EVALUATION OF RELATIVE POSE ESTIMATION METHODS FOR MULTI-CAMERA SETUPS

Transcription

1 EVALUAION OF RELAIVE POSE ESIMAION MEHODS FOR MULI-CAMERA SEUPS Volker Rodehorst *, Matthas Henrchs and Olaf Hellwch Computer Vson & Remote Sensng, Berln Unversty of echnology, Franklnstr. 8/9, FR 3-, D-587 Berln, Germany (vr, matzeh, KEY WORDS: Camera moton estmaton, mult-vew geometry, relatve orentaton, algorthm comparson, performance analyss ABSRAC: he fully automatc and relable calculaton of the relatve camera pose from mage correspondences s one of the challengng tasks n photogrammetry and computer vson. he problem has not been solved satsfactorly, e.g. n case of crtcal camera motons or when observng specal pont confguratons. Furthermore, some methods provde multple solutons for the relatve orentaton. In ths paper we compare varous technques, analyze ther dffcultes and gve results on synthetc and real data. We show that n case of nosy data pose estmaton of a sngle camera remans dffcult. he use of multple calbrated cameras that are fxed on a rg leads to addtonal constrants, whch sgnfcantly stablze the pose estmaton process.. INRODUCION Camera pose estmaton from mage correspondences s one of the central tasks n photogrammetry and computer vson. he recovery of the poston and orentaton of one camera relatve to another can be used for bnocular stereo or ego-moton estmaton. From an algebrac pont of vew the fundamental matrx descrbes the proectve relaton between two uncalbrated vews. he essental matrx s mportant for moton analyss wth a calbrated camera, as t contans the rotaton and translaton up to an unknown scale factor. he automated vsual-based moton estmaton usng thousands of vdeo frames requres an extremely accurate and relable relatve orentaton method to handle extensve and ntrcate camera paths. We would lke to pont out that wthout the presence of nose all tested methods perform well. Nevertheless, t s very dffcult to obtan stable results for all frames under real condtons wth nosy mage measurements. herefore, we propose addtonal constrants to avod false estmatons. In lterature several algorthms for drect relatve orentaton exst. In (McGlone et al., 4) Förstner and Wrobel gve an overvew of varous methods, whch s updated n able. A detaled descrpton of recent developments can be found n (Stewénus et al., 6) and degenerate confguratons, lke a) coplanar obect ponts and ruled quadrc contanng the proecton centers or b) orthogonal ruled quadrc, especally cylnder contanng the proecton centers, are dscussed n (Phlp, 998). Method Ponts Deg. Solutons Hartley, a) Hartley & Zsserman, 4 7 a) 3 Phlp, 996/98 6 a) Pzarro et al., 3 6 b) 6 Nster, 4 Stewénus et al., 6 L & Hartley, 6 5 b) able : Drect solvers for relatve orentaton An nterestng aspect of the three mnmal 5-pont algorthms and the 6-pont algorthm of (Pzarro et al., 3) s ther stablty, even f ponts from coplanar obects are observed. hs s especally convenent n archtectural envronment, where many coplanar obects appear. In ths paper we compare all drect solvers of able and a non-lnear solver (Batra et al., 7) to determne ther strengths and weaknesses targetng an automatc approach for real world setups. Four mplementatons are based on the orgnal MALAB code provded by (Stewénus, 4). hs paper s organzed as follows: Frst, the used methods are shortly ntroduced, followed by some notes on data condtonng. Secton 3 deals wth the selecton of a unque soluton from multple results. he evaluaton of the solver usng synthetc data s descrbed n secton 4, followed by a dscusson of the results. Secton 6 ntroduces addtonal constrants for multple cameras. he results usng real data are shown n secton 7. Fnally, a dscusson and concluson of the results closes the paper.. RELAIVE POSE RECOVERY In general, all methods analyze the moton or relatve orentaton of calbrated cameras usng the essental matrx E. he man property of E s the coplanarty or eppolar-constrant n terms of the normalzed coordnates u Eu = () u = K x = and u K x () of correspondng mage ponts x x ' wth known calbraton matrces K and K'. hs lnear relaton s also known as the Longuet-Hggns equaton (Longuet-Hggns, 98). he essental matrx has addtonal algebrac propertes, e.g. the cubc rank-constrant det( E ) = (3) and the cubc trace-constrant (Demazure, 988) EE E trace( EE ) E = (4) 35

