Paracatadioptric Camera Calibration Using Lines

Transcription

1 Prctdioptric Cmer Clibrtion Using Lines Joo P Brreto, Helder Arujo Institute of Systems nd Robotics - Dept of Electricl nd Computer Engineering University of Coimbr, 33 Coimbr, Portugl Abstrct Prctdioptric sensors combine prbolic shped mirror nd cmer inducing n orthogrphic projection Such configurtion provides wide field of view while keeping single effective viewpoint Previous work in centrl ctdioptric sensors proved tht line projects into conic curve nd tht three line imges re enough to clibrte the system However the estimtion of the conic curves where lines re mpped is hrd to ccomplish In generl only smll rc of the conic is visible in the imge nd conventionl conic fitting techniques re unble to correctly estimte the curve The present work shows tht set of conic curves corresponds to prctdioptric line imges if, nd only if, certin properties re verified These properties re used to constrint the serch spce nd correctly estimte the curves The ccurte estimtion of minimum of three line imges llows the complete clibrtion of the prctdioptric cmer If the cmer is skewless nd the spect rtio is known then the conic fitting problem is solved nturlly by n eigensystem For the generl sitution the conic curves re estimted using non-liner optimiztion 1 Introduction Omnidirectionl vision is becoming n incresingly importnt sub-re in computer vision reserch The pproch of combining mirrors with conventionl cmers to enhnce sensor field of view is referred to s ctdioptric imge formtion Ctdioptric sensors with n unique effective viewpoint re of primry interest The entire clss of ctdioptric configurtions verifying the fixed viewpoint constrint is derived in [1] Pnormic centrl ctdioptric systems cn be built by combining n hyperbolic mirror with perspective cmer nd, prbolic mirror with n orthogrphic cmer (prctdioptric sensor) The construction of the former requires creful lignment between the mirror nd the imging device The cmer projection center must be positioned in the outer focus of the hyperbolic reflective surfce The prctdioptric cmer is esier to construct being brodly used in vision pplictions In [], Geyer nd Dniilidis introduce for the first time n unifying theory for generl centrl ctdioptric imge formtion A modified version of this mpping model is proposed in [3] It is shown tht centrl ctdioptric projection is isomorphic to projective mpping from sphere, centered in the effective viewpoint, to plne with projection center on the perpendiculr to the plne For the prticulr cse of prctdioptric sensors the projection center lies on the sphere nd the projective mpping is stereogrphic projection The plne nd the finl ctdioptric imge re relted by collinetion depending on the mirror nd cmer intrinsic prmeters The system is clibrted when this collinetion is known It hs lredy been proved tht ny centrl pnormic system cn be fully clibrted from the imge of three lines in generl position [3] However, since lines re mpped into conic curves which re only prtilly visible, the ccurte estimtion of ctdioptric line imges is fr from being trivil tsk [7] The present pper focuses on prctdioptric cmer clibrtion using lines in generl position If it is true tht ny line mps into conic in the ctdioptric imge plne, it is not true tht ny conic is the imge of line We derive for the first time the necessry nd sufficient conditions tht must be verified by set of conic curves to be the prctdioptric projection of lines We lso show tht the derived conditions cn be used to ccurtely estimte the line imges by non-liner optimiztion Moreover if the system is skewless nd the spect rtio is known then the lines cn be computed by solving n eigensystem Given the imge of t lest three lines the prctdioptric cmer is esily clibrted using the results presented in [3] Previous Work The imge formtion model for prctdioptric systems nd the clibrtion lgorithm using three or more lines re introduced For further detils plese consult [3] Consider prctdioptric system combining prbolic mirror with ltus rectum h, nd n orthogrphic cmer The principl xis of the cmer must be ligned with the symmetry xis of the prboloid The prctdioptric projection cn be modeled by stereogrphic projection from n unitry sphere, centered in the effective /3 $17 3 IEEE

