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Tham khảo tài liệu 'vision systems - applications part 11', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Omnidirectional Vision-Based Control From Homography 391 2.3 Polar lines The quadratic equation 5 is defined by five coefficients. Nevertheless the catadioptric image of a 3D line has only two degrees of freedom. In the sequel we show how we can get a minimal representation using polar lines. Let and A be respectively a 2D conic curve and a point in the definition plane of . The polar line l of A with respect to is defined by l a A. Now consider the principal point Oi u0 v0 1 T K 0 0 1 T and the polar line li of Oi with respect to i li a i Oi then -1 -1 .1T li K 1 QK 1Oi K 1 QK 1K 0 0 1 _.-Tr K h Moreover equation 6 yields T KTli h M 6 7 It is thus clear that the polar line li contains the coordinates of the projection of the 3D line L in an image plane of an equivalent virtual perspective camera defined by the frame Fv Fm see Figure 2 with internal parameters chosen equal to the internal parameters of the catadioptric camera i.e Kv Kc M . This result is fundamental since it allows us to represent the physical projection of a 3D line in a catadioptric camera by a simple polar line in a virtual perspective camera rather than a conic. Knowing only the optical center Oi it is thus possible to use the linear pin-hole model for the projection of a 3D line instead of the non linear central catadioptric projection model. 3. Scaled Euclidean reconstruction Several methods were proposed to obtain Euclidean reconstruction from two views Faugeras et al 1988 . They are generally based on the estimation of the fundamental matrix Faugeras et al 96 in pixel space or on the estimation of the essential matrix Longuet and Higgins 1981 in normalized space. However for control purposes the methods based on the essential matrix are not well suited since degenerate configurations can occur such as pure rotational motion . Homography matrix and Essential matrix based approaches do not share the same degenerate configurations for example pure rotational motion is not a degenerate .