A Revisit of Methods for Determining the Fundamental Matrix with Planes

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1 A Revst of Methods for Determnng the Fundamental Matrx wth Planes Y Zhou 1,, Laurent Knep 1,, and Hongdong L 1,,3 1 Research School of Engneerng, Australan Natonal Unversty ARC Centre of Excellence for Robotc Vson 3 NICA Canberra Labs {yzhou, laurentknep, hongdongl}@anueduau Abstract Determnng the fundamental matrx from a collecton of nter-frame homographes (more than two s a classcal problem he compatblty relatonshp between the fundamental matrx and any of the deally consstent homographes can be used to compute the fundamental matrx Usng the drect lnear transformaton (DL, the compatblty equaton can be translated nto a least squares problem and can be easly solved va SVD decomposton However, ths soluton s extremely susceptble to nose and moton nconsstences, hence rarely used Inspred by the normalzed eght-pont algorthm, we show that a relatvely smple but non-trval two-step normalzaton of the nput homographes acheves the desred effect, and the results are at last comparable to the less attractve hallucnated ponts method he algorthm s theoretcally justfed and verfed by experments on both synthetc and real data I INRODUCION he eppolar geometry of two perspectve mages can be descrbed by a sngular 3 3 matrx When the camera s calbrated, the matrx s known as the essental matrx E For uncalbrated systems, t s known as the fundamental matrx F he estmaton of the fundamental matrx s a classcal and thoroughly studed topc whch plays an essental role n many applcatons nvolvng multple-vew geometry, such as vsual odometry (VO, structure from moton (SfM, and vsual SLAM, etc he most popular method for estmatng the fundamental matrx s based on sparse correspondences between local nvarant keyponts, for nstance gven by the popular SIF algorthm [1] Seven ponts consttute the mnmal confguraton because the fundamental matrx has 7 degrees of freedom (DoF Compared to the eght-pont algorthm, the seven pont algorthm needs an addtonal step to calculate the lnear combnaton factor of the obtaned two-dmensonal null-space Whle seven pont correspondences represent the mnmum for estmatng the fundamental matrx [18], the 8- pont algorthm [11] s the most popular method because of ts lnear nature and thus smplcty to mplement However, t was only after Hartley publshed hs semnal work [1] on usng data normalzaton that the eght-pont algorthm became truly useful n practce It s beleved that the reconstructon performance can be mproved by ncorporatng addtonal geometrc constrants lke coplanarty of certan ponts Luong and Faugeras [13, 14] are the frst who propose to estmate the fundamental matrx wth multple homographes n a lnear way hey compared the lnear soluton wth other non-lnear ones concludng that none of the developed methods s stable under nose In other words, though the drect lnear method s qute smple and straghtforward, t has lmted practcal usefulness Zhang [4] gave a thorough revew on the technques of fundamental matrx estmaton and ts uncertanty he bad performance of the Drect Lnear ransformaton (DL appled to the compatblty relaton between the homography H and the fundamental matrx F was however not dscussed n much detal Szelsk and orr [19] thoroughly dscussed three methods used for solvng structure from moton (SfM wth planes hey presented an analyss of the robustness of each method and then suggested to estmate the fundamental matrx wth hallucnated ponts (HP that le on planes nstead of the compatblty equaton (and thus the homograhes drectly Anubhav et al [1] demonstrates that the compatblty constrant s an mplct equaton n H and F hey also concluded that an explct expresson lke F = [e ] H s more sutable for a computatonal algorthm Vncent and Laganere [1] proposed a detecton algorthm for planar homographes workng on a par of uncalbrated mages hey clamed that the estmaton of the fundamental matrx from pont correspondences derved from homographes allows to use data normalzaton technques, and thus performs much better than usng the homographes drectly A method was ntroduced to estmate the fundamental matrx wth a homology n [15, 17, 9] heoretcally, a homology has two dentcal egenvalues and another unque one whch s correspondng to the eppole e However, n practcal stuatons, the homographes are never perfect, whch s why a double egenvalue s never guaranteed It s hard n practce to choose whch egenvalue corresponds to the unque one; the real parts of the egenvalues are often equally spread and/or very close to each other he goal of ths paper s to present a drect method for computng the fundamental matrx from a set of homographes estmated ndependently between two vew-ponts of a rgd, pece-wse planar scene Key to our method s a two-step normalzaton procedure leadng to a rescaled lnear soluton of the compatblty equaton We fnally acheve comparable results to the hallucnated ponts method, however wthout

