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 ae’ = H e

Compatibility constraints 2 Fig. 12.2 b HT le

’ = le

Compatibility constraints 3 Fig. 12.2 c x)H ( x x F ,'

e l

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(

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