2 he Internatonal Archves of the Photogrammetry, Remote Sensng and Spatal Informaton Scences. Vol. XXXVII. Part B3b. Beng 8 to ensure that the two non-zero sngular values are equal. A complete lst of necessary constrants can be found n (Batra et al., 7). he lnear 8-pont algorthm for the computaton of the fundamental matrx (Hartley, 997) can be used to estmate the essental matrx as well. Frst, the algorthm uses eght lnear equatons of () wth normalzed coordnates to estmate E and afterwards, the addtonal constrants (3) and (4) must be enforced. If the sngular value decomposton s E= U dag( σ, σ, σ3) V for σ σ σ3 (5) then the closest essental matrx that mnmzes obtaned as follows E E % can be E % = U dag( σσ,,) V wth σ + σ σ =. (6) However, the later nserton of the constrants may provde wrong estmatons. (Stewénus et al., 6) and (Šegvć et al., 7) menton that the 8-pont algorthm has a forward bas, whch leads to undesred camera motons. he 7-pont solver (Hartley & Zsserman, 4) uses seven eppolar-constrants () on the nne components of the essental matrx. An orthogonal bass for the two-dmensonal null-space of these constrants s computed usng sngular value decomposton. hus E can be wrtten as E= α E + α E (7) where E are the null-vectors for the eppolar-constrants rowwse n matrx form. Snce E can only be solved up to scale, we are free to set the scalar multpler α =. Subsequently, the soluton space s reduced usng the rank-constrant (3), whch gves a thrd-order polynomal equaton n α wth three possble solutons for the essental matrx. he lnear 6-pont solver (Phlp, 998) composes the nne thrd-order polynomal equatons from the trace-constrant (4) nto a 9 matrx and solves for the unknowns lnearly. It provdes a unque soluton but s very senstve to nose (Stewénus et al., 6). he 6- pont method of (Pzarro et al., 3) also composes the nne equatons of (4) nto a 9 matrx from whch four rows correspondng to the largest sngular values are selected. From these equatons, a sxth-degree polynomal s computed wth sx possble solutons for E. he mnmal 5-pont algorthms (Nster, 4), (Stewénus et al., 6), (L & Hartley, 6) need only fve pont correspondences. In general, the soluton n the four-dmensonal null-space derved from the eppolarconstrant () 4 E= αe (8) = s found usng the nne polynomal equatons from the traceconstrant (4) and the polynomal equaton from the rankconstrant (3). he real-valued zero crossngs of ths tenth-order polynomal ndcate possble solutons for the essental matrx E. hey can be found usng Sturm sequences to bracket the roots (Nster, 4) or an egen-decomposton (Stewénus et al., 6), whch produces slghtly better results. he approach of (L & Hartley, 6) computes the unknown parameters smultaneously nstead of back-substtutng and solvng all the unknowns sequentally. Fnally, a non-lnear solver from fve ponts (Batra et al., 7) was evaluated. hs technque also extracts the four-dmensonal null-space (8). o avod egen-decompostons, ths approach suggests a nonlnear optmzaton technque, e.g. Levenberg-Marquardt. hs technque extracts the translaton vector t = (t x,t y,t z ) from the essental matrx E usng sngular value decomposton (Wang & su, ) te=. (9) Note, that t s related to the second proecton center C' wth t = RC. () he translaton vector s used to parameterze a cost functon, whch enforces necessary constrants for the essental matrx. he state vector b = ( α, α, α3, α4, tx, ty, tz) () conssts of the