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Unformatted text preview: 9 Epipolar Geometry and the Fundamental Matrix The epipolar geometry is the intrinsic projective geometry between two views. It is independent of scene structure, and only depends on the cameras’ internal parameters and relative pose. The fundamental matrix F encapsulates this intrinsic geometry. It is a 3 × 3 matrix of rank 2. If a point in 3-space X is imaged as x in the first view, and x ′ in the second, then the image points satisfy the relation x ′ T F x = 0 . We will first describe epipolar geometry, and derive the fundamental matrix. The properties of the fundamental matrix are then elucidated, both for general motion of the camera between the views, and for several commonly occurring special motions. It is next shown that the cameras can be retrieved from F up to a projective transformation of 3-space. This result is the basis for the projective reconstruction theorem given in chapter 10. Finally, if the camera internal calibration is known, it is shown that the Eu- clidean motion of the cameras between views may be computed from the fundamental matrix up to a finite number of ambiguities. The fundamental matrix is independent of scene structure. However, it can be com- puted from correspondences of imaged scene points alone, without requiring knowl- edge of the cameras’ internal parameters or relative pose. This computation is de- scribed in chapter 11. 9.1 Epipolar geometry The epipolar geometry between two views is essentially the geometry of the inter- section of the image planes with the pencil of planes having the baseline as axis (the baseline is the line joining the camera centres). This geometry is usually motivated by considering the search for corresponding points in stereo matching, and we will start from that objective here. Suppose a point X in 3-space is imaged in two views, at x in the first, and x ′ in the second. What is the relation between the corresponding image points x and x ′ ? As shown in figure 9.1a the image points x and x ′ , space point X , and camera centres are coplanar. Denote this plane as π . Clearly, the rays back-projected from x and x ′ intersect at X , and the rays are coplanar, lying in π . It is this latter property that is of most significance in searching for a correspondence. 239 240 9 Epipolar Geometry and the Fundamental Matrix C C / π x x X epipolar plane / x e X ? X X ? l e epipolar line for x / / a b Fig. 9.1. Point correspondence geometry. (a) The two cameras are indicated by their centres C and C ′ and image planes. The camera centres, 3-space point X , and its images x and x ′ lie in a common plane π . (b) An image point x back-projects to a ray in 3-space defined by the first camera centre, C , and x . This ray is imaged as a line l ′ in the second view. The 3-space point X which projects to x must lie on this ray, so the image of X in the second view must lie on l ′ ....
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- Fall '08