On the Spacetime Geometry of Galilean Cameras

Transcription

1 On he Spaceime Geomery of Galilean Cameras Yaser Sheikh Roboics Insie Carnegie Mellon Uniersiy Alexei Griai Comper Vision Laboraory Uniersiy of Cenral Florida Mbarak Shah Comper Vision Laboraory Uniersiy of Cenral Florida Absrac In his paper, a projecion model is presened for cameras moing a consan elociy (which we refer o as Galilean cameras). To ha end, we inrodce he concep of spaceime projecion and show ha perspecie imaging and linear pshbroom imaging are specializaions of he proposed model. The epipolar geomery beween wo sch cameras is deeloped and we derie he Galilean fndamenal marix. We show how six differen fndamenal marices can be direcly recoered from he Galilean fndamenal marix inclding he classic fndamenal marix, he Linear Pshbroom (LP) fndamenal marix and a fndamenal marix relaing Epipolar Plane Images (EPIs). To esimae he parameers of his fndamenal marix and he mapping beween ideos in he case of planar scenes we describe linear algorihms and repor experimenal performance of hese algorihms.. Inrodcion A camera is normally hogh of as a deice ha generaes wo dimensional images of a hree dimensional world. This projecie engine ([]) akes a single snapsho of he world from a pariclar posiion a a pariclar ime insan. The camera, howeer, is ofen dynamic and he op of cameras is beer considered a ideo raher han an image. We reexamine he process of projecion, no beween he saic world and an image plane b insead where an ncalibraed camera is moing wih (nknown) consan elociy. We refer o sch a camera as a Galilean camera and model he projecion of he world ono he ideo hyperplane. The epipolar geomery of a pair of saic cameras has been exhasiely sdied (more han wo decades of research smmarized by Harley and Zisserman in [9], Fageras and Long in []), and we show ha his concep can be generalized for Galilean cameras. Paying homage o he obserers in Galileo s principle of relaiiy. Eqaion 4 frher jsifies his choice. Or work is relaed o ha of Wolf and Shasha in [6], where hey inesigae higher-dimensional mappings beween k-spaces and 2-spaces ha arise from differen problem insances for 3 k 6. Howeer, where hey proide six problem definiions describing arios configraions of poins moing in sraigh lines, we sdy he geomery of cameras moing wih consan elociy. Baroli in [], Srm in [5] and Han and Kanade in [] all also make similar assmpions abo objecs moing along sraigh lines. In his paper, we describe a spaceime projecion model for a Galilean camera and propose a mapping fncion beween he ideos of wo Galilean cameras when he scene is planar. We hen presen he epipolar geomery for his case and describe a normalized linear algorihm for esimaing he parameers of he fndamenal marix relaing Galilean cameras. We show how he original fndamenal marix, he LP fndamenal marix, he orho-perspecie fndamenal marix and hree, as ye nknown, fndamenal marices can be direcly recoered from his Galilean fndamenal marix. The res of his paper is organized as follows. In Secion 2 we inrodce he Galilean projecion model sed in he remainder of his paper, followed by Secion 3 where we presen he mapping beween Galilean images when he scene is planar. A descripion of he relaie geomery beween wo Galilean cameras and he resling fndamenal marix is presened in Secion 4 and specializaions of his marix o seeral known and nknown fndamenal marices are described in Secion 5. Finally, experimenal alidaion is presened in Secion. 2. Galilean Projecion Model We define a worldpoin as X = [T X Y Z] T R 4, on a world coordinae U = [T λx λy λz λ] T and a ideopoin as x = [ ] T R 3 on a ideo coordinae sysem = [ w w w] T R 4. Noe ha an addiional ime dimension has been added o he sal spaial erms. When he world and camera coordinae sysems are aligned, he mapping describing cenral projecion for he spaial coordinaes and orhographic projecion for he emporal coor-

