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