Structure and Motion from Uncalibrated Catadioptric Views

Transcription

1 Structure and Motion from Uncalibrated Catadiotric Views Christoher Geyer and Kostas Daniilidis Λ GRASP Laboratory, University of Pennsylvania, Philadelhia, PA 94 fcgeyer,kostasgseas.uenn.edu Abstract In this aer we resent a new algorithm for structure from motion from oint corresondences in images taken from uncalibrated catadiotric cameras with arabolic mirrors. We assume that the unknown intrinsic arameters are three the combined focal length of the mirror and lens and the intersection of the otical axis with the image. We introduce a new reresentation for images of oints and lines in catadiotric images which we call the circle sace. This circle sace includes imaginary circles, one of which is the image of the absolute conic. We formulate the eiolar constraint in this sace and establish a new 4 4 catadiotric fundamental matrix. We show that the image of the absolute conic belongs to the kernel of this matrix. This enables us to rove that Euclidean reconstruction is feasible from two views with constant arameters and from three views with varying arameters. In both cases, it is one less than the number of views necessary with ersective cameras.. Introduction During the last years there has been a considerable effort in studying the reconstruction of scenes from uncalibrated ersective views given oint corresondences. This is considered now a thoroughly understood roblem. Solutions and insights gained from these studies boosted alications in video rocessing and image based rendering. Two books [] and [5] contain comrehensive treatments of the subject. In the meantime, comuter vision researchers realized that ersective cameras are just one modality among many. Motivated by the need for a anoramic field of view, catadiotric cameras have been designed and can be already urchased off-the-shelf. For an extensive coverage the reader is referred to the recent book by Benosman and Kang [2] and the roceedings of the Worksho for Omnidirectional Vision [4]. Among several designs, the catadiotric systems with a single effective viewoint, called central Λ This work has been suorted by NSF IIS-992 (sub-contracted to UNC), NSF IIS-8329, NSF CDS , Penn Research Foundation grants, and a GAANN fellowshi. catadiotric, attracted secial attention due to their elegant and useful geometric roerties. Several authors have studied the roerties of central catadiotric cameras and the image formation in them [5, 2, 3, 22, 2, 8]. Kang [2] roosed a single view aroach from the image of the circular mirror boundary of a araboloid mirror. Geyer and Daniilidis showed [7, 8] how calibration of a arabolic catadiotric system can be achieved from a single view of three lines in sace or from a single view of two sets of arallel lines. In this aer, we study the recovery of motion and scene structure from multile arabolic catadiotric views. Such views can be obtained from a reflective surface of revolution of arabolic rofile and an orthograhic lens. We assume that the otical axes of the lens and the mirror are arallel. They do not have to coincide but to avoid aberrations and enable maximal coverage of the CCD-chi they should be close to each other. We assume, thus, that the catadiotric system is correctly aligned. We further assume that the asect ratio and skew arameter are known leaving only the focal length (combined scaling factor of mirror, lens, and CCD-chi) and the image center (intersection of the otical axis with the image lane) as unknown. It is already known that in such arabolic catadiotric systems lines roject onto circles. We introduce a new reresentation for circles in the image lane the circle sace of three dimensions. This sace is divided into two arts by an abstract araboloid. The exterior of the araboloid reresents all circles with real radius and the interior all circles with imaginary radius. The sace does not contain circles with comlex radii but the araboloid itself reresents all circles with zero radius which are just oints on the lane. By lifting each image oint to a oint of the araboloid and each image circle to a oint outside the araboloid we have one sace for both oints and circles. The fact that we can reresent imaginary circles enables us to reresent the image of the absolute conic. In the calibrated case, the image of the absolute conic is the focus of the abstract araboloid in the circle sace. In the noncalibrated case, the imaginary image of the absolute conic is a oint inside the abstract araboloid that is vertically symmetric to the oint reresenting the real image of the fronto-arallel horizon.

