Scene planes and homographies Homographies given the plane

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 H T l e’ = l e

Compatibility constraints 3 Fig. 12. 2 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

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 = H 2 -1 H 1 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

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