Robust Computation and Parametrization of Multiple View. Relations. Oxford University, OX1 3PJ. Gaussian).
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1 Robust Computaton and Parametrzaton of Multple Vew Relatons Phl Torr and Andrew Zsserman Robotcs Research Group, Department of Engneerng Scence Oxford Unversty, OX1 3PJ. Abstract A new method s presented for robustly estmatng multple vew relatons from mage pont correspondences. There are three new contrbutons, the rst s a general purpose method of parametrzng these relatons usng pont correspondences. The second contrbuton s the formulaton of a common Maxmum Lkelhood Estmate (MLE) for each of the multple vew relatons. The parametrzaton facltates a constraned optmzaton to obtan ths MLE. The thrd contrbuton s a new robust algorthm, MLESAC, for obtanng the pont correspondences. The method s general and ts use s llustrated for the estmaton of fundamental matrces, mage to mage homographes and quadratc transformatons. Results are gven for both synthetc and real mages. It s demonstrated that the method gves results equal or superor to prevous approaches. 1 Introducton It s well known that rgd 3D moton mposes relatons on the mages of ponts. For example, correspondng ponts are related over two vews by the eppolar constrant, and over three vews by a trfocal constrant. These mage relatons are used for many purposes ncludng: matchng; recovery of structure; moton segmentaton; and moton model selecton. Ths paper presents a new methodology for estmatng these relatons whch s both robust, detectng and dscardng msmatches, and fully automatc. The mportant ssues are: rst, due to the frequent occurrence of msmatches a robust estmator s necessary to dentfy nlers; second, the correct cost functon should be mnmzed n order to generate the optmal estmate; thrd, a consstent parametrzaton should be used for the multple vew relaton. These ssues are explaned n the followng sectons. It s shown that the mnmal correspondence set provded by the robust estmator can be used as a consstent parametrzaton for the cost functon mnmzaton. The method s llustrated here for the two-vew relatons. The applcaton to other multple vew relatons s descrbed n Secton 7. Notaton A 3D scene pont projects to x and x 0, n the rst and second mages respectvely, where x = (x; y; 1) > s a homogeneous three vector. The correspondence x $ x 0 s also denoted concsely as x 1;2. Underlnng, x, ndcates the perfect or nose-free quantty, dstngushng t from x = x + x, whch s the measured value corrupted by nose (assumed Gaussan). 2 The Two Vew Relatons In ths secton the relatons on correspondng mage ponts are summarzed. Three examples are consdered n detal: (a) the fundamental matrx F, (b) a planar homography H, (c) a quadratc transformaton Q. All these two vew relatons are estmable from mage correspondences alone. The fundamental matrx [2, 5] represents the eppolar constrant. Ths relaton apples for uncalbrated cameras undergong general moton wth non-zero translaton between vews. Correspondng mage ponts satsfy the relaton x 0> Fx = 0 wth F a 3 3 matrx. If all the world ponts le on a plane, or the camera rotates about ts centre and does not translate, then mage pont correspondences are related by a homography, x 0 = Hx, wth H a 3 3 matrx. If all the world ponts and both camera centres le on a crtcal surface, whch s a quadrc surface then correspondng mage ponts are consstent wth two (or more) F's. Then x 0 > F1x = 0 and x 0 > F2x = 0, and correspondng ponts are related by a quadratc transformaton [11] x 0 = F1x F2x, whch s represented by a 3 6 matrx. Although the exstence of the crtcal surface s well known, lttle research has been put nto eectvely estmatng quadratc transformatons. Degrees of freedom The fundamental matrx has 9 elements, but only 7 degrees of freedom (DOF): Only the 8 ratos of elements of the homogeneous matrx are sgncant, and the elements are related by a cubc constrant because det F = 0. If a fundamental matrx s parametrzed by the elements of a 3 3 matrx t s over parametrzed (more parameters than DOF). Furthermore, a general 3 3 matrx wll not satsfy the det F = 0 constrant. If ths constrant s not mposed then the eppolar lnes do not all ntersect, and the eppole s not unquely dened. The homography has 9 elements and 8 DOF as these elements are only dened up to a scale. The quadratc transformaton has 18 elements and 14 degrees of freedom [10] (ths can be seen by the fact that t can be parametrzed by two Fs). Problems arse f the constrants between the elements are not enforced; e.g. the crtcal surface cannot correctly be recovered from the quadratc transformaton.
