A Toolbox for Easily Calibrating Omnidirectional Cameras

Transcription

1 A oolbox for Easly Calbratng Omndretonal Cameras Davde Saramuzza, Agostno Martnell, Roland Segwart Autonomous Systems ab Swss Federal Insttute of ehnology Zurh EH) CH-89, Zurh, Swtzerland {davdesaramuzza, Abstrat - In ths paper, we present a novel tehnque for albratng entral omndretonal ameras he proposed proedure s very fast and ompletely automat, as the user s only asked to ollet a few mages of a heker board, and lk on ts orner ponts In ontrast wth prevous approahes, ths tehnque does not use any spef model of the omndretonal sensor It only assumes that the magng funton an be desrbed by a aylor seres expanson whose oeffents are estmated by solvng a four-step least-squares lnear mnmzaton problem, followed by a non-lnear refnement based on the maxmum lkelhood rteron o valdate the proposed tehnque, and evaluate ts performane, we apply the albraton on both smulated and real data Moreover, we show the albraton auray by proetng the olor nformaton of a albrated amera on real 3D ponts extrated by a 3D sk laser range fnder Fnally, we provde a oolbox whh mplements the proposed albraton proedure Index erms atadoptr, omndretonal, amera, albraton, toolbox I IRODUCIO An omndretonal amera s a vson system provdng a 36 panoram vew of the sene Suh an enhaned feld of vew an be aheved by ether usng atadoptr systems, whh opportunely ombne mrrors and onventonal ameras, or employng purely doptr fsh-eye lenses [] Omndretonal ameras an be lassfed nto two lasses, entral and non-entral, dependng on whether they satsfy the sngle effetve vewpont property or not [] As shown n [], entral atadoptr systems an be bult by ombnng an orthograph amera wth a parabol mrror, or a perspetve amera wth a hyperbol or ellptal mrror Conversely, panoram ameras usng fsh-eye lenses annot n general be onsdered as entral systems, but the sngle vewpont property holds approxmately true for some amera models [8] In ths paper, we fous on albraton of entral omndretonal ameras, both doptr and atadoptr After desrbng our novel proedure, we provde a pratal Matlab oolbox [4], whh allows the user to qukly estmate the ntrns model of the sensor n a very pratal way II REAED WOR Prevous works on omndretonal amera albraton an be lassfed nto two dfferent ategores he frst one nludes methods whh explot pror knowledge about the sene, suh as the presene of albraton patterns [3, 4] or plumb lnes [5] he seond group overs tehnques that do not use ths knowledge hs nludes albraton methods from pure rotaton [4] or planar moton of the amera [6], and self-albraton proedures, whh are performed from pont orrespondenes and eppolar onstrant through mnmzng an obetve funton [7, 8, 9, ] All mentoned tehnques allow obtanng aurate albraton results, but prmarly fous on partular sensor types eg hyperbol and parabol mrrors or fsh-eye lenses) Moreover, some of them requre speal settng of the sene and ad-ho equpment [4, 6] In the last years, novel albraton tehnques have been developed, whh apply to any knd of entral omndretonal ameras For nstane, n [], the authors extend the geometr dstorton model and the self-albraton proedure desrbed n [8], nludng mrrors, fsh-eye lenses and non-entral ameras In [5, 7, 8], the authors desrbe a method for entral atadoptr ameras usng geometr nvarants hey show that any entral atadoptr system an be fully albrated from an mage of three or more lnes In [6], the authors present a unfed magng model for fsheye and atadoptr ameras Fnally, n [9], they present a general magng model whh enompasses most proeton models used n omputer vson and photogrammetry, and ntrodue theory and algorthms for a gener albraton onept In ths work, we also fous on albraton of any knd of entral omndretonal ameras, but we want to provde a tehnque, whh s very pratal and easy to apply he result of ths work s a Matlab oolbox, whh requres a mnmum user nteraton In our work, we use a heker board as a albraton pattern, whh s shown at dfferent unknown postons he user s only asked to ollet a few mages of ths board and lk on ts orner ponts o a pror knowledge about the mrror or the amera model s requred he work desrbed n ths paper reexamnes the generalzed parametr model of a entral system, whh we presented n our prevous work [] hs model assumes that the magng funton, whh manages the relaton between a

