Optimal Combination of Stereo Camera Calibration from Arbitrary Stereo Images.

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1 Tna Memo No Image and Vson Computng, 9(1), 27-32, Optmal Combnaton of Stereo Camera Calbraton from Arbtrary Stereo Images. N.A.Thacker and J.E.W.Mayhew. Last updated 6 / 9 / 2005 Imagng Scence and Bomedcal Engneerng Dvson, Medcal School, Unversty of Manchester, Stopford Buldng, Oxford Road, Manchester, M13 9PT.

2 Optmal Combnaton of Stereo Camera Calbraton from Arbtrary Stereo Images. Nel A. Thacker, John E. W. Mayhew. whle at AI Vson Research Unt, Unversty of Sheffeld Abstract. Many stereo correspondence algorthms requre relatve camera geometry, as the eppolar constrant s fundamental to ther matchng processes. We ntend to buld a eye/head camera rg to mount on the moble platform COMODE to enhance the abltes of the TINA system to recover 3D geometry from ts envronment. Thus we wll need to be able to assocate camera geometry wth partcular head confguratons. Generc calbraton of such a system would requre the ablty to compute camera geometry from arbtrary stereo mages. Ths paper descrbes a system whch solves ths problem usng an establshed corner detector combned wth a robust stereo matchng algorthm and a varatonal soluton for the camera geometry. Keywords. Calbraton, Corner detecton, Stereo, Stereo matchng. Introducton. We wsh to develop a stereo eye/head camera rg whch wll support smlar low level vson competences to prmates, these are: foveaton, vergence, saccades and trackng. Ths head confguraton s currently under constructon [Fgure 1] and a smulaton of the hardware has been used for the work presented here. We wsh to be able to use ths head wth the TINA [1] vson system to recover stereo geometry and generate a 3D representaton of the world. These low level vson competences wll requre stereo correspondence of well located mage features. We show here that we can also use these correspondences to compute the relatve camera geometry necessary to provde eppolar geometry for other stereo matchng algorthms. Identfcaton of such features can be acheved usng an nterest operator smlar to that developed by Moravec [2]. The Plessey group [3] developed ths dea further and the resultng edge and corner detector was used to obtan structure from moton [4]. Thus t seems natural to use the Moravec/Plessey corner detector as our startng pont. In order to use corners to generate the necessary camera translaton and rotaton parameters, we need to robustly match the sets of corners obtaned. We cannot use the Plessey algorthm here as there may be substantal translatons between vews from two stereo cameras. Also, we cannot make much use of eppolar constrants as ths would requre the camera geometry whch we are tryng to obtan. Ths s not a dffcult problem to solve provded we only requre a subset of the total number of corners matched. (Fgure 1 about here ) Estmaton of the camera geometry needs to be robust and unbased, we would prefer to use the varatonal method proposed by Trved [5]. However, we would requre n excess of 100 data ponts to provde suffcent calbraton accuracy, whch s large compared to the number found and matched n most scenes. For ths reason we have appled standard statstcal methods for data combnaton to the resultng calbraton. We have extended ths dea further to the calbraton of a movng camera system whch moves on a one dmensonal trajectory n a space descrbed by the calbraton parameters. Corner Detecton and Matchng. The corner detector we use s that suggested by Harrs and Stephens [2] whch calculates an nterest operator defned accordng to an auto-correlaton of local patches of the mage. [ ] ( I/ u) M uv = 2 w I/ u I/ v w I/ u I/ v w ( I/ v) 2 w where u and v are mage coordnates and w mples a convoluton wth a gaussan mage mask. Any functon of the egenvalues α and β of the matrx M wll have the property of rotaton nvarance. What s found s that the trace of the matrx T r(m) = α + β s large where there s an edge n the mage and the determnant Det(M) = αβ s large where there s an edge or a corner. Thus edges are gven when ether α or β are large and corners can be dentfed where both are large. Corner strength s defned as C uv = Det(M) kt r(m) 2 2

