computation of fundamental matrix f

Computation of Fundamental matrix F

Basic equations

• x’T F x = 0• x’= (x’, y’, 1)T x = (x, y, 1)T

0333231232212131211 fyfxffyyfyxfyfxyfxxfx

Basic equations 2

Basic equations 2

The singularity constraint

The singularity constraint 2

The singularity constraint 3

Fig. 10.1 Epipolar lines

Epipolar lines

10.2 The normalized 8 point algorithm

The normalized 8 point algorithm

The normalized 8 point algorithm

Computing F: Recommendations

Image pairs with epipoles far from the image centres Fig 10.2

Image pairs with epipoles close to the image centres Fig. 10.2

Automatic computation of F

Automatic computation of F 2

Automatic computation of F 3

Automatic computation of fundamental matrix using RANSAC 640 x 480 pixels

Detected corners(500) superimposed on the images

188 putative matches shown by the line linking corners, 89 are outliers

Inliners –99 correspondences consistent with the estimated F

Final set of 157 correspondence after guided matching using MLE, with a few mismatches

(e.g. the long line on the left)

Special cases of F-computation

Fig. 10.5 for a pure translation, the epipole can be estimated from the image motion of

two points

Translational motion

10. 7.2 Planar motion

10.7.3 The calibrated case

10.7.3 The calibrated case 2

10.8 Correspondence of other entitiesLine 1

10.8 Correspondence of other entitiesLine 2

10.8 Correspondence of other entitiesSpace curves

10.8 Correspondence of other entitiesSurfaces

Epipolar tangency

Fig.10.6 Epipolar tangency

10.9 Degeneracies

Table 10.1

10.9.1 Points on a ruled quadric

10.9.1 Points on a ruled quadric 2

10.9.2 Points on a plane

10.9.2 Points on a plane 2

10.9.2 Points on a plane 3

10.9.3 No translation:The epipolar geometry is not defined.

Two images are related by a 2D homography

10.12 Image rectification

10.12 Image rectification 2

Mapping the epipole to infinity

Force the transformation H to be rigid transformation in the neighborhood of x0

A good choice of x0 be the image centre

X0 is the origin

X0 is arbitrary placed point of interest

10.12.2 Matching transformations

The strategy

Result 10.3 for matching transform

An Affine transform

Corollary 10.4 A special case for matching transform

Rectification algorithm outline

Rectification algorithm outline 2

Image rectification Fig 10.11 aA pair of images of a house

Image rectification Fig 10.11 bResampled images, corresponding points match

horizontally

Example 10.5 Model house images

Affine rectification

Fig.10.12 a Image rectification using affinities : a pair of images

Fig. 10.12 b Affine rectification:The average y-disparity is of the order of 3

pixels in a 512 x 512 image