scalars α defnng the soluton wthn the fourdmensonal null-space and the three translaton components. he cost functon can be derved from the equaton EE [ t] [ t] =, () where [] denotes the skew-symmetrc matrx of vector t. he nne elements of E dependng on seven elements of b are stored n a 7 9 matrx A. Overall, nne of those matrces can be formed, three from equaton (9) and sx from (). he nonlnear mnmzaton task 9 = mn bab wth b= b (3) starts wth random values for α. Snce there are up to possble solutons and the null-space s generally non convex, ths optmzaton should be terated several tmes. For real-tme applcatons ths may not be sutable, but the technque can be used to mprove the results obtaned by drect solvers, whch provde good approxmaton values. Hartley proved (Hartley, 997) that the lnear 8-pont algorthm performs sgnfcantly better, f the nput data s condtoned. hs nsght should be stll mportant for mnmal solvers (L & Hartley, 6). he normalzaton s done by translatng the centrod of the measured mage ponts to the orgn and scalng them to a mean Eucldean dstance of, whch can be combned nto a smlarty transformaton for the frst and ' for the second mage. Note, that the resultng relatve orentaton must be decondtoned before the essental constrants are enforced. In case of the mnmum solvers or the non-lnear algorthm, the four resultng null-vectors E must be decondtoned E % = E (4) before the root searchng step (Šegvć et al., 7). One mght thnk that the data could be decondtoned as a fnal step, but ths leads to false solutons as shown n (Hartley, 997). 3. CHOOSING HE RIGH SOLUION Most drect solvers provde multple solutons for the relatve orentaton, except the lnear solvers (Hartley, 997) and (Phlp, 998). Up to dstnct physcally vald solutons are possble (see able ). Although, n most cases the number of solutons vares between one and four, the correct soluton s dffcult to dentfy. Snce the 7-pont solver (Hartley & Zsserman, 4) doesn t enforce the trace-constrant (4), the smlarty of the two non-zero sngular values may ndcate the soluton. Furthermore, the 6-pont method of (Pzarro et al., 3) doesn t employ the rank-constrant (4), so that the smallest determnant value maybe analyzed. 36

3 he Internatonal Archves of the Photogrammetry, Remote Sensng and Spatal Informaton Scences. Vol. XXXVII. Part B3b. Beng 8 If no addtonal assumptons can be made, a possble crteron for choosng the rght soluton are the number of ponts, whch le n front of both cameras. A method to recover the twsted par ambguty and extract the proecton matrces from E s descrbed n (Nster, 4; Hartley & Zsserman, 4). hen, spatal obect ponts are trangulated and ther cheralty can be tested. Snce several solutons may have all ponts n front of both cameras, ths crteron s not suffcent. Furthermore, the cheralty-condton suffers from three dsadvantages. Frst, f there s no camera translaton, the ponts can not be trangulated. o overcome ths ssue, a threshold t mov for detectng enough moton (Weng et al., 989) can be ntroduced: u Ru P= I, P = Rt, < tmov u u (5) Second, the trangulaton of obect ponts can only be performed wth a suffcent baselne to depth rato. For example, nosy mage ponts may cause the trangulated obect pont to flp behnd the camera. hs can be avoded, by testng ponts wth a certan proxmty to the camera. hrd, f many ponts are used for a cheralty test, the trangulaton s computatonally ntensve. If more than fve correspondences are avalable, the addtonal nformaton should be used to fnd the rght soluton. We compute the frst-order geometrc error (Sampson-dstance) for all pont correspondences u u ' d error = ( ueu ) ( Eu) + ( Eu) + ( E u ) + ( E u ) x y (6) x y evaluaton of the selecton crtera s done wth the 5-pont algorthm of (Nstér, 4). Here, addtonal random 5- tuples are selected n each dataset and the best sample s taken as result. he experments for over-determned