Scene planes and homographies
Homographies given the plane and vice versa
Proof of result 12.1
Example 12.2 A calibrated stereo rig
A calibrated stereo rig 2
A calibrated stereo rig 3
The homography induced by a plane Fig.12.1
Fig 12.1 Legend
Homographies compatible with epipolar Geometry
Two sets of 4 arbitrary points from 2 images
Epipolar geometry define conditions on homographies
Counting degrees of freedom
Compatibility constraints Fig.12.2 a e’ = H e
Compatibility constraints 2 Fig b H T l e ’ = l e
Compatibility constraints 3 Fig c
Fig 12.2 Compatibility constraints
Result 12.3
Homographies are compatible with fundamental matrix
Corollary 12.4
Result 12.5
13.6 Plane induced homographies given F and image correspondences: (a) 3 points, (b) a line and a point
Three points
Three points
The first (explicit) method is preferred
Degenerate geometry for an implicit computation of the homography Fig. 12.3
Fig Legend
Determining the points X i is not necessary in first method All that is important
Result 12.6
Proof
Proof 2
Consistency conditions
Consistency conditions 2
Algorithms 12.1 The optimal estimate of homography induced by a plane defined by 3 points
A point and line
A one parameter family of homographies Fig 12.4 (a), (b)
Fig 12.4 Legend
Result 12.7
Proof of result 12.7
Proof of result 12.7 (2)
Proof of result 12.7 (3)
Result 12.8
Result
Geometric interpretation of the point map H( Explore the
A homography between corresponding line images Fig. 12.5
Fig Legend
Degenerate homographies
Degenerate homographies 2
A degenerate homography Fig. 12.6
Fig Legend
12.3 Computing F given the homography induced by a plane
Plane induced parallax
Plane induced parallax Fig. 12.7
Fig Legend
Plane induced parallax 2 Fig. 12.8
Fig Legend
Plane induced parallax 2
Algorithm 12.2 Computing F given the correspondence of 6 points, 4 of which are coplanar
Fundamental matrix from 6 points of which 4 are coplanar Fig. 12.9
Fig Legend
Projective Depth
Example 12.9
Binary space partition: left and right images Fig a,b
(c ) Points with known correspondence (d) A triplet of points selected from ( c ) and this triplet defines a plane Fig c,d
(e) Points on one side of the plane (f) Points on the other side Fig e, f
Fig Legend
Two planes
Two planes 2
The action of the map H = H 2 -1 H 1 on x Fig
Fig Legend
Two planes 3 Up to this points, the results of this chapter have been entirely projective
12.4 The infinite homography H inf
The infinite homography H inf 2
The infinite homography H inf 3
Vanishing points and lines
The infinite homography H inf maps vanishing points between images Fig
Affine and metric reconstruction
Affine and metric reconstruction 2
Affine and metric reconstruction 3
Stereo Correspondence
Reducing the search region using H inf Fig 12.13
Fig Legend