Epipolar Geometry and the Fundamental Matrix

Transcription

1 9 Epipolar Gomtry and th Fundamntal Matrix Th pipolar gomtry is th intrinsic projctiv gomtry btwn two viws. It is indpndnt of scn structur, and only dpnds on th camras intrnal paramtrs and rlativ pos. Th fundamntal matrix F ncapsulats this intrinsic gomtry. It is a 3 3 matrix of rank 2. If a point in 3-spac X is imagd as x in th first viw, and x in th scond, thn th imag points satisfy th rlation x T Fx =0. W will first dscrib pipolar gomtry, and driv th fundamntal matrix. Th proprtis of th fundamntal matrix ar thn lucidatd, both for gnral motion of th camra btwn th viws, and for svral commonly occurring spcial motions. It is nxt shown that th camras can b rtrivd from F up to a projctiv transformation of 3-spac. This rsult is th basis for th projctiv rconstruction thorm givn in chaptr 10. Finally, if th camra intrnal calibration is known, it is shown that th Euclidan motion of th camras btwn viws may b computd from th fundamntal matrix up to a finit numbr of ambiguitis. Th fundamntal matrix is indpndnt of scn structur. Howvr, it can b computd from corrspondncs of imagd scn points alon, without rquiring knowldg of th camras intrnal paramtrs or rlativ pos. This computation is dscribd in chaptr Epipolar gomtry Th pipolar gomtry btwn two viws is ssntially th gomtry of th intrsction of th imag plans with th pncil of plans having th baslin as axis (th baslin is th lin joining th camra cntrs). This gomtry is usually motivatd by considring th sarch for corrsponding points in stro matching, and w will start from that objctiv hr. Suppos a point X in 3-spac is imagd in two viws, at x in th first, and x in th scond. What is th rlation btwn th corrsponding imag points x and x? As shown in figur 9.1a th imag points x and x, spac point X, and camra cntrs ar coplanar. Dnot this plan as π. Clarly, th rays back-projctd from x and x intrsct at X, and th rays ar coplanar, lying in π. It is this lattr proprty that is of most significanc in sarching for a corrspondnc. 239

2 240 9 Epipolar Gomtry and th Fundamntal Matrix pipolar plan π X X X? X? x x x l C C pipolar lin for x a b Fig Point corrspondnc gomtry. (a) Th two camras ar indicatd by thir cntrs C and C and imag plans. Th camra cntrs, 3-spac point X, and its imags x and x li in a common plan π. (b) An imag point x back-projcts to a ray in 3-spac dfind by th first camra cntr, C, and x. This ray is imagd as a lin l in th scond viw. Th 3-spac point X which projcts to x must li on this ray, so th imag of X in th scond viw must li on l. π l l X baslin baslin a b Fig Epipolar gomtry. (a) Th camra baslin intrscts ach imag plan at th pipols and. Any plan π containing th baslin is an pipolar plan, and intrscts th imag plans in corrsponding pipolar lins l and l. (b) As th position of th 3D point X varis, th pipolar plans rotat about th baslin. This family of plans is known as an pipolar pncil. All pipolar lins intrsct at th pipol. Supposing now that w know only x, w may ask how th corrsponding point x is constraind. Th plan π is dtrmind by th baslin and th ray dfind by x. From abov w know that th ray corrsponding to th (unknown) point x lis in π, hnc th point x lis on th lin of intrsction l of π with th scond imag plan. This lin l is th imag in th scond viw of th ray back-projctd from x. Itisthpipolar lin corrsponding to x. In trms of a stro corrspondnc algorithm th bnfit is that th sarch for th point corrsponding to x nd not covr th ntir imag plan but can b rstrictd to th lin l. Th gomtric ntitis involvd in pipolar gomtry ar illustratd in figur 9.2. Th trminology is Th pipol is th point of intrsction of th lin joining th camra cntrs (th baslin) with th imag plan. Equivalntly, th pipol is th imag in on viw

3 9.2 Th fundamntal matrix F 241 a b c Fig Convrging camras. (a) Epipolar gomtry for convrging camras. (b) and (c) A pair of imags with suprimposd corrsponding points and thir pipolar lins (in whit). Th motion btwn th viws is a translation and rotation. In ach imag, th dirction of th othr camra may b infrrd from th intrsction of th pncil of pipolar lins. In this cas, both pipols li outsid of th visibl imag. of th camra cntr of th othr viw. It is also th vanishing point of th baslin (translation) dirction. An pipolar plan is a plan containing th baslin. Thr is a on-paramtr family (a pncil) of pipolar plans. An pipolar lin is th intrsction of an pipolar plan with th imag plan. All pipolar lins intrsct at th pipol. An pipolar plan intrscts th lft and right imag plans in pipolar lins, and dfins th corrspondnc btwn th lins. Exampls of pipolar gomtry ar givn in figur 9.3 and figur 9.4. Th pipolar gomtry of ths imag pairs, and indd all th xampls of this chaptr, is computd dirctly from th imags as dscribd in sction 11.6(p290). 9.2 Th fundamntal matrix F Th fundamntal matrix is th algbraic rprsntation of pipolar gomtry. In th following w driv th fundamntal matrix from th mapping btwn a point and its pipolar lin, and thn spcify th proprtis of th matrix. Givn a pair of imags, it was sn in figur 9.1 that to ach point x in on imag, thr xists a corrsponding pipolar lin l in th othr imag. Any point x in th scond imag matching th point x must li on th pipolar lin l. Th pipolar lin

