Line-based Camera Movement Estimation by Using Parallel Lines in Omnidirectional Video

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1 01 IEEE Internatonal Conference on Robotcs and Automaton RverCentre, Sant Paul, Mnnesota, USA May 14-18, 01 Lne-based Camera Movement Estmaton by Usng Parallel Lnes n Omndrectonal Vdeo Ryosuke kawansh, Atsush amashta, Toru Kaneko and Hajme Asama Abstract In ths paper, we propose an effcent estmaton method of an omndrectonal camera movement. The proposed method s based on Structure from Moton utlzng a constrant of parallel lnes. In an envronment havng manmade structures, parallel lnes can be extracted from an omndrectonal mage easly and constantly, because of ts wde feld of vew. Parallel lnes provdes a valuable constrant for camera movement estmaton. The proposed method can estmate 3-D camera movements by solvng one degree of freedom problem three tmes wthout regard to the number of vewponts. Expermental results show the effectveness of our proposed method. I. INTRODUCTION To estmate camera movement accurately and effcently s a essental assgnment for scene reconstructon by monocular stereo, such as an approach based on Structure from Moton SfM) [1], [] or vslam [3], [4]. Camera movement estmaton from an mage sequence acqured by a sngle camera s dffcult, because a camera movement matrx, such as the essental matrx, has at least 6 degrees of freedom DOFs) rotaton s 3 DOFs and translaton s 3 DOFs) and s a non-lnear matrx. The processng cost wll be hgh and the calculaton of the optmal camera matrces wll be complex f vewponts to be estmated ncrease. A lmted feld of vew of cameras equpped wth a typcal lens makes t dffcult to estmate camera movement, too. Self-localzaton method by monocular stereo often needs feature trackng KLT tracker [5], SIFT [6], and so on) to obtan correspondence between dfferent vewponts. However, features wll be lost easly due to camera swng. Consequently, the camera movement wll be lmted f we use a such camera for scene reconstructon. Therefore, we use an omndrectonal camera equpped wth a hyperbolod mrror. An omndrectonal camera s sutable for self-localzaton [7], because the acqured mage has a panoramc feld of vew Fg. 1). Self-localzaton and scene reconstructon method by usng an omndrectonal camera have been proposed [8], [9], [10]. In prevous monocular stereo, there are some approaches by usng feature ponts [], [3], [8], [9], [10], straght lnes Ths work was not supported by any organzaton R. Kawansh and Toru Kaneko are wth Department of Mechancal Engneerng, Faculty of Engneerng, Shzuoka Unversty, Johoku, Naka-ku, Hamamatsu-sh, Shzuoka , Japan f @pc.shzuoka.ac.jp, tmtkane@pc.shzuoka.ac.jp A. amashta and H. Asama s wth Department of Precson Engneerng, The Unversty of Tokyo, Hongo, Bunkyo-ku, Tokyo , Japan yamashta@robot.t.u-tokyo.ac.jp, asama@robot.t.u-tokyo.ac.jp Hyperbolod Mrror Camera Fg. 1. The omndrectonal camera whch we use s shown n the left fgure. The camera attaches a hyperbolod mrror on the front of the camera lens. An omndrectonal mage s shown n the rght fgure. The mage has a 360-degree horzontal feld of vew. [11], [1], [13] or both features [1]. Pont-based methods have the beneft of fast calculaton by the lnear soluton 8-pont algorthm [14], 5-pont algorthm [], and so on). However, there may be not suffcent number of feature ponts n ndoor envronments. On the other hand, the method needs to solve hgher DOF problem than a pont-based method, because a lne-based method has to use more than 3 vewponts for camera movement estmaton. There are lnear computaton algorthms for a lne-based SfM by usng trfocal tensor [15], [16]. However, these methods are only for mage trplets. We propose an effcent camera movements estmaton method by usng correspondences of parallel lnes between not less than three mages. The constrant obtaned from parallel lnes s useful for reducton n the degree of freedom of the camera movement estmaton. As a prevous method, SLAM by usng