3 Corners are dentfed as local maxma n corner strength whch are ftted to a two dmensonal quadratc n order to mprove postonal accuracy whch has been estmated as 0.3 pxels. Gven 5 or more correspondence ponts n the two mages t s possble to compute the camera translaton/rotaton parameters for the left to rght camera transformaton. There are generally an order of magntude more corners than ths n even a relatvely smple mage. The corners are matched usng a robust stereo matchng algorthm whch dentfes relable matches. Image tokens can be matched n some cases usng the followng heurstcs; (a) restrcted search strateges (eg eppolars n the case of stereo). (b) local mage propertes (eg mage correlaton). (c) unqueness. (d) dsparty gradent ( or smoothness ) constrants. For stereo matchng potental matches are sought n a varable eppolar band, wth a wdth determned by the accuracy of stereo calbraton. As the corner detector fnds local maxma n an auto-correlaton measure t makes sense to compare possble matches between ponts on the bass of local mage cross correlaton. Lsts of possble matches are generated, for corners n the left mage to the rght and rght to left, and ordered n terms of the local mage correlaton measure; wth A = M = w uv I 2 uv du dv A 2 w uv I uv I uv du dv nfty w uv I 2 uv du dv where w s a gaussan weghtng functon. Ths measure vares between 0 and 1 (close to 1 for good agreement), agan the assumpton has been made that there s lttle rotaton about the vewng axs. Ths measure s nvarant to the scale of the regstered mage ntensty ( assumng that no pror knowledge of the lghtng condtons and ndvdual camera aperture settngs s avalable). Weak dependence on the absolute mage ntensty can be rentroduced usng an asymmetry cut on the relatve corner strength. C 1 C 2 C 1 + C 2 > η A value of 0.85 s generally chosen for η, ths wll allow a dfference of 12 n relatve corner strength or a factor of 1.8 n mage ntensty. Only f the absolute value of the correlaton measure s hgh ( M max > ρ) s the match accepted and added to the lst of possble matches. rho can be set arbtrarly hgh to ensure that the underlyng mages are essentally dentcal and a value of 0.99 s generally used. We accept that ths wll nevtably result n some bas n matchng ablty for front-to-parallel surfaces. Canddate matches are only consdered further f they nvolve the best correlaton measure M max found for that par of ponts matched both ways between the left and rght mages. Ths algorthm mplctly enforces one to one matchng and also elmnates ncorrect matches resultng when a feature has only been detected n one mage. Due to the sparseness of corner data n many regons of an mage t s dffcult to mpose a smoothness or dsparty gradent constrant. However, t may be possble n future to constran possble matchng usng the results from less sparse matchng prmatves such as edges. On real mages corner detecton can be very nosy and settng a generc threshold for corner detecton s problematc. Also hgh frequency textured regons generally gve rse to many corners whch, on the bass of the above heurstcs, are unmatchable, as there are many smlar canddate matches for each feature. Thus n real mages t s dffcult to automate the generaton of a relable set of correspondences, potentally preventng successful ego-moton calculaton. What s requred s a method of dentfyng those features whch may be unrelably matched. Unrelable features can be defned as those whch have many canddate matches and consequently may be expected to be ambguously matched. Ambguous matches can be excluded by selectng matches where nether lst of other canddate matches has an entry whch s above a value of M max δ. The requred value of δ s defned by the expected varablty of the cross-correlaton value for correct matches and can be expected to be relatvely constant for all mages. δ can be defned so that only very unque matches are accepted as good, a value of has been found generally to be suffcent. Such a relablty heurstc reduces the effects of feature detecton thresholds on the matchng of hgh frequency features. 3