2 We formulate the calibration roblem as the question for a linear transformation that will ma uncalibrated oints on the abstract araboloid to calibrated oints on a araboloid and the image of the absolute conic to its focus. Indeed, such a linear transformation K exists and encodes all three intrinsic arameters (focal length and image center). The question is now to find this maing from multile views. It turns out that we can formulate the eiolar constraint using rojective coordinates of the circle sace we have been working on. A new 4 4 catadiotric fundamental matrix is comosed from the essential matrix E and an induced rojection following the maing K above. We rove that the circle reresentation of the images of the absolute conic in the left and the right view resectively lie in the left and right nullsaces of the catadiotric fundamental matrix. Because the catadiotric fundamental matrix is rank 2, the image of the absolute conic is in the intersection of the left and right nullsace if the intrinsic arameters are constant and rotation does not vanish and is not about the translation direction. For three views, it is even ossible to determine the image of the three different absolute conics in the case of varying intrinsics. Thus, the main result of this aer is that, with unknown focal length and image center, Euclidean reconstruction from arabolic catadiotric views is feasible. From two views with the same camera arameters. 2. From three views with varying camera arameters. In both cases, it is one view less, than in the case of ersective views with the same unknowns (focal length and image center) Three views are necessary for constant arameters [4, 3] and four views are necessary for varying arameters []. In articular the fundamental matrix has seven degrees of freedom whereas the intrinsics have three and the motion has five for a total of eight. In the three view case the trifocal tensor has 9 free arameters whereas the three intrinsincs have nine lus for the motion, yielding a total of 2. We are not going to review here the vast amount of literature on uncalibrated Euclidean reconstruction which has been comrehensively summarized in the two recent books [, 5]. The main result [4] is that three views suffice for Euclidean reconstruction with all intrinsics unknown but constant. The results still hold for known asect ratio and skew. Hartley [] showed that a varying focal length can be recovered from two views with all other intrinsic arameters fixed. Sturm [8] studied the degenerate configurations for the same assumtion. Heyden and Astrom [] roved that four views suffice for unknown varying focal length and image center but known asect ratio and skew. Pollefeys et al. [7] studied several configurations of unknown and varying arameters. In the omnidirectional vision literature, there are very few aroaches dealing with structure from motion. Gluckman and Nayar [9] studied ego-motion estimation by maing the catadiotric image to the shere. Svoboda et al [2] first established the eiolar geometry for all central catadiotric systems. Kang [2] roosed a direct selfcalibration by minimizing the eiolar constraint. Fermueller and Aloimonos [6] roved the sueriority of the shere over the lane regarding stability. Teller [] showed how to comute ego-motion from sherical mosaics. Multile view algorithms for the ersective case which assume iecewise lanar environments are simler when modified for catadiotric imagery. [2, 9]. In the next section we mention introductory facts about catadiotric geometry. We introduce the notion of circle sace and we find the image of the absolute conic on that sace. We finish the second section with the recovery of the image of the absolute conic from the catadiotric fundamental matrix. In the third section we resent reconstruction algorithms for two and three views. In the fourth section a real exeriment is described. 2. Prearations 2.. Known Facts We recall from [7] some facts about the rojection induced by arabolic mirror. Fact. In a coordinate system whose origin is the focus of the araboloid and axis of symmetry coincides with the z-axis, the rojection of a sace oint (x; y; z; ) is in image coordinates 2fx c x + u v A = B c y + z+ x 2 +y 2 +z 2 2fy z+ x 2 +y 2 +z 2 C A ; () where f is the combined focal length of the mirror and camera, and (c x ;c y ) is the image center, the intersection of the axis of the arabola with the image lane. We assume that the asect ratio is and that there is no skew. The image oint is obtained by intersecting the ray through the focus and the sace oint with the arabola, then orthograhically rojecting the intersection to a lane erendicular to the axis of the araboloid. Fact 2. The horizon of the fronto-arallel lane, the lane erendicular to the axis of the araboloidal mirror, is the circle (c x u) 2 +(c y v) 2 =4f 2 (2) This circle of radius 2f centered about the image center is the equivalent of the calibrating conic which we call! since we call the image of the absolute conic!.