computatons are performed wth the whole dataset of ponts. Fnally, the mpact of data condtonng s evaluated for the 5- and 8-pont algorthms wth random n-tuples n each dataset. he best soluton accordng to the cheralty test and Sampson dstance s selected. he devaton error of translaton and rotaton are measured n degrees. he ncluded angle between the orgnal and estmated translaton drecton gves an nterpretable result. For rotaton evaluaton three unt vectors to the three axs drectons e x, e y and e z are rotated usng the orgnal and the estmated rotaton matrx. he error value s averaged over the three ncludng angles of the resultng vectors: r (( ) ) = acos Re Re error 3 x, y, z (7) We count all translatons wth an error less than degrees and all rotatons wth an error less than degrees (see able ). where ( ) x represents the square of the vectors x-component. Fnally, d error should be mnmal for the correct soluton. In our approach we used a combnaton of these crtera: Frst the translaton s examned accordng to (5) and then the fve ponts are tested to le n front of both cameras. If there are multple solutons wth all ponts n front, the eppolar dstance of all avalable correspondences s evaluated. 4. EVALUAION OF HE ALGORIHMS Subsequently, we analyze all technques wth respect to ther behavor under Gaussan nose, the selecton strategy for multple solutons, over-determned estmaton and data condtonng. he evaluaton was performed usng synthetcally generated data wth ground truth. he camera moton between two vews s randomly chosen from a unform dstrbuton. o generate the random numbers, we use the advanced mersenne twster (Matsumoto & Nshmura, 998). he camera translaton s scaled to and the three rotaton angles are constraned between - and degrees. hen, spatal obect ponts are randomly generated n general poston and proected nto the mages usng the smulated cameras. If the known calbraton matrces are appled (), the normalzed coordnates range from - to. he mage coordnates are dsplaced wth Gaussan nose. he standard devaton σ corresponds to an mage wth 4 4 pxels and the maxmum Eucldean dsplacement s.4σ. We ensure that the selected pont correspondences are not collnear and avod degenerate confguratons of mnmal sets wth the constrants proposed by (Werner, 3). For allowed confguratons, the eppoles must le n domans wth pecewseconc boundares. An example for estmatng the relatve orentaton s shown n Fgure. o obtan statstcally sgnfcant results, every technque s examned tmes. he Fgure : Estmated relatve orentaton usng normalzed mage pars wth overlad eppolar rays (correct reference soluton n red) a) Mnmal 5-pont b) Over-determned 5-pont 9 5 c) Sampson Selecton d) Cheralty and Sampson Fgure : ranslaton error n degrees of the drect 5-pont solver (Nstér, 4) for runs. blue: σ =.7, yellow: σ =.5, red: σ =.9, green: σ =.3. 37

4 he Internatonal Archves of the Photogrammetry, Remote Sensng and Spatal Informaton Scences. Vol. XXXVII. Part B3b. Beng 8 Method Ground ruth Rotaton Errors σ =.7 σ =.5 σ =.9 σ =.3 cnt mean cnt mean cnt mean cnt mean.8.74 Evaluaton of Dfferent Algorthms Pont Pont Lnear Pzzaro Nstér Stewénus L Non-lnear Selecton Strategy for Multple Solutons Sampson S Cheralty C C-5 + S C-all + S Over-determned Soluton 8-Pont Nstér Data Condtonng 8-Uncond Cond Uncond Cond Method Ground ruth ranslaton Errors σ =.7 σ =.5 σ =.9 σ =.3 cnt mean cnt mean cnt mean cnt mean.3. Evaluaton of Dfferent Algorthms Pont Pont Lnear Pzzaro Nstér * Stewénus L Non-lnear Selecton Strategy for Multple Solutons Sampson S * Cheralty C C-5 + S * C-all + S Over-determned Soluton 8-Pont Nstér * Data Condtonng 8-Uncond Cond Uncond Cond able : Percentage of correct solutons and mean errors. (* ndcates method n Fgure ) 5. DISCUSSION OF HE SIMULAION RESULS In general, all algorthms show one common effect: he estmaton of camera rotaton s much more stable and accurate, than