parallel lnes and ther vanshng pont [17], SfM n urban envronment buldng scene) [18], rotaton estmaton by vdeo compass [19] and so on have been proposed. However, the computaton tme of these method ncreases exponentally as the number of features and vewponts ncrease [17], [18]. In addton, these methods assume that a camera movement s only horzontal movement [18], [19]. The proposed method can estmate 3-D camera movements of more than 3 vewponts effcently. Rotaton matrces and translaton vectors estmaton are dvded n two phases. Each phase s solved as a 1 DOF problem wthout regard to the number of vewponts and features. Consequently, although these are non-lnear problems, the calculaton cost s extremely low. The proposed method can fnd the global optmal soluton easly even f an ntal value near to the ground truth s not gven. The effectveness of our proposed method s shown n expermental results. The proposed method needs at least 3 mages and 6 lnes. Three of these lnes have to be parallel. Parallel lnes are easly extracted from a man-made structure n an ndoor /1/$ IEEE 3573

2 Straght-lne trackng Detecton of parallel lnes and vanshng pont Hyperbolod mrror + α β = 1 Focus of hyperbolod mrror r : Ray vector Rotaton calculaton by VPV drecton γ = α + β Estmaton of rotaton about VPV Estmaton of translaton on vertcal plane Estmaton of translaton along parallel lnes Image plane γ x Lens u, v ) y f : focal length Fg.. Procedure of our proposed method. Estmaton of rotaton matrces and translaton vectors are dvded n two phases. Each phase s solved as a 1 DOF problem wthout regard to the number of vewponts and features. envronment, because an omndrectonal camera has a wde feld of vew. The others must have dfferent drecton from the parallel lnes. A lne whch connects two feature ponts can be used as the other lne. Therefore, the assumpton s relevant for general ndoor envronments. An omndrectonal camera s calbrated n advance. The procedure of our proposed method s shown n Fg.. Straght-lnes are extracted and tracked along an omndrectonal vdeo. Parallel lnes and the vanshng pont are detected from these lnes. The vector drected to the vanshng pont from the vewpont s calculated. It s called a VPV n ths paper. Camera rotaton estmaton s dvded n two phases. In the frst phase, the method calculates a camera rotaton matrx whch makes a VPV at each vewpont have at the same 3-D drecton. In the second phase, a rotaton matrx about a VPV s estmated. If a rotaton matrx between two vewponts s determned, rotaton matrces at the other vewponts are calculated by solvng a quartc functon about the rotaton angle. Therefore, n the proposed method, rotaton estmaton can be solved as a 1 DOF problem wthout regard to the number of vewponts and features. Camera translatons are estmated by two phases. In the frst phase, translatons n a plane are estmated. The plane s vertcal to 3-D drecton of parallel lnes. If a translaton drecton between two vewponts s determned, translatons at the other vewponts are calculated by solvng a smultaneous equaton. Therefore, the estmaton s a 1 DOF problem about the translaton drecton. In the second phase, translatons along 3-D drecton of parallel lnes are estmated. In ths phase, f a translaton between two vewponts s determned, translatons at the other vewponts are obtaned by algebrac calculatons. Consequently, the proposed method estmates camera movements by solvng 1 DOF problems. II. COORDINATE SSTEM OF OMNIDIRECTIONAL CAMERA The coordnate system of our omndrectonal camera s shown n Fg. 3. A hyperbolod mrror reflects a ray headng to mage coordnates u, v) from the camera lens. In ths paper, the reflected ray s called a ray vector. The extenson lnes of all ray vectors ntersect at the focus of Fg. 3. The coordnate system of the omndrectonal camera. Ray vector s defned as a unt vector whch starts from the focus of a hyperbolod mrror. a) b) c) Fg. 4. Edge segment extracton. a) Input mage. b) Detected canny edge ponts. c) Edge ponts whch are