2 pxel pont and the 3D half-ray emanatng from the sngle vewpont, an be desrbed by a aylor seres expanson, whose oeffents are the parameters to be estmated he ontrbutons of the present work are the followng Frst, we smplfy the amera model by redung the number of parameters ext, we refne the albraton output by usng a 4-step least squares lnear mnmzaton, followed by a nonlnear refnement, whh s based on the maxmum lkelhood rteron By dong so, we mprove the auray of the prevous tehnque and allow albraton to be done wth a smaller number of mages hen, n ontrast wth our prevous work, we no longer need the rular boundary of the mrror to be vsble n the mage In that work, we used the appearane of the mrror boundary to ompute both the poston of the enter of the omndretonal mage and the affne transformaton Conversely, here, these parameters are automatally omputed usng only the ponts the user seleted In ths paper, we evaluate the performane and the robustness of the albraton by applyng the tehnque to smulated data hen, we albrate a real atadoptr amera, and show the auray of the result by proetng the olor nformaton of the mage onto real 3D ponts extrated by a 3D sk laser range fnder Fnally, we provde a Matlab oolbox [4] whh mplements the proedure desrbed here he paper s organzed n the followng way For the sake of larty, we report n seton III the amera model ntrodued n our prevous work, and provde ts new smplfed verson In seton IV, we desrbe our amera albraton tehnque and the automat deteton of both the mage enter and the affne transformaton Fnally, n seton V, we show the expermental results, on both smulated and real data, and present our Matlab oolbox III A PARAMERIC CAMERA MODE For maor larty, we ntally report the entral amera model ntrodued n [], then, we provde ts new smplfed verson We wll use the notaton gven n [8] In the general entral amera model, we dentfy two dstnt referenes: the amera mage plane u', v') and the sensor plane u'', v '') he amera mage plane ondes wth the amera CCD, where the ponts are expressed n pxel oordnates he sensor plane s a hypothetal plane orthogonal to the mrror axs, wth the orgn loated at the plane-axs nterseton In Fg, the two referene planes are shown n the ase of a atadoptr system In the doptr ase, the sgn of u would be reversed beause of the absene of a refletve surfae All oordnates wll be expressed n the oordnate system plaed n O, wth the z axs algned wth the sensor axs see Fg a) et be a sene pont hen, assume u' ' = [u'', v ''] be the proeton of onto the sensor plane, and u' = [u', v'] ts mage n the amera plane Fg b and ) As observed n [8], the two systems are related by an affne transformaton, whh norporates the dgtzng proess and small axes x x msalgnments; thus u' ' = Au' +t, where A R and t R At ths pont, we an ntrodue the magng funton g, whh aptures the relatonshp between a pont u' ', n the sensor plane, and the vetor p emanatng from the vewpont O to a sene pont see Fg a) By dong so, the relaton between a pxel pont u and a sene pont s: λ p = λ g u' ') = λ gau'+t) = P, λ >, ) 4 where R s expressed n homogeneous oordnates 3x4 and P R s the perspetve proeton matrx By albraton of the omndretonal amera we mean the estmaton of the matres A and t, and the non-lnear funton g, so that all vetors gau'+t) satsfy the proeton equaton ) We assume for g the followng expresson gu'',v'' ) = u'',v'' f u'',v'' )), ), We assume that the funton f depends on u and v only through ρ ' ' = u ' ' +v ' ' hs hypothess orresponds to assume that the funton g s rotatonally symmetr wth respet to the sensor