the estmate of camera translaton. o show the nfluence of nose, the camera pose s calculated from the obect ponts and nosy mage ponts va spatal resecton and compared wth the ground truth. Surprsngly, all drect 5-pont solvers produce exactly the same results up to nose of σ =.9. he methods (Nstér, 4) and (Stewenus et al., 6) produce exactly the same results n all tests. he supposable hgher accuracy of the second algorthm can not be verfed by evaluatng the frst fve sgnfcant dgts. he method of (L & Hartley, 6) has problems wth large nose and can not reach the qualty of the other two 5-pont technques at σ =.3. Opposed to (Batra et al., 7), the numercal nstablty of the egen-decomposton s not lmtng the fve-pont algorthm. Even worse, the non-lnear 5-pont technque suffers from fndng the soluton and a perfect startng value does not ensure convergence of the non-lnear optmzaton technque. hs can be seen by the low number of correct solutons, even for small amount of nose. Nevertheless, f a soluton was found, t s hghly accurate. Both 6-pont algorthms produce results not as good as the 5- pont technques, especally f nose ncreases. he method of (Pzzaro, 3) s a bt more relable, than the lnear one of (Phlp, 998). he 7- and 8-pont algorthms show a smlar behavor. If nose ncreases to realstc amounts, the results become even worse than the 6-pont algorthms. In general, the 5-pont algorthms outperform every other technque. he cheralty test alone s not suffcent to select a good soluton, because many estmates of essental matrces have all ponts n front of both cameras. he Sampson dstance of addtonal ponts s a better crteron, but wth ncreasng nose t has a hgher probablty to select a wrong soluton. Combnng the two crterons can be done n two ways: Frst, the cheralty s tested only for the 5 ponts used for the computaton. In ths case for every set of solutons at least one has all ponts n front, but some solutons can be gnored. hs makes the selecton very robust and needs only moderate computaton tme. Second, the cheralty test can be performed over all avalable ponts. hs lead to a bt more accurate results, but for larger nose the number of acceptable solutons decreases. In addton, the trangulaton of all ponts s computatonally ntensve. he best trade off between robustness, accuracy and computatonal effort s to compare the Sampson dstances of all ponts, whch have the 5 ponts n front of both cameras. We also nvestgated weather the algorthms can mprove the accuracy of the essental matrx, f more than the mnmal number of ponts s used. Surprsngly, both over-determned 5- and 8-pont algorthms decrease n accuracy. he comparson of results wth and wthout data condtonng shows that an addtonal condtonng of the already normalzed coordnates s not necessary for the computaton of the essental matrx. he average values are so close to each other, that almost no nfluence can be measured. 6. MULIPLE-CAMERA POSE ESIMAION As mentoned before, the man drawback of the 5-pont algorthms are the possble solutons. Selectng and detectng the rght soluton s not trval, especally n the presence of nose. Snce cheralty tests and eppolar dstance flterng are nstable, addtonal constrants must be mposed. he stuaton becomes easer, f two or more cameras move n a fxed relaton to each other, e.g. f they are mounted on the same vehcle. her moton s not ndependent of each other (see Fgure 3). hs fxed relatonshp can be used to select the rght soluton par of the two soluton sets. 38