4 242 9 Epipolar Gomtry and th Fundamntal Matrix at infinity at infinity a b c Fig Motion paralll to th imag plan. In th cas of a spcial motion whr th translation is paralll to th imag plan, and th rotation axis is prpndicular to th imag plan, th intrsction of th baslin with th imag plan is at infinity. Consquntly th pipols ar at infinity, and pipolar lins ar paralll. (a) Epipolar gomtry for motion paralll to th imag plan. (b) and (c) a pair of imags for which th motion btwn viws is (approximatly) a translation paralll to th x-axis, with no rotation. Four corrsponding pipolar lins ar suprimposd in whit. Not that corrsponding points li on corrsponding pipolar lins. is th projction in th scond imag of th ray from th point x through th camra cntr C of th first camra. Thus, thr is a map x l from a point in on imag to its corrsponding pipolar lin in th othr imag. It is th natur of this map that will now b xplord. It will turn out that this mapping is a (singular) corrlation, that is a projctiv mapping from points to lins, which is rprsntd by a matrix F, th fundamntal matrix Gomtric drivation W bgin with a gomtric drivation of th fundamntal matrix. Th mapping from a point in on imag to a corrsponding pipolar lin in th othr imag may b dcomposd into two stps. In th first stp, th point x is mappd to som point x in th othr imag lying on th pipolar lin l. This point x is a potntial match for th point x. In th scond stp, th pipolar lin l is obtaind as th lin joining x to th pipol. Stp 1: Point transfr via a plan. Rfr to figur 9.5. Considr a plan π in spac not passing through ithr of th two camra cntrs. Th ray through th first camra cntr corrsponding to th point x mts th plan π in a point X. This point X is thn projctd to a point x in th scond imag. This procdur is known as transfr via th plan π. Sinc X lis on th ray corrsponding to x, th projctd point x must li on th pipolar lin l corrsponding to th imag of this ray, as illustratd in

5 9.2 Th fundamntal matrix F 243 X π x Hπ x l Fig A point x in on imag is transfrrd via th plan π to a matching point x in th scond imag. Th pipolar lin through x is obtaind by joining x to th pipol. In symbols on may writ x = H π x and l =[ ] x =[ ] H π x = Fx whr F =[ ] H π is th fundamntal matrix. figur 9.1b. Th points x and x ar both imags of th 3D point X lying on a plan. Th st of all such points x i in th first imag and th corrsponding points x i in th scond imag ar projctivly quivalnt, sinc thy ar ach projctivly quivalnt to th planar point st X i. Thus thr is a 2D homography H π mapping ach x i to x i. Stp 2: Constructing th pipolar lin. Givn th point x th pipolar lin l passing through x and th pipol can b writtn as l = x =[ ] x (th notation [ ] is dfind in (A4.5 p581)). Sinc x may b writtn as x = H π x,whav l =[ ] H π x = Fx whr w dfin F =[ ] H π, th fundamntal matrix. This shows Rsult 9.1. Th fundamntal matrix F may b writtn as F =[ ] H π, whr H π is th transfr mapping from on imag to anothr via any plan π. Furthrmor, sinc [ ] has rank 2 and H π rank 3, F is a matrix of rank 2. Gomtrically, F rprsnts a mapping from th 2-dimnsional projctiv plan IP 2 of th first imag to th pncil of pipolar lins through th pipol. Thus, it rprsnts a mapping from a 2-dimnsional onto a 1-dimnsional projctiv spac, and hnc must hav rank 2. Not, th gomtric drivation abov involvs a scn plan π, but a plan is not rquird in ordr for F to xist. Th plan is simply usd hr as a mans of dfining a point map from on imag to anothr. Th connction btwn th fundamntal matrix and transfr of points from on imag to anothr via a plan is dalt with in som dpth in chaptr Algbraic drivation Th form of th fundamntal matrix in trms of th two camra projction matrics, P, P, may b drivd algbraically. Th following formulation is du to Xu and Zhang [Xu-96].

6 244 9 Epipolar Gomtry and th Fundamntal Matrix Th ray back-projctd from x by P is obtaind by solving PX = x. Th onparamtr family of solutions is of th form givn by (6.13 p162) as X(λ) =P + x + λc whr P + is th psudo-invrs of P, i.. PP + = I, and C its null-vctor, namly th camra cntr, dfind by PC = 0. Th ray is paramtrizd by th scalar λ. In particular two points on th ray ar P + x (at λ = 0), and th first camra cntr C (at λ = ). Ths two points ar imagd by th scond camra P at P P + x and P C rspctivly in th scond viw. Th pipolar lin is th lin joining ths two projctd points, namly l =(P C) (P P + x). Th point P C is th pipol in th scond imag, namly th projction of th first camra cntr, and may b dnotd by. Thus, l =[ ] (P P + )x = Fx, whr F is th matrix F =[ ] P P +. (9.1) This is ssntially th sam formula for th fundamntal matrix as th on drivd in th prvious sction, th homography H π having th xplicit form H π = P P + in trms of th two camra matrics. Not that this drivation braks down in th cas whr th two camra cntrs ar th sam for, in this cas, C is th common camra cntr of both P and P, and so P C = 0. It follows that F dfind in (9.1) is th zro matrix. Exampl 9.2. Suppos th camra matrics ar thos of a calibratd stro rig with th world origin at th first camra Thn and F = [P C] P P + P = K[I 0] [ ] K P + 1 = 0 T P = K [R t]. C = ( 0 1 = [K t] K RK 1 = K T [t] RK 1 = K T R[R T t] K 1 = K T RK T [KR T t] (9.2) whr th various forms follow from rsult A4.3(p582). Not that th pipols (dfind as th imag of th othr camra cntr) ar ( R = P T ) ( ) t 0 = KR T t = P = K t. (9.3) 1 1 Thus w may writ (9.2) as F =[ ] K RK 1 = K T [t] RK 1 = K T R[R T t] K 1 = K T RK T []. (9.4) Th xprssion for th fundamntal matrix can b drivd in many ways, and indd will b drivd again svral tims in this book. In particular, (17.3 p412) xprsss F in trms of 4 4 dtrminants composd from rows of th camra matrics for ach viw. )