2 dinae are, (T, X, Y, Z) T (α T, fx/z + p, fy/z + p ) T () where f is he focal lengh of he camera and α is he reciprocal of he frame-rae of he camera (casing an effec akin o ime dilaion) and (p, p ) are he coordinaes of he principal poin. This can be expressed in marix form as T X Y Z w w w = α f p f p T X Y Z, (2) or more concisely = KX where K is he calibraion marix. When he spaial world and camera coordinae sysems are no aligned hey are relaed by roaion and ranslaion. The emporal coordinaes are relaed by a ranslaion (e.g. he world ime index when camera begins recording). These ransformaions can be capred by a 4 4 orhogonal marix Q and a 4 5 displacemen marix D, where Q = R,D = D D x D y D z, (3) where C = [D, D x, D y, D z ] T is he posiion of he camera cener and R is a 3 3 roaion marix represening he orienaion of he camera coordinae sysem. The 4 5 projecion marix relaes he world and ideo coordinae sysems, = PU. This projecion marix can be decomposed as P = KQD = KQ[I C] or simply P = K[Q QC]. Now if he cameras are moing a consan elociy according o C = [, D x, D y, D z ] T, we hae he following series if we consider only he spaial dimensions 2, û() = ˆKR[I Ĉ]Û û() = ˆKR[I (Ĉ + Ĉ)]Û. û(t) = ˆKR[I (Ĉ + T Ĉ)]Û. If we inclde he emporal dimension ino he objec ecor we can rewrie hese compacly as = KQ[G C]U, where G = D x D y D z (4) 2 Eniies wih a ha denoe he spaial enries of he eniy, e.g. Û = [X, Y, Z,] T, ˆQ = R ec. T Z ' ' ' Camera Camera 2 Y X rajecory ideoline (a) (b) Figre. Galilean Cameras.(a) Projecion ono he ideo hyperplane (b) The ideoline of a poin chars o a cre in spaceime. is a Galilean ransformaion. We refer o he 4 5 marix M = KQ[G C] (5) as he Galilean projecion marix. As wih he spaial projecion marix, he nll ecor of M corresponds o he spaceime locaion of he camera cener in he world a =. In addiion, m 2 = m 3 = m 4 =, where M = {m ij }. The ideo aken by a Galilean camera can herefore be properly considered a hree-dimensional image projeced from a for-dimensional world. I is a mliperspecie (noncenral) camera in he sense described in [4] and [3], he generaor being a line in 4D spaceime. Analogos o worldlines in spaceime geomery [3], we refer o he cre chared o by sccessie world eens from a (spaially) saic poin as ideolines. I was shown by Bolles e al. in [2] ha hese cres are described hyperbolic fncions on EPIs, b in he ideo hyperplane (assming ha he world reference frame is aligned wih he camera reference frame) hey follow he parameric form, (T) = p + f D xt + fx D z T + Z (6) (T) = p + f D yt + fy D z T + Z () (T) = α T. (8) I shold be noed hen ha sraigh lines in he spaceime world are no mapped o sraigh lines in he ideo hyperplane, excep when he principal axis is orhogonal o he elociy ecor (in which case D z = and Z is consan and herefore Eqaions 6 and are linear). As a resl, spaial inarians sch as he cross-raio are no presered in spaceime. 3. Planar Geomery In his secion, we describe a ransformaion analogos o he planar homography relaing wo images of a world plane. By choosing wo orhogonal basis ecors ha span he scene plane as he X and Y axes of he world coordinae