2 Ω^ Ο^ 11 H c ^ x 1 Prctdioptric Imge Plne Step 1 Step Determine the ctdioptric line imges ˆΩ i for i =1,, 3 K For ech pir of conics ˆΩ i, ˆΩ j, compute the intersection points ˆF ij, ˆB ij nd determine the cor- Ξ world point x Ω x P n X Y O Z O 11 1 Z c Xc 11 1 O Yc c Π responding line ˆµ ij = ˆF ij ˆB ij Step 3 Estimte the imge center Ô which is the intersection point of lines ˆµ ij Step 4 For ech conic ˆΩ i compute the polr line ˆπ i of the imge center Ô (i =1,, 3 K) Step For ech conic curve obtin the points Î i nd Ĵ i where line ˆπ i intersects ˆΩ i (i =1,, 3 K) Step 6 Estimte the conic ˆΩ going through points Î i, Ĵ i (i =1,, 3 K) Step 7 Compute the Cholesky decomposition of ˆΩ to derive mtrix H c Figure 1: Model for prctdioptric imge formtion Tble 1: Clibrting prctdioptric system using K lines viewpoint, into plne Ξ s shown in Fig 1 The world point shown in Fig 1 is imged t point ˆx in the prctdioptric imge plne To ech visible scene point corresponds n oriented projective ry x =(x, y, z) t, joining the 3D point with the projection center O The projective ry intersects the unit sphere in single point P Consider point O c, with coordintes (,, 1) t, which lies in the unitry sphere To ech x corresponds n oriented projective ry x joining O c with the intersection point P The non-liner mpping F (eqution 1) corresponds to projecting the scene in the unity sphere surfce nd then re-projecting the points on the sphere into plne Ξ from novel projection center O c Points in ctdioptric imge plne ˆx re obtined fter collinetion H c of D projective points x Eqution shows tht H c depends on the intrinsic prmeters K c of the orthogrphic cmer, nd on the ltus rectum of the prbolic mirror F(x) =(x, y, z + x + y + z ) t (1) h ˆx = K c h x () 1 H c Consider the plne Π =(n, ) t going through the effective viewpoint O s depicted in Fig 1 (n =(n x,n y,n z ) t ) The prctdioptric imge of ny line lying on Π is the conic curve ˆΩ The line in the scene is projected into gret circle in the sphere surfce This gret circle is the curve of intersection of plne Π, contining both the line nd the projection center O, nd the unit sphere The projective rys x, joining O c to points in the gret circle, form centrl cone surfce The centrl cone, with vertex in O c, projects into the conic Ω in plne Ξ (eqution 3) Since the imge plne nd Ξ re relted by collinetion H c,the result of eqution 4 comes in strightforwrd mnner Notice tht conic Ω degenerte to line whenever n z = Ω = ˆΩ = n z n x n z n z n y n z n x n z n y n z n z b d b c e d e f = H t 1 c ΩH c (3) The ctdioptric system is clibrted whenever the collinetion H c is known Assume tht the imge center is C =(c x,c y ) t, nd tht α, f o nd s k re the spect rtio, the focl length nd the skew of the orthogrphic imging device The projective trnsformtion H c is given in eqution where f c = f o h/ is mesurement in pixels of the combined focl length of the cmer nd the mirror H c = αf c s k c x α 1 f c c y 1 (4) () Any centrl ctdioptric system cn be clibrted from the imge of K lines with K 3 Tble 1 summrizes the steps of the clibrtion method proposed in [3] The first step is to estimte the conic curves ˆΩ i corresponding to the prctdioptric projection of K lines in the scene Any two line imges ˆΩ i, ˆΩ j intersect in two rel points ˆB ij, ˆF ij It cn be shown tht the imge center Ô must lie in the line ˆµ ij going through the two intersection points Consider the originl line in the scene nd the plne Π i going through the effective viewpoint nd contining the imged line (see Fig /3 $17 3 IEEE

3 1) If the line is projected in the conic ˆΩ i, then the polr line ˆπ i of the center Ô is the imge of the vnishing line of Π i The polr line ˆπ i intersects the conic curve ˆΩ i in two points Î i, Ĵ i It cn be proved tht these two points lie on the conic ˆΩ, which is the locus where the bsolute conic is mpped by collinetion H c Conic ˆΩ cn be estimted using the K pirs of points Î i, Ĵ i Since H c is n upper tringulr mtrix nd ˆΩ = H c t H c 1, then H c cn be determined from the Cholesky decomposition of ˆΩ The clibrtion of the prctdioptric system is strightforwrd whenever the conic curves corresponding to the line imges re known However the estimtion of the these conics using imge points is hrd to ccomplish There re severl lgorithms to fit