axs a) b) ) Fg a) oordnate system n the atadoptr ase b) Sensor plane, n metr oordnates ) Camera mage plane, expressed n pxel oordnates b) and ) are related by an affne transformaton Funton f an have varous forms related to the mrror or the lens onstruton hese funtons an be found n [,, ] Unlke usng a spef model for the sensor n use, we hoose to apply a generalzed parametr model of f, whh s sutable to dfferent knds of sensors he reason for dong so, s that we want ths model to ompensate for any msalgnment between the fous pont of the mrror or the fsheye lens) and the amera optal enter Furthermore, we desre our generalzed funton to approxmately hold wth the sensors where the sngle vewpont property s not exatly verfed eg gener fsheye ameras) In our earler work, we proposed the followng polynomal form for f,,,,,, f u'',v'' ) = a + a ρ + a ρ + + a ρ, 3) where the oeffents a, =,,,, and the polynomal degree are the parameters to be determned by the albraton hus, ) an be rewrtten as

3 u '' λ v '' w '' = λ g A u'+ t)= Au'+ t) λ = P, λ > 4) f u '', v '') pattern, M = [,, Z ] the 3D oordnates of ts ponts n the pattern oordnate system, and m = [ u, v ] the orrespondent pxel oordnates n the mage plane Sne we assumed the pattern to be planar, wthout loss of generalty we have Z = As mentoned n the ntroduton, n ths paper we want to hen, equaton 7) beomes redue the number of albraton parameters hs an be done by observng that all defntons of f, whh hold for hyperbol and parabol mrrors or fsheye ameras [,, u ], always satsfy the followng: v λ p = λ = P = [ r r r 3 t ] = [ r ] r t 8) df a = 5) + + a ρ dρ ρ = hs allows us to assume a =, and thus 3) an be rewrtten as: herefore, n order to solve for amera albraton, the extrns parameters have to be determned for eah pose of,,,, f u'',v'' )= a + a ρ + + a ρ 6) the albraton pattern IV CAMERA CAIBRAIO A Solvng for amera extrns parameters By albraton of an omndretonal amera we mean the Before desrbng how to determne the extrns estmaton of the parameters [A, t, a, a,, a ] In order to parameters, let us elmnate the dependene from the depth estmate A and t, we ntrodue a method, whh, unlke other sale λ hs an be done by multplyng both sdes of prevous works, does not requre the vsblty of the rular equaton 8) vetorally by p external boundary hs method s based on an teratve proedure Frst, t starts by settng A to the untary matrx Its u elements wll be estmated usng a non-lnear refnement λ p p = p [ r t ] v [ t r = r r ] = 9) hen, our method assume the enter of the omndretonal a + + a ρ mage O to onde wth the mage enter I, that s O = I and thus t = α O I ) = Observe that, for A, the ow, let us fous on a partular observaton of the albraton assumpton to be untary s reasonable beause the eentrty pattern From 9), we have that eah pont p on the pattern of the external boundary, n the omndretonal mage, s usually lose to Conversely, O an be very far from the ontrbutes three homogeneous equatons mage enter I he method we wll dsuss does not are v r3 + r 3 + t 3 ) f ρ ) r + r + t ) = ) about ths In setons IVD and IVE, we wll dsuss how to f ρ ) ) u r ) ompute the orret values of A and O r + r + t 3 + r 3 + t3 ) = u r + r + t ) v r + r + t ) = 3) o resume, from now on we assume u' ' α = u' hus, by substtutng ths relaton n 4) and usng 6), we have the followng proeton equaton Here, and Z are known, and so are u, v Also, observe that only 3) s lnear n the unknown r, t u '' α u ' u ' hus, by stakng all the unknown entres of 3) nto a λ v '' = λ g α u' ) = λ α v ' = λ α v ' = P, λ,α > 7) vetor, we rewrte the equaton 3) for ponts of the w '' f α ρ ') a albraton pattern as a system of lnear equatons + + a ρ ' M H =, ) where now u' and v' are the pxel oordnates of an mage where pont wth respet to the mage enter, and ρ ' s the Euldean H = [r ], dstane Also, note that the fator α an be dretly ntegrated n the depth fator λ ; thus, only parameters a v v u u v u, a,, a ) need to be estmated M = : : : : : : ) Durng the albraton proedure, a planar pattern of v v u u v u known geometry s shown at dfferent unknown postons, whh are related to the sensor oordnate system by a rotaton A lnear estmate of H an be obtaned by mnmzng the matrx R = [ r ] and a translaton t, alled extrns 3 least-squares rteron mn M H, subet to H = hs parameters et I be an observed mage of the albraton s aomplshed by usng the SVD he soluton of ) s