2 Parametrzatons If a parametrzaton enforces the constrants of a relaton t s termed consstent. In Secton 3 a consstent parametrzaton s ntroduced; and Secton 4 descrbes varatons whch result n a mnmal parametrzaton. A mnmal parametrzaton has the same number of parameters as the degrees of freedom (the number of ndependent elements) of the relaton. A relaton can be over parametrzed, and the parametrzaton can stll be consstent e.g. representng a homography by a 33 matrx. Luong's parametrzaton of the fundamental matrx [9] by the eppoles (4 parameters) and eppolar homography (3 parameters) s both mnmal and consstent. 3 What should be mnmzed? When computng a multple vew relaton from mage correspondences there are often far more correspondences than DOF n the relaton. Ths means that the ponts wll not exactly satsfy the relaton and the relaton s estmated by mnmzng a sutable cost functon. In ths secton the cost functon s formulated for a Maxmum Lkelhood Estmate (MLE) of the relaton. It s assumed that the nose n the two mages s Gaussan on each mage coordnate wth zero mean and unform standard devaton. Thus gven a true correspondence x 1;2 (pont over vews 1,2), the probablty densty functon of the nose perturbed data s P = Y P exp? (x j? )2 + j xj (y j? y )2 j 2 2 (1) over all the correspondences x 1;2, = 1::n, where n s the number of correspondences. The log lkelhood of all the correspondences s: L = P P j=1;2 (x j? )2 + xj (y j? y )2 j ; dscountng the constant term. Gven two vews wth assocated relaton for each correspondence x 1;2 the maxmum lkelhood estmate, ^x 1;2 of the true ponts x 1,x 2, are ponts ^x 1;2 whch satses the relaton and mnmze d 2 = P 2 2. P j=1;2 ^x j? xj + ^y j? yj Thus C = d2 provdes the cost functon, and the maxmum lkelhood estmate of the relaton (fundamental matrx, homography or quadratc transformaton) s the one for whch C s a mnmum. A geometrc nterpretaton of ths cost functon s now gven. Each par of correspondng ponts x, x 0 de- nes a sngle pont n a measurement space R 4, formed by consderng the coordnates n each mage. Ths space s the `jont mage space'. The mage correspondences fx g $ fx 0 g; = 1; : : : n; nduced by a rgd moton have an assocated algebrac varety V n R 4. Fundamental matrces dene a three dmensonal varety n R 4, whereas projectvtes and quadratc transformatons are only two dmensonal. Gven a set of correspondences the (unbased) mnmum varance soluton for the relaton s that whch x mage 1 mage 2 x e x e e / H -1 H x / e / mage 1 mage 2 Fgure 1: The derence between transfer error and reprojecton error n the case of a homography. The ponts x and x 0 are the measured (nosy) ponts, and ^x and ^x 0 ML estmated ponts whch correspond by the homography ^x 0 = H^x. Usng the notaton e(x; y) for Eucldean mage dstance between x and y, the transfer error s d 2 = e(x; H?1 x 0 ) 2 + e(x 0 ; Hx) 2 ; the t reprojecton error s d 2 = e(x; ^x) 2 + e(x 0 ; ^x 0 ) 2. The reprojecton error s the error of the MLE. In general d d t. mnmzes the sum of squares of dstances orthogonal to the varety from each pont (x; y; x 0 ; y 0 ) n R 4 [8],.e. the pont (^x; ^y; ^x 0 ; ^y 0 ) s on the varety, and (x; y; x 0 ; y 0 ) s the closest pont, and so s orthogonal to the tangent plane at (^x; ^y; ^x 0 ; ^y 0 ). Ths orthogonal dstance s equvalent to the reprojecton error of the back projected 3D projectve pont, whch s the dstance d mnmzed n C. In prevous work the transfer error has often been used as the error functon. The transfer dstance s dfferent from the orthogonal dstance as shown n Fgure 1 for the homography. In the case of the fundamental matrx the orthogonal dstance (reprojecton error) s derent to the error mnmzed by Luong [9]. H Hartley and Sturm [7] show that gven F then d, ^x 1 and ^x 2 may be found as the soluton of a degree 6 polynomal of one varable. A computatonally ecent rst order approxmaton based on Sampson's/Taubn's method s gven n [17]. In ths paper the dstance s found as part of the non-lnear mnmzaton. Mnmzaton of the MLE cost functon requres a parametrzaton of the relaton. Ths s descrbed n the followng secton. 4 A consstent parametrzaton Each of the multple vew relatons can be computed from a number of pont correspondences: four for a homography; seven for the fundamental matrx (1 or 3 real solutons) [16]; seven for the quadratc transformaton. In each case the computed relaton satses the approprate constrant. The key dea here s that the relaton can be consstently parametrzed by ths mnmum number of correspondences. For example, n the case of the homography by xng x; y and varyng x 0 ; y 0 for each of the four pont cor- x / x /
