C E C C. (17.83

31 17. Coordnate-Free Projectve Geometry for Computer Vson 445 Corollary For any 1 j l, l 3, where 1 l 4, v C v C = vc 3 4 v C 3 4 ; v C v C = vc j 3j 4 v C j 3j 4. ( For any 1 l, j, j l, l 3 where 1 l 2, v C v C = uc C 1 2j 1 2j. (17.85 Notce that (17.84 and (17.85 are groups of degree 6 constrants on T. (17.84 s closely related to Faugeras and Mourran s thrd group of degree 6 constrants: T.1 T.1l 2 T.l1l 2 T.1 T.l1 T.l1l 2 = T.l1 T.1l 2 T.l1l 2 T.1 T.l1 T.1l 2, (17.86 where T.1 = (T 1 =1..3. Let us express (17.86 n terms of Grassmann-Cayley algebra. Usng the fact that T.1 s the coordnates of C ((C ê C (C ê C, we get T.1 T.1l 2 T.l1l 2 C ( ( = C (C ê C (C ê C ( C (C (C ê C (C ê C l 2 ( C (C (C ê C l 1 (C ê C l 2 ( ( = C C (C ê C (C ê C ( ( C (C ê C (C ê C l 2 By (17.31, we have (C ê C l 1 (C ê C l 2. T.1 T.1l 2 T.l1l 2 ( ( = C (C ê C (C ê C (C ê C (C ê C l 2 (C ê C l 1 (C ê C l 2 C = (C ê C C (C ê C l 2 C (C ê C (C ê C l 1 (C ê C (C ê C l 2. (17.87 Defne a mappng V C : (C G 2 4 (C G 2 4 (C G 2 4 (C G 2 4 R as follows:

32 446 Hongbo L, Gerald Sommer V C (L C 1, L C 2, L C 1, L C 2 = (L C 1 C(L C 2 C(L C 1 L C 2 L C 1 L C 2. (17.88 Then V C (C ê C, C ê C l 1, C ê C, C ê C l 2 = v C l 1l 2, (17.89 Accordng to (17.87, T.1 T.1l 2 T.l1l 2 = v C l 1l 2. (17.90 Smlarly, we have T.1 T.l1 T.l1l 2 = vl C 1l 2, T.l1 T.1l 2 T.l1l 2 = vl C 1l 2, T.1 T.l1 T.1l 2 = v C 1l 1l 2. Now (17.86 s equvalent to (17.91 v C l 1l 2 v C l 1l 2 = vc l 1l 2 v C l 1l 2, (17.92 or more explctly, the followng dentty: L C 1 C LC 2 C L C 1 LC 2 LC 1 L C 2 L C 1 C LC 1 C L C 1 LC 2 LC 2 L C 1 = LC 2 C LC 2 C L C 2 LC 1 LC 1 L C 2 L C 2 C LC 1 C L C 2 LC 1 LC 2 L C 1 (17.84 s a straghtforward generalzaton of t.. ( A Unfed Treatment of Degree-sx Constrants In ths secton we mae a comprehensve nvestgaton of Faugeras and Mourran s three groups of degree-sx constrants. We have defned 1 2 n (17.47 to derve and generalze the frst group of constrants. We are gong to follow the same lne to derve and generalze the other two groups of constrants. The trfocal tensor T nduces 6 nds of lnear mappngs as shown n table We have defned two lnear mappngs t C C and t CC j n (17.19 and (17.30 respectvely, whch are generated by the T s. There are 6 such lnear mappngs as shown n table Let m C = L C 1 L C 2, m C = L C 1 L C 2, L C = m C 1 C m C 2. Here t C j = T 1T 2 T 1T 2, (17.94 where 1 2 and j. t C j and (17.25 respectvely. and tc j have been defned n (17.13

33 17. Coordnate-Free Projectve Geometry for Computer Vson 447 Table Lnear mappngs nduced by T Mappng Defnton Matrx T C C C G 2 4 C 4 ( T j j,=1..3 L C C ((C e C L C T C C C G 2 4 C 4 ( T j T j,=1..3 L C C ((C e C L C T CC j C G 2 4 C G 2 4 ( T j,=1..3 L C C ((C ê C j L C T C C j C 4 C 4 ( T j T,=1..3 m C C (m C (C ê C j T CC C G 2 4 C G 2 4 ( T j,j=1..3 L C C (L C (C ê C T C C C 4 C 4 ( T j T,j=1..3 m C C (m C (C ê C The mappngs t s are well-defned because t C C (m C = (C e C mc C C e C, t C C (m C = (C e C mc C C e C, t CC j (m C = (C ê C j C C ((C ê C j mc, t C C j (L C = (C ê C j C C (L C (C ê C j, t CC t C C (m C = (C ê C C C (C ê (mc C, (L C = (C ê C C C (L C (C ê C. (17.95 For any 1 1, 2,,,, 3, let