5 he Internatonal Archves of the Photogrammetry, Remote Sensng and Spatal Informaton Scences. Vol. XXXVII. Part B3b. Beng 8 As the rotaton s much more stable than the translaton (see secton 4 and 5), some weghtng factors w trans and w rot should be used to balance the cost functon, e.g. w trans = 5w rot. e = w r + w r (4) trans rot error Fgure 3: Constraned multple-camera moton We denote a camera at tme wth P and assume that the reference camera P s located at orgn. As shown n Fgure 8, the proecton matrces P and P have a fxed relatve orentaton defned by Δ. If the cameras move to the postons of P and P, the essental matrces for both motons and are determned. Inspred by (Esquvel et al., 7), the followng constrant s mposed: Δ =Δ (8) Each transformaton can be separated nto a translaton vector C and a rotaton matrx R. herefore, a correct par of solutons must also fulfll: R Δ R =ΔR R λ = Δ 3 3 Δ 3 ΔC3 (9) Unfortunately, the essental matrx contans the translatonal component only up to scale and therefore, the rato of the moton lengths and C s unknown: C R Δ C + μc = λδ R C +ΔC () o estmate the relatve scale factors μ and λ, the lnear equaton system μ C Δ R C = ( Δ Δ ) λ C R C, () of the form A x = b can be used to solve for the unknown n x. An extenson for multple cameras s straght forward: μ C ΔR C ΔC R ΔC C R C κ C R () he mssng constrant between P and P 3 requres one camera as reference at orgn. Nevertheless, we recommend ths extra coordnate transformaton, because the thrd constrant mproves sgnfcantly the estmaton of the translaton factors. he equaton systems () and () are over-determned, snce every camera ntroduces one unknown scale factor and each par of cameras ntroduces three constrants. herefore, the resdual errors r = Ax b (3) are used as qualty measure for the pose estmaton. We combned the condtons (9) and () nto a cost functon, whch s computed for every possble par of solutons. If temporally tracked ponts n the camera pars ( P, P ) and ( P, P ) are addtonally matched between the cameras, the mssng scale factors μ and λ can be computed accordng to the fxed camera pose Δ. he trangulaton of obect ponts X usng the spatal par ( P, P ) defnes a reference scale. Now the trangulated ponts X and X derved from the temporal pars ( P, P ) and ( P, P ) respectvely can be scaled to the reference ponts X. he dstance between the 3Dcoordnates and the proecton centers gves the relaton of the scale factors used n Δ, and. In case of napproprate moton the trangulaton of temporally tracked ponts s less accurate than spatally matched ponts on the camera rg. herefore, the average of the nearest fve obect ponts for each trangulaton par s used to compute the scale factors μ and λ:. X μ =, =. X X X 5 5 λ = = 7. EXPERIMENS WIH MULIPLE CAMERAS (5) he multple-camera setup s tested on a real data sequence. he sequence conssts of three ndependent camera streams, whch are mounted on a calbrated rg. Every frame has at least tracked features. he essental matrces are computed wth a RANSAC technque usng the mnmal 5-pont solver. o ensure a certan moton of the cameras, frames wth an average trackng dsparty below pxels are omtted. he camera paths are reconstructed fully automatcally. o compare the robustness of the proposed mult-camera technque, paths of the sngle track of the reference camera and the lnked track are shown n Fgures 4 and 5. he path n Fgure 4 suffers from a mscalculaton n the mddle of the track, whch results n orthogonal camera placement. After ths dscontnuty the scale s wrong, because the scale factor s calculated on the last camera par. In Fgure 5 the path s smooth and correctly scaled. he gaps n the path ndcate skpped mages wth nsuffcent camera moton. he whole mult-camera path conssts of 9 frames. he reference poston was manually set and the extracted path s shown as a red lne n Fgure 6. Please note, that the camera path has not been optmzed by bundle adustment or any further reference ponts. Fgure 4: Reconstructed sngle-camera path over frames 39