3 Π ß Figure. A circle fl is reresented by the oint ~fl. The lane ß is the olar lane of ~fl with resect to Π. fl is obtained by rojecting the intersection of ß with Π to the lane. ~fl fl to one arameter systems of coaxial circles. Planes in the sace corresond to two arameter systems of circles which intersect a single circle antiodally. See Figure in which a circle is obtained from a oint in sace by taking the olar of the oint with resect to the araboloid, and rojecting to a lane the intersection of the olar lane with the araboloid; this rojection will be a circle. We call the araboloid Π; it is given by the quadratic form C Π = ψ! 2 (5) 2 4 Its focus is at the origin and has a focal length equal to 4. So, Π=Φ T C Π = Ψ n = x; y; x 2 + y To 2 =4; Fact 3. The rojection of a line is an arc of a circle. If ο is the center and R the radius of the circle and if d 2 = (c x ο x ) 2 +(c y ο y ) 2 then 4f 2 + d 2 = R 2 (3) This condition is equivalent to the condition that the circle intersect! antiodally. Fact 4. The image! of the absolute conic Ω is the circle (c x u) 2 +(c y v) 2 = 4f 2 ; (4) centered at the image center with radius 2if. This can be derived by solving for x and y in the rojection formula () after substituting x 2 + y 2 + z 2 =, x = (u cx)z 2f y = (v cy)z 2f Substitute the right hand sides into x 2 + y 2 + z 2 =, obtaining z 2 2 4f 2 +(c 4f 2 x u) 2 +(c y v) = Dividing by z 2 =4f 2 leaves (4). Thus, knowledge of either the absolute conic or the calibrating conic yields the intrinsic arameters Parabolic Circle Sace In the next few aragrahs we consider an abstract araboloid which is different from the hysical araboloid of the mirror. Following Pedoe [6], we use this surface to describe a corresondence between oints in sace and circles in the lane. Lines in this circle sace corresond Definition. radius R Suose fl is the circle centered at (; q) with ( x) 2 +(q y) 2 = R 2 ; (6) where R is ossibly zero or imaginary, but never comlex. Let the oint reresentation of fl be the the rojective oint ~fl = ; q; 2 + q 2 R 2 4 ; T (7) Note that the circle s radius is real iff it lies outside of Π. Its radius is imaginary iff it lies inside of (above) Π. If R =then fl is a single oint and ~fl lies on Π. Thesetof oints f~flg is the arabolic circle sace. When fl is a oint, because ~fl has the same x and y coordinates as fl but lying on Π, we say that ~fl is the lifting of fl to Π. Proosition. If ß is the olar lane of the oint ~fl with resect to the araboloid Π, the orthograhic rojection in the direction of the z-axis of the intersection of ß with Π is the circle fl. Proof The imlicit equation of the olar lane ß of ~fl is = ~fl T C Π x y z T = q 2 +2x +2q z Substitute z = x 2 + y 2 =4, yielding (6). Therefore the oint (; q; r; ) reresents the circle ( x) 2 +(q y) 2 = 2 + q 2 r 4 (8) Λ

4 We can extend the definition to encomass lines as well; they are reresented by oints on the lane at. The olar lane of a oint (; q; r; ) at infinity is the lane = r + x + qy ; 2 which is indeendent of z andsothelineinthelanehas the same equation Alication of Circle Reresentation First, note the oint reresentations of the calibrating conic, ~! = c x ;c y ;c 2x + c2y 4f 2 4 ; T ; (9) which, because it has a real radius, lies outside of Π; and the absolute conic, ~! = c x ;c y ;c 2x + c2y +4f 2 4 ; T ; () which, because it has an imaginary radius, lies inside of Π. The oints ~! and ~! lie the same vertical distance, 4f 2, away from Π. Proosition. The oint reresentations of circles which are images of lines in a arabolic rojection lie in a lane whose ole with resect to Π is ~!. Proof If (; q; r; ) is a circle which is the arabolic rojection of a line it must satisfy (3). Using (8), 4f 2 +(c x ) 2 +(c y q) 2 = 2 + q 2 r 4 4f 2 + c 2 x + c2 y + 4 =2c x +2qc y r; () which, in the variables, q,andr, is the equation of a lane. This lane is reresented by the row vector ß = The oint cx ;c y ; =2; 2f 2 c 2 x =2 c2 y =2 =8 C Π ßT = cx ;c y ;c 2 x + c2 y +4f 2 =4; =~!; is the ole of the lane ß. Λ The araboloid Π was defined so that its focus is the origin. The oint ~! is located at the origin when c x ;c y = and f =. The olar lane of this oint () reduces to 4 r =. In this case, image oints lifted to the arabola 2 exactly corresond to calibrated rays. When these intrinsics hold, the lifting of a sace oint rojected by formula () is a oint on the arabola which is collinear with the focus and the oint in sace. In articular, the rojection of the oint (x; y; z; ) T in sace is ψ 2 x z + x 2 + y 2 + z ; 2 z + x 2 + y 2 + z 2 ;! T ; 2 y according to (). The lifting of this oint is, B ( z+ 2 x z+ x 2 +y 2 +z 2 2 y z+ x 2 +y 2 +z 2 4 x2 + 4 y2 x 2 +y 2 +z 2 ) 2 4 C A / B 2z +2 x y z x 2 + y 2 + z 2 which lies on the line through the focus and the oint (x; y; z; ) T. Is there a linear transformation which transforms oint reresentations of uncalibrated image oints, in which ~! is in general osition, to calibrated rays, in which ~! is the origin? In the next section we show that this is indeed the case Transformations Fixing Π In this section we find linear transformations under which Π is invariant. The four transformations, R = T fi = cos sin sin cos fix fiy 2fix 2fiy fi x 2 + fi2 y ; S ff = ; H = ψ ff ff ff 2 ff2 4 are such that for any choice of, ff, and vectors fi, R T C Π R / C Π ; Sff T C Π S ff / C Π ; Tfi T C Π S fi / C Π ; H T C Π H / C Π ;! C A where C Π was reviously defined in (5) and is the quadratic form of Π. Therefore these transformations affect the arabolic circle sace such that they take oints to oints, as oosed to say oints to circles. The transformations have the following effect on oints in the image lane R induces a rotation of about the origin; S ff induces a scale of ff also about the origin; T fi translates oints by fi; andh reflects about the line x =. Any comosition of these transformations will also leave Π invariant. Note that these transformations also leave ß invariant. They are therefore affine transformations, and also they send lines to lines. These transformations act as similarity transformations on the oints. Do they change the image of the absolute conic and the line image lane so as to correctly reflect the transformation induced on the oints? In other words, say ; ;

5 c x, c y,andf are fixed, alying T fi would induce a translation of fi on oints; it should therefore transform ~! into (c x + fi x;c y + fi y; (c x + fi x) 2 +(c y + fi y) 2 +4f 2 4 ; )T ; and the line image lane () to 2(c x +fi x )+2q(c y +fi y ) r =(c x +fi x ) 2 +(c y +fi y ) ; so that the new image center is (c x + fi x ;c y + fi y ) as desired; any rotation or scaling should act similarly. One can verifythatall fourtransformationstransform ~! and the line image lane in a manner consistent with the way in which the transformations affect oints. Thus, there is a linear transformation taking oint reresentations of image oints obtained from a camera with intrinsic arameters c x, c y,andf, to calibrated rays. This transformation is the 4 4 matrix, K = S = 4f T ( cx; c y) (2) 4f 4cxf 4f 4c yf 2c x 2c y 4 + c2 + x c2 4f 2 y 6f 2 A (3) This is an imortant oint, for if q = (u; v; ) T is the arabolic rojection (with intrinsics c x, c y, f) ofthesace oint =(x; y; z; ) T