geometric optimization problems in computer vision
Post on 20-Jan-2016
226 views
TRANSCRIPT
Geometric Optimization Problems in Computer
Vision
X
x1 x2 x3
Computation of the Fundamental Matrix
b
AxSpan(A)
O
1D Gauss-Newton (Newton) iteration.
1D Gauss-Newton (Newton) iteration (failure)
x0
x1
x2
First step minimizes on line.
Second step minimizes function in the plane.
X0
Subdivision search
Gradient Descent
Conjugate Gradient
Newton
Levenberg-Marquardt
Gauss-Newton (without line search)
Conjugate gradient
Gradient descent Newton
Model 1
Conjugate gradient Gauss-Newton Gradient descent
Levenberg Newton
Model 2
Conjugate gradient Gauss-Newton Gradient descent
Levenberg Newton
Model 3
Conjugate gradient Gauss-Newton Gradient descent
Levenberg Newton
Model 4
Conjugate gradient Gauss-Newton Gradient descent
Levenberg Newton
Model 5
Conjugate gradient Gauss-Newton Gradient descent
Levenberg Newton
Model 6
Bundle-adjustment
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]
• 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
• 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
Correlation matching results
• Many wrong matches (10-50%), but enough to compute F
Correspondences consistent with epipolar geometry
Computed epipolar geometry
h