2 ntroducng addtonal, vrtual correspondences he rest of the paper s organzed as follows Secton quckly revews the three methods dscussed n [19] Our two-step lnear (SL method wth all theoretcal dervatons s descrbed n Secton 3 In Secton 4, we compare DL, HP and SL by separately runnng them on synthetc and real data An analyss of numercal stablty and algorthmc complexty s also gven II QUICK REVIEW OF HE HREE MEHODS Szelsk and orr dscussed three methods that can be used for estmatng the fundamental matrx gven several ( homographes n [19], whch are revewed n the followng Hallucnatng addtonal correspondences: Hallucnated ponts refer to augmented sample ponts on planes heses ponts are also called vrtual control ponts Hallucnated correspondences are generated by frst creatng several vrtual D ponts x on mage one whch are assumed to be the projecton of vrtual ponts on the plane her correspondng ponts x are then found by applyng the correspondng homography to ponts x hen the fundamental matrx F s computed by applyng normalzed 8-pont algorthm on the obtaned hallucnated correspondences Drect lnear method: he mplct compatblty relatonshp between nterframe homographes and the fundamental matrx can be drectly used for computng the fundamental matrx he compatblty equaton F H + H F = gves sx constrants [13] (for whch only 5 are lnearly ndependent herefore, at least homographes are needed for computng the fundamental matrx he queston can be translated to a least squares problem by DL and can be easly solved by SVD decomposton However, ths straghtforward method s unstable for naccurate homographes, sometmes leadng to completely meanngless results he reason gven by Szelsk and orr s that usng the compatblty equaton drectly corresponds to samplng homographes at locatons where ther predctve power s very weak he samples are far from havng the normal dstrbuton requred for total least squares to work reasonably well Plane plus parallax: Plane plus parallax technques are always used to recover the depth (projectve or Eucldean of the scene o compute the fundamental matrx, one of the homographes s choosen and used to unwarp all ponts n the current frame he eppole e s computed by mnmzng the sum of the weghted dstance between eppole and lnes passng through correspondng ponts x and x hen the fundamental matrx F can be computed by F = [e ] H hs method cannot work well when ponts are evenly dstrbuted over several planes he computaton s also more complcated and expensve compared to the former two methods III ROBUS WO-SEP LINEAR SOLUION he compatblty equaton F H + H F = gves only 6 lnear equatons [13] In fact, as shown later, only 5 of them are ndependent herefore, at least homographes are needed for computng the fundamental matrx Applyng the DL transformaton to the compatblty equaton leads to the least squares problem, W 1 W Af = f =, (1 W n where f = (f 11, f 1, f 31, f 1, f, f 3, f 13, f 3, f 33 denotes a vector obtaned by rearrangng the entres of the fundamental matrx n a column vector Matrx A s made up of several sub matrces W of same dmenson whch s defned as, W = h π 11 h π 1 h π 31 h π 1 h π 11 h π h π 1 h π 3 h π 31 h π 13 h π 11 h π 3 h π 1 h π 33 h π 31 h π 1 h π h π 3 h π 13 h π 1 h π 3 h π h π 33 h π 3 h 13 h π 3 h π 33 ( he entres of the matrx W orgnate from the homography H = hπ 11 h π 1 h π 13 h π whch s nduced by plane π As 1 h π h π 3 h π 31 h π 3 h π 33 dscussed n Secton, the least squares problem descrbed n Eq (1 s serously ll-condtoned, whch means that even under a tny perturbaton of any entry of matrx A, the soluton quckly dverges from the groundtruth result hus, the matrx A should be re-condtoned n order to stablze ts null space We follow the dea of [1] and ntroduce normalzaton n order to stablze the result However, t s not trval to drectly normalze the matrx A as t has been done n pror work for estmatng the fundamental matrx or even the homography from pont correspondences