then for some scalar, = K u; v; u 2 + v 2 4 ; T Imlying that if P = A ; then PK u; v; u 2 + v 2 4 ; T / x z ; y z ; (4) which is the ersective rojection of (x; y; z; ) with image center (; ; ) and focal length f =. Note that K is different from the usual camera matrix it is not actually a rojection; P induces the rojection. Leaving K nonsingular (i.e. not incororating P ) will make it easier to rove that a matrix, a fundamental matrix, created with it has a certain rank The Catadiotric Fundamental Matrix Let m and n be calibrated rays ointing to the same oint (x; y; z; ) in sace taken from two views related by a rotation R and translation t. The oints m and n must satisfy the eiolar constraint which is secified by n T [t] Rm = n T Em =; (5) where E = [t] R is called the essential matrix. Say = (u ;v ; ) T and q = (u 2 ;v 2 ; ) T are two arabolic catadiotric rojections of the sace oint, and say the camera matrices are K and K, with ~! and ~! the oint reresentations of the image of the absolute conic. If ~ and ~q are their liftings to Π, then using equation (4), so that m = PK~ and n = PK ~q, the eiolar constraint (5) becomes, Let the 4 4 matrix ~q T K T P T EPK~ = (6) F = K T P T EPK (7) be called the catadiotric fundamental matrix. Then the eiolar constraint for arabolic catadiotric cameras is ~q T F ~ = (8) Theorem. The catadiotric fundamental matrix defined in (7) has rank 2. Let ~! be the oint reresentation of the image of the absolute conic in the first image, corresonding to K, and similarly for ~! 2 corresonding to K in the second image. Then, ~! 2 F = and F ~! = (9) E Proof The essential matrix E is known to be of rank 2, thus P T EP = has rank 2. Since K and K are non-singular then F must also have rank 2. Let us calculate the left and right null vectors of F.First,lettand t be the images of the viewoints from each camera, t T E =; and Et = Then by insection, linearly indeendent left and right null vectors of P T EP are f = t T ; f 2 = and f = t T ; f 2 = T Hence g i=;2 = K f i are vectors sanning the right nullsace of F and g i=;2 = f i T K T are vectors sanning the left nullsace. Note that g 2 = ~! and g 2 = ~!T 2. Therefore, ~! T 2 F = and F ~! = Corollary. If K = K and t 6= t then, ker F ker F T = f ~!g The condition t 6= t is true when the rotation is not trivial and when the axis of rotation is not the translation vector.

6 3. Algorithm The algorithm roceeds in three stes. First estimate the fundamental matrix, from the fundamental matrix extract the intrinsic arameters via the image of the absolute conic, and reconstruct using well known ersective methods. 3.. Estimating F We use a non-linear method to estimate F. An algorithm based on singular value decomosition which is similar to the the 8-oint algorithm for the ersective case exists for arabolic catadiotric rojections but is equally sensitive.. Obtain images i;j =(u i;j ;v i;j ; ) T of the same oint q j=;;n in sace in two catadiotric views i = ; 2. Let ~ i;j = 4 u i;j ;v i;j ;u 2i;j + v2i;j ; T 2. Minimize the sum of first-order geometric errors, X j (~ 2;j F ~ ;j ) 2 (F ~ 2;j ) ffi (F ~ 2;j )+(F T ~ ;j ) ffi (F T ~ ;j ) ; where the minimization is over F and using the notation ffi q = q. F is arameterized as in B a b ffa + fib fla + ffib c d ffc + fid flc + ffid C e f ffe + fif fle + ffifa ; g h ffg + fih flg + ffih where one of a;;f is held constant at. This ensures that F has rank 2. Initial estimates for F can be obtained using the singular value decomosition method since the comonents of F are linear in coordinates of the lifted image oints Estimating! In the case where K = K the left and