4 known up to a sale fator, whh an be determned unquely lnear mnmzaton In subseton E, we wll apply a nonlnear refnement based on the maxmum lkelhood rteron sne vetors r are orthonormal Beause of the orthonormalty, the unknown entres r 3 3 an also be he struture of the lnear refnement algorthm s the followng: omputed unquely he frst step uses the amera model a, a,, a ) o resume, the frst albraton step allows fndng the extrns parameters r 3 estmated n B, and reomputes all extrns parameters 3, t for eah pose of by solvng all together equatons ), ) and 3) the albraton pattern, exept for the translaton parameter t 3 he problem leads to a lnear homogeneous system, hs parameter wll be omputed n the next step, whh whh an be solved, up to a sale fator, usng SVD onerns the estmaton of the mage proeton funton hen, the sale fator s determned unquely by explotng the orthonormalty between vetors r B Solvng for amera ntrns parameters In the seond stage, the extrns parameters reomputed n the prevous step are substtuted n equatons ) In the prevous step, we exploted equaton 3) to fnd and ) to ulterorly refne the ntrns amera model the amera extrns parameters ow, we substtute the he problem leads to a lnear system, whh an be estmated values n the equatons ) and ), and solve solved as usual by usng the pseudonverse for the amera ntrns parameters a, a,, a that desrbe the shape of the magng funton g At the same tme, we also D Iteratve enter deteton ompute the unknown t 3 for eah pose of the albraton pattern As done above, we stak all the unknown entres of As stated at the begnnng of seton IV, we want our ) and ) nto a vetor and rewrte the equatons as a albraton toolbox to be as automat as possble, and so, we system of lnear equatons But now, we norporate all desre the apablty of dentfyng the enter of the observatons of the albraton board We obtan the followng omndretonal mage O Fg ) even when the external system boundary of the sensor s not vsble n the mage a o ths end, observe that our albraton proedure a orretly estmates the ntrns parametr model only f O s A A ρ A ρ B v : taken as orgn of the mage oordnates If ths s not the ase, C C ρ C ρ u a D by bak-proetng the 3D ponts of the heker board nto the : : : : : : = :,3) t 3 mage, we would observe a large reproeton error wth A A ρ A ρ v t B respet to the albraton ponts see Fg a) Motvated by 3 C ρ C ρ : D ths observaton, we performed many trals of our albraton C u proedure for dfferent enter loatons, and, for eah tral, we t 3 omputed the Sum of Squared Reproeton Errors SSRE) where As a result, we verfed that the SSRE always has a global A = r + r + t, B = v r 3 + r ), C = r + r + t mnmum at the orret enter loaton 3, D = u r 3 + r 3 ) Fnally, the least-squares soluton of the overdetermned system s obtaned by usng the pseudonverse hus, the ntrns parameters a, a,, a, whh desrbe the model, are now avalable In order to ompute the best polynomal degree, we atually start from = hen, we nrease by untary steps and we ompute the average value of the reproeton error of all albraton ponts he proedure stops when a mnmum error s found C near refnement of ntrns and extrns parameters o resume, the seond lnear mnmzaton step desrbed n part B fnds out the ntrns parameters of the amera, and smultaneously estmates the remanng extrns t 3 he next two steps, whh are desrbed here, am at refnng ths prmary estmaton hs refnement s stll performed by O a) Fg When the poston of the enter s wrong, the 3D ponts of the heker board do not orretly bak-proet green rounds) onto the albraton ponts red rosses) a) Conversely, b) shows the reproeton result when the enter s orret hs result leads us to an teratve searh of the enter O, whh stops when the dfferene between two potental enter loatons s less than a ertan fraton of pxel ε we reasonably set ε=5 pxels): At eah step of ths teratve searh, a partular mage regon s unformly sampled n a ertan number of ponts b)

