3D reconstruction of cameras and structure x i = PX i x’ i = P’X i
Outline of Reconstruction method 1. Compute the fundamental matrix from point correspondences 2. Compute the camera matrices from the fundamental matrix 3. For each point correspondence x i x’ i, compute the point in space that projects to these 2 image points
Computation of the fundamental matrix x’ i F x i = 0 With the x’ I and x i known, this equation is linear in the unknown entries of the matrix F. Thus 8 pairs of corresponding points is sufficient to solve for the entries of F up to scale. Usually, more than 8 point correspondences are used in a least square solution.
Computation of the camera matrices
Triangulation
Reconstruction ambiguity (a)
Reconstruction ambiguity (b)
Fig 9.2 Reconstruction ambiguity
Ambiguity for calibrated camera
Projective ambiguity
Projective reconstruction theorem
Relationship between projective and Euclidean reconstructions
Projective reconstruction
Projective Reconstruction 2 views of a house Fig. 9.3 a
Two views of a 3D projective reconstruction ( camera calibration matrices and scene geometry are not required) Fig 9.3b
Stratified reconstruction
The step to affine reconstruction
The essence of affine reconstruction is to locate the plane at infinity
Translation motion, Scene constraints
Parallel lines, distance ratios on a line
Projective reconstruction can be upgraded to affine using parallel scene lines
Affine reconstruction
Affine reconstruction 2
Affine reconstruction 3
The infinite homography
Result 9.3
One of the cameras is affine
The step to metric reconstruction
Proof
Proof 2
Constraints
Constraints 2
Constraints from the same cameras in all images
Direct metric reconstruction uisng
Metric Reconstruction Fig. 9.5
Metric Reconstruction Texture mapped piecewise planar model
Metric Reconstruction 2
Direct Reconstruction Fig 9.6
Direct Reconstruction
Direct reconstruction Fig. 9.6
Direct reconstruction 2
Direct reconstruction 3
Table 9.1