4 If we have temporal match nformaton, a more drect method of selectng relable matches can be used. Temporal matches are sought usng three dmensonal postons of corner features combned wth odometry nformaton specfyng the expected moton of COMODE. Match lsts are generated between temporal pars of mages n exactly the same way as for the stereo matcher. The result s a set of possble matchng lsts for each pont n each mage to ts stereo and temporal counterpart. A subset of correct matches s then selected by checkng that the matchng between all sets of stereo and temporal mages s consstent. After removal of non-unque matches there were generally between 20 and 100 matches fewer than 2 % of these were ncorrect. Ths s enough to obtan an estmate of the camera rotaton sutable for ep-polar matchng, though generally too poor to obtan good geometrcal accuracy. For ths reason a method of combnng the results from successve calbratons was requred. Camera Calbraton. It s possble to formulate the soluton for an arbtrary camera rotaton/translaton (RT ) from two sets of correspondng vector ponts n the mages x and x usng a varatonal prncple [5]. The small shfts δx and deltax needed to move these correspondences n each mage, so that they satsfy an estmate of the transformaton, can be approxmated to lnear order n an expanson about the current soluton [Appendx 1] gvng; δx = δx = F S F T F S F T F S F T + F S F T F S F T + F S F T F F F T = x T (RT )x = x T (RT ) = (RT )x Where the rotaton/translaton constrant equaton F uses the matrx formulaton frst suggested by Longuet- Hggns [6], whch s a matrx alternatve to wrtng the vector constrant equaton; F = x.t Rx ( = 0) where t s the translaton vector. Ths follows drectly from the coordnate transformaton equaton whch s vald for both ponts n the real world and mage coordnates. The transformaton matrx T and error matrx S are gven by T = 0 e 6 e 5 e 6 0 e 4 e 5 e 4 0 S = σ 2 x σ 2 y σ 2 z where e 4 e 5 e 6 are the drecton cosnes ( xyz ) of the translaton between the optcal centres of the cameras n the left camera frame. Many of the constrants between elements of the rotaton matrx can be mposed n a way that permts a unque reconstructon of the rotaton matrx. Ths s done by parametersng the rotaton matrx R n terms of Euler parameters (a quaternon representaton [Appendx 2]). The error matrx allows proper account to be made of the asymetrc nature of the x and y corner locaton accuracy ntroduced by the camera aspect rato a wth σ x = aσ y. The error n the z drecton σ z s set to zero as per the orgnal mplementaton by Trved. Ths s a relatvely smple model for the expected errors on the locaton of corners and a more prncpled one could be used f known. In our experence all corner locatons are determned wth the same accuracy wthn a factor of two. An approprately weghted sum of the mnmum shfts requred for each pont to be ndependently consstent wth the current transformaton can be formed. note also that E = E = (δx S 1 δx + δx T S 1 δx ) E = F S F T F 2 + F S F T 4 = F 2 /σ2

5 The transformaton matrx whch s most consstent wth the poston of the observed correspondences can be obtaned. Ths s done by mnmsng ths sum wth respect to the fve free rotaton and translaton parameters e 1, e 2, e 3, e 5, e 6 whle at the same tme enforcng the followng constrants. e 2 0 = 1 e 2 1 e 2 2 e 2 3 e 2 4 = 1 e 2 5 e 2 6 Dervatve nformaton can be computed for each correspondence pont [Appendx 3]. However, t was found that mnmsaton routnes whch could make use of ths nformaton were not very effcent or robust when used on ths partcular mnmsaton task. Mnmsaton s best done usng a robust numercal mnmsaton routne as for example the smplex mnmsaton algorthm of Nelder and Mead (see for example [5]). Ths method lends tself to robust statstcal methods should the ftted data be found to have a dstrbuton whch s non-normal. The Trved algorthm has no adjustable parameters and yelds errors n terms of mage varables whch can be used to judge the accuracy of the result. Ths nformaton combned wth knowledge of the corner detecton accuracy allows rogue ponts to be teratvely removed from the fttng process. The number of corners located n a par of stereo mages may not be suffcent to calbrate the camera geometry accurately. For ths reason we need to be able to combne the estmates of the calbraton varables e from several mages. Ths can be done usng the covarance matrx [C] (as estmated as n Appendx 3) as follows e t = C t (C 1 t 1 e t 1 + C 1 e) and C 1 t = C 1 t 1 + C 1 Flexblty can be obtaned by lmtng the sze of C t to that whch provdes the requred calbraton accuracy. Ths then allows the calbraton to track any systematc changes n the camera system. Calbratng a Moveable Head. For a system whch follows a one dmensonal trajectory n a hgh dmensonal calbraton space we can approxmate ths trajectory locally usng lnear nterpolaton between data ponts. The calbraton parameters must follow such a trajectory n the case of our smulated head when we restrct the control vergence rotaton angles to be symetrcal (Here symetrcal s defned only n terms of control sgnals and places no restrcton on the actual orentaton of the cameras or ther rotaton axes). We can parameterze ths curve usng one free parameter φ the control vergence angle of both cameras obtaned from accurate odometry. Usng ths parameter t s possble to nterpolate calbraton parameters across a range of camera angles ê = (e (φ φ ) + e (φ φ))/(φ φ ) where e and e the camera transformaton parameters at φ and φ. These estmates can be concatenated nto one calbraton vector g whch can be estmated from successve observatons of e at known φ gven the covarance C usng a kalman flter. g t = g t 1 + C gt ( g ê) T C 1 (e ê) wth C 1 gt = C 1 gt 1 + ( gê) T C 1 ( g ê) The ntrnsc parameters of the camera system, focal lengths and mage centres are requred as nput parameters and could be assumed to be fxed for our camera rg. These can be determned ndependently usng a combnaton of optcal methods and alternatve calbraton algorthms [8]. We are currently workng on several calbraton systems whch are to be unfed wthn one statstcal framework. The current mplementaton gnores radal dstortons but these could easly be ncorporated should t become necessary. The algorthm s ndependent of the magntude of nterocular separaton and only the drecton of translaton between the cameras s determned. Ths s suffcent for obtanng eppolar geometry sutable for stereo matchng but the nterocular separaton s needed for absolute depth measurements. Ths would mply that calbraton for our movng head would be made smpler f the cameras were to rotate about ther optcal centre. 5