conic curve to dt points [7] A robust conic fitting lgorithm hs to cope with noisy dt points, bising due to curvture nd prtil occlusions The occlusion problem is of prticulr importnce for our purposes By occlusion we men tht the vilble dt points lie on smll rc of the curve It is intuitive tht in this circumstnces, even for smll mounts of noise, it is very hrd to obtin the correct conic curve The present work ims to to cope with this problem using the properties of prctdioptric line projection 3 The Line Imge ˆω In generl conic curve hs DOF nd it cn be represented by symmetric 3 3 mtrix ˆΩ in P (eqution 4), or by point ˆω =(, b, c, d, e, f) t in P From eqution 4, replcing Ω, H c by the result of equtions 3, nd ssuming Υ=αs k c y f c c x, yields b c d e f = αn yn z f c n z αcynynz f c n z fc n xn z αf c n z α f c n z s k αf 3 c ( s k f c + α ) n z Υ α f 3 c + α n z cy s kn xn z fc α3 c y n z nxnzυ αfc + s kn z Υ αf 4 c n z Υ α f 4 c The prctdioptric imge of line depends on the intrinsic prmeters of the system nd on the orienttion of the 3D plne Π (Fig 1) After some lgebric mnipultion the previous result cn be rewritten in the form of eqution 6 If the clibrtion is known then the conic curve ˆω is only described by prmeters, d nd e These three prmeters encode the scle informtion nd the orienttion of plne Π contining the imged line Considering tht conic ˆω hs DOF, we my conclude tht 3 DOF depend on the prbolic system prmeters, nd the remining DOF re relted with the line tht is projected ˆω = b c d e f = ( α s k f c αs k f c + α 4 ) d e α fc c x d c y e 4 Prctdioptric Line Estimtion Assume tht we hve prctdioptric imge of K lines in spce Ech line is projected in conic curve ˆω i prmeterized hs point in P (eqution 7) The gol is to correctly estimte the set of conic curves knowing neither the system clibrtion nor the position of the lines in the scene (6) ˆω i =( i,b i,c i,d i,e i,f i ) t, i=1,,3 K (7) 41 Estimtion of ˆω i from Imge Points Consider the imge points ˆx i j = (ˆx j, ŷ j ) t with j = 1, M i nd M i, lying on conic ˆω i Neglecting the effects of noise it is strightforwrd tht A iˆω i must be null (eqution 8) ˆx 1 ˆx 1 ŷ 1 ŷ 1 ˆx 1 ŷ 1 1 ˆx ˆx ŷ ŷ ˆx ŷ 1 ˆω i = } ˆx M i ˆx Mi ŷ Mi ŷm {{ i ˆx Mi ŷ Mi 1 } A i (8) Eqution 9 provides the design mtrix A for the entire set of conic curves p =(ˆω 1, ˆω ˆω K ) t A 1 A A 3 } {{ A K } A ˆω 1 ˆω ˆω 3 ˆω K } {{ } p = (9) The equlity of eqution 9 is only verified in idel circumstnces In generl the dt points re corrupted with noise nd Ap One wy to estimte the set of conic curves is to find the solution p tht minimizes the lgebric distnce φ (eqution 1) under the constrint p t p =1 The minimizer is the normlized eigenvector of A t A corresponding to the smllest eigenvlue φ(p) =p t A t Ap (1) /3 $17 3 IEEE

4 As referred bove the problem is tht in generl the conic curves corresponding to the prctdioptric projection of lines re strongly occluded in the imge We re only ble to obtin points lying on smll rc of the curve There re severl other conic fitting lgorithms which minimizes other distnces thn the lgebric one [7] However none of them works properly under these circumstnces since the dt points do not provide enough informtion to correctly estimte the conics This pper proposes to solve the estimtion problem by constrining the serch spce using the properties of prctdioptric line projection 4 Constrints Verified by Set of Prctdioptric Line Imges Assume K lines in the scene tht re projected into K conic curves in the prctdioptric imge plne (K 3) These conic curves cn be represented by points of P s shown in eqution 7 From the result of eqution 6 yields b 1 = b = b 3 = = b K = αs k 1 3 K f c c 1 = c = c 3 = = c K = α s k 1 3 K fc + α 4 From the first expression rises tht η i = for i =, 3 K with η i provided by eqution 11 Moreover, using the second expression in similr mnner, comes tht χ i =for i =, 3 K where χ i is given by eqution 1 η i = 1 b i i b 1, i=,3 K (11) χ i = 1 c i i c 1, i=,3 K (1) From eqution 6 rises tht ech line imge ˆω i must verify α f c i + c x d i + c y e i + f i = Consider the conic curves ˆω 1, ˆω nd ˆω 3, which re the first three elements of the set of line imges α f c, c x nd c y cn be determined s follows α f c c x c y = 1 d 1 e 1 d e 3 d 3 e 3 } {{ } Φ 1 f 1 f f 3 Γ If K>3 then