computation of fundamental matrix f
DESCRIPTION
Computation of Fundamental matrix F. Basic equations. x’ T F x = 0 x’= ( x’, y’, 1) T x = ( x, y, 1) T. Basic equations 2. Basic equations 2. The singularity constraint. The singularity constraint 2. The singularity constraint 3. Fig. 10.1 Epipolar lines. - PowerPoint PPT PresentationTRANSCRIPT
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