5 For eah of these ponts, albraton s performed by usng that pont as a potental enter loaton, and SSRE s omputed 3 he pont gvng the mnmum SSRE s assumed as a potental enter 4 he searh proeeds by refnng the samplng n the regon around that pont, and steps, and 3 are repeated untl the stop ondton s satsfed Observe that the omputatonal ost of ths teratve searh s so low that t takes only 3 seonds to stop E on- lnear refnement he lnear soluton gven n the prevous subsetons A, B and C s obtaned through mnmzng an algebra dstane, whh s not physally meanngful o ths end, we hose to refne t through maxmum lkelhood nferene et us assume we are gven mages of a model plane, eah one ontanng orner ponts ext, let us assume that the mage ponts are orrupted by ndependent and dentally dstrbuted nose hen, the maxmum lkelhood estmate an be obtaned by mnmzng the followng funtonal: Fg 3 A pture of our smulator showng several albraton patterns and the vrtual omndretonal amera at the axs orgn ^ E = m = = m R, O,A, a,a,,a,m, ), 4) ^ where m R,,A O, a,a,,a,m, ) s the proeton of the pont M of the plane aordng to equaton ) R and are the rotaton and translaton matres of eah plane pose; R s parameterzed by a vetor of 3 parameters related to R by the Rodrgues formula Observe that now we norporate nto the funtonal both the affne matrx A and the enter of the omndretonal mage O By mnmzng the funtonal defned n 4), we atually ompute the ntrns and extrns parameters whh mnmze the reproeton error In order to speed up the onvergene, we deded to splt the non-lnear mnmzaton nto two steps he frst one refnes the extrns parameters, gnorng the ntrns ones hen, the seond step uses the extrns parameters ust estmated, and refnes the ntrns ones By performng many smulatons, we found that ths splttng does not affet the fnal result wth respet to a global mnmzaton o mnmze 4), we used the evenberg-marquadt algorthm, as mplemented by the Matlab funton lsqnonln he algorthm requres an ntal guess of the ntrns and extrns parameters hese parameters are obtaned usng the lnear tehnque desrbed n the prevous subsetons As a frst guess for A, we used the untary matrx, whle for O we used the poston estmated through the teratve proedure explaned n subseton D V EPERIMEA RESUS In ths seton, we present the expermental results of the proposed albraton proedure on both omputer smulated and real data A Smulated Experments he reason for usng a smulator s that we an montor the atual performane of the albraton, and ompare the results wth a known ground truth he smulator we developed allows hoosng both the ntrns parameters e the magng funton g) and extrns ones e the rotaton and translaton matres of the smulated heker boards) Moreover, t permts to fx the sze of the vrtual pattern, and also the number of albraton ponts, as n the real ase A ptoral mage of the smulaton senaro s shown n Fg 3 As a vrtual albraton pattern we set a heker plane ontanng 6x8=48 orner ponts he sze of the pattern s 5x mm As a amera model, we hoose a 4 th order polynomal, whose parameters are set aordng to those obtaned by albratng a real omndretonal amera hen, we set to 9x pxels the mage resoluton of the vrtual amera A Performane wth respet to the nose level In ths smulaton experment, we study the robustness of our albraton tehnque n ase of nauray n detetng the albraton ponts o ths end, we use 4 poses of the albraton pattern hen, Gaussan nose wth zero mean and standard devaton σ s added to the proeted mage ponts We vary the nose level from σ= pxels to σ=3 pxels, and, for eah nose level, we perform ndependent albraton trals he results shown are the average Fg 4 shows the plot of the reproeton error vs σ We defne the reproeton error as the dstane, n pxels, between the bak-proeted 3D ponts and orret mage ponts Fgure 4 shows both the plots obtaned by ust usng the lnear mnmzaton method, and the non-lnear refnement As you an see, the average error nreases lnearly wth the nose level n both ases Observe that the reproeton error n the non-lnear estmaton s always less than that n the lnear