7 9.2 Th fundamntal matrix F Corrspondnc condition Up to this point w hav considrd th map x l dfind by F. W may now stat th most basic proprtis of th fundamntal matrix. Rsult 9.3. Th fundamntal matrix satisfis th condition that for any pair of corrsponding points x x in th two imags x T Fx =0. This is tru, bcaus if points x and x corrspond, thn x lis on th pipolar lin l = Fx corrsponding to th point x. In othr words 0=x T l = x T Fx. Convrsly, if imag points satisfy th rlation x T Fx =0thn th rays dfind by ths points ar coplanar. This is a ncssary condition for points to corrspond. Th importanc of th rlation of rsult 9.3 is that it givs a way of charactrizing th fundamntal matrix without rfrnc to th camra matrics, i.. only in trms of corrsponding imag points. This nabls F to b computd from imag corrspondncs alon. W hav sn from (9.1) that F may b computd from th two camra matrics, P, P, and in particular that F is dtrmind uniquly from th camras, up to an ovrall scaling. Howvr, w may now nquir how many corrspondncs ar rquird to comput F from x T Fx =0, and th circumstancs undr which th matrix is uniquly dfind by ths corrspondncs. Th dtails of this ar postpond until chaptr 11, whr it will b sn that in gnral at last 7 corrspondncs ar rquird to comput F Proprtis of th fundamntal matrix Dfinition 9.4. Suppos w hav two imags acquird by camras with non-coincidnt cntrs, thn th fundamntal matrix F is th uniqu 3 3 rank 2 homognous matrix which satisfis x T Fx =0 (9.5) for all corrsponding points x x. W now brifly list a numbr of proprtis of th fundamntal matrix. Th most important proprtis ar also summarizd in tabl 9.1. (i) Transpos: IfF is th fundamntal matrix of th pair of camras (P, P ), thn F T is th fundamntal matrix of th pair in th opposit ordr: (P, P). (ii) Epipolar lins: For any point x in th first imag, th corrsponding pipolar lin is l = Fx. Similarly, l = F T x rprsnts th pipolar lin corrsponding to x in th scond imag. (iii) Th pipol: for any point x (othr than ) th pipolar lin l = Fx contains th pipol. Thus satisfis T (Fx) =( T F)x =0for all x. It follows that T F = 0, i.. is th lft null-vctor of F. Similarly F = 0, i.. is th right null-vctor of F.

8 246 9 Epipolar Gomtry and th Fundamntal Matrix F is a rank 2 homognous matrix with 7 dgrs of frdom. Point corrspondnc: Ifx and x ar corrsponding imag points, thn x T Fx =0. Epipolar lins: l = Fx is th pipolar lin corrsponding to x. l = F T x is th pipolar lin corrsponding to x. Epipols: F = 0. F T = 0. Computation from camra matrics P, P : Gnral camras, F =[ ] P P +, whr P + is th psudo-invrs of P, and = P C, with PC = 0. Canonical camras, P =[I 0], P =[M m], F =[ ] M = M T [], whr = m and = M 1 m. Camras not at infinity P = K[I 0], P = K [R t], F = K T [t] RK 1 =[K t] K RK 1 = K T RK T [KR T t]. Tabl 9.1. Summary of fundamntal matrix proprtis. (iv) F has svn dgrs of frdom: a 3 3 homognous matrix has ight indpndnt ratios (thr ar nin lmnts, and th common scaling is not significant); howvr, F also satisfis th constraint dt F =0which rmovs on dgr of frdom. (v) F is a corrlation, a projctiv map taking a point to a lin (s dfinition (p59)). In this cas a point in th first imag x dfins a lin in th scond l = Fx, which is th pipolar lin of x. Ifl and l ar corrsponding pipolar lins (s figur 9.6a) thn any point x on l is mappd to th sam lin l. This mans thr is no invrs mapping, and F is not of full rank. For this rason, F is not a propr corrlation (which would b invrtibl) Th pipolar lin homography Th st of pipolar lins in ach of th imags forms a pncil of lins passing through th pipol. Such a pncil of lins may b considrd as a 1-dimnsional projctiv spac. It is clar from figur 9.6b that corrsponding pipolar lins ar prspctivly rlatd, so that thr is a homography btwn th pncil of pipolar lins cntrd at in th first viw, and th pncil cntrd at in th scond. A homography btwn two such 1-dimnsional projctiv spacs has 3 dgrs of frdom. Th dgrs of frdom of th fundamntal matrix can thus b countd as follows: 2 for, 2 for, and 3 for th pipolar lin homography which maps a lin through to a lin through. A gomtric rprsntation of this homography is givn in sction 9.4. Hr w giv an xplicit formula for this mapping.

9 9.3 Fundamntal matrics arising from spcial motions 247 l 3 l2 l 1 l 3 l 2 l 1 p a b Fig Epipolar lin homography. (a) Thr is a pncil of pipolar lins in ach imag cntrd on th pipol. Th corrspondnc btwn pipolar lins, l i l i, is dfind by th pncil of plans with axis th baslin. (b) Th corrsponding lins ar rlatd by a prspctivity with cntr any point p on th baslin. It follows that th corrspondnc btwn pipolar lins in th pncils is a 1D homography. Rsult 9.5. Suppos l and l ar corrsponding pipolar lins, and k is any lin not passing through th pipol, thn l and l ar rlatd by l = F[k] l. Symmtrically, l = F T [k ] l. Proof. Th xprssion [k] l = k l is th point of intrsction of th two lins k and l, and hnc a point on th pipolar lin l call it x. Hnc, F[k] l = Fx is th pipolar lin corrsponding to th point x, namly th lin l. Furthrmor a convnint choic for k is th lin, sinc k T = T 0, so that th lin dos not pass through th point as is rquird. A similar argumnt holds for th choic of k =. Thus th pipolar lin homography may b writtn as l = F[] l l = F T [ ] l. 9.3 Fundamntal matrics arising from spcial motions A spcial motion ariss from a particular rlationship btwn th translation dirction, t, and th dirction of th rotation axis, a. W will discuss two cass: pur translation, whr thr is no rotation; and pur planar motion, whr t is orthogonal to a (th significanc of th planar motion cas is dscribd in sction 3.4.1(p77)). Th pur indicats that thr is no chang in th intrnal paramtrs. Such cass ar important, firstly bcaus thy occur in practic, for xampl a camra viwing an objct rotating on a turntabl is quivalnt to planar motion for pairs of viws; and scondly bcaus th fundamntal matrix has a spcial form and thus additional proprtis Pur translation In considring pur translations of th camra, on may considr th quivalnt situation in which th camra is stationary, and th world undrgos a translation t. In this situation points in 3-spac mov on straight lins paralll to t, and th imagd intrsction of ths paralll lins is th vanishing point v in th dirction of t. This is illustratd in figur 9.7 and figur 9.8. It is vidnt that v is th pipol for both viws, and th imagd paralll lins ar th pipolar lins. Th algbraic dtails ar givn in th following xampl.

