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