Download - Computation of Fundamental matrix F
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Computation of Fundamental matrix F
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Basic equations
• x’T F x = 0• x’= (x’, y’, 1)T x = (x, y, 1)T
0333231232212131211 fyfxffyyfyxfyfxyfxxfx
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Basic equations 2
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Basic equations 2
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The singularity constraint
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The singularity constraint 2
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The singularity constraint 3
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Fig. 10.1 Epipolar lines
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Epipolar lines
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10.2 The normalized 8 point algorithm
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The normalized 8 point algorithm
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The normalized 8 point algorithm
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Computing F: Recommendations
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Image pairs with epipoles far from the image centres Fig 10.2
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Image pairs with epipoles close to the image centres Fig. 10.2
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Automatic computation of F
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Automatic computation of F 2
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Automatic computation of F 3
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Automatic computation of fundamental matrix using RANSAC 640 x 480 pixels
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Detected corners(500) superimposed on the images
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188 putative matches shown by the line linking corners, 89 are outliers
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Inliners –99 correspondences consistent with the estimated F
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Final set of 157 correspondence after guided matching using MLE, with a few mismatches
(e.g. the long line on the left)
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Special cases of F-computation
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Fig. 10.5 for a pure translation, the epipole can be estimated from the image motion of
two points
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Translational motion
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10. 7.2 Planar motion
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10.7.3 The calibrated case
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10.7.3 The calibrated case 2
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10.8 Correspondence of other entitiesLine 1
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10.8 Correspondence of other entitiesLine 2
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10.8 Correspondence of other entitiesSpace curves
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10.8 Correspondence of other entitiesSurfaces
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Epipolar tangency
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Fig.10.6 Epipolar tangency
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10.9 Degeneracies
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Table 10.1
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10.9.1 Points on a ruled quadric
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10.9.1 Points on a ruled quadric 2
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10.9.2 Points on a plane
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10.9.2 Points on a plane 2
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10.9.2 Points on a plane 3
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10.9.3 No translation:The epipolar geometry is not defined.
Two images are related by a 2D homography
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10.12 Image rectification
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10.12 Image rectification 2
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Mapping the epipole to infinity
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Force the transformation H to be rigid transformation in the neighborhood of x0
A good choice of x0 be the image centre
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X0 is the origin
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X0 is arbitrary placed point of interest
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10.12.2 Matching transformations
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The strategy
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Result 10.3 for matching transform
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An Affine transform
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Corollary 10.4 A special case for matching transform
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Rectification algorithm outline
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Rectification algorithm outline 2
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Image rectification Fig 10.11 aA pair of images of a house
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Image rectification Fig 10.11 bResampled images, corresponding points match
horizontally
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Example 10.5 Model house images
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Affine rectification
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Fig.10.12 a Image rectification using affinities : a pair of images
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Fig. 10.12 b Affine rectification:The average y-disparity is of the order of 3
pixels in a 512 x 512 image