6 method Furthermore, note that for σ =, whh s larger than the normal nose n a pratal albraton, the average reproeton error of the non-lnear method s less than 4 pxels Fg 4 he reproeton error vs the Fg 5 Auray of the extrns nose level wth the lnear parameters: the absolute error mm) of mnmzaton dashed lne n blue) the translaton vetor vs de nose and after the non-lnear refnement level pxels) he error along the x, y sold lne n red) Both unts are n and z oordnates s represented pxels respetvely n red, blue and green 6 B Real Experments Usng the Proposed oolbox Followng the steps outlned n the prevous setons, we developed a Matlab oolbox [4], whh mplements our new albraton proedure hs tool was tested on a real entral atadoptr system, whh s made up of a hyperbol mrror and a amera havng the resoluton of 4x768 pxels Only three mages of a heker board taken all around the mrror were used for albraton Our oolbox only asks the user to lk on the orner ponts he lkng s faltated by means of a Harrs orner detetor havng sub-pxel auray he enter of the omndretonal mage was automatally found as explaned n IVD After albraton, we obtaned an average reproeton error less than 3 pxels Fg b) Furthermore, we ompared the estmated loaton of the enter wth that extrated usng an ellpse detetor, and we found they dffer by less than 5 pxels B Mappng Color Informaton on 3D ponts Fg 6 An mage of the albraton pattern, proeted onto the smulated omndretonal mage Calbraton ponts are affeted by nose wth σ =3 pxels blue rounds) Ground truth red rosses) Reproeted ponts after the albraton red squares) One of the hallenges we are gong to fae n our laboratory onssts n gettng hgh qualty 3D maps of the envronment by usng a 3D rotatng sk laser range fnder SIC MS [3]) Sne ths sensor annot provde the olor nformaton, we used our albrated omndretonal amera to proet the olor onto eah 3D pont he results are shown n Fg 7 In order to perform ths mappng both the ntrns and extrns parameters have to be aurately determned Here, the extrns parameters desrbe poston and orentaton of the amera frame wth respet to the sk frame ote that even small errors n estmatng the orret ntrns and extrns parameters would produe a large offset nto the output map In ths experment, the olors perfetly reproeted onto the 3D struture of the envronment, showng that the albraton was aurately done In Fg 6, we show the 3D ponts of a heker board bakproeted onto the mage he ground truth s hghlghted by red rosses, whle the blue rounds represent the albraton ponts perturbed by nose wth σ=3 pxels Despte the large amount of nose, the albraton s able to ompensate for the error ntrodued In fat, after albraton, the reproeted albraton ponts are very lose to the ground truth red squares) We also want to evaluate the auray n estmatng the extrns parameters R and of eah albraton plane o ths end, Fgure 5 shows the plots of the absolute error measured n mm) n estmatng the orgn oordnates x, y and z) of a gven heker board he absolute error s very small beause t s always less than mm Even f we do not show the plots here, we also evaluated the error n estmatng the orret plane orentatons, and we found an average absolute error less than VI COCUSIOS In ths paper, we presented a novel and pratal tehnque for albratng any entral omndretonal ameras he proposed proedure s very fast and ompletely automat, as the user s only asked to ollet a few mages of a heker board, and to lk on ts orner ponts hs tehnque does not use any spef model of the omndretonal sensor It only assumes that the magng funton an be desrbed by a aylor seres expanson, whose oeffents are the parameters to be estmated hese parameters are estmated by solvng a four-step least-squares lnear mnmzaton problem, followed by a non-lnear refnement, whh s based on the maxmum lkelhood rteron

