Epipolar geometry

# Epipolar geometry - Epipolar geometry Three questions(i...

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Epipolar geometry

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(i) Correspondence geometry: Given an image point x in the first view, how does this constrain the position of the corresponding point x’ in the second image? (i) Camera geometry (motion): Given a set of corresponding image points {x i ↔x’ i }, i=1,…,n, what are the cameras P and P’ for the two views? Or what is the geometric transformation between the views? (iii) Scene geometry (structure): Given corresponding image points x i ↔x’ i and cameras P, P’, what is the position of the point X in space? Three questions:
The epipolar geometry C,C’,x,x’ and X are coplanar

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The epipolar geometry All points on π project on l and l’
The epipolar geometry The camera baseline intersects the image planes at the epipoles e and e’ . Any plane π conatining the baseline is an epipolar plane . All points on π project on l and l’.

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The epipolar geometry Family of planes π and lines l and l’ Intersection in e and e’
The epipolar geometry epipoles e,e’ = intersection of baseline with image plane = projection of projection center in other image = vanishing point of camera motion direction an epipolar plane = plane containing baseline (1-D family) an epipolar line = intersection of epipolar plane with image (always come in corresponding pairs)

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Example: converging cameras
Example: motion parallel with image plane

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Example: forward motion e e’
Matrix form of cross product [ ] b a b a a a a a a b a b a b a b a b a b a b a × = - - - = - - - = × 0 0 0 1 2 1 3 2 3 1 2 2 1 3 1 1 3 2 3 3 2 0 ) ( 0 ) ( = × = × b a b b a a

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Geometric transformation ] | [ ' with ' ' ' ] 0 | [ with ' t R M P M p I M MP p t RP P = = = = + =
Calibrated Camera = = = × T T v u p v u p Rp t p ) 1 , ' , ' ( ' ) 1 , , ( with 0 )] ( [ ' 0 ' = Ep p [ ] SR R t E Ep p = = = × with 0 ' Essential matrix

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Uncalibrated Camera 0 ' = p E p a a s coordinate camera in ' and to ing correspond s coordinate pixel in points ' and p p p p a a p' M p p M p ' ' and 1 int 1 int - - = = with 0 ' 1 int ' int - - = = M E M F F p p T T Fundamental matrix
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• Winter '08
• staff
• Singular value decomposition, Fundamental matrix, Epipolar geometry, Projective geometry, Geometry in computer vision

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