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Geometric Optimization Problems in Computer
Vision
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X
x1 x2 x3
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Computation of the Fundamental Matrix
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b
AxSpan(A)
O
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1D Gauss-Newton (Newton) iteration.
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1D Gauss-Newton (Newton) iteration (failure)
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x0
x1
x2
First step minimizes on line.
Second step minimizes function in the plane.
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X0
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Subdivision search
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Gradient Descent
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Conjugate Gradient
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Newton
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Levenberg-Marquardt
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Gauss-Newton (without line search)
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Conjugate gradient
Gradient descent Newton
Model 1
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Conjugate gradient Gauss-Newton Gradient descent
Levenberg Newton
Model 2
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Conjugate gradient Gauss-Newton Gradient descent
Levenberg Newton
Model 3
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Conjugate gradient Gauss-Newton Gradient descent
Levenberg Newton
Model 4
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Conjugate gradient Gauss-Newton Gradient descent
Levenberg Newton
Model 5
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Conjugate gradient Gauss-Newton Gradient descent
Levenberg Newton
Model 6
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Bundle-adjustment
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Robust line estimation - RANSACFit a line to 2D data containing outliers
There are two problems
1. a line fit which minimizes perpendicular distance
2. a classification into inliers (valid points) and outliers
Solution: use robust statistical estimation algorithm RANSAC
(RANdom Sample Consensus) [Fishler & Bolles, 1981]
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• Repeat1. Select random sample of 2 points
2. Compute the line through these points
3. Measure support (number of points within threshold distance of the line)
• Choose the line with the largest number of
inliers
– Compute least squares fit of line to inliers
(regression)
RANSAC robust line estimation
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• Repeat1. Select random sample of 7 correspondences
2. Compute F (1 or 3 solutions)
3. Measure support (number of inliers within threshold distance of epipolar line)
• Choose the F with the largest number of
inliers
Algorithm summary – RANSAC robust F estimation
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Correlation matching results
• Many wrong matches (10-50%), but enough to compute F
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Correspondences consistent with epipolar geometry
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Computed epipolar geometry
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h
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