lne-lke. the hyperbolod mrror. The ray vector s calculated by the followng equatons. r = λu c x)p x λv c y )p y, 1) λf γ α f α + β + β ) u + v + f λ = α f β u + v, ) ) where c x and c y are the center coordnates of the omndrectonal mage, p x and p y are pxel sze, f s the focal length of a camera lens, α, β and γ are hyperbolod parameters. III. STRAIGHT-LINE TRACKING Straght-lnes are extracted from a dstorted omndrectonal mage. The proposed method obtans edge ponts detected by Canny edge detector [0]. An example of edge pont detecton s shown n Fg. 4a) and b). To separate each straght-lnes, corner ponts are rejected as shown n Fg. 4c). Corner ponts are detected by usng two egenvalues of the Hessan of the mage. If the rato of egenvalues s hgh enough, the edge pont s regarded as lne-lke. Other edge ponts are rejected as corner ponts. The rato s set to 10 by the tral-and-error method. A least square plane s calculated from ray vectors of segmented edge ponts. If the edge segment conssts of a straght-lne, these ray vectors are located on a plane Fg. 5). Therefore, an edge segment whch has a small least square error s regarded as a straght-lne. The proposed method can extract straght-lnes, even f an edge segment looks lke 3574

3 Focus of hyperbolod mrror Omndrectonal mage Lens Edge pont Ray vector Plane Straght lne Fg. 5. The relatonshp between a straght-lne and a ray vector. Ray vectors whch belongs to the lne are located on the plane ncludng the lne. a) b) Ponts extracted at constant ntervals Edge ponts n the next frame Tracked pont c) d) e) Correspondng edge ponts Correspondng edge segment Fg. 6. Searchng for a correspondng edge segment n the next frame. a) Ponts extracted at constant ntervals. b) Edge segments n the next frame. c) Ponts a) are tracked between the current frame and the next frame. d) Correspondng edge ponts. e) Correspondng edge segment. a curve n an omndrectonal mage. If over half the edge ponts of the edge segment satsfy the followng equaton, the segment s determned as a straght-lne. r,j T n ) < lth, 3) where l th s a threshold. r,j s a ray vector headng to the edge pont j ncluded n the lne. n s the normal vector of the least square plane calculated from the lne. n s an unt vector. The vector s called NV n ths paper. In the detecton, edge ponts whch do not consttute the lne are rejected as nose by RANSAC [1]. The threshold l th s determned from the mage resoluton. Straght-lnes are tracked along the omndrectonal mage sequence. The proposed method obtans ponts located on a straght-lne. Ponts located on the lne are extracted at constant ntervals Fg. 6a)). Edge segments are extracted n the next frame Fg. 6b)). The ponts extracted n Fg. 6a) are tracked to the next frame by KLT tracker Fg. 6c)). The edge pont closest to the tracked pont s selected as a correspondng edge pont Fg. 6d)). The edge segment whch has the maxmum number of correspondng edge ponts s regarded as a correspondng edge segment Fg. 6e)). If an edge segment corresponds to several lnes, a lne whch has larger number of correspondng edge ponts s selected. Matchng pont serach on a lne has the aperture problem []. However, t s not dffcult for the proposed method to obtan correspondng edges, because t does not requre pont-to-pont matchng. By contnung the above processes, straght-lnes are tracked along the omndrectonal mage sequence. IV. DETECTION OF PARALLEL LINES AND VANISHING POINT Parallel lnes and ther vanshng pont are detected from tracked lnes. A VPV v c and NV n c about parallel lnes at a vewpont c satsfy the followng equaton. n T c v c = 0. 