3 sysem and ignoring he perpendiclar Z coordinae (since all Z ales will eqal zero), we hae, T T = w w = Mˆ4 X Y, = w w = M ˆ4 X Y w w (9) where Mˆ4 and are nonsinglar 4 4 marices, consrced by remoing he forh colmn from M and M M ˆ4 respeciely. There exiss a ransformaion relaing and, H = M M where ˆ4 ˆ4 = H. Considering ime independenly we see ha, = m T + m 4, = m T + m 4, m 4 = m 4 m m = T, from which we ge he mapping = h +h 4, or in oher words, h 2 = h 3 =. As a resl, we ge he following fncions o deermine, and, =h + h 4, = h 2 + h 22 + h 23 + h 24 h 4 + h 42 x + h 43 y + h 44, = h 3 + h 32 + h 33 + h 34 h 4 + h 42 x + h 43 y + h 44, Ths, a nonsinglar 4 4 marix H relaes he spaceime coordinaes of wo ideos capred by Galilean cameras obsering a planar scene. Definiion 3. (Planar Galilean Mapping) A planar Galilean mapping is a linear ransformaion of = [ ], represenable as a nonsinglar 4 4 marix H, w w w = h h 4 h 2 h 22 h 23 h 24 h 3 h 32 h 33 h 34 h 4 h 42 h 43 h 44. () This marix H is an inhomogeneos marix ha can be diided ino an inhomogeneos par, i.e. he firs row, h and a homogeneos par, i.e. he second, hird and forh rows, h 2,h 3 and h 4. Unlike he planar homography, his mapping does no form a grop or in oher words he prodc of wo planar Galilean mappings is no, in general, a planar Galilean mapping. To esimae he parameers of his mapping, he homogeneos and inhomogeneos pars can be comped separaely. The Direc Linear Transformaion Algorihm (see [9]) can be sed o esimae he homogeneos par since, x i y i w i i h2 i h3 i h4 =. () An oer-deermined homogeneos sysem of eqaions can be consrced as, w i i y i i h 2 w i i x i i h 3 =, (2) y i i x i i h 4 and he solion can be fond sing SVD (see Secion 4. in [9] for frher deails). For he inhomogeneos par, he following linear sysem of eqaions can be soled sing leas sqares, 4. Two View Geomery [ i ] [h h 3 ] = i. (3) Consider a pair of Galilean cameras ha moe in differen direcions a differen elociies 3. The coordinaes of he corresponding projecions in firs and second camera are = (, w, w, w) T and = (, w, w, w ) T respeciely. The imaged coordinaes in he wo cameras are = MU and = M U. This pair of eqaions may be rewrien as Ag = (4) where, A = m m 5 m 2 m 22 m 23 m 24 m 25 m 3 m 32 m 33 m 34 m 35 m 42 m 42 m 43 m 44 m 45 m m 5 m 2 m 22 m 23 m 24 m 25 m 3 m 32 m 33 m 34 m 35 m 4 m 42 m 43 m 44 m 45, (5) m ij are he elemens of M and g = [T, X, Y, Z,, w, w ] T. Since A in he homogeneos sysem of Eqaion 4 is a 8 marix, i ms hae a rank of a mos six for a solion o exis. As a resl, any minor ms hae a zero deerminan. There are eigh differen ways o choose he minor o sole he sysem, b only wo ineresing ariaions. The firs selecion ses boh rows conaining he emporal indices (, ) and fie rows conaining he spaial indices (,,, ) and he second selecion ses one row conaining he emporal indices and six rows conaining he spaial indices. As in [6], de(a i ) = will prodce he fndamenal polynomial ha has ineracion erms (beween,,,, and b no sqared erms. Hence, here are exiss a 6 6 marix called he Galilean fndamenal marix, (,,,,, )Γ(,,,,, ) T =. (6) 3 By he principal of relaiiy we can assme one of he cameras o be saionary, b o mainain a symmeric formlaion beween boh cameras we do no make ha assmpion here.