10 248 9 Epipolar Gomtry and th Fundamntal Matrix paralll lins vanishing point imag camra cntr Fig Undr a pur translational camra motion, 3D points appar to slid along paralll rails. Th imags of ths paralll lins intrsct in a vanishing point corrsponding to th translation dirction. Th pipol is th vanishing point. C C a b c Fig Pur translational motion. (a) undr th motion th pipol is a fixd point, i.. has th sam coordinats in both imags, and points appar to mov along lins radiating from th pipol. Th pipol in this cas is trmd th Focus of Expansion (FOE). (b) and (c) th sam pipolar lins ar ovrlaid in both cass. Not th motion of th postrs on th wall which slid along th pipolar lin. Exampl 9.6. Suppos th motion of th camras is a pur translation with no rotation and no chang in th intrnal paramtrs. On may assum that th two camras ar

11 9.3 Fundamntal matrics arising from spcial motions 249 P = K[I 0] and P = K[I t]. Thn from (9.4) (using R = I and K = K ) F =[ ] KK 1 =[ ]. If th camra translation is paralll to th x-axis, thn =(1, 0, 0) T,so F = Th rlation btwn corrsponding points, x T Fx =0, rducs to y = y, i.. th pipolar lins ar corrsponding rastrs. This is th situation that is sought by imag rctification dscribd in sction 11.12(p302). Indd if th imag point x is normalizd as x =(x, y, 1) T, thn from x = PX = K[I 0]X, th spac point s (inhomognous) coordinats ar (X, Y, Z) T = ZK 1 x, whr Z is th dpth of th point X (th distanc of X from th camra cntr masurd along th principal axis of th first camra). It thn follows from x = P X = K[I t]x that th mapping from an imag point x to an imag point x is x = x + KtZ. (9.6) Th motion x = x + KtZ of (9.6) shows that th imag point starts at x and thn movs along th lin dfind by x and th pipol = = v. Th xtnt of th motion dpnds on th magnitud of th translation t (which is not a homognous vctor hr) and th invrs dpth Z, so that points closr to th camra appar to mov fastr than thos furthr away a common xprinc whn looking out of a train window. Not that in this cas of pur translation F =[ ] is skw-symmtric and has only 2 dgrs of frdom, which corrspond to th position of th pipol. Th pipolar lin of x is l = Fx =[] x, and x lis on this lin sinc x T [] x =0, i.. x, x and = ar collinar (assuming both imags ar ovrlaid on top of ach othr). This collinarity proprty is trmd auto-pipolar, and dos not hold for gnral motion. Gnral motion. Th pur translation cas givs additional insight into th gnral motion cas. Givn two arbitrary camras, w may rotat th camra usd for th first imag so that it is alignd with th scond camra. This rotation may b simulatd by applying a projctiv transformation to th first imag. A furthr corrction may b applid to th first imag to account for any diffrnc in th calibration matrics of th two imags. Th rsult of ths two corrctions is a projctiv transformation H of th first imag. If on assums ths corrctions to hav bn mad, thn th ffctiv rlationship of th two camras to ach othr is that of a pur translation. Consquntly, th fundamntal matrix corrsponding to th corrctd first imag and th scond imag is of th form ˆF =[ ], satisfying x TˆFˆx =0, whr ˆx = Hx is th corrctd point in th first imag. From this on dducs that x T [ ] Hx =0, and so th fundamntal matrix corrsponding to th initial point corrspondncs x x is F =[ ] H. This is illustratd in figur 9.9.

12 250 9 Epipolar Gomtry and th Fundamntal Matrix H C C Fig Gnral camra motion. Th first camra (on th lft) may b rotatd and corrctd to simulat a pur translational motion. Th fundamntal matrix for th original pair is th product F = [ ] H, whr [ ] is th fundamntal matrix of th translation, and H is th projctiv transformation corrsponding to th corrction of th first camra. Exampl 9.7. Continuing from xampl 9.2, assum again that th two camras ar P = K[I 0] and P = K [R t]. Thn as dscribd in sction 8.4.2(p204) th rquisit projctiv transformation is H = K RK 1 = H, whr H is th infinit homography (s sction 13.4(p338)), and F =[ ] H. If th imag point x is normalizd as x = (x, y, 1) T, as in xampl 9.6, thn (X, Y, Z) T = ZK 1 x, and from x = P X = K [R t]x th mapping from an imag point x to an imag point x is x = K RK 1 x + K tz. (9.7) Th mapping is in two parts: th first trm dpnds on th imag position alon, i.. x, but not th point s dpth Z, and taks account of th camra rotation and chang of intrnal paramtrs; th scond trm dpnds on th dpth, but not on th imag position x, and taks account of camra translation. In th cas of pur translation (R = I, K = K ) (9.7) rducs to (9.6) Pur planar motion In this cas th rotation axis is orthogonal to th translation dirction. Orthogonality imposs on constraint on th motion, and it is shown in th xrciss at th nd of this chaptr that if K = K thn Fs, th symmtric part of F, has rank 2 in this planar motion cas (not, for a gnral motion th symmtric part of F has full rank). Thus, th condition that dt Fs =0is an additional constraint on F and rducs th numbr of dgrs of frdom from 7, for a gnral motion, to 6 dgrs of frdom for a pur planar motion. 9.4 Gomtric rprsntation of th fundamntal matrix This sction is not ssntial for a first rading and th radr may optionally skip to sction 9.5. In this sction th fundamntal matrix is dcomposd into its symmtric and skwsymmtric parts, and ach part is givn a gomtric rprsntation. Th symmtric and