he reason s twofold Frst, the normalzaton ncludes two parts, translaton and scalng he translaton operaton can only be performed by a lnear transformaton when the normalzed object s descrbed n homogeneous form Second, the normalzaton should be performed to data whch has the same meanng he key to deal wth above two ssues comes from the specal structure of the matrx F H he compatblty equaton requres that F H s a skew-symmetrc matrx, and thus s of the form F H = a 3 a a 3 a 1 (3 a a 1 he dagonal entres gve three equatons whch descrbe an orthogonal relatonshp between correspondng column vectors of the fundamental matrx and a homography, f h =, = 1,, 3 (4 f and h denote the th column vector of the fundamental matrx F = ( f 1 f f 3 and the homography H =

3 ( h1 h h 3 he other three equatons enforce the skew symmetrc property However, only two of them are ndependent hs makes sense because a homography has 8 degrees of freedom (DoF For the uncalbrated case, the ntrnsc matrx s unknown whch removes three constrants hus, only fve ndependent constrants can be obtaned from one homography, three from the orthogonal relatonshp descrbed n Eq (4 and the other two from the skew-symmetrc property Our two-step recondtonng method realzes the non-trval normalzaton by fully usng the specal structure of matrx F H Frst, by utlzng the orthogonal relatonshp, we decompose the orgnal least squares problem Af = nto three sub least squares problems A f =, where matrx A = ( h π 1 h π h π n and = 1,, 3 Each column f of F s estmated ndvdually he relatve scale factor for each estmated soluton f can then be recovered by usng the skew-symmetrc property of matrx F H n Eq (3 Wth ths formulaton, every column of matrx A has the same meanng Besdes, n order to do the translaton, the matrx A should be extended by an addtonal column = (1 1 1 whch leads to à = [A ] Accordngly, ( the extended λ 1 soluton vector f s defned as f = f, where λ denotes the relatve scale factor of the ndvdually estmated soluton he mathematcal proof s gven after the whole algorthm s ntroduced hs extenson turns each row of matrx A nto homogeneous form he normalzaton s then performed by nsertng a 4 4 lnear transformaton matrx Q and ts nverse n between à and f, resultng n à Q Q 1 f =  ˆf =, (5 where  = à Q and ˆf = Q 1 f he lnear transformaton Q ncludes a translaton and a scalng We regard each h π j as a 3d pont Followng the dea of [1], the coordnates are translated such that the centrod c of the set of all such ponts becomes the orgnhe coordnates are then scaled by applyng an sotropc scalng factor s to all three coordnates of each pont Fnally, we choose to scale the coordnates such that the average dstance of a pont h π j from the orgn s equal to 3 he lnear transformaton Q and scalng related varables are defned as below Q = s s s (6 c 1 s c s c 3 s 1 c = ( m j=1 c 1 c c hπ j 3 = m m s = 3 d, d j=1 = hπ j c F m he soluton of the three sub least square problems  ˆf = can be easly obtaned va SVD hen f = Q ˆf he only remanng task s to fnd the scale factor λ (7 (8 he skew-symmetrc property of matrx F H can be translated nto another least squares problem A λ λ = va DL, where λ = ( λ 1 λ λ 3 and Aλ s gven by A λ = f 1,1:3 hπ 1 f,1:3 hπ 1 1 f 1,1:3 hπ 1 3 f 3,1:3 hπ 1 1 f,1:3 hπ 1 3 f 1,1:3 hπ m f,1:3 hπ m f 3,1:3 hπ 1 1 f 1,1:3 hπm 3 f 3,1:3 hπm 1 f,1:3 hπm 3 f 3,1:3 hπm (9 f,1:3 n A λ s defned as the frst three rows of vector f h π j s defned same as before he full two-step lnear method (SL s descrbed n Algorthm 1 Algorthm 1 wo-step Lnear Method (SL 1: Input: A collecton of ndependently estmated homographes H s : for = 1:3 do 3: à = [A ] 4:  à Q 5: ˆf solveâ ˆf = 6: f Q ˆf 7: λ = ( λ 1 λ λ 3 solveaλ λ = 8: end for 9: Output: F = ( f 1 f f 3 It should be noted that n order to apply the normalzaton, the orgnal least squares problem s modfed However, we wll see n the followng that solvng the modfed problem à f = s equvalent to solvng the orgnal problem A f = herefore, two questons need to be answered n order to prove ths clam: 1 After extendng the matrx A by an addtonal column = (1 1 1, what s the null-space confguraton of Ã? Why does the soluton ( of problem à f = have the λ 1 structure as f = f? he answers are gven by provng the followng two clams: Property 1 Rank(A =, 1 dm(n(ã when the number of planes m 3,, where N( denotes the null space of ( Property N(à = ( N(A Proof Property 1 Assumng that two camera matrces are gven by P 1 = [I ] and P = [B b], each homography nduced by a plane π j = [ vj, 1] observed by the two cameras can be denoted as H π j B + bv j (1