right nullsaces of F contain the oint reresentation of the image of the absolute conic. In the resence of noise the nullsaces will not intersect. Once we have calculated the two-dimensional nullsaces, we choose the oint equidistant to the two lines as the estimate of ~!. When the intrinsics vary and we have images from three views, with three matrices K i=;2;3 and oint reresentations ~! i=;2;3,wethenhave F 2 = K T 2 P T E 2 PK ; F 23 = K T 3 P T E 23 PK 2 ; F 3 = K T P T E 3 PK 3 Then once we have estimated the three fundamental matrices we calculate say ~! from the fact that, ker F 2 ker F T 3 = f~! g Again, the estimate of ~! is the oint equidistant to the two nullsaces Reconstruction Reconstruction roceeds as in the calibrated ersective case. Once we have determined ~! and consequently!, we can transform the image oints into calibrated rays with which we determine the essential matrix E using a nonlinear otimization and then back-roject the rays into sace using a linear algorithm, both algorithms described in []. 4. Exeriments We use the algorithm to erform a reconstruction of a scene from two views. The two ictures in Figure 2 are of a building on the camus of our institution and are assumed to have the same intrinsic arameters. First we manually choose and corresond oints in the two images. We calculate the fundamental matrix F between the two views from the oint corresondences using the algorithm described in the revious section. We estimate the oint reresentation of the image of the absolute conic by finding the left and right nullsaces of F and finding the oint equidistant to each. Using the intrinsic arameters we back-roject the image oints to calibrated rays. Using the calibrated rays weestimatetheessentialmatrixe, decomosee into translation and rotation, and determine the ersective camera rojection matrices P and P 2. We then back-roject the rays and use homogeneous linear triangulation to estimate scene oints. The reconstruction is shown in the to and bottom of Figure 3. In the reconstruction we have fitted a lane to the oints on the front facade of the building and to oints on the ground lane, these are highlighted in Figure 2 and shaded differently in Figure 3. The viewoints and oses are also dislayed in the figures. The triangulation is manually added and shown for visualization uroses only. The ground lane and front facade were reconstructed to almost lanar surfaces and are close to erendicular. The other facade of the building, on the left in the images, did not reconstruct true to the scene, this is because this lane is erendicular to the axis of motion which makes estimating deth more error-rone. In two views with such small motion, the reconstruction erforms remarkably well.

7 Figure 2. Two images taken with the same arabolic catadiotric camera. Points are those used for corresondence. Points highlighted in white are on the ground lane; oints highlighted in black are on one side of the building facade. Figure 3. Reconstruction from two images. Black oints are in the ground lane. Darkly shaded oints are on the front facade of the building; lightly shaded oints are on the other facade (which is on the left in the images). Planes are fitted to the facade and ground lane (and translated slightly so oints are made visible). The coordinate systems at the oints are the ose estimates. Tilt of the fitted lane is irrelevant to the results of the reconstruction. The to view is taken looking straight at the front facade; the bottom view is from the side. Note that the mirror reverses the orientation; this has been accounted for in the reconstruction.