13 9.4 Gomtric rprsntation of th fundamntal matrix 251 skw-symmtric parts of th fundamntal matrix ar Fs = ( F + F T) 2 Fa = ( F F T) 2 so that F = Fs + Fa. To motivat th dcomposition, considr th points X in 3-spac that map to th sam point in two imags. Ths imag points ar fixd undr th camra motion so that x = x. Clarly such points ar corrsponding and thus satisfy x T Fx =0, which is a ncssary condition on corrsponding points. Now, for any skw-symmtric matrix A th form x T Ax is idntically zro. Consquntly only th symmtric part of F contributs to x T Fx =0, which thn rducs to x T Fsx =0. As will b sn blow th matrix Fs may b thought of as a conic in th imag plan. Gomtrically th conic ariss as follows. Th locus of all points in 3-spac for which x = x is known as th horoptr curv. Gnrally this is a twistd cubic curv in 3-spac (s sction 3.3(p75)) passing through th two camra cntrs [Maybank-93]. Th imag of th horoptr is th conic dfind by Fs. W rturn to th horoptr in chaptr 22. Symmtric part. Th matrix Fs is symmtric and is of rank 3 in gnral. It has 5 dgrs of frdom and is idntifid with a point conic, calld th Stinr conic (th nam is xplaind blow). Th pipols and li on th conic Fs. To s that th pipols li on th conic, i.. that T Fs =0, start from F = 0. Thn T F = 0 and so T Fs + T Fa =0. Howvr, T Fa =0, sinc for any skw-symmtric matrix S, x T Sx =0. Thus T Fs =0. Th drivation for follows in a similar mannr. Skw-symmtric part. Th matrix Fa is skw-symmtric and may b writtn as Fa = [xa], whr xa is th null-vctor of Fa. Th skw-symmtric part has 2 dgrs of frdom and is idntifid with th point xa. Th rlation btwn th point xa and conic Fs is shown in figur 9.10a. Th polar of xa intrscts th Stinr conic Fs at th pipols and (th pol polar rlation is dscribd in sction 2.2.3(p30)). Th proof of this rsult is lft as an xrcis. Epipolar lin corrspondnc. It is a classical thorm of projctiv gomtry du to Stinr [Smpl-79] that for two lin pncils rlatd by a homography, th locus of intrsctions of corrsponding lins is a conic. This is prcisly th situation hr. Th pncils ar th pipolar pncils, on through and th othr through. Th pipolar lins ar rlatd by a 1D homography as dscribd in sction Th locus of intrsction is th conic Fs. Th conic and pipols nabl pipolar lins to b dtrmind by a gomtric construction as illustratd in figur 9.10b. This construction is basd on th fixd point proprty of th Stinr conic Fs. Th pipolar lin l = x in th first viw dfins an pipolar plan in 3-spac which intrscts th horoptr in a point, which w will call X c. Th point X c is imagd in th first viw at x c, which is th point at which l intrscts th conic Fs (sinc Fs is th imag of th horoptr). Now th imag of X c is also x c in th scond viw du to th fixd-point proprty of th horoptr. So x c is th

14 252 9 Epipolar Gomtry and th Fundamntal Matrix Fs x c F s x l l a a x a b Fig Gomtric rprsntation of F. (a) Th conic Fs rprsnts th symmtric part of F, and th point xa th skw-symmtric part. Th conic Fs is th locus of intrsction of corrsponding pipolar lins, assuming both imags ar ovrlaid on top of ach othr. It is th imag of th horoptr curv. Th lin la is th polar of xa with rspct to th conic Fs. It intrscts th conic at th pipols and. (b) Th pipolar lin l corrsponding to a point x is constructd as follows: intrsct th lin dfind by th points and x with th conic. This intrsction point is x c. Thn l is th lin dfind by th points x c and. imag in th scond viw of a point on th pipolar plan of x. It follows that x c lis on th pipolar lin l of x, and consquntly l may b computd as l = x c. Th conic togthr with two points on th conic account for th 7 dgrs of frdom of F: 5 dgrs of frdom for th conic and on ach to spcify th two pipols on th conic. Givn F, thn th conic Fs, pipols, and skw-symmtric point xa ar dfind uniquly. Howvr, Fs and xa do not uniquly dtrmin F sinc th idntity of th pipols is not rcovrd, i.. th polar of xa dtrmins th pipols but dos not dtrmin which on is and which on Pur planar motion W rturn to th cas of planar motion discussd abov in sction 9.3.2, whr Fs has rank 2. It is vidnt that in this cas th Stinr conic is dgnrat and from sction 2.2.3(p30) is quivalnt to two non-coincidnt lins: Fs = l h l T s + l s l T h as dpictd in figur 9.11a. Th gomtric construction of th pipolar lin l corrsponding to a point x of sction 9.4 has a simpl algbraic rprsntation in this cas. As in th gnral motion cas, thr ar thr stps, illustratd in figur 9.11b: first th lin l = x joining and x is computd; scond, its intrsction point with th conic x c = l s l is dtrmind; third th pipolar lin l = x c is th join of x c and. Putting ths stps togthr w find It follows that F may b writtn as l = [l s ( x)] = [ ] [l s ] [] x. F =[ ] [l s ] []. (9.8) Th 6 dgrs of frdom of F ar accountd for as 2 dgrs of frdom for ach of th two pipols and 2 dgrs of frdom for th lin.

15 9.5 Rtriving th camra matrics 253 x c l s l l x s l xs lh xa a imag b imag Fig Gomtric rprsntation of F for planar motion. (a) Th lins l s and l h constitut th Stinr conic for this motion, which is dgnrat. Compar this figur with th conic for gnral motion shown in figur (b) Th pipolar lin l corrsponding to a point x is constructd as follows: intrsct th lin dfind by th points and x with th (conic) lin l s. This intrsction point is x c. Thn l is th lin dfind by th points x c and. Th gomtry of this situation can b asily visualizd: th horoptr for this motion is a dgnrat twistd cubic consisting of a circl in th plan of th motion (th plan orthogonal to th rotation axis and containing th camra cntrs), and a lin paralll to th rotation axis and intrscting th circl. Th lin is th scrw axis (s sction 3.4.1(p77)). Th motion is quivalnt to a rotation about th scrw axis with zro translation. Undr this motion points on th scrw axis ar fixd, and consquntly thir imags ar fixd. Th lin l s is th imag of th scrw axis. Th lin l h is th intrsction of th imag with th plan of th motion. This gomtry is usd for autocalibration in chaptr Rtriving th camra matrics To this point w hav xamind th proprtis of F and of imag rlations for a point corrspondnc x x. W now turn to on of th most significant proprtis of F, that th matrix may b usd to dtrmin th camra matrics of th two viws Projctiv invarianc and canonical camras It is vidnt from th drivations of sction 9.2 that th map l = Fx and th corrspondnc condition x T Fx =0ar projctiv rlationships: th drivations hav involvd only projctiv gomtric rlationships, such as th intrsction of lins and plans, and in th algbraic dvlopmnt only th linar mapping of th projctiv camra btwn world and imag points. Consquntly, th rlationships dpnd only on projctiv coordinats in th imag, and not, for xampl on Euclidan masurmnts such as th angl btwn rays. In othr words th imag rlationships ar projctivly invariant: undr a projctiv transformation of th imag coordinats ˆx = Hx, ˆx = H x, thr is a corrsponding map ˆl = ˆFˆx with ˆF = H T FH 1 th corrsponding rank 2 fundamntal matrix. Similarly, F only dpnds on projctiv proprtis of th camras P, P. Th camra matrix rlats 3-spac masurmnts to imag masurmnts and so dpnds on both th imag coordinat fram and th choic of world coordinat fram. F dos not