4 6 -axis x-axis 2 Figre 2. Epipolar Srface. The ideopoin in he second ideo corresponding o a space-ime poin in he firs ideo ms lie on his srface. Howeer, in all he eigh ariaions (of differen minors), nine ineracion erms do no exis all, i.e., o of a oal of 36 possible ineracion erms only 2 appear. Definiion 4. (Galilean Fndamenal Marix) If and are ideopoins corresponding o he same worldline nder wo Galilean cameras, here exiss a 6 6 marix Γ referred o as he Galilean Fndamenal Marix sch ha, T f f 2 f 3 f 4 f 5 f 6 f f 8 f 9 f f f 2 f 3 f 4 f 5 f 6 f f 8 f 9 f 2 f 2 f 22 f 23 f 24 f 25 f 26 f y-axis 4 6 =. Γ can be wrien more compacly as, ( ) F Γ =, () F F where F is he fndamenal marix beween he image in he firs ideo a ime = and he image in he second ideo a ime =, and ( F, F ) are marices ha capre informaion abo he elociy of each camera as will be seen presenly. 4.. Epipolar Geomery Unless here is zero moion, no epipoles (single image poins of he opposie camera cener) in he sal sense exis. In general, epipolar lines (or cres) in he sal sense do no exis eiher, insead here are epipolar srfaces in one camera corresponding o a poin in he oher camera. These srfaces are defined by seing a spaioemporal poin in one camera, i.e. (,, ) and applying he Galilean fndamenal marix. The srface is defined by he s + s 2 + s 3 + s 4 + s 5 + s 6 = where s = [s,...s 6 ] is comped as s = [,,,,, ]Γ. An example of his srface is ploed in Figre 2. I can be seen ha his srface is rled, since he inersecion wih each ime plane is a line (corresponding o he classic epipolar line of ha image). 5. Specializaions The work by Feldman e al. in [5] on Crossed-Slis projecion, considers he algebraic consrains beween differen slices of he spaceime olme. We show ha seeral sch specializaions can be direcly deried from he consrains described in his paper. The differen specializaions are shown in Figre 3(a), (b) and (c) for he original fndamenal marix, he orhoperspecie fndamenal marix and he linear pshbroom fndamenal marix respeciely. Proceeding similarly i is sraighforward o recoer he fndamenal marices for he configraions in Figre 3(d), (e) and (f). 5.. Beween Perspecie Images The classic fndamenal marix beween wo ncalibraed perspecie images was deried independenly by Fageras in [] and Harley in [8]. For corresponding poins, his singlar 3 3 marix saisfies he consrain [,, ]F[,, ] =. This marix can be direcly recoered from he Galilean fndamenal marix. For he (, ) pair image we can recoer he fndamenal marix F by parially collapsing Γ and plgging in he ales of (, ). Ths F = ( ) f + f + f 3 f 2 + f + f 4 f 3 + f 2 + f 5 f 4 + f 6 + f 9 f 5 + f + f 2 f 6 + f 8 + f 2 f + f 22 + f 25 f 8 + f 23 + f 26 f 9 + f 24 + f 2 or simply F = F + F + F. (8) Ths, F + F is he fndamenal marix beween he image in he firs ideo a ime = and he second ideo a ime = and F + F is he fndamenal marix beween he image in he firs ideo a ime =, and he second ideo a ime =. We can infer he following propery from Eqaion 8. Theorem. (Fndamenal Boos Marix) The marices ( F, F ) are rank-2 marices. As a resl, he rank of Γ is a mos 5. Proof. A = and =, a fndamenal marix F can be decomposed ino ˆK T [ RĈ] RˆK. A =, if he second camera is displaced by Ĉ hen he fndamenal marix becomes, ˆK T [ R(Ĉ + Ĉ)] RˆK = ˆK T [ RĈ] RˆK ˆK T [R Ĉ] RˆK.,

5 (a) (b) (c) (d) (e) (f) Figre 3. Specializaions. Fndamenal marices can be recoered beween (a) a pair of perspecie images, (b) an EPI and a perspecie images, (c) a pair of EPIs, (d) a pair of LP images, (e) a LP and an EPI, (f) a LP and a perspecie images From Eqaion 8, = F + F = F. Since [ Ĉ] is a skew-symmeric marix, i follows ha F is a rank 2 marix. The lef 6 3 sbmarix or he pper 3 6 sbmarix of Γ are boh herefore rank 2 marices, and Γ has a rank of a mos Beween Linear Pshbroom Images If he camera moion saisfies he condiions described in [6], linear pshbroom images can be recoered from a horizonal slice of he ideo olme. Beween wo sch images, he LP fndamenal marix was deried by Gpa and Harley in [6]. The relaionship capred by his marix is expressed as (,,, )F (,,, ) T =. This 4 4 marix oo can be