4) Three lnes are requred for parallel lnes detecton from an mage. The proposed method selects three lnes from tracked lnes randomly. A VPV s calculated from selected lnes by solvng 5) by the least squares method. n T,cv c ) mn, 5) where n l s the number of lnes. If selected lnes satsfy the followng equaton at all of an nput mage sequence, these are regarded as parallel lnes. 3 n T,cv c < p th, 6) where p th s a threshold. Lnes whch are regarded as parallel each other are ntegrated. A lne group whch has the maxmum number of lnes s used for the followng process as parallel lnes. The VPV v c at the vewpont c s calculated from the ntegrated parallel lnes by 5). V. ROTATION ESTIMATION Camera rotaton estmaton s dvded n two phases. In the frst phase, the method calculates a camera rotaton matrx whch makes a VPV at each vewpont have the same 3-D drecton. In the second phase, a rotaton matrx about a VPV s estmated by usng at least 3 lnes. These lnes must have dfferent 3-D drecton from parallel lnes. A. Rotaton calculaton by VPV drecton Ths phase requres VPVs only. A VPV at each vewpont should have the same 3-D drecton n the world coordnate, because a vanshng pont s theoretcally at an nfnte dstance away from vewponts. Therefore, the proposed method calculates a rotaton matrx R m c satsfyng the followng equaton. = R mt c v c. 7) where R m c s a rotaton matrx between the ntal vewpont c 0 and a vewpont c. In ths paper, the ntal camera coordnate system s equal to the world coordnate system. R m c s calculated as a rotaton matrx about a vector m c by Rodrgues rotaton formula. The rotaton angle θ c s 3575

4 Vewpontc 0 Vewpont c v c m c θ c v c v c Vewpont c m R T Vewpontc c vc 0 φ c v T R c Fg. 7. The relatonshp among VPVs, v c, a rotaton axs vector m c and the rotaton angle θ c. Fg. 8. Rotaton about VPV. calculated A relatonshp between m c and θ c are shown n Fg. 7. They are defned as the followng equatons. m c = v c, 8) θ c = arccos v c ). 9) The 3-D drecton of parallel lnes and the VPV are the same. Therefore, the VPV also means 3-D drecton of parallel lnes n the followng explanaton. B. Estmaton of rotaton around parallel lne axs In the second phase, a rotaton matrx about the VPV s estmated. Ths phase needs at least three lnes. The 3-D drecton of these lnes must not be equal to the VPV. In the proposed method, unknown parameter of 3-D rotaton remans only a rotaton R v c about the VPV, because the other two parameters can be obtaned from a constrant of VPV drecton. Therefore, ths phase estmates a rotaton matrx R v c, namely, the rotaton angle ϕ c Fg. 8). The true rotaton matrx R c between the ntal vewpont c 0 and a vewpont c s defned as the followng equaton. R c = R m c R v c. 10) The true rotaton matrx R c and 3-D lne drecton d satsfy the followng equaton. ) R T T c n,c d = 0, 11) where n c, s the NV of the lne at the vewopnt c. d s an unt vector. If a rotaton angle ϕ c s gven, the 3-D drecton d of the lne s calculated by the folowng equaton. d = n c0 R c T n c ). 1) By usng the 3-D lne drecton, a rotaton matrx R k between the ntal vewpont c 0 and the other vewpont k s calculated by solvng the followng equaton. e rot ϕ k ) = R T ) T k n,k d mn. 13) where, n l s the number of non-parallel lnes. Although the functon s non-lnear, t can be solved easly because e rot ϕ k ) s just a quartc functon about ϕ k. Consequently, f a rotaton angle ϕ c at a vewpont c s gven, rotaton angles ϕ k at the other vewponts k are determned. The proposed optmzaton of rotatons s represented as the followng equaton. n c E rot ϕ c ) = e rot ϕ k ) mn, 14) k where n c s the number of vewponts. In the proposed method, rotaton estmaton s 1 DOF problem about the rotaton angle ϕ c wthout regard to the number of vewponts and features. In the followng explanaton, v expresses the VPV. v also means 3-D drecton of parallel lnes. VI. TRANSLATION ESTIMATION Camera translatons are estmated by two phases. In the frst phase, translatons on the plane vertcal to v are estmated. In the second phase, translatons drected along parallel lnes are estmated. As same as the rotaton estmaton, translatons can be optmzed by solvng 1 DOF problems. A. Estmaton of translaton on vertcal plane In the frst phase, translatons on the plane vertcal to the VPV v are estmated. Ths phase requres parallel lnes. Translatons n the plane and locatons of parallel lnes are optmzed smultaneously. Here, the method ntroduces bass vectors a and b a b, a v, and b v). A unt vector g p,c drected to the parallel lne from the vewpont c are calculated by 15). The vector g,c s perpendcular to the VPV. g p,c = v RT cn,c ). 