7 Fg 7 he panoram pture shown n the upper wndow was taken by usng a hyperbol mrror and a perspetve amera, the sze of 64x48 pxels After ntrns amera albraton, the olor nformaton was mapped onto the 3D ponts extrated from a sk laser range fnder In the lower wndows are the mappng results he olors are perfetly reproeted onto the 3D struture of the envronment, showng that the amera albraton has been aurately done In ths work, we also presented a method to teratvely ompute the enter of the omndretonal mage wthout explotng the vsblty of the rular feld of vew of the amera he enter s automatally omputed by usng only the ponts the user seleted Furthermore, we used smulated data to study the robustness of our albraton tehnque n ase of nauray n detetng the albraton ponts We showed that the nonlnear refnement sgnfantly mproves the albraton auray, and that aurate results an be obtaned by usng only a few mages hen, we albrated a real atadoptr amera he albraton was very aurate as we obtaned an average reproeton error les than 3 pxels n an mage the resoluton of 4x768 pxels We also showed the auray of the result by proetng the olor nformaton from the mage onto real 3D ponts extrated by a 3D sk laser range fnder Fnally, we provded a Matlab oolbox [4], whh mplements the entre albraton proedure ACOWEDGEMES hs work was supported by the European proet COGIRO the Cogntve Robot Companon) We also want to thank Jan Wengarten, from EPF, who provded the data from the 3D sk laser range fnder [3] REFERECES Baker, S and ayar, S 998 A theory of atadoptr mage formaton In Proeedngs of the 6th Internatonal Conferene on Computer Vson, Bombay, Inda, IEEE Computer Soety, pp 35 4 BMusk, Padla Autoalbraton & 3D Reonstruton wth onentral Catadoptr Cameras CVPR 4, Washngton US, June 4 3 C Cauhos, E Brassart, Delahohe, and Delhommelle Reonstruton wth the albrated sylop sensor In IEEE Internatonal Conferene on Intellgent Robots and Systems IROS ), pp , akamatsu, Japan, 4 H Baksten and Padla Panoram mosang wth a 8 feld of vew lens In Pro of the IEEE Workshop on Omndretonal Vson, pp 6 67, 5 C Geyer and Danlds Paraatadoptr amera albraton PAMI, 45), pp , May 6 J Glukman and S ayar Ego-moton and omndretonal ameras ICCV, pp 999-5, S B ang Catadoptr self-albraton CVPR, pp -7, 8 B Musk and Padla Estmaton of omndretonal amera model from eppolar geometry CVPR, I: 48549, 3 9 BMusk, Padla Para-atadoptr Camera Auto-albraton from Eppolar Geometry ACCV 4, orea January 4 J umler and M Bauer Fsheye lens desgns and ther relatve performane BMusk, DMartne, Padla 3D Metr Reonstruton from Unalbrated Omndretonal Images ACCV 4, orea January 4 Svoboda, Padla Eppolar Geometry for Central Catadoptr Cameras IJCV, 49), pp 3-37, luwer August 3 Wengarten, J and Segwart, R EF-based 3D SAM for Strutured Envronment Reonstruton In Proeedngs of IROS 5, Edmonton, Canada, August -6, 5 4 Google for OCAMCAIB 5 ng, Z Hu, Catadoptr Camera Calbraton Usng Geometr Invarants, IEEE rans on PAMI, Vol 6, o : 6-7, Otober 4 6 ng, Z Hu, Can We Consder Central Catadoptr Cameras and Fsheye Cameras wthn a Unfed Imagng Model?, ECCV'4, Prague, May 4 7 J Barreto, H Arauo, Geometr Propertes of Central Catadoptr ne Images and ther Applaton n Calbraton, PAMI-IEEE rans on PAMI, Vol 7, o 8, pp , August 5 8 J Barreto, H Arauo, Geometr Propertes n Central Catadoptr ne Images, ECCV', Copenhagen, Denmark, May 9 P Sturm, S Ramalgam, A Gener Conept for Camera Calbraton, ECCV'4, Prague, 4 D Saramuzza, A Martnell, R Segwart, A Flexble ehnque for Aurate Omndretonal Camera Calbraton and Struture from Moton Proeedngs of IEEE Internatonal Conferene on Computer Vson Systems ICVS 6), ew ork, January 6