direcly deried from Γ. Ths, gien (, ), we can compe F = (9) f f 8 + f 6 f + f 23 f 22 + f 24 + (f + f 2) f 5 f 4 + f 6 f 2 + f 8 f + f 9 + (f + f 3) f 2 f 9 + f 2 f 4 + f 26 f 25 + f 2 + (f 3 + f 5). I can be obsered ha he srcre of he marix is he same as he one deried in [6] Beween Epipolar Plane Images Epipolar plane images were defined by Bolles e al. in [2] as he collecion of epipolar lines ha correspond o one epipolar plane in he world. We can recoer he fndamenal marix beween wo EPIs. In his case i has a form similar o he LP fndamenal marix, (2) f f 2 + f f 6 + f 22 f 23 + f 24 + (f + f 8) f f 2 + f 3 f 2 + f f 8 + f 9 + (f 5 + f 6) f 3 f 4 + f 5 f 9 + f 25 f 26 + f 2 + (f 2 + f 2) Beween a Pshbroom and a Perspecie Image Recenly in [], Khan e al. deried he 4 3 perspecie-orhoperspecie fndamenal marix beween a pshbroom image and a perspecie image. The relaionship capred by his marix is expressed as (,,, )F (,, ) T =. This marix can also be direcly deried from Γ. Ths, gien (, ), we can compe F = f 4 f 5 f + f f 2 + f 8 f 6 + f 9 f + f 2 f + f 22 + f 3 + f 25 f + f 23 + f 4 + f 26 f 6 f 3 + f 9 f 8 + f 2 f 2 + f 24 + f 5 + f 2 (, ( ) ) φ F or simply, where φ = φ F + F. Similarly, i is sraighforward o recoer F he fndamenal marices beween an EPI and an LP image, and F beween an EPI and a perspecie image. 6. Normalized Linear Algorihm A linear algorihm can sed o esimae he parameers of Γ. Eqaion can be rewrien as he homogeneos sysem Aγ = where γ = [f,, f 2 ] T is a 2-ecor, consrced from he non-zero elemens of Γ and A is a

6 Objecie Gien n 26 maches from corresponding ideolines, esimae he Galilean fndamenal marix Γ sch ha p T Γp =. -coordinae 2 3 -coordinae e Algorihm. Normalizaion: Normalize he coordinaes hrogh an appropriae scaling and ranslaion. 2. Linear Solion: Perform singlar ale decomposiion on A and deermine Γ by selecing he singlar ecor corresponding o he smalles singlar ale of A and reconsrcing a 6 6 marix. 3. Rank Consrain: Se he smalles singlar ale of Γ o zero, enforce he rank 5 consrain. 4. Denormalizaion: Denormalize Γ according o he original scaling and ranslaion. Figre 4. A linear algorihm for esimaing Γ marix consrced from spaioemporal coordinaes of corresponding poins. If noiseless poins correspond exacly o each oher he rank of A is 26 and he nll-ecor of A corresponds o an esimae of γ. In he presence measremen noise, he 2 h singlar ale will be non zero. In ha case, he singlar ecor corresponding o he smalles singlar ale of A can be sed as an esimae of γ. The rank consrains on Γ can be enforced pos faco by seing he smalles singlar ales of Γ o zero and reconsrcing he marix. Of corse, as wih oher linear algorihms of his sor, o obain good esimaes i is imporan o appropriaely normalize he daa (see Secion 4.4 of [9]). Lasly, o obain a meaningfl solion, A has o hae a rank of more han 26. I is emphasized ha o ensre he rank of A is greaer han, correspondences from differen ideopoins of he same worldpoin ms be sed. For insance, if ideolines of a saic world poin in he scene of lengh n are associaed in boh cameras, here are n 2 rows ha shold be added o A.. Experimenaion An experimen was condced where wo cameras were placed on a moing walkway 8 fee apar, looking a differen angles and moing a approximaely 2 miles per hor in he same direcion. A pair of frame seqences were recorded a a resolion of by wo SONY HDV cameras (images were downsampled) and 22 ideopoins were racked across 6 frames in each of he wo ideos (frames 9 o 96 in boh seqences). The moion of he cameras was no perpendiclar o he opical axis of eiher camera. A 3 fps, he disance raersed by boh cameras dring his period was abo 98 fee, and he disance raeled in beween sccessie frames was approximaely.2 inches. Three slices of his ideo are shown in -coordinae coordinae coordinae coordinae Figre 5. Three slices of a ideo aken from a Galilean camera. Figre 5 (he resls in his paper are bes seen in color). These poins were sed o esimae he Galilean Fndamenal marix sing he linear algorihm presened in his paper. To ealae his esimae, differen ime slices of he ideo were analyzed sing he differen