16 254 9 Epipolar Gomtry and th Fundamntal Matrix dpnd on th choic of world fram, for xampl a rotation of world coordinats changs P, P, but not F. In fact, th fundamntal matrix is unchangd by a projctiv transformation of 3-spac. Mor prcisly, Rsult 9.8. If H is a 4 4 matrix rprsnting a projctiv transformation of 3-spac, thn th fundamntal matrics corrsponding to th pairs of camra matrics (P, P ) and (PH, P H) ar th sam. Proof. Obsrv that PX =(PH)(H 1 X), and similarly for P. Thus if x x ar matchd points with rspct to th pair of camras (P, P ), corrsponding to a 3D point X, thn thy ar also matchd points with rspct to th pair of camras (PH, P H), corrsponding to th point H 1 X. Thus, although from (9.1 p244) a pair of camra matrics (P, P ) uniquly dtrmin a fundamntal matrix F, th convrs is not tru. Th fundamntal matrix dtrmins th pair of camra matrics at bst up to right-multiplication by a 3D projctiv transformation. It will b sn blow that this is th full xtnt of th ambiguity, and indd th camra matrics ar dtrmind up to a projctiv transformation by th fundamntal matrix. Canonical form of camra matrics. Givn this ambiguity, it is common to dfin a spcific canonical form for th pair of camra matrics corrsponding to a givn fundamntal matrix in which th first matrix is of th simpl form [I 0], whr I is th 3 3 idntity matrix and 0 a null 3-vctor. To s that this is always possibl, lt P b augmntd by on row to mak a 4 4 non-singular matrix, dnotd P. Now ltting H = P 1, on vrifis that PH =[I 0] as dsird. Th following rsult is vry frquntly usd Rsult 9.9. Th fundamntal matrix corrsponding to a pair of camra matrics P = [I 0] and P =[M m] is qual to [m] M. This is asily drivd as a spcial cas of (9.1 p244) Projctiv ambiguity of camras givn F It has bn sn that a pair of camra matrics dtrmins a uniqu fundamntal matrix. This mapping is not injctiv (on-to-on) howvr, sinc pairs of camra matrics that diffr by a projctiv transformation giv ris to th sam fundamntal matrix. It will now b shown that this is th only ambiguity. W will show that a givn fundamntal matrix dtrmins th pair of camra matrics up to right multiplication by a projctiv transformation. Thus, th fundamntal matrix capturs th projctiv rlationship of th two camras. Thorm Lt F b a fundamntal matrix and lt (P, P ) and ( P, P ) b two pairs of camra matrics such that F is th fundamntal matrix corrsponding to ach of ths pairs. Thn thr xists a non-singular 4 4 matrix H such that P = PH and P = P H.

17 9.5 Rtriving th camra matrics 255 Proof. Suppos that a givn fundamntal matrix F corrsponds to two diffrnt pairs of camra matrics (P, P ) and ( P, P ). As a first stp, w may simplify th problm by assuming that ach of th two pair of camra matrics is in canonical form with P = P =[I 0], sinc this may b don by applying projctiv transformations to ach pair as ncssary. Thus, suppos that P = P =[I 0] and that P =[A a] and P =[à ã]. According to rsult 9.9 th fundamntal matrix may thn b writtn F =[a] A =[ã] Ã. W will nd th following lmma: Lmma Suppos th rank 2 matrix F can b dcomposd in two diffrnt ways as F =[a] A and F =[ã] Ã; thn ã = ka and à = k 1 (A + av T ) for som non-zro constant k and 3-vctor v. Proof. First, not that a T F = a T [a] A = 0, and similarly, ã T F = 0. Sinc F has rank( 2, it follows that ã = ka as rquird. Nxt, from [a] A =[ã] à it follows that [a] ) kã A = 0, and so kã A = av T for som v. Hnc, à = k 1 (A + av T ) as rquird. Applying this rsult to th two camra matrics P and P shows that P =[A a] and P =[k 1 (A+av T ) ka] if thy ar to gnrat th sam F. It only rmains [ now to show k that ths camra pairs ar projctivly rlatd. Lt H b th matrix H = 1 ] I 0 k 1 v T. k Thn on vrifis that PH = k 1 [I 0] =k 1 P, and furthrmor, P H =[A a]h =[k 1 (A + av T ) ka] =[à ã] = P so that th pairs P, P and P, P ar indd projctivly rlatd. This can b tid prcisly to a counting argumnt: th two camras P and P ach hav 11 dgrs of frdom, making a total of 22 dgrs of frdom. To spcify a projctiv world fram rquirs 15 dgrs of frdom (sction 3.1(p65)), so onc th dgrs of frdom of th world fram ar rmovd from th two camras = 7 dgrs of frdom rmain which corrsponds to th 7 dgrs of frdom of th fundamntal matrix Canonical camras givn F W hav shown that F dtrmins th camra pair up to a projctiv transformation of 3-spac. W will now driv a spcific formula for a pair of camras with canonical form givn F. W will mak us of th following charactrization of th fundamntal matrix F corrsponding to a pair of camra matrics: Rsult A non-zro matrix F is th fundamntal matrix corrsponding to a pair of camra matrics P and P if and only if P T FP is skw-symmtric. Proof. Th condition that P T FP is skw-symmtric is quivalnt to X T P T FPX =0 for all X. Stting x = P X and x = PX, this is quivalnt to x T Fx =0, which is th dfining quation for th fundamntal matrix.

