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. 12.2 b
HT le’ = le
Compatibility constraints 3
Fig. 12.2 c
le'  F x  x,  ( H x)
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
12.2.1 Three points
Three points
The first (explicit) method is
preferred
Degenerate geometry for an implicit
computation of the homography Fig. 12.3
Fig. 12.3 Legend
Determining the points Xi 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
12.2.2 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 12.8 2
Geometric interpretation of the point map
H(m)
Explore the
A homography between corresponding line
images Fig. 12.5
Fig. 12.5 Legend
Degenerate homographies
Degenerate homographies 2
A degenerate homography Fig. 12.6
Fig. 12.6 Legend
12.3 Computing F given the homography
induced by a plane
Plane induced parallax
Plane induced parallax
Fig. 12.7
Fig. 12.7 Legend
Plane induced parallax 2 Fig. 12.8
Fig. 12.8 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. 12.9 Legend
Projective Depth
Example 12.9
Binary space partition: left and right images
Fig. 12.10 a,b
(c ) Points with known correspondence
(d) A triplet of points selected from ( c ) and
this triplet defines a plane Fig. 12.10 c,d
(e) Points on one side of the plane
(f) Points on the other side Fig. 12.10 e, f
Fig 12.10 Legend
Two planes
Two planes 2
The action of the map H = H2-1 H1 on x
Fig. 12.11
Fig. 12.11 Legend
Two planes 3
Up to this points, the results of this chapter have
been entirely projective
12.4 The infinite homography Hinf
The infinite homography Hinf 2
The infinite homography Hinf 3
Vanishing points and lines
The infinite homography Hinf maps vanishing points
between images Fig. 12.12
Affine and metric reconstruction
Affine and metric reconstruction 2
Affine and metric reconstruction 3
Stereo Correspondence
Reducing the search region using Hinf
Fig 12.13
Fig. 12.13 Legend