6 he Internatonal Archves of the Photogrammetry, Remote Sensng and Spatal Informaton Scences. Vol. XXXVII. Part B3b. Beng 8 Fgure 5: Reconstructed mult-camera path over frames Fgure 6: Reconstructed mult-camera path over 9 frames. 8. CONCLUSIONS In ths paper we have shown that robust camera pose estmaton on real data s stll a dffcult task. In general, the estmaton of camera rotaton s more relable than the translaton. he mnmal 5-pont solvers produce better results than all other methods, especally n presence of nose. In case of multple solutons, the best selecton crteron s a combnaton of a precedng cheralty test wth mnmal ponts followed by the computaton of the Sampson dstance over all avalable ponts. Furthermore, usng over-determned varants of the mnmal solver not necessarly ncrease the accuracy of the essental matrx. Instead, the result should geometrcally be mproved wth standard technques lke bundle adustment. Fnally, data condtonng for the computaton of the essental matrx s not necessary. However, the essental matrx computaton produces wrong estmates from tme to tme. If a camera path over several hundred frames needs to be reconstructed, one mscalculaton corrupts the whole path. We have shown that addtonal mult-camera constrants can be mposed to gves stable results for extensve camera path reconstructons. ACKNOWLEDGEMENS hs work was partally supported by grants from the German Research Foundaton DFG. REFERENCES Batra, D., Nabbe, B. and Hebert, M., 7. An alternatve formulaton for fve pont relatve pose problem, IEEE Workshop on Moton and Vdeo Computng WMVC '7, 6 p. Demazure, M., 988. Sur deux problemes de reconstructon, echncal Report No 88, INRIA, Rocquencourt, France. Esquvel, S., Woelk, F., Koch, R., 7. Calbraton of a multcamera rg from non-overlappng vews, DAGM Symposum, LNCS 473, Hedelberg, pp Hartley, R., 997. In defense of the eght-pont algorthm, IEEE rans. on Pattern Analyss and Machne Intellgence, vol. 9, no. 6, pp Hartley, R. and Zsserman, A., 4. Multple vew geometry n computer vson, Cambrdge Unversty Press,. edton, 67 p. L, H.D. and Hartley, R.I., 6. Fve-pont moton estmaton made easy, Int. Conf. on Pattern Recognton, vol., pp Longuet-Hggns, H.C., 98. A computer algorthm for reconstructng a scene from two proectons, Nature, vol. 93, pp Matsumoto, M. and Nshmura,., 998. Mersenne twster: a 63-dmensonally equdstrbuted unform pseudo-random number generator, ACM ransactons on Modelng and Computer Smulaton, vol. 8, no., pp 3-. McGlone, J.C., Mkhal, E.M., Bethel, J., Mullen, R. (Eds.), 4. Manual of Photogrammetry, 5 th edton, Amercan Socety of Photogrammetry and Remote Sensng. Nster, D., 4. An effcent soluton to the fve-pont relatve pose problem, IEEE Conf. on Computer Vson and Pattern Recognton, vol., pp Phlp, J., 996. A non-teratve algorthm for determnng all essental matrces correspondng to fve pont pars, Photogrammetrc Record, vol. 5(88), pp Phlp, J., 998. Crtcal pont confguratons of the 5-, 6-, 7-, and 8-pont algorthms for relatve orentaton, echncal Report RIA-MA-998-MA-3, Dept. of Mathematcs, Royal Inst. of ech., Stockholm. Pzarro, O., Eustce, R., Sngh, H., 3. Relatve pose estmaton for nstrumented, calbrated platforms, 7 th Dgtal Image Computng: echnques and Applcatons. Šegvć, S., Schweghofer, G. and Pnz A., 7. Influence of numercal condtonng on the accuracy of relatve orentaton, IEEE Conf. on Computer Vson and Pattern Recognton, 7, 8 p. Stewénus, H., 4. Matlab code for solvng the fvepont problem, (accessed 8. Nov. 7). Stewénus, H., Engels, C. and Nstér, D., 6. Recent developments on drect relatve orentaton, ISPRS Journal of Photogrammetry and Remote Sensng, vol., no. 4, June 6, pp Wang, W. and su, H.,. An SVD decomposton of the essental matrx wth eght solutons for the relatve postons of two perspectve cameras, Int. Conf. on Pattern Recognton, vol., pp Weng, J.Y., Huang,.S. and Ahua, N., 989. Moton and structure from two perspectve vews: algorthms, error analyss, and error estmaton, rans. on Pattern Analyss and Machne Intellgence, vol., no. 5, pp Werner,., 3. Constrant on fve ponts n two mages, IEEE Conf. on Computer Vson and Pattern Recognton, vol. II, pp