15) By usng these vectors, a and b elements of the true translatons can be estmated. A translaton vector t p c between the ntal vewpont c 0 and a vewpont c, the locaton l p of parallel lnes and a vector g p,c satsfy the followng equaton. The relatonshp among these vectors s shown n Fg. 9. δ,c g p,c + tp c l p = 0, 16) where δ,c s a fxed number whch means the depth of the lne at the vewpont. δ,c s calculated by the followng equaton. δ,c = tp c l p )T g p,c g,ct. 17) p g p,c Here, the translaton t p c s expressed by 18). t p c = a cos ψ c + b sn ψ c, 18) 3576

5 Vewpont c 0 p t c Vewpont c p l δ,g p c, c v Parallel lnes b a Vewpont c 0 t c Vewpont c o g,c o l q,c τ d Straght-lne d, c Fg. 9. The relatonshp among VPV v, bass vector a and b, a lne locaton l p, a vector gp,c and a translaton tp c on the plane. Fg. 10. Reprojecton error of straght-lne. where ψ c means translaton drecton from the ntal vewpont c 0 to the vewpont c. The absolute scale s unknown n the SfM approach. Thus, the dstance between these two vewponts s set to 1 n the proposed method. When the drecton ψ c s gven, the locatons of parallel lnes are calculated by the followng equaton. l p = ζ,c0 g p,c 0 + η,c g p,c + tp c) /, 19) where ζ,c0 and η,c are fxed factors whch mean the depth of the lne from each vewpont. Fxed factors ζ,c0 and η,c are calculated as factors satsfyng the followng expresson. ζ,c0 g p,c 0 η,c g p,c tp c mn. 0) By usng the locatons of parallel lnes, translatons at the other vewpont k are calculated by mnmzng the followng equatons. E t1 λ k, µ k ) = δ,k g p,k + tp k lp, 1) t p k = λ ka + µ k b. ) When the lne locaton l p, namely, the translaton drecton ψ c s gven, E t1 λ k, µ k ) can be solved as smultaneous equatons about λ k and µ k. Therefore, translatons on the plane vertcal to parallel lnes are optmzed by mnmzng the followng functon about ψ c. n c E t1 ψ c ) = E t1 λ k, µ k ). 3) B. Estmaton of translaton along parallel lnes k In the secons phase, translatons drected along parallel lnes are optmzed by the method based on Bundle adjustment [3]. At least 3 lnes are requred n ths phase. The 3-D drecton of the lnes used for the estmaton must be dfferent from VPV. The true translaton vector t c s represented as the follows: t c = t p c + ω c v, 4) where ω c s an unknown parameter about translaton dstance to be estmated n ths phase. Translatons and lne locatons are optmzed by mnmzng the sum of reprojecton errors as shown n Fg. 10 and 5)-7). E t = 1 T q,c g,c o ), 5) q,c = lo t c + τ,c d l o t c + τ,c d, 6) τ,c = t c l o )T d d T d, 7) where q,c s a unt vector crossed at a rght angle to the straght-lne. g,c o s a unt vector headng to the lne from the vewpont c. The vector s calculated by the same way as 15). If there are no errors, these vectors wll be the same. However, n fact, these have dfferent drecton because of varous errors. The angle error s almost the same to a reprojecton error. When ω c, namely, the translaton from the ntal vewpont c 0 to the vewpont c s determned, 3-D lne locatons l o are calculated n the same way as 16). By usng the locatons, 5) at the other vewpont k can be solved as a quartc functon about ω k. Therefore, the translaton estmaton s a non-lnear 1 DOF problem about ω c. The optmum translatons are estmated by mnmzng the followng equaton. n c E t ω c ) = E t ω k ). 8) k VII. EPERIMENT We demonstrate proposed camera movement estmaton by usng smulaton data. In these experments, the true values of NVs n,c acqured at each vewpont c are gven. It s known that whch lnes are parallel. The vector headng to ther vanshng pont s calculated from gven normal vectors. The camera movement ncludes 3-D rotatons and translatons. In these experments, we estmate camera movements by exhaustve searches wthout usng any precondtons. At frst, we verfed that the proposed method can estmate camera movements from 6 lnes. Three lnes of these are parallel. The others have dfferent drecton. The number of vewponts s 10. The poston relatonshp between the vewponts and lnes s shown n Fg. 11. Red, yellow and orange axes show a camera coordnate system at each vewpont. Parallel lnes are represented by green color. Other 3577