4 Each row of matrx A contans the th column of one homography, whch gves h π j B + v j, b, (11 where B denotes the th column of the matrx B and v j, the th element of the vector v j It s obvous to see that f we regard each row of matrx A as a general 3D pont, all the ponts h π j are lyng on the lne wth the drecton of v j, b passng pont B hus Rank(A = Snce matrx à s obtaned by addng an addtonal column to A, t s also obvous to see that Because thus we fnally have Rank(A Rank(à 3 (1 Rank(à + dm(n(ã = 4, (13 1 dm(n(ã 1 (14 Proof Property Assumng x N(A, and x N(Ã, we have A x =, and à x = ( ( x x Obvoulsy, x N(A, à = [A ] = ( N(A hus, N(à Necessary condton QED ( x On the other hand, assume x =, ω ω Snce à x =, A x + ω1 3 1 =, ω1 3 1 =, ω =, contradcton, ( N(A hus, N(à Suffcent condton QED Summarzng, N(à = ( N(A Above mathematcal proofs explan why we can get the soluton to the orgnal problem by solvng the recondtoned least squares problems One drawback of the proposed method s that at least 3 planes (homographes are needed for computng the fundamental matrx he reason les n the normalzaton he matrx A s extended by an addtonal column = (1 1 1 hus, wth only two homographes, the rank of à s always, and the normalzaton cannot be appled 1 If Rank(à = 3, à has only a one dmensonal null space whch s the egen vector correspondng to the smallest egenvalue of matrx à Otherwse, f Rank(à =, the fnal soluton of problem à f resdes n a two dmensonal null space However, durng our experment, we never observed the case of Rank(à = IV EXPERIMENAL EVALUAION In ths secton, we compare the performance of DL, HP and SL on both synthetc and real data Numercal stablty of DL and SL as well as algorthmc complexty of the three methods are also dscussed he nput homographes can be derved from ether pont or lne features as they are dual geometrc enttes [6, 7, 4] We use lne features durng the synthetc experments, and pont correspondences durng the experment on real data A Synthetc experment For each sngle experment, we construct two artfcal vews observng planes n a 3D envronment Groundtruth moton and structure (planes s generated n the same way as n [] Wthout loss of generalty, the camera pose of the frst vew s assumed to be dentcal wth the world frame he absolute pose of the second vew s defned by moton parameters lyng wthn a certan range he rotaton angles along each axs (roll, ptch, yaw le wthn ( 5, 5 and the translaton n each drecton (X, Y, Z s wthn ( 1, 1 he structure s randomly generated by creatng N = 5 planes wth known homographes 4 groups of Gaussan nose (µ =, σ [, 5] corrupted ponts are created on each plane, whch are used for fttng the lne features he mage sze s and the focal length s f x = f y = 5 he relatve moton parameters are extracted from the estmated fundamental matrx F (n fact from essental matrx E = K FK As shown n Fg 1, both HP and SL outperform DL n the accuracy of the estmated fundamental matrx and the moton parameters We use max norm of the dfference between F groundtruth and F estmated as a crteron for assessng the accuracy of the estmated F he estmated rotaton matrx s compared to groundtruth by computng the anlge Θ = arccos( trace(r groundtruth R estmated 1 And the estmated translaton s compared aganst groundtruth by computng the angle between two translaton vectors t groundtruth and t estmated SL s more nose reslent n terms of fundamental matrx estmaton n comparson to HP Concernng the accuracy of the extracted moton parameters, SL and HP perform equally well B Experment on real mages he algorthm s tested on the famous Oxford Corrdor sequence Homographes are estmated from Harrs corner correspondences [8] Ponts on each plane are grouped manually and outlers are rejected by applyng the Random sample consensus (RANSAC technque [5] As shown n Fg, the eppole estmated by SL (e the ntersecton pont of blue lnes s closest to groundtuth he eppole e s extracted from the null space of the fundamental As shown n [3], when the lne s close to or passng through the orgn of the coordnate frame, the qualty of the estmated homographes decreases dramatcally hs problem can be solved by proceedng a pror normalzaton of the lne parameters For the sake of smplcty and wthout losng generalty, the lnes generated n our experment are forced to be away from the orgn of the coordnate frame by at least 1 pxels

