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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, inf , 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 inf , depend on both K and camera rotation.
• The image of the absolute conic, , depends only on K; it is unaffected by camera rotation and position.
• = ( KKT )-1
Camera properties 2
• 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.
Perspective image of points on a plane
Action of a projective camera on planes
Action of a projective camera on lines
Action of a projective camera on lines
Line projection
Action of a projective camera on conics
Action of a projective camera on conics 2
On conics
Images of smooth surfaces
Images of smooth surfaces 2
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 7.8
On quadrics
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.
The cone rays of a quadric
The cone rays with vertex the camera centre
Example 7.10
The importance of the camera centre
The camera centre
Moving image plane
Moving image plane 2
Moving image plane 3
Camera rotation
Example
(a), (b) camera rotates about camera centre. (c) camera rotates about camera centre and translate
Synthetic views
Synthetic views. (a) Source image(b) Frontal parallel view of corridor floor
Synthetic views. (a) Source image(c) Frontal parallel view of corridor wall
Planar panoramic mosaicing
Three images acquired by a rotating camera may be registered to the frame of
the middle one
Planar panoramic mosaicing 1
Planar panoramic mosaicing 2
Planar panoramic mosaicing 3
Projective (reduced) notation
Moving camera centre
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 inf are related by planar homography for any camera motion.
Motion parallax
Camera calibration and image of the absolute conic
The angles between two rays
The angle between two rays
Relation between an image line and a scene plane
The image of the absolute conic
The image of the absolute conic 2
The image of the absolute conic 3
The image of the absolute conic 4
The image of the absolute conic 5
Example: A simple calibration device
Calibration from metric planes
100
395.91097.80
525.89.8-1108.3
K
Outline of the calibration algorithm
Orthogonality in the image
Orthogonality in the image 2
Orthogonality represented by pole- polar relationship
Reading the internal parameters K from the calibrated conic
To construct the line perpendicular to the ray through image point x
Vanishing point formation(a) Plane to line camera
Vanishing point formation:3-space to plane camera
Vanishing line formation(a)
Vanishing line formation (b)
Vanishing points and lines
Image plane and principal point
The principal point is the orthocentre of an orthongonal triad of vanishing points
in image (a)
The principal point is the orthocentre of the triangle with the vanishing point as the vertices
The calibrating conic computed from the three orthogonal vanishing point
The calibrating conic for the image (a)