6 Other lnes Camera moton Parallel lnes Rotaton error [deg].56 Maxmum 1.8 Mnmum 0.64 Average Nose [deg] a) Rotaton. a) Top vew. Translaton error [-] Maxmum Mnmum Average Nose [deg] b) Translaton. b) Brd s-eye vew. Fg. 1. Estmaton errors wth nosy data. Fg. 11. Estmated camera movement and lne locatons. lnes are represented by blue color. The estmaton errors of camera movement and lne measurement are wthn a roundoff error of the computaton n ths experment. We verfed the robustness of the proposed method n nosy data. In ths experment, nosy NVs are gven. The nose means an angle error between a gven vector and the true one. The nose follows normal gaussan dstrbuton. We evaluated estmaton errors of camera rotatons and translatons by usng nosy data ncludng 0.01 to 1.8 degrees angle errors on average. Input data s 40 lnes ncludng 0 parallel lnes acqured at 0 vewponts. A dfferent nose s added to a NV randomly n 10 tmes tral runs. The estmaton results are shown n Fg. 1. The rotaton error n Fg. 1 a) means an angle errors between the axs of estmated camera coordnate system and the true one. The translaton error n Fg. 1 b) s a dstance between an estmated camera locaton and the true one. These values are the average of 10 tmes tral runs. In ths experment, translaton dstance between the begnnng vewpont and the end vewpont s set to 1, because translaton scale s ndefnte n the SfM approach. The translaton error mean a dstance error between the true locaton and the estmated locaton. These results show that the proposed method performs well n nosy data, because these estmaton errors are nearly wthn the gven nose. The proposed method can fnd the global optmal soluton easly because the proposed method can estmate camera movement by solvng 1 DOF problem three tmes. It contrbutes mplovement of robustness of selflocalzaton. We verfed the proposed method by usng real mages. 00 mages are acqured by a moble robot equpped wth an omndrectonal camera. The movement dstance s about 1.5 meters. The mage sze s An nput mage s shown n Fg. 13. The expermental envronment s an ndoor texture-less hallway). Ths experment s done wth off-lne processng. CPU s Intel Core GHz). Extracted lnes are shown n Fg. 14. Blue lnes show parallel lnes. Red lnes show the other lnes. Correspondence of 10 parallel lnes and 5 other lnes were obtaned from the nput omndrectonal vdeo. The computaton tme of lne trackng s 5 ms per frame. That of parallel lnes detecton s 35 ms. The estmaton result of camera movement and lne measurement s shown n Fg. 15. Ths fgure shows a top vew of hallway. Although camera rotatons and translatons at 00 vewponts are estmated, ths fgure shows 0 postons of all vewponts for easer vewng. An average of the reprojecton errors s wthn 6 pxels. The major cause of the error s correspondence accuracy of lnes. The computaton tme of rotaton estmaton s ms. That of translaton estmaton s 16 ms. VIII. CONCLUSION In ths paper, we proposed an effcent estmaton method of 3-D camera movement. Camera rotatons and translatons are estmated as a reasonable 1 DOF problem. Although these are non-lnear problems, the calculaton cost s extremely low and to fnd the global optmal soluton s easy 3578

