Epipolar Geometry - dhoiem.cs.illinois.edu
Transcript of Epipolar Geometry - dhoiem.cs.illinois.edu
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Epipolar Geometry Slides mostly from Hartley-Zisserman
CVPR’99 tutorial
Presented by: Varsha HedauDate : 08/04/2010
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Last Class
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Example: converging cameras
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Example: motion parallel to image plane
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Fundamental Matrix•x
•p ( )
• ( )
•p
• is 3X3 matrix of rank 2
•det( )=0
< − >
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Fundamental Matrix in terms of camera matrices
• is projection of ray in camera
• t
•
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Reconstruction Ambiguity
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Reconstruction takes place in following steps
• Compute fundamental matrix F from point correspondences
• Decompose F to get camera projection matrices
• Compute points in 3D by triangulation
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Camera projection matrix from F
• P and P’can be obtained upto projective transformation due to projective ambiguity
• t
• Canonical pair of cameras
• Factor the fundamental matrix F as
• Get from svd(F)eigenvector with minimum eigen value
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Reconstructing the points in 3D
• Back project rays and compute intersection
• Rays do not intersect in presence of noise
• Estimate by minimizing projection error
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•F has 9 entries but defined upto scale
•Singularity constraint det(F)=0
•F has 7 degrees of freedom
• 7 point algorithm - nonlinear equations
• 8 point algorithm -linear solution
• In presence of noise 7 point algorithm could be more efficient
- constraint enforcement
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8 Point Algorithm
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