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##### More single view geometry

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**More single view geometry**Describes the images of planes, lines,conics and quadrics under perspective projection and their forward and backward properties**Camera properties**• Images acquired by the cameras with the same centre are related by a plane projective transformation • Image entities on the plane at infinity, pinf , do not depend on camera position, only on camera rotation and internal parameters, K**Camera properties 2**• The image of a point or a line on pinf , depend on both K and camera rotation. • The image of the absolute conic, w , depends only on K; it is unaffected by camera rotation and position. • w = ( KKT )-1**Camera properties 2**• w defines the angle between the rays back-projected from image points • Thus camera rotation can be computed from vanishing points independent from camera position. • In turn, K may be computed from the known angle between rays; in particular, K may be computed from vanishing points corresponding to orthogonal scene directions.**Contour generator and apparent contour: for parallel**projection**Contour generator and apparent contour: for central**projection**Action of a projective camera on quadrics**• Since intersection and tangency are preserved, the contour generator is a (plane) conic. Thus the apparent contour of a general quadric is a conic, so is the contour generator.**Result 7.9**• The cone with vertex V and tangent to the quadric is the degenerate quadric • QCO = (VT QV) Q – (QV)(QV)T • Note that QCOV = 0, so that V is the vertex of the cone as assumed.**(a), (b) camera rotates about camera centre. (c) camera**rotates about camera centre and translate**Synthetic views. (a) Source image(b) Frontal parallel view**of corridor floor**Synthetic views. (a) Source image(c) Frontal parallel view**of corridor wall**Three images acquired by a rotating camera may be registered**to the frame of the middle one**Parallax**• Consider two 3-space points which has coincident images in the first view( points are on the same ray). If the camera centre is moved (not along that ray), the iamge coincident is lost. This relative displacement of image points is termed Parallax. • An important special case is when all scene points are coplanar. In this case, corresponding image points are related by planar homography even if the camera centre is moved. Vanishing points, which are points on pinf are related by planar homography for any camera motion.