6 Results. The Trved algorthm was frst tested on smulated data. The head was smulated assumng that when fxatng on objects t always adopted symetrc vergence so that the transformaton between cameras would be a functon of the verge angle. The parameters used to montor the resultng calbraton accuracy of the method were the error on the obtaned vergence angle and the sum of the squared mnmum shfts requred to make the smulated data consstent wth the estmated transformaton. The frst of these gves a drect estmate of the lmtng accuracy of depth measurement [Fgure 2]. (Fgure 2 about here ) Verge error can be estmated usng the covarance matrx and mproves as the results from several fts are combned. For a unform dstrbuton of n data ponts ths was found to vary as approxmately (n 5) 1/2. The second parameter s drectly related to the accuracy of the eppolar geometry whch was generally found to less than 0.1 pxels 2 for n > 20. The Trved algorthm was found to delver an unbased estmate of the true transformaton when the correct ntrnsc camera parameters were suppled (see below). When calbratons were combned (as above) the resultng accuracy was consstent wth that whch would have been determned usng the whole data set. The correct calbraton was also recovered followng a shft n the smulated camera system [Fgure 3] (correspondng to a knock on the real system). Recovery to a useable estmate was found to be an exponental functon of the number of data ponts, as expected. The varaton of transformaton parameters e wth camera vergence angle was found to be suffcently lnear to allow calbraton over a 15 degree range usng the method outlned above. The results ndcate that only twce the number of correspondences requred n the fxed camera method would be needed to obtan the same accuracy on reconstructed geometrcal data. (Fgure 3 about here ) The performance of the algorthm was also nvestgated n the case where ncorrect aspect ratos and mage centres were provded. Errors on these parameters appear to provde the real lmt on the accuracy of obtaned stereo data, errors n the mage centres of only 10 pxels can produce systematc depth errors of as much as 5 %. The effects of these errors are compounded when usng the Tsa calbraton algorthm wth ncorrect ntrnsc parameters to determne the camera focal-lengths and nterocular separaton. The algorthms were used to calbrate the camera geometry wth several real scenes, usng focal lengths and nterocular separaton obtaned from the Tsa algorthm and corner matched correspondences. The estmate of the vergence measurement accuracy calculated from the covarance matrx can be seen n Table 1. The value of χ 2 was entrely domnated by the expected error n the y drecton (by two orders of magntude) correspondng to a reproduceablty n poston of 0.3 pxels. Ponts whch were not consstent wth the obtaned camera geometry were excluded teratvely untl the χ 2 was observed to be consstent wth the corner locaton accuracy. (Table 1 about here) The overall accuracy of the rotaton parameters was found to be n agreement wth [5]. The new algorthm was found to be better than Tsa at determnng the eppolar geometry on the same set of data ponts. There was agreement between both methods wthn the smulated errors for each process gven the uncertantes on the ntrnsc parameters. Concluson It has been shown that a subset of robustly matched corner correspondences can be obtaned from real mages sutable for calbraton purposes. A general purpose calbraton algorthm has been demonstrated whch enables optmal combnaton of calbraton over a sequence of mages. The method can be used to calbrate ether fxed or movng head confguratons (wth symetrc vergence). We beleve that the method should be extendable to asymetrc vergence confguratons by nterpolatng on a plane defned between three calbraton ponts. 6