18 256 9 Epipolar Gomtry and th Fundamntal Matrix On may writ down a particular solution for th pairs of camra matrics in canonical form that corrspond to a fundamntal matrix as follows: Rsult Lt F b a fundamntal matrix and S any skw-symmtric matrix. Dfin th pair of camra matrics P =[I 0] and P =[SF ], whr is th pipol such that T F = 0, and assum that P so dfind is a valid camra matrix (has rank 3). Thn F is th fundamntal matrix corrsponding to th pair (P, P ). To dmonstrat this, w invok rsult 9.12 and simply vrify that [ F [SF ] T F[I 0] = T S T ] [ F 0 F T = T S T ] F 0 F 0 0 T 0 (9.9) which is skw-symmtric. Th skw-symmtric matrix S may b writtn in trms of its null-vctor as S =[s]. Thn [[s] F ] has rank 3 providd s T 0, according to th following argumnt. Sinc F = 0, th column spac (span of th columns) of F is prpndicular to. But if s T 0, thn s is not prpndicular to, and hnc not in th column spac of F. Now, th column spac of [s] F is spannd by th cross-products of s with th columns of F, and thrfor quals th plan prpndicular to s. So[s] F has rank 2. Sinc is not prpndicular to s, it dos not li in this plan, and so [[s] F ] has rank 3, as rquird. As suggstd by Luong and Viévill [Luong-96] a good choic for S is S =[ ], for in this cas T 0, which lads to th following usful rsult. Rsult Th camra matrics corrsponding to a fundamntal matrix F may b chosn as P =[I 0] and P =[[ ] F ]. Not that th camra matrix P has lft 3 3 submatrix [ ] F which has rank 2. This corrsponds to a camra with cntr on π. Howvr, thr is no particular rason to avoid this situation. Th proof of thorm 9.10 shows that th four paramtr family of camra pairs in canonical form P =[I 0], P =[A + av T ka] hav th sam fundamntal matrix as th canonical pair, P =[I 0], P =[A a]; and that this is th most gnral solution. To summariz: Rsult Th gnral formula for a pair of canonic camra matrics corrsponding to a fundamntal matrix F is givn by P =[I 0] P =[[ ] F + v T λ ] (9.10) whr v is any 3-vctor, and λ a non-zro scalar.

19 9.6 Th ssntial matrix Th ssntial matrix Th ssntial matrix is th spcialization of th fundamntal matrix to th cas of normalizd imag coordinats (s blow). Historically, th ssntial matrix was introducd (by Longut-Higgins) bfor th fundamntal matrix, and th fundamntal matrix may b thought of as th gnralization of th ssntial matrix in which th (inssntial) assumption of calibratd camras is rmovd. Th ssntial matrix has fwr dgrs of frdom, and additional proprtis, compard to th fundamntal matrix. Ths proprtis ar dscribd blow. Normalizd coordinats. Considr a camra matrix dcomposd as P = K[R t], and lt x = PX b a point in th imag. If th calibration matrix K is known, thn w may apply its invrs to th point x to obtain th point ˆx = K 1 x. Thn ˆx =[R t]x, whr ˆx is th imag point xprssd in normalizd coordinats. It may b thought of as th imag of th point X with rspct to a camra [R t] having th idntity matrix I as calibration matrix. Th camra matrix K 1 P =[R t] is calld a normalizd camra matrix, th ffct of th known calibration matrix having bn rmovd. Now, considr a pair of normalizd camra matrics P =[I 0] and P =[R t]. Th fundamntal matrix corrsponding to th pair of normalizd camras is customarily calld th ssntial matrix, and according to (9.2 p244) it has th form E =[t] R = R [R T t]. Dfinition Th dfining quation for th ssntial matrix is ˆx T Eˆx =0 (9.11) in trms of th normalizd imag coordinats for corrsponding points x x. Substituting for ˆx and ˆx givs x T K T EK 1 x =0. Comparing this with th rlation x T Fx =0for th fundamntal matrix, it follows that th rlationship btwn th fundamntal and ssntial matrics is E = K T FK. (9.12) Proprtis of th ssntial matrix Th ssntial matrix, E =[t] R, has only fiv dgrs of frdom: both th rotation matrix R and th translation t hav thr dgrs of frdom, but thr is an ovrall scal ambiguity lik th fundamntal matrix, th ssntial matrix is a homognous quantity. Th rducd numbr of dgrs of frdom translats into xtra constraints that ar satisfid by an ssntial matrix, compard with a fundamntal matrix. W invstigat what ths constraints ar. Rsult A 3 3 matrix is an ssntial matrix if and only if two of its singular valus ar qual, and th third is zro.

20 258 9 Epipolar Gomtry and th Fundamntal Matrix Proof. This is asily dducd from th dcomposition of E as [t] R = SR, whr S is skw-symmtric. W will us th matrics W = and Z = (9.13) It may b vrifid that W is orthogonal and Z is skw-symmtric. From Rsult A4.1- (p581), which givs a block dcomposition of a gnral skw-symmtric matrix, th 3 3 skw-symmtric matrix S may b writtn as S = kuzu T whr U is orthogonal. Noting that, up to sign, Z = diag(1, 1, 0)W, thn up to scal, S = U diag(1, 1, 0)WU T, and E = SR = U diag(1, 1, 0)(WU T R). This is a singular valu dcomposition of E with two qual singular valus, as rquird. Convrsly, a matrix with two qual singular valus may b factord as SR in this way. Sinc E = U diag(1, 1, 0)V T, it may sm that E has six dgrs of frdom and not fiv, sinc both U and V hav thr dgrs of frdom. Howvr, bcaus th two singular valus ar qual, th SVD is not uniqu in fact thr is a on-paramtr family of SVDs for E. Indd, an altrnativ SVD is givn by E =(Udiag(R 2 2, 1)) diag(1, 1, 0)(diag(R T 2 2, 1))V T for any 2 2 rotation matrix R Extraction of camras from th ssntial matrix Th ssntial matrix may b computd dirctly from (9.11) using normalizd imag coordinats, or ls computd from th fundamntal matrix using (9.12). (Mthods of computing th fundamntal matrix ar dfrrd to chaptr 11). Onc th ssntial matrix is known, th camra matrics may b rtrivd from E as will b dscribd nxt. In contrast with th fundamntal matrix cas, whr thr is a projctiv ambiguity, th camra matrics may b rtrivd from th ssntial matrix up to scal and a four-fold ambiguity. That is thr ar four possibl solutions, xcpt for ovrall scal, which cannot b dtrmind. W may assum that th first camra matrix is P =[I 0]. In ordr to comput th scond camra matrix, P, it is ncssary to factor E into th product SR ofaskwsymmtric matrix and a rotation matrix. Rsult Suppos that th SVD of E is U diag(1, 1, 0)V T. Using th notation of (9.13), thr ar (ignoring signs) two possibl factorizations E = SR as follows: S = UZU T R = UWV T or UW T V T. (9.14) Proof. That th givn factorization is valid is tru by inspction. That thr ar no othr factorizations is shown as follows. Suppos E = SR. Th form of S is dtrmind by th fact that its lft null-spac is th sam as that of E. Hnc S = UZU T. Th rotation R may b writtn as UXV T, whr X is som rotation matrix. Thn U diag(1, 1, 0)V T = E = SR =(UZU T )(UXV T )=U(ZX)V T from which on dducs that ZX = diag(1, 1, 0). Sinc X is a rotation matrix, it follows that X = W or X = W T, as rquird.