34 448 Hongbo L, Gerald Sommer Table Lnear mappngs nduced by t Mappng Defnton Matrx t C C t C C C m C T C C (L C C 4 C G 2 4 (t C j j,= C T C C (L C 2 4 C G 2 4 (t C j T j,=1..3 m C T C C (L C 1 C T C C (L C 2 t CC j C m C Tj CC (L C 4 C 4 1 C Tj CC (L C 2 (t C j,=1..3 t C C j C G4 2 C G4 2 (t C j T,=1..3 L C T C C j (m C 1 C T C C j (m C 2 t CC C 4 C m C T CC (L C 1 C T CC (L C 2 4 (t C j,j=1..3 t C C C G4 2 C G4 2 (t C j T,j=1..3 L C T C C (m C 1 C T C C (m C = 3 =1 1 2 = 3 =1 u C C 1 2 = 3 j=1 u C C 1 2 = 3 j=1 u CC = 3 =1 u CC = 3 =1 t C 1 T 2, t C 1 T 2, t C 1j T 2j, t C 1j T 2j, t C T j2, t C T j2. (17.96 Then

35 17. Coordnate-Free Projectve Geometry for Computer Vson C = t C C 1 (C e C T C C 2 (C ê C, 1 2 C = t C C (C ê C 1 T C C (C e C 2, 1 2 C = t C C(C e C T C C(C ê C, u C C 1 u C C 1 2 C = t C C (C ê C 1 T C C (C e C 2, 2 (17.97 C = t CC (C e C T CC (C ê C, u CC u CC C = t CC (C e C T CC (C ê C. Expandng the rght-hand sde of the above equaltes, we can get a factored form of the u s, from whch we get the followng constrants. Constrants from 1 2 : (see also subsecton , f 1 = 2 ; 1 2 = F CC 1 E C C E CC, f 1 2 ; F CC E CC, f E C C (17.98 Two constrants can be obtaned from 1 2 : 1. For any 1 l, j l 3, where 1 l 4, 1 2 C 3 4j 4 = uc 1 2j 3 3 4j 4j 3. ( Let 1 2. Then for any 1 j l 3 where 1 l 4, 1 2 C 1 2j 3j 4 = uc 1 1j 3j 4. ( Defne U C C : (C R 4 (C R 4 (C R 4 (C G 2 4 R by U C C (m C 1, mc 2, mc, L C = (m C 1 m C (C L C (C (m C 1 C m C 2. ( Then U C C (C e C 1, C e C 2, C e C, C ê C = 1 2. ( Constrants from 1 2 : If 1 2, then

36 450 Hongbo L, Gerald Sommer 1 2 = E C C E C C 0, f = ; E CC 1 F CC 2j, f j ; E CC 1 F CC 2j, f j. ( Two constrants can be obtaned from 1 2 : 1. Let 1 2, 3 4. Then for any 1, 3, = uc C ( Let 1 2. Then for any 1 j l 3 where 1 l 4, = uc C 1 2j 3j Defne U C C 2j 3j 4. ( : (C G4 2 (C R4 (C G4 2 (C G4 2 R by U C C (L C, m C, L C 1, LC 2 = (LC C (L C 1 C(mC L C 1 LC 2. ( When 1 2, U C C (C ê C 1, C e C 2, C ê C, C ê C = 1 2. ( Constrants from u C C u C C 1 2 = 1 2 : E C C E C C 0, f 1 = 2 ; E CC F CC 1, f 1 2 ; E CC F CC 1, f 2 1. ( Two constrants can be obtaned from u C C 1 2 : 1. For any 1 l, l 3 where 1 l 4, u C C 1 2 u C C C = uc u C C ( Let 1 2. Then for any 1 l 3 where 1 l 4, u C C 1 2 u C C C = uc 1 u C C (17.110