ech conic curve ˆω i with i =4Kmust verify the constrint ν i =(eqution 13) ν i = [ i d i e i f i ] [ Φ 1 Γ 1 ], i=4 K (13) It is cler tht if set of K conic curves corresponds to the prctdioptric projection of K lines, then η i, χ i nd ν i, provided in equtions 11, 1 nd 13, must be equl to zero We hve derived 3K independent conditions which re necessry for the conic curves to be prctdioptric line imges However it hs not been proved tht these conditions re lso sufficient By sufficient we men tht, if certin set of conic curves verify these conditions then it cn be the prctdioptric projection of set of lines Consider the unclibrted imge of K lines tht re mpped in the sme number of conics Since ech conic hs DOF then set of K conics hs totl of K DOF Ech line introduces unknowns (DOF), which correspond to the orienttion of the ssocited plne Π (see Fig 1) Moreover the prmeters of mtrix H c re lso unknown (eqution ) Thus there re totl of K +unknowns (DOF) Since K >K +then it is obvious tht there re sets of conic curves tht cn never be the prctdioptric projection of lines The conics tht cn correspond to the imge of the lines lie in subspce of dimension K + This mens tht there re 3K independent constrints, which proves the sufficiency of the conditions derived bove 43 Estimtion of Line Imges for Clibrtion Purposes Section 41 shows how to estimte the set of K prctdioptric line imges using imge points The method consists in finding solution p for the conic curves which minimizes the lgebric distnce to the dt (eqution 1) Since the curves pper strongly occluded in the imge plne, the estimtion results re in generl very inccurte Section 4 shows tht set of K conic curves corresponds to the prctdioptric projection of K lines, if nd only if, it verifies the constrints provided by equtions 11, 1 nd 13 Our pproch consists in using the necessry nd sufficient conditions derived bove to constrint s much s possible the serch spce 431 Generl Cse Assume the imge of K lines cquired by n unclibrted prctdioptric cmer Nothing is known bout the prmeters of mtrix H c The skew cn be non null nd the spect rtio different from one Function φ provides the lgebric distnce between the set of conic curves p nd the dt points (eqution 1) We im to minimize of the lgebric distnce under the constrints η i =, χ i =nd ν i =(equtions 11, 1 nd 13) One wy to chieve this gol is to find the solution p which minimizes the function ɛ provided in eqution 14 K K K ɛ(p) =φ(p)+λ( ηi + χ i + νi ) (14) i= i= i= /3 $17 3 IEEE

5 The constrints re introduced s penlty terms weighted by prmeter λ The minimiztion of the function ɛ cn be stted s nonliner lest squres problem The solution cn be found using Guss-Newton or Levenberg-Mrqurdt lgorithms Notice tht the Jcobin mtrix cn be explicitly derived 43 Skewless Imges with Known Aspect Rtio Assume tht the orthogrphic cmer is skewless nd tht the spect rtio α is known Replcing s k by in eqution 6 yields b =nd c = α 4 The constrints η i =nd χ i =for i =K, become b i =nd c i α 4 i = for i =1K Notice tht there re two dditionl constrints becuse now two of the clibrtion prmeters re known The new function ɛ is given by eqution 1 K K K ɛ(p) =φ(p)+λ( b i + (c i α 4 i ) + νi ) (1) i=1 i=1 i=4 Mking b i =nd c i = α 4 i in eqution 8 yields ˆx 1 + α 4 ŷ1 ˆx 1 ŷ 1 1 ˆx + α 4 ŷ ˆx ŷ 1 ˆx M i + α 4 ŷm i ˆx Mi ŷ Mi 1 à i i d i e i f i ω i = A novel design mtrix à cn be obtined by replcing A i by à i for i = 1K in eqution 9 Consider φ( p) = p t à t à p with p =( ω 1 t ω K t )t The minimum of function φ( p) is the set of conic curves which minimizes the lgebric distnce to the dt points nd verifies the constrints b i =nd c i α 4 i = Thus we cn use the objective function ɛ insted of the one of eqution 1 ɛ( p) = p t à t à p + φ( p) [ λ K νi i=4 ] (16) In generl the minimiztion of function ɛ is still nonliner lest squres problem However if K =3then the second term of eqution 16 disppers nd the problem becomes closed from The solution is given by the eigenvector corresponding to the smllest eigenvlue of à t ÃEven when K > 3 the eigenvector solution is in generl quite ccurte In this cse the conditions of eqution 13 re neglected nd the serch spce is not fully constrined Nevertheless it is constrined enough to provide good results Performnce Evlution Other uthors hve lredy