3 respondences, a parametrzaton wth 8 DOF (2 varables for each of the 4 correspondences) s obtaned. A number of varatons on the free/xed partton wll now be dscussed, together wth constrants on the drecton of movement of the free parameters. In all cases the parametrzaton s consstent, but may not be mnmal. Although a non-mnmal parametrzaton over parametrzes the relaton, the man detrmental eect s on the speed of the numercal mnmzaton. The rst parametrzaton, P1, s made mnmal by restrctng the number of free coordnates. For H and Q: x; y are xed for each pont and x 0, y 0 vared. For F: x; y; x 0 are xed and the seven y 0 coordnates are vared. Ths parametrzaton s both mnmal and consstent, but has the dsadvantage that certan relatons cannot be reached. Ths can be seen most easly for F f the eppolar lnes n mage 2 are parallel to the y axs then movement of y 0 wll not change F. Parametrzaton P2 does not have ths problem. In P2 the coordnates are vared n a drecton orthogonal to the surface (varety) dened by the mage relaton n R 4. For the fundamental matrx there s a sngle drecton orthogonal to the varety and the parametrzaton s mnmal wth 7 DOF. For H and Q there are two drectons orthogonal to the varety. Agan the parametrzaton s mnmal. Parametrzaton P3 vares all the coordnates gvng a 28 DOF parametrzaton for F and Q, and 16 for H. For comparson, P4 s the lnear method for each relaton, whch s used as a benchmark. In parametrzaton P5 the homography and quadratc transformatons are estmated by xng one of the elements (the largest) of the matrx. For the homography ths s mnmal whereas t s not for the quadratc transformaton. P6 s Luong's parametrzaton [9] for the fundamental matrx (eppoles and eppolar homography). Non-lnear mnmzaton: The mnmzaton of the cost functon C s a constraned optmzaton because a soluton for F, H or Q s sought whch enforces the constrants between the elements of the matrx. The cost functon s mnmzed over all the n correspondences. The bass ponts used to parametrze the relaton wll generally be selected from these correspondences (as descrbed n secton 5). It s best to thnk of the ponts parametrzng the relaton as vrtual ponts. Intally ther postons correspond to real ponts, but after the mnmzaton the free varables wll not correspond n general to ther ntal (real) postons. If the 7 ponts used to estmate F produce three real solutons, then the soluton wth lowest cost s used. The non-lnear mnmzaton s conducted usng the method descrbed n Gll and Murray [4], whch s a modcaton of the Gauss-Newton method. If the data s over parametrzed the algorthm has an eectve strategy for dscardng redundant combnatons of the varables, and choosng ecent subsets of drecton to search n parameter space. Ths makes t deal for comparng mnmzatons conducted wth derent amounts of over parametrzaton. The termnaton crtera for the mnmzaton s a threshold on the change n the cost functon. Methods for F DOF Evaluatons p P1 Vary y P2 Orthogonal varaton P3 Vary x, y, x 0, y P4 Lnear P5 Fx largest element of F P6 Luong Method for H DOF Evaluatons p P1 Vary x 0 and y P2 Orthogonal varaton P3 Vary x, y, x 0, y P4 Lnear P5 Fx largest element of H Method for Q DOF Evaluatons p P1 Vary x 0 and y P2 Orthogonal varaton P3 Vary x, y, x 0, y P4 Lnear P5 Fx largest element of Q Table 1: The DOF, average number of evaluatons of the total cost functon C n the gradent descent algorthm, and the standard devaton p for the perfect synthetc pont data for the two vew relatons. Results: comparng parametrzatons Two measures are compared for synthetc data: The rst assesses the accuracy of the soluton by ts standard devaton, p. It s 2 p = P j e ^x j ; xj 2 =n 2 where ^x j s the