37 17. Coordnate-Free Projectve Geometry for Computer Vson 451 Defne U C C : (C R 4 (C R 4 (C R 4 (C G 2 4 R by Then U C C (m C 1, mc 2, mc, L C = (m C 1 mc (C L C (C (m C 1 C m C 2. ( U C C (C e C 1, C e C 2, C e C, C ê C = u C C 1 2. ( Constrants from u C C 1 2 : If 1 2, then 0, f 1 = ; u C C 1 2 = E C C E C C E CC 1 F CC, f ; E CC 1 F CC, f. ( Two constrants can be obtaned from u C C 1 2 : 1. Let 1 2 and 3 4. Then for any 1, 3, = uc C u C C u C C 1 2 u C C 3 4. ( Let 1 2. Then for any 1 l 3 where 1 l 4, = uc C u C C u C C 1 2 u C C ( Defne U C C : (C G4 2 (C R 4 (C G4 2 (C G4 2 R by U C C (L C, m C, L C 1, L C 2 = (L C C (L C 1 C(m C L C 1 L C 2. ( When 1 2, U C C (C ê C 1, C e C 2, C ê C, C ê C = u C C 1 2. ( Constrants from u CC : If, then 0, f j 1 = ; u CC = E C C E C C F C C j, f j ; E C C E C C F C C j, f j. ( Two constrants can be obtaned from u CC :

38 452 Hongbo L, Gerald Sommer 1. Let and 3 4. Then for any 1, 3, = ucc 3 4 u CC u CC u CC 3 4. ( Let, then for any 1 j l 3 where 1 l 4, u CC u CC = ucc j 3j 4 u CC j 3j 4. ( Defne U CC : (C G4 2 (C G4 2 (C R 4 (C G4 2 R by U CC (L C 1, L C 2, m C, L C = (L C 1 C(LC C(L C 1 LC 2 mc. ( When, U CC (C ê C, C ê C, C e C, C ê C = u CC. ( Constrants from u CC : If, then 0, f 1 = ; u CC = E C C E C C F C C, f ; E C C E C C F C C, f. ( Two constrants can be obtaned from u CC : 1. Let and j 3 j 4. Then for any 1, 3, = ucc j 3j 4 u CC u CC u CC j 3j 4. ( Let j, then for any 1 l 3 where 1 l 4, = ucc 3 4 j u CC u CC u CC j 3 4. ( Defne U CC : (C R 4 (C G4 2 (C G4 2 (C G4 2 R by U CC (m C, L C, L C 1, LC 2 = (LC 1 C(L C C (m C L C 1 L C 2. ( When, U CC (C e C, C ê C, C ê C, C ê C = u CC. (17.127

39 17. Coordnate-Free Projectve Geometry for Computer Vson 453 We have V C (m C 1, mc 2, LC = Thus 1, LC 2 U C C (m C 1, mc 2, LC 1 LC 2, LC 2, f LC 1 LC 2 0; U C C (m C 1 C m C 2, mc 1, LC 2, LC 1, f mc 1 mc 2 0. v C 1 2 = 1 2j, f j ; 1 2j, f j ; 1, f 1 2 ; 1, f 2 1. ( ( Comparng these constrants, we fnd that the constrants (17.65, (17.66 from V C are equvalent to the constrants (17.104, ( from U C C, and are ncluded n the constrants (17.99, ( from U C C. Faugeras and Mourran s frst group of constrants s a specal case of any of (17.65, ( and ( Smlarly, Faugeras and Mourran s second group of constrants s a specal case of any of (17.109, ( We also have = Thus V C (L C 1, LC 2, LC 1, LC 2 U CC (L C 1, LC 2, LC 1 L C 2, LC 2, f LC 1 L C 2 0; U CC (L C 1 C L C 2, L C 1, L C 2, L C 1, f L C 1 L C 2 0. vj C 1 = u CC j, f j ; u CC j, f j ; u CC, f ; ( ( u CC, f. The constrants (17.84, (17.85 from V C are equvalent to the constrants (17.124, ( from U C C, and are also equvalent to the constrants (17.119, ( from U C C. Faugeras and Mourran s thrd group of constrants s a specal case of any of (17.84, ( and ( Concluson In ths chapter we propose a new algebrac representaton for mage ponts obtaned from a pnhole camera, based on Hestenes and Zegler s dea of projectve splt. We reformulate camera modelng and calbraton, eppolar and

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