5 DL HP SL F gt -F est max Nose Level (a Synthetc experment confguraton (b Max norm of dfference between groundtruth and estmated F 5 4 DL HP SL 35 3 DL HP SL 15 5 Degree Degree Nose Level (c Error n rotaton Nose Level (d Error n translaton Fg 1 Fgure (a shows the confguraton of the experment he accuracy of fundamental matrx estmaton s shown n Fg (b wth max norm as assessng crtera Fgures (c (d separately depcts rotaton and translaton error of DL,HP,SL matrx A small error n any entry of the fundamental matrx can easly cause the resultng eppole to severly devate from the groundtruth locaton We can easly see that our conclusons from the synthetc experment are verfed, namely that the proposed method clearly outperforms DL and shows advantages over HP as well C Numercal stablty and algorthmc complexty It s easy to understand why the performance of DL can be dramatcally mproved by ncludng normalzaton Wthout the normalzaton, as shown n Eq (1 and Eq (, some of the entres are smaller than the others by several magntude whch drectly causes the serous ll-condtonng of the orgnal least squares problem We record the numercal stablty of DL and SL As can be seen n Fg 3, the condton numbers of the three normalzed sub least squares problem are far smaller than the one of the DL soluton he average varance of the condton number also demonstrates that SL s numercally more stable A smple complexty s gven n ab I In our experment, N = 8 and M = 5 SL and HP lead to smlar performance under these condtons, whle SL however needs less computatonal resources than HP ABLE I ALGORIHM COMPLEXIY COMPARISON Method Input Matrx sze to be solved HP N ponts (not coplanar, N 8 A N 9 SL M planes (M 3 3 A M 4 + A 3M 3 It s worth pontng out that, durng the experment, we dscovered that f the consstency among the nter-frame homographes s guaranteed, the estmated fundamental matrx s always accurate and robust no matter whch method s used Usually, perfect consstency constrants are avalable only n mplct form whch can only be acheved by teratve nonlnear methods, eg Jont Bundle Adjustment (BA-Jont and AML [, 3] Explct methods lke [16,, ] use a low-rank approxmaton under the Frobenus norm or the Mahalanobs

6 (a Grouped pont features n mage one (b Grouped pont features n mage two (c Eppolar lnes of groundtruth and all three methods Fg Grouped pont features whch are used for estmatng the homographes are shown n Fg (a and (b Eppolar lnes obtaned by DL(yellow, HP(green, SL(blue and groundtruth (red are shown n Fg (c Average Condton Number DL SL1 SL SL3 Average Varance of Condton Number DL SL1 SL SL Nose Level (a Average condton number of DL and SL Nose Level (b Average varance of condton number Fg 3 he average condton number under each nose level s shown n Fgure (a SL1, SL and SL3 are the three sub least squares problems of SL Fgure (b shows the correspondng average varance of the condton number norm to enforce the rank-four constrant However, the explct form s derved from a relaxed consstency constrant whch means the consstency cannot be perfectly guaranteed hs dscovery n fact gves an alternatve explanaton to why the drect estmaton of the fundamental matrx by the compatblty equaton s not stable V CONCLUSION In ths paper, we revsted an old topc: accurately and robustly estmatng the fundamental matrx gven a collecton of ndependently estmated homograhes We frst revew three classcal methods and then show that a smple but nontrval two-step normalzaton wthn the drect lnear method acheves smlar performance than the less attractve and more computatonally ntensve hallucnated ponts based method We verfy the correctness and robustness of our method by gvng a mathematcal proof and an expermental evaluaton on both synthetc and real data he numercal stablty analyss and algorthm complexty dscusson fnally demonstrates our mprovement and further advantages of the proposed technque ACKNOWLEDGMEN he research s supported by the ARC Centre of Excellence for Robotc Vson he work s furthermore supported by ARC grants DP and DE Y Zhou s funded by the Chnese Scholarshp Councl to be a PhD student at the Australan Natonal Unversty We also thank Dr Yuchao Da for hs comments and suggestons to mprove ths work REFERENCES [1] Anubhav Agarwal, CV Jawahar, and PJ Narayanan A survey of planar homography estmaton technques Centre for Vsual Informaton echnology, ech Rep III/R/5/1, 5

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