8 5. Conclusion We have established a new reresentation for images of lines and oints in arabolic catadiotric cameras. Based on this reresentation we found a natural reresentation for the image of the absolute conic if asect ratio and skew are assumed known. Writing the eiolar constraint in this new sace yields a new catadiotric fundamental matrix. It turns out that the image of the absolute conic belongs to the twodimensional kernel of this matrix. Alying thus only subsace recovery and intersection we can obtain Euclidean reconstructions ffl from two views with the same camera ffl from three views with three different cameras. The corresonding minimal views for the ersective case are three and four, resectively. This aroach oened new questions which we address in our current work What is the number of indeendent conditions on F to be decomosable? What is the degree of the manifold of all catadiotric fundamental matrices? Which oint configurations make the recovery of the fundamental matrix degenerate? What is the minimal number of oints for directly comuting motion and the intrinsics? Sensor resolution of commercial catadiotric cameras is increasing every year. We believe that geometrically intuitive algorithms working directly on catadiotric images can rovide flexible solutions for anoramic image-based rendering and visualization. References [] M. Antone and S. Teller. Automatic recovery of relative camera rotations in urban scenes. In IEEE Conf. Comuter Vision and Pattern Recognition, Hilton Head Island, SC, June 3-5, 2. [2] R. Benosman and S.B. Kang. Panoramic Vision. Sringer- Verlag, 2. [3] A. Bruckstein and T. Richardson. Omniview cameras with curved surface mirrors. In IEEE Worksho on Omnidirectional Vision, Hilton Head, SC, June 2, ages 79 86, 2. originally ublished as Bell Labs Technical Memo, 996. [4] K. Daniilidis, editor. IEEE Worksho on Omnidirectional Vision, Hilton Head Island, SC, June 2, 2. [5] O. Faugeras, Q.-T. Luong, and T. Paadooulo. The Geometry of Multile Images The Laws That Govern the Formation of Multile Images of a Scene and Some of Their Alications. MIT Press, 2. [6] C.Fermüller and Y. Aloimonos. Ambiguity in structure from motion Shere vs. la. International Journal of Comuter Vision, , 998. [7] C. Geyer and K. Daniilidis. Catadiotric camera calibration. In Proc. Int. Conf. on Comuter Vision, ages , Kerkyra, Greece, Se. 2-23, 999. [8] C. Geyer and K. Daniilidis. A unifying theory for central anoramic systems. In Proc. Sixth Euroean Conference on Comuter Vision, ages , Dublin, Ireland, 2. [9] J. Gluckman and S.K. Nayar. Ego-motion and omnidirectional cameras. In Proc. Int. Conf. on Comuter Vision, ages 999 5, Bombay, India, Jan. 3-5, 998. [] R. Hartley and A. Zisserman. Multile View Geometry. Cambridge Univ. Press, 2. [] A. Heyden and K. Aström. Euclidean reconstruction from image sequences with varying and unknown focal length and rincial oint. In IEEE Conf. Comuter Vision and Pattern Recognition, ages , 997. [2] S.B. Kang. Catadiotric self-calibration. In IEEE Conf. Comuter Vision and Pattern Recognition, ages I 2 27, Hilton Head Island, SC, June 3-5, 2. [3] Y. Ma, S. Soatto, J. Kosecka, and S.S. Sastry. Euclidean reconstruction and rerojection u to subgrous. International Journal of Comuter Vision, , 2. [4] S.J. Maybank and O.D. Faugeras. A theory of selfcalibration of a moving camera. International Journal of Comuter Vision, 823 5, 992. [5] S. Nayar. Catadiotric omnidirectional camera. In IEEE Conf. Comuter Vision and Pattern Recognition, ages , Puerto Rico, June 7-9, 997. [6] D. Pedoe. Geometry A comrehensive course. Dover Publications, New York, NY, 97. [7] M. Pollyfeys, R. Koch, and L. van Gool. Self-calibration and metric reconstruction in site of varying and unknown internal camera arameters. In Proc. Int. Conf. on Comuter Vision, ages 9 95, Bombay, India, Jan. 3-5, 998. [8] P. Sturm. Critical motion sequences for the self-calibration of cameras and stereo systems with variable focal length. In BMVC, 999. [9] P. Sturm. A method for 3d-reconstruction of iecewise lanar objects from single anoramic images. In IEEE Worksho on Omnidirectional Vision, Hilton Head, SC, June 2, ages 9 26, 2. [2] T. Svoboda, T. Pajdla, and V. Hlavac. Eiolar geometry for anoramic cameras. In Proc. 5th Euroean Conference on Comuter Vision, ages 28 23, 998. [2] C.J. Taylor. Video lus. In IEEE Worksho on Omnidirectional Vision, Hilton Head, SC, June 2, ages 3, 2. [22] Y. Yagi, S. Kawato, and S. Tsuji. Real-time omnidirectional image sensor (cois) for vision-guided navigation. Trans. on Robotics and Automation, 22, 994.