pont satsfyng the estmated relaton and closest to the nose free datum x j, and s the th pont xj n the jth mage. Ths provdes a measure of how far the estmated relaton s from the true data.e. the t of the relaton estmated from the nosy data s tested aganst the known ground truth correspondences. The second measure s the number of tmes C s evaluated. Synthetc data s produced by the followng procedure: Ponts are randomly generated n three space and projected to mage pars usng xed camera matrces. The mage data are perturbed by Gaussan nose, standard devaton 1:0, and then quantzed to the nearest 0.1 pxel. A 100 sets of space ponts are generated. The results for the synthetc tests usng parametrzaton P1-P6 are summarzed for each of the two vew relatons n Table 1. For all the relatons the orthogonal varaton parametrzaton P2 produces the lowest standard devaton. Apart from the fundamental matrx relaton ths parametrzaton also requres the least number of functon evaluatons. Note, that P2 produces a lower standard devaton than Luong's parametrzaton P6, and wth fewer evaluatons. P3, whch s a consstent over parametrzaton, requres more functon evaluatons and (n these examples) results n a hgher standard devaton. In the homography case, the results of all non-lnear parametrzatons are almost dentcal, wth standard
4 devatons of the error wthn 0:04 pxels of each other. The smlarty can be explaned by the lack of complex constrants between the elements of the homography matrx. The reason why the orthogonal parametrzaton provde the best results s uncertan. It seems that movement perpendcular to the varety produces speedy convergence to a good soluton, perhaps because t moves the parameters n the drecton that wll change the relaton the most. 5 Robust Estmaton: MLESAC We have now descrbed all the ngredents requred to estmate the two vew relatons apart from obtanng the correspondences and a bass set to parametrze the relaton. If there are outlers n the data the Gaussan assumpton of errors must be altered. Rather than mnmze C, the cost actually mnmzed s D = X d (e) = e 2 e < 2 4:0 e 2 : (2) Here d are pont errors and (e) s a robust functon such that outlers to a gven model are gven a xed cost, reectng that they probably arse from a duse or unform dstrbuton, the log lkelhood of whch s a constant, whereas nlers conform to a Gaussan model. The formulaton of the cost functon D allows the mnmzaton to be conducted on all correspondences whether they are outlers or nlers. Typcally, as the mnmzaton progresses many outlers are redesgnated nlers. Prevously, corner (pont) correspondences and eppolar geometry have been estmated smultaneously usng RANSAC [3, 16]. A robust estmator such as RANSAC s essental because many msmatches are produced when correspondences are estmated by smple proxmty and anty measures. After nlers have been dented, a MLE t can be obtaned as descrbed n the prevous sectons. The denton of the maxmum lkelhood cost suggests that an mprovement over RANSAC may be made. As the am s to mnmze (2) we suggest the followng change to RANSAC, rather than selectng the soluton that maxmzes the number of nlers, we choose the soluton that mnmzes D, and we dub ths process MLESAC, for maxmum lkelhood sample consensus. The output of MLESAC (as wth RANSAC) s an ntal estmate of the relaton, together wth a set of nler correspondences whch are consstent wth the relaton. MLESAC tes n perfectly wth the \ponts as parametrzaton" because the output of MLESAC s a mnmal set of correspondences wth the most support, and the relaton computed from these correspondences. The correspondences can then be used to parametrze the relaton from there on, becomng vrtual ponts n the mnmzaton. Algorthm summary: There are three stages to our new proposed algorthm: 1. Putatve correspondences are obtaned by matchng of corners over the two mages usng proxmty and neghbourhood anty measures. 2. MLESAC: random samples of mnmal sets are selected from the putatve correspondences. The set whch mnmzes D over all correspondences s selected. The outputs are the bass ponts used for the mnmal set; and a classcaton nto nler correspondences consstent wth the relaton, and outlers. 