specializaions of he Galilean Fndamenal marix. A se of 6 poins (differen from he ones sed dring compaion) were seleced in boh frames and he poins and he epipolar lines of heir correspondences beween wo perspecie images are ploed in Figre. Despie he fac ha he ideopoins correspondences were aken owards he end of he seqence and he frames in his figre were aken owards he beginning (frame from ideo and frame 2 from ideo 2) he fndamenal marix recoered is accrae. This demonsraes ha Γ may be sed o predicing fndamenal marices in he fre wiho haing poin correspondences for hose frames. The epipolar cres indced by poins in frames 96, 98 and ono he pshbroom image in ideo generaed from colmn 2 are shown in Figre 8. As he corresponding frames moe, he asympoe ranslaes in he LP image ranslaes oo. Figre shows frame 98 wih he poins sed o plo he cres in Figres 8 and. The corresponding epipolar lines from he EPI are missing since i is difficl o find poin correspondences on EPIs. Nine poins lay on a plane and were sed o compe he planar Galilean mapping shown in Figre 6. The figre shows he firs seqence where he yellow boxes show he posiions of he nine poins in ha seqence and he black poins indicae he posiions of he poins in Seqence 2 (a) before and (b) afer warping. 8. Discssion In his work, we presen a spaceime projecion model for cameras moing a consan elociies. In pracice, he assmpion of consan elociy is ofen reasonable for shor draions of ime, especially when he camera is moned on a robo, eleaor or on a ehicle sch as an aircraf, rain, car or a spacecraf. An imporan applicaion for he ideas described in his paper is for predicion of relaie camera

7 Seqence (Perspecie Perspecie) Seqence 2 (Perspecie Perspecie) Seqence (EPI EPI) Seqence 2 (EPI EPI) Seqence (Pshbroom Pshbroom) Seqence 2 (Pshbroom Pshbroom) Seqence (EPI Pshbroom) Seqence 2 (EPI Pshbroom) Seqence (EPI Perspecie) Seqence 2 (EPI Perspecie) Seqence (Pshbroom Perspecie) Seqence 2 (Pshbroom Perspecie) (a) (b) (c) (d) (e) (f) Figre. Corresponding poins and heir epipolar cres. (a) Beween wo perspecie images (b) Beween wo LP images (c) Beween wo EPIs (d) Beween a LP image and an EPI (e) Beween a LP image and a perspecie (f) Beween an EPI and a perspecie (a) (b) Figre 6. Videopoins mapped sing he planar Galilean mapping. Yellow sqares indicae he posiion of he poins in Seqence, black poins indicae he posiion of (a) corresponding poins in Seqence 2 and (b) corresponding poins afer warping Figre 8. The epipolar cres indced by frames 96, 98 and of camera 2 on he pshbroom images of camera corresponding o colmn Figre. Recoering he fndamenal marix beween frame in seqence and frame 2 in seqence 2 from he Galilean fndamenal marix. The Galilean fndamenal marix was comped from ideopoins in six frames (9 o 96 in boh seqences). posiion. When cameras moe, he degree of oerlap beween heir fields of iew sally changes and when he fields of iew become disjoin, esimaion of relaie camera posiion becomes impossible. Howeer, if he moion of he cameras follow some srcred moion (like consan elociy) he ideas presened here can be sed o predic he fndamenal marix relaing iews een when heir fields of iew are disjoin. We inesigae he relaie geomery relaing a pair of sch cameras in planar and general scenes. We show how hree known fndamenal marices are specializaions of his marix and cold be readily recoered from he proposed fndamenal marix, proiding a nify- ing link beween he classic fndamenal marix and he LP fndamenal marix. In addiion we describe hree new fndamenal marices ha can also be recoered. In he fre, we inend o inesigae he applicaion of differen moion models, sch as a consan acceleraion model, and sdy he relaionships beween hree or more Galilean cameras. Acknowledgemens The ahors hank Takeo Kanade for his sefl commens and sggesions. This work was fnded by he Disrpie Technologies Office, Video Analysis and Conen Exracion (VACE) Program - Phase III, Conrac No. NBCHC65 issed by he Deparmen of he Inerior. The iew and conclsions are hose of he ahors, no of he US Goernmen or is agencies.

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