Geometric Optimization Problems in Computer Vision

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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
Span(A)
Ax
O
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1D Gauss-Newton (Newton) iteration.
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1D Gauss-Newton (Newton) iteration (failure)
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x1
x0
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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Newton
Conjugate gradient
Gradient descent
Model 1
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Conjugate gradient
Gauss-Newton
Gradient descent
Model 2
Levenberg
Newton
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Conjugate gradient
Gauss-Newton
Gradient descent
Model 3
Levenberg
Newton
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Conjugate gradient
Gauss-Newton
Gradient descent
Model 4
Levenberg
Newton
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Conjugate gradient
Gauss-Newton
Gradient descent
Model 5
Levenberg
Newton
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Conjugate gradient
Gauss-Newton
Gradient descent
Model 6
Levenberg
Newton
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Bundle-adjustment
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Robust line estimation - RANSAC
Fit a line to 2D data containing outliers

• There are two problems
• a line fit which minimizes perpendicular distance
• 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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RANSAC robust line estimation

• Repeat
• Select random sample of 2 points
• Compute the line through these points
• 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)

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Algorithm summary RANSAC robust F estimation

• Repeat
• Select random sample of 7 correspondences
• Compute F (1 or 3 solutions)
• Measure support (number of inliers within
  threshold distance of epipolar line)
• Choose the F with the largest number of inliers

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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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