3. A non lnear mnmzaton of D (2) over all correspondences, usng the pont bass provded by step 2 as the parametrzaton, usng one of the parametrzatons P1-3. Convergence problems mght arse f ether the chosen bass set s exactly degenerate, or the data as a whole are degenerate. The dscusson of degeneracy s beyond the scope of ths paper, see [15]. 6 Results RANSAC LMS 0.6 MLESAC Fgure 3: Varance ( 2 p) of three robust estmators of the fundamental matrx: RANSAC, LMS, MLESAC; plotted aganst the percentage of outlers. Each varance was estmated from 30 tests on 200 correspondences contanng varyng percentages of outlers. We have rgorously tested the varous parametrzatons on real and synthetc data. Synthetc data s generated as descrbed n secton 4. Msmatched features are then ntroduced to make a gven percentage of the total, between 10% and 50%, nto outlers. A comparson was made between RANSAC and MLESAC by evaluatng p, before applyng the gradent descent stage, for varous percentages of outlers. The results show an mprovement wth MLESAC over RANSAC. In the rst test the data conformed to a homography. The standard devaton was reduced from 1.20 (RANSAC) to 0.80 (MLESAC) when estmatng the homography. After the non-lnear stage the standard devaton of p drops to 0.22.
5 Fgure 2: Row 1 Frst mage; Row 2 Second mage wth ntal correspondences supermposed. The correspondences are obtaned by matchng corners on cross-correlaton and proxmty; Row 3 Bass selected by MLESAC; Row 4 Inlers after MLE estmaton. The two-vew relatons tted are: left column, F; mddle H; rght Q. In the second test the data conformed to a fundamental matrx. Fgure 3 shows p for the tted fundamental matrx for three random samplng style robust estmators: RANSAC [3], LMS [12] and MLESAC. It can be seen that for outler percentages below 40% MLESAC outperforms the other two estmators, provdng a 5? 10% mprovement. Ths s because MLE- SAC evaluates the least squares error of all the nlers detected as a score functon, whereas LMS uses only the medan, and RANSAC counts only the number of nlers. Consequently, for no extra computatonal cost MLESAC can more accurately assess the goodness of a putatve t. In the case of real data the accuracy s assessed from the the standard devaton of the nlers r =?P d2 =n 1 2. In general the non-lnear mnmzaton requres approxmately the same number of functon evaluatons as the random samplng stage. However, the number requred vares wth parametrzaton, and s an addtonal measure (over standard devaton) on whch to assess the parametrzaton. Fgures 2 shows ntal matches, pont bases, and nlers for three mage pars. Chapel Data Estmatng F. column 1 are for two mages of a chapel, the camera moves around the chapel rotatng to keep t n vew. As the mnmzaton progresses the bass ponts move only a few hundredths of a pxel each, but the soluton s much mproved: the standard devaton of the nlyng data r decreases from 0.67 to Cup Data Estmatng H. column 2, are two mages from a camera undergong cyclotorson about ts
6 optc axs combned wth an mage zoom. It can be seen that outlers to the cyclotorson are clearly dent- ed. The non lnear step does not produce any new nlers as the MLESAC step has successfully elmnated all msmatches. The standard devaton of the nlyng data r s reduced by 1% under a mnmzaton wth parametrzaton P1. Model house data Estmatng Q. column 3 are two mages from a camera rotatng and translatng whlst xatng on a model house. The standard devaton of the nlers r mproves from 0.39 after the estmaton by MLESAC to 0.35 after the non-lnear mnmzaton. The good t of the quadratc transformaton ndcates that these vews are on a crtcal surface. 