21 9.7 Closur 259 Th factorization (9.14) dtrmins th t part of th camra matrix P, up to scal, from S = [t]. Howvr, th Frobnius norm of S = UZU T is 2, which mans that if S = [t] including scal thn t = 1, which is a convnint normalization for th baslin of th two camra matrics. Sinc St = 0, it follows that t = U (0, 0, 1) T = u 3, th last column of U. Howvr, th sign of E, and consquntly t, cannot b dtrmind. Thus, corrsponding to a givn ssntial matrix, thr ar four possibl choics of th camra matrix P, basd on th two possibl choics of R and two possibl signs of t. To summariz: Rsult For a givn ssntial matrix E = U diag(1, 1, 0)V T, and first camra matrix P =[I 0], thr ar four possibl choics for th scond camra matrix P, namly P =[UWV T +u 3 ] or [UWV T u 3 ]or[uw T V T +u 3 ] or [UW T V T u 3 ] Gomtrical intrprtation of th four solutions It is clar that th diffrnc btwn th first two solutions is simply that th dirction of th translation vctor from th first to th scond camra is rvrsd. Th rlationship of th first and third solutions in rsult 9.19 is a littl mor complicatd. Howvr, it may b vrifid that [ VW [UW T V T u 3 ]=[UWV T u 3 ] T W T V T ] 1 and VW T W T V T = V diag( 1, 1, 1)V T is a rotation through 180 about th lin joining th two camra cntrs. Two solutions rlatd in this way ar known as a twistd pair. Th four solutions ar illustratd in figur 9.12, whr it is shown that a rconstructd point X will b in front of both camras in on of ths four solutions only. Thus, tsting with a singl point to dtrmin if it is in front of both camras is sufficint to dcid btwn th four diffrnt solutions for th camra matrix P. Not. Th point of viw has bn takn hr that th ssntial matrix is a homognous quantity. An altrnativ point of viw is that th ssntial matrix is dfind xactly by th quation E =[t] R, (i.. including scal), and is dtrmind only up to indtrminat scal by th quation x T Ex =0. Th choic of point of viw dpnds on which of ths two quations on rgards as th dfining proprty of th ssntial matrix. 9.7 Closur Th litratur Th ssntial matrix was introducd to th computr vision community by Longut- Higgins [LongutHiggins-81], with a matrix analogous to E apparing in th photogrammtry litratur,.g. [VonSandn-08]. Many proprtis of th ssntial matrix hav bn lucidatd particularly by Huang and Faugras [Huang-89], [Maybank-93], and [Horn-90]. Th ralization that th ssntial matrix could also b applid in uncalibratd situations, as it rprsntd a projctiv rlation, dvlopd in th arly part of th 1990s,

22 260 9 Epipolar Gomtry and th Fundamntal Matrix A B B A (a) (b) A B B A (c) (d) Fig Th four possibl solutions for calibratd rconstruction from E. Btwn th lft and right sids thr is a baslin rvrsal. Btwn th top and bottom rows camra B rotats 180 about th baslin. Not, only in (a) is th rconstructd point in front of both camras. and was publishd simultanously by Faugras [Faugras-92b, Faugras-92a], and Hartly t al. [Hartly-92a, Hartly-92c]. Th spcial cas of pur planar motion was xamind by [Maybank-93] for th ssntial matrix. Th corrsponding cas for th fundamntal matrix is invstigatd by Bardsly and Zissrman [Bardsly-95a] and Viévill and Lingrand [Vivill-95], whr furthr proprtis ar givn Nots and xrciss (i) Fixating camras. Suppos two camras fixat on a point in spac such that thir principal axs intrsct at that point. Show that if th imag coordinats ar normalizd so that th coordinat origin coincids with th principal point thn th F 33 lmnt of th fundamntal matrix is zro. (ii) Mirror imags. Suppos that a camra viws an objct and its rflction in a plan mirror. Show that this situation is quivalnt to two viws of th objct, and that th fundamntal matrix is skw-symmtric. Compar th fundamntal matrix for this configuration with that of: (a) a pur translation, and (b) a pur planar motion. Show that th fundamntal matrix is auto-pipolar (as is (a)). (iii) Show that if th vanishing lin of a plan contains th pipol thn th plan is paralll to th baslin. (iv) Stinr conic. Show that th polar of xa intrscts th Stinr conic Fs at th pipols (figur 9.10a). Hint, start from F = Fs + Fa = 0. Sinc lis on

23 9.7 Closur 261 th conic Fs, thn l 1 = Fs is th tangnt lin at, and l 2 = Fa =[xa] = xa is a lin through xa and. (v) Th affin typ of th Stinr conic (hyprbola, llips or parabola as givn in sction 2.8.2(p59)) dpnds on th rlativ configuration of th two camras. For xampl, if th two camras ar facing ach othr thn th Stinr conic is a hyprbola. This is shown in [Chum-03] whr furthr rsults on orintd pipolar gomtry ar givn. (vi) Planar motion. It is shown by [Maybank-93] that if th rotation axis dirction is orthogonal or paralll to th translation dirction thn th symmtric part of th ssntial matrix has rank 2. W assum hr that K = K. Thn from (9.12), F = K T EK 1, and so Fs =(F + F T )2 =K T (E + E T )K 1 2=K T EsK 1. It follows from dt(fs) =dt(k 1 ) 2 dt(es) that th symmtric part of F is also singular. Dos this rsult hold if K K? (vii) Any matrix F of rank 2 is th fundamntal matrix corrsponding to som pair of camra matrics (P, P ) This follows dirctly from rsult 9.14 sinc th solution for th canonical camras dpnds only on th rank 2 proprty of F. (viii) Show that th 3D points dtrmind from on of th ambiguous rconstructions obtaind from E ar rlatd to th corrsponding 3D points dtrmind from anothr rconstruction by ithr (i) an invrsion through th scond camra cntr; or (ii) a harmonic homology of 3-spac (s sction A7.2(p629)), whr th homology plan is prpndicular to th baslin and through th scond camra cntr, and th vrtx is th first camra cntr. (ix) Following a similar dvlopmnt to sction 9.2.2, driv th form of th fundamntal matrix for two linar pushbroom camras. Dtails of this matrix ar givn in [Gupta-97] whr it is shown that affin rconstruction is possibl from a pair of imags.