proposed lgorithms to clibrte prctdioptric cmer [4, 6, ] The pproch presented in [6] requires sequence of prctdioptric imges The system is clibrted using the consistency of pirwise trcked point fetures cross the sequence, bsed on the chrcteristics of ctdioptric imging In [], the center nd focl length re determined by fitting circle to the imge of the mirror boundry The method is simple nd cn be esily utomted, however it is not very ccurte nd requires the visibility of the mirror boundry Its mjor drwbck is tht it is only pplicble for the sitution of skewless cmer with unitry spect rtio Geyer nd Dniilidis propose clibrtion lgorithm using line imges [4] They present closed-form solution for focl length, imge center, nd spect rtio for skewless cmers, nd polynomil root solution in the presence of skew The line imges re estimted tking into ccount the properties of prbolic projections Nevertheless the conic curves verifying those properties re not necessrily the prctdioptric projection of lines In this section we use simulted imges to evlute the performnce of our lgorithm nd compre it with the one proposed by Geyer nd Dniilidis 1 Simultion Scheme Assume prctdioptric cmer with field of view (FOV) of 18 nd pre-defined intrinsic prmeters The imge of set of K lines is generted s follows To ech line in the scene corresponds plne Π with norml n (Fig 1) The K normls re unitry nd rndomly chosen from n uniform distribution in the sphere Ech norml defines plne tht intersects the unit sphere in gret circle Notice tht hlf of the gret circle is within the cmer field of view An ngle θ, less or equl to the FOV, is chosen to be the mplitude of the rc tht is ctully visible in the prctdioptric imge The rc is rndomly nd uniformly positioned long the prt of the gret circle which is within the FOV The visible rc is uniformly smpled by fixed number N of smple points The ech smple point corresponds projective ry x The smple rys re projected using formul 1 nd trnsformed using with the chosen intrinsic prmeters Two dimensionl gussin noise with zero men nd stndrd devition σ is dded to ech imge point ˆx As finl remrk notice tht the mplitude of the visible rc is mesured in the gret circle where plne Π intersects the sphere, nd not in the conic curve where the line is projected In generl the visible ngle of the prctdioptric line imge is much less thn θ /3 $17 3 IEEE

6 4 N = 3 ; Θ = 17 N = 8 ; Θ = 17 N = 4 ; Θ = 9 N = ; Θ = Lines Lines 7 Lines 9 Lines 4 Focl Length RMS Error (pixel) 3 3 Skew MEDIAN Error (pixel) Noise Stndrd Devition (pixel) 11 N = 3 ; Θ = 17 1 N = 8 ; Θ = 17 N = 4 ; Θ = 9 9 N = ; Θ = Noise Stndrd Devition (pixel) 3 3 Lines Lines 7 Lines 9 Lines Center RMS Error (pixel) Focl Length MEDIAN Error (pixel) Noise Stndrd Devition (pixel) Noise Stndrd Devition (pixel) Figure : Clibrtion of skewless system with known spect rtio using three line imges Figure 3: Clibrtion results for skew nd focl length using 3,, 7 nd 9 lines Clibrtion of Skewless Cmer with Known Aspect Rtio Consider prbolic cmer such tht the skew is (s k = ), the spect rtio is 11 (α = 1), nd both re ssumed to be known We wish to determine the focl length (f c = 4) nd the imge center ((c x,c y ) = (33, 38)) using the imge of three lines (K =3) The line imges re estimted using the closed-form solution proposed t the end of section 43 nd the system is clibrted using the lgorithm presented in Tb 1 The dt points re rtificilly generted using the simultion scheme explined bove The estimted clibrtion prmeters re compred with the ground truth nd the RMS error is computed over 1 runs of ech experiment Fig shows the results for different choices of θ (mplitude of the visible rc) nd N (number of smple points) For ech sitution the stndrd devition of the dditive gussin noise vries between nd 6 pixels by increments of pixels For θ = 17 the lgorithms presents n excellent performnce The decrese on the number of smple points from 3 to 8 only slightly ffects the ro- bustness to noise Since we re only using three lines, the decrese on the mplitude of the visible rc θ nd on the number of points N hs strong impct on the performnce Even so the clibrtion using rcs of 9 is still prcticble The sitution of θ =7 nd N =is very extreme leding to bd estimtion of the intrinsic prmeters The performnce of method proposed in [4] is evluted using similr simultion conditions A direct comprison cn be mde between the results presented in here nd the ones presented on their pper In generl terms they estimte the conic curves by exploiting the fct tht the imge center must lie in the line going through the intersection points of ny two line imges As discussed in section 4, this condition is necessry, but not sufficient, for set of conic curves to be the prctdioptric projection of lines Since the serch spce is not fully constrined, they need much more thn three line imges to clibrte the sensor The results presented in Fig re obtined using the minimum theoreticl number of lines for clibrtion [3] Even so, nd s fr s we re ble to judge from the results presented in [4], the performnce of our pproch seems to be significntly better /3 $17 3 IEEE

7 3 Lines 4 Lines Lines 6 Lines α men std f c men std s k men std c x men std c y men std Figure 4: Line imges estimtion for clibrtion Tble : Clibrting prctdioptric system using K lines 3 Generl Clibrtion Consider tht nothing is known bout the clibrtion prmeters We wish to determine the spect rtio, skew, focl length nd imge center using line imges The mplitude of the visible rc nd the number of smple points re respectively θ = 14 nd N = 14 (section 1) The present experiment compres the performnce of the clibrtion lgorithm for 3,, 7 nd 9 line imges The set of line imges is estimted by minimizing the function ɛ provided by eqution 14 The solution is found using n itertive grdient descent method The strting point is the minimizer of the lgebric distnce to the dt points (eqution 1) The clibrtion prmeters re computed (Tb 1), the results re compred with the ground truth nd the medin error is computed over 1 runs The performnce of the clibrtion lgorithm cn be observed in Fig 3 As expected the increse in the number of lines improves the robustness of the clibrtion 6 Experiments with Rel Imges Five imges re tken using prctdioptric cmer with resolution nd the FOV 18 Fig 4 depicts one of the clibrtion imges For ech frme set of points lying on 6 different line imges is selected The points re used to estimte the conic loci where the lines re imged Since nothing is known bout the sensor clibrtion, the estimtion of the set of line imges must be performed by minimizing function ɛ (eqution 14) The sensor is clibrted using 3, 4, nd 6 lines Tb presents the men nd the stndrd devition of the clibrtion results obtined with ech imge Notice tht the estimted vlues for the clibrtion prmeters re more or less the sme for the different K (number of lines) The stndrd devition cts s mesure of confidence If the results obtined for ech imge re very different, the vrince is high nd the chieved clibrtion is not trustble As expected the stndrd devition decreses when the number of lines increses To evlute the correctness of the clibrtion we performed the perspective rectifiction of prctdioptric imge of six pirs of prllel lines The lines were estimted in the rectified imge using norml lest squres The ngle between ech two directions ws determined using the corresponding vnishing points nd the imge of the bsolute conic (which is known since the perspective is rtificilly generted) The ngles between ech two directions were computed nd the results were compred with the ngles mesured in the scene The men of the error ws 1 nd the stndrd devition ws 3 References [1] S Bker nd S Nyr, A theory of ctdioptric imge formtion, in IEEE Interntionl Conference on Computer Vision, Bomby, 1998 [] C Geyer nd K Dniilidis, An unifying theory for centrl pnormic systems nd prticl implictions, in Europen Conference on Computer Vision, Dublin, [3] Joo P Brreto nd Helder Arujo, Geometric properties of centrl ctdioptric line imges, in Europen Conference on Computer Vision, Copenhgen, My [4] C Geyer nd K Dniilidis, Prctdioptric cmer clibrtion, IEEE Trns on Pttern Anlysis nd Mchine Intelligence, vol 4, no 4, April [] Y Ygi, S Kwto nd S Tsuji, Rel-time Omnidirectionl Vision Sensor(copis) for Vision Guided Nvigtion, IEEE Trns on Robotics nd Automtion, vol 1, pp 11-, 1994 [6] S B Kng, Ctdioptric self clibrtion, in IEEE Int Conf on Computer Vision nd Pttern Recognition, USA, [7] A Fitzgibbon nd R Fisher, A buyer s guide to conic fitting, in British Mchine Vision Conference, Birminghm, /3 $17 3 IEEE