7 Dscusson Why does the pont parametrzaton work so well? One reason s that the ponts ntally selected by MLESAC are known to provde a good estmate of the relaton (because there s a lot of support for ths soluton). Hence ntal estmated ponts are qute close to the true soluton and consequently the non-lnear mnmzaton typcally avods local mnma. Secondly the parametrzaton s consstent whch means that durng the gradent descent phase only relatons that mght actually arse are searched for. The methodology of the algorthm descrbed here s qute general. Further applcatons can be dvded nto two types. There are two vew relatons wth addtonal constrants; and there are relatons nvolvng more than two vews. Examples of the rst type nclude: 1. Planar moton fundamental matrx where the axs of rotaton of the camera s perpendcular to the drecton of translaton. The constrant s that det(f + F > ) = 0 [1]. F has sx DOF and can be determned (up to 9 solutons) from sx pont correspondences n general poston; 2. Domnant plane fundamental matrx where four or more ponts n 3D le on the plane, and two o the plane. Ths s a general F wth seven DOF but can be recovered (unquely) from sx ponts: two n general poston and four on a plane. Ths estmaton technque s most ecacous when a large proporton of the data are maged from a plane. 3. The ane fundamental matrx wth four DOF and determned unquely by four pont correspondences n general poston; 4. The pure translaton fundamental matrx wth two DOF and determned unquely by two pont correspondences n general poston; nally, 5. An ane homography wth sx DOF and determned unquely by three pont correspondences n general poston. Examples of the second type nclude: 1. The trfocal tensor [6, 13, 14] over three vews wth 18 DOF and determned (1 or 3 real solutons) by sx pont correspondences n general poston; 2. Camera projecton matrces wth 11 DOF. A camera matrx can be computed from the 3D-2D correspondences of 5.5 ponts. Ths s mplemented by varyng 11 of the mage coordnates. All of these relatons have been mplemented n our current multple hypothess system that estmates each relaton for a par of vews and uses ths to gude matchng [15]. The estmaton of the trfocal tensor s descrbed n detal n [17] ncludng the extenson of ML estmaton to lne correspondences over three vews. Generally the orthogonal parametrzaton gves the best results, and outperforms nconsstent and over parametrzatons. The general method (MLE parametrzed va bass ponts found from MLESAC) could be used for other estmaton problems n vson, for nstance estmatng the quadrfocal tensor over four vews, complex polynomal curves etc. Acknowledgements We would lke to thank Stephane Laveau for dscussons. Ths research was funded by EU ACTS Project VANGUARD. References [1] Beardsley, P. and Zsserman, A. Ane calbraton of moble vehcles. Europe-Chna workshop on Geometrcal Modellng and Invarants for Computer Vson, pages 214{ [2] O.D. Faugeras. What can be seen n three dmensons wth an uncalbrated stereo rg? ECCV2, pages 563{ [3] M. Fschler and R. Bolles. Random sample consensus Commun. Assoc. Comp. Mach., vol. 24:381{95, [4] P. E. Gll and W. Murray. Algorthms for the soluton of the nonlnear least-squares problem. SIAM J Num Anal, 15(5):977{992, [5] R. I. Hartley. Estmaton of relatve camera postons for uncalbrated cameras. ECCV2, pages 579{ [6] R. I. Hartley. Lnes and ponts n three vews { a uned approach. ARPA IUW, [7] R. I. Hartley and P. Sturm. Trangulaton. In IUW, pages 957{966, [8] K. Kanatan. Statstcal Optmzaton for Geometrc Computaton. Elsever Scence, [9] Q. T. Luong, R. Derche, O. D. Faugeras, and T. Papadopoulo. On determnng the fundamental matrx: analyss of derent methods and expermental results. Report 1894, INRIA (Sopha Antpols), [10] Q. T. Luong and O. D. Faugeras. A stablty analyss for the fundamental matrx. ECCV3 pages 577{ [11] S.J. Maybank. Theory of Reconstructon From Image Moton. Sprnger-Verlag, [12] P. J. Rousseeuw. Robust Regresson and Outler Detecton. Wley, [13] A. Shashua. Trlnearty n vsual recognton by algnment. ECCV3 pages 479{484, [14] M. Spetsaks and J. Alomonos. A mult-frame approach to vsual moton percepton. IJCV, 6:245{255, [15] P. H. S. Torr, A. FtzGbbon, and A. Zsserman. Mantanng multple moton model hypotheses through many vews to recover matchng and structure. In ICCV6. IEEE, [16] P. H. S. Torr and D. W. Murray. A revew of robust methods to estmate the fundamental matrx. To Appear IJCV, [17] P. H. S. Torr and A Zsserman. Robust parametrzaton and computaton of the trfocal tensor. Image and Vson Computng, 15:591{607, 1997.
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