7 Fg. 13. Input mage. The mage sequence s acqured n an ndoor envronment texture-less hall way). Fg. 14. Extracted lnes. Blue lnes show parallel lnes. Red lnes show the other lnes. even f an ntal value near to the ground truth s not gven. Experments show the effectveness of the proposed method. As future works, effcent and robust search for obtanng the soluton should be consdered. We should verfy the proposed method n a large envronment. I. ACKNOWLEDGMENTS Ths work was n part supported by MET KAKENHI, Grant-n-Ad for oung Scentst A), REFERENCES [1] D. Nster: Reconstructon From Uncalbrated Sequences wth a Herarchy of Trfocal Tensors, Proceedngs of the 6th European Conference on Computer Vson, Vol. 1, pp , 000. Shape of hallway Parallel lnes Other lnes Camera coordnate system [] D. Nster: An Effcent Soluton to the Fve-Pont Relatve Pose Problem, IEEE Transactons on Pattern Analyss and Machne Intellgence, Vol. 6, No. 6, pp , 004. [3] A. J. Davson: Real-Tme Smultaneous Localsaton and Mappng wth a Sngle Camera, Proceedngs of the 9th IEEE Internatonal Conference on Computer Vson, Vol., pp , 003. [4] M. Tomono: 3D Object Mappng by Integratng Stereo SLAM and Object Segmentaton Usng Edge Ponts, Advances n Vsual Computng, Lecture Notes n Computer Scence, Vol. 5875, pp , 009. [5] J. Sh and C. Tomas: Good Features to Track, Proceedngs of the 1994 IEEE Computer Socety Conference on Computer Vson and Pattern Recognton, pp , [6] D. G. Lowe: Dstnctve Image Features from Scale-Invarant Keyponts, Internatonal Journal of Computer Vson, Vol. 60, No., pp , 004. [7] J. Gluckman and S. K. Nayar: Ego-moton and Omn-drectonal Cameras, Proceedngs of the 6th Internatonal Conference on Computer Vson, pp , [8] R. Bunschoten and B. Krose: Robust Scene Reconstructon from an Omndrectonal Vson System, IEEE Transactons on Robotcs and Automaton, Vol. 19, No., pp , 003. [9] C. Geyer and K. Danlds: Omndrectonal Vdeo, The Vsual Computer, Vol. 19, No. 6, pp , 003. [10] A. Murllo, J. Guerrero, and C. Sagues, SURF Features for Effcent Robot Localzaton wth Omndrectonal Images, Proceedngs of the 007 IEEE Internatonal Conference on Robotcs and Automaton, pp , 007. [11] A. Bartol and P. Sturm: Mult-Vew Structure and Moton from Lne Correspondences, Proceedngs of the 9th IEEE Internatonal Conference on Computer Vson, pp. 07-1, 003. [1] A. Bartol and P. Sturm: Structure-from-moton usng lnes: Representaton, Trangulaton, and Bundle Adjustment, Computer Vson and Image Understandng, Vol. 100, Issue 3, pp , 005. [13] P. Smth, I. Red and A. Davson: Real Tme Monocular SLAM wth Straght lnes, Proceedngs of the 006 Brtsh Machne Vson Conference, pp. 17-6, 006. [14] R. Hartley: In Defense of the Eght-pont Algorthm, IEEE Transactons on Pattern Analyss and Machne Intellgence, Vol. 19, No. 6, pp , [15] P. H. S. Torr and A. sserman: Robust Parameterzaton and Computaton of the Trfocal Tensor, Image and Vson Computng, Vol. 15, Issue 8, pp , [16] R. Hartley: Lnes and Ponts n Three Vews and the Trfocal Tensor, Internatonal Journal of Computer Vson, Vol., No., pp , [17] M. Bosse, R. Rkosk, J. Leonard, S. Teller: Vanshng Ponts and 3D Lnes from Omndrectonal Vdeo, Proceedngs of the 00 Internatonal Conference on Image Processng, Vol. 3, pp , 00. [18] G. Schndler, P. Krshnamurthy and F. Dellaert: Lne-Based Structure from Moton for Urban Envronments, Proceedngs of the 3rd Internatonal Symposum on 3D Data Processng, Vsualzaton, and Transmsson, pp , 006. [19] G.L. Marottn and D. Prattchzzo: Uncalbrated Vdeo Compass for Moble Robots from Paracatadoptrc Lne Images, Proceedngs of the 007 IEEE/RSJ Internatonal Conference on Intellgent Robots and Systems, pp. 6-31, 007. [0] J. F. Canny: A Computatonal Approach to Edge Detecton, IEEE Transactons on Pattern Analyss and Machne Intellgence, Vol. PAMI-8, No. 6, pp , [1] M. A. Fschler and R. C. Bolles: Random Sample Con-sensus: A Paradgm for Model Fttng wth Applcatons to Image Analyss and Automated Carto-graphy, Communcatons of the ACM, Vol. 4, No. 6, pp , [] K. Nakayama and G. Slverman: The Aperture Problem I. Spatal Integraton of Velocty Informaton Along Contours, Vson Research, Vol. 8, No. 6, pp , [3] B. Trggs, P. McLauchlan, R. Hartley and A. Ftzgbbon: Bundle Adjustment -A Modern Synthess, Vson Algorthms: Theory and Practce, Lecture Notes n Computer Scence, Vol pp , 000. Fg. 15. Estmaton result of camera movement and lne measurement. 3579