7 Appendx 1. To obtan the mnmum shft deltaxsub needed to make the observed data consstent wth a constrant F sub we can use the method of Lagrange. Here we mnmse the expresson; E = (δx T S 1 δx Ths can be done analytcally as follows; gvng + δx T S 1 δx ) + E/ δx = 2δx T S 1 + λ F = 0 δx = λ S F T /2 thus expandng the constrant equaton about the pont xsub gvng and hence δx and smlarly for δx. 2F + F λ S F T + F λ S F T = 0 λ /2 = F S F T F λ (F + F.δx + F.δx ) + F S F T Appendx 2. The quaternon representaton for the rotaton of a coordnate frame can be wrtten as follows where and q = (e 0, e 1, e 2, e 3 ) e 0 = cos(θ/2) e 1 = r 0 sn(θ/2) e 2 = r 1 sn(θ/2) e 3 = r 2 sn(θ/2) where r s a vector defnng the axs of rotaton and θ s the angle of rotaton about that axs. The rotaton matrx s then reparametersed as R = e2 0 + e 2 1 e 2 2 e 2 3 2(e 1 e 2 + e 0 e 3 ) 2(e 1 e 3 e 0 e 2 ) 2(e 1 e 2 e 0 e 3 ) e 2 0 e2 1 + e2 2 e2 3 2(e 2 e 3 + e 0 e 1 ) 2(e 1 e 3 + e 0 e 2 ) 2(e 2 e 3 e 0 e 1 ) e 2 0 e 2 1 e 2 2 e 2 3 Appendx 3. The elements of the nverse covarance matrx are defned from the chsup2 varable by α nm = χ 2 / e n e m whch can be constructed n our case from ndvdual contrbutons from each data pont. α nm = 1/(2σ 2 c) 2 E / e n e m where σ c s the estmated corner locaton accuracy and the frst dervatve s gven by E = E / e n F S F T F 2 + F S F T = 2F F n / e n σ 2 = F 2 /σ2 F 2 σ2 / e n σ 4 At the mnmum the second term s found to be three orders of magntude smaller than the frst, allowng the second dervatves to be approxmated to around the same accuracy usng; 2 E / e n e m = 2 F n/ e n F m / e m σ 2 7

8 Acknowledgments. We gratefully acknowledge the grant holders Dr. John E.W. Mayhew Dr. Paul Dean and Prof. John Frby and the support of ESRC/MRC/SERC for the fundng of ths project. References. [1] Porrll, J., S.B.Pollard, T.P.Prdmore, J.B.Bowen TINA: The Sheffeld AIVRU Vson System Proc. 9th IJCAI. Vol.2 pp [2] Moravec, H.P. Obstacle avodance and navgaton n the real world by a seeng robot rover Ph.D Thess, Stanford Unv., Sept., [3] Harrs, C. and M.Stephens A Combned Corner and Edge Detector. Proceedngs of the Fourth Alvey Vson Conference. pp August [4] Charnley, D. and R.Blsset Surface Reconstructon from Outdoor Image Sequences. Proceedngs of the Fourth Alvey Vson Conference, pp August [5] Trved,H.P. Estmaton of Stereo and Moton Parameters usng a Varatonal Prncple. Image and Vson Computng 5,2,pp May [6] Longuet-Hggns, H.C. A Computer Algorthm for Reconstructng a Scene from Two Projectons. Nature, Vol 293 pp September [7] Press,W.H., B.P.Flannery, S.A.Teukolsky, W.T.Vetterlng, Numercal Recpes n C. Cambrdge Unversty Press [8] Tsa,R.Y. An effcent and Accurate Camera Calbraton Technque for 3D Machne Vson. IEEE Computer Vson and Pattern Recognton, pp Fgure Legends. Fgure 1. The Robot Head. Fgure 2. Percentage depth error on absolute depth measurement for specfc verge angle accuraces. For relatve depth errors smply multply by two. Fgure 3. Varaton of the optmal estmate of verge angle wth tme. 20 new data ponts were combned at each tme step whle the covarance matrx for the estmate was lmted to a sze whch specfed an error of 0.05 degress (generally requrng 400 data ponts). The fgure shows how the estmate recovers after a shft n the camera system. Table 1. Results from real scenes showng the mprovng calbraton accuracy wth ncreasng numbers of data ponts. 8

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