Epipolar Geometry from Two Correspondences

1
Epipolar Geometry from Two Correspondences

• Michal Perdoch, Jirí Matas, Ondrej Chum
• Center for Machine Perception
• Czech Technical University, Prague
• http//cmp.felk.cvut.cz

2
Problem formulation

• Problem Wide-baseline stereo with severe
  viewpoint and scale change.
• Message of this paper Two correspondences of
  Local Affine Frames suffice to estimate epipolar
  geometry.

3
Local Affine Frame correspondences

• Local Affine Frames (LAFs), Matas ICPR2002
• Affine-covariantly detected local coordinate
  systems.
• One matching LAF pair provides three
  point-to-point correspondences.

4
The correspondence problem

• Wide-baseline stereo framework (Matas ICPR2002)

• Detect regions. MSERs are used here.
• Build Local Affine Frames, define Measurement
  Region.
• Geometric and photometric normalisation of MRs.
• Establish correspondences
  by
  nearest neighbour search.
• Verify correspondences,
• estimate model using
• RANSAC.

5
RANSAC Algorithm

• How to find correct model of EG in presence of
  outliers?
• RANSAC - widely used robust estimator proposed by
  Fishler and Bolles 1988.
• Hypothesise verify search

6
RANSAC Algorithm

• Average number of RANSAC samples k
• Example (h0 0.01)
• Chum DAGM2003, has shown that the theoretical
  number of samples can be achieved by introducing
  Local Optimisation step.

probability of finding a better model
proportion of inliers size of the sample
2 LAFs
3 LAFs
7 points
7
LO-RANSAC Algorithm

• Proposed by Chum DAGM2003, used with 3LAFs and
  seven point-to-point correspondences.
• Input set of data points T, confidence h0.
• Output model M of EG with the largest support
  S.
• Repeat
• Draw a random sample of minimal size m from data
  points.
• Compute model parameters Mi and its support Si.
• If new maximum was detected (i.e., Si gt Sj
  for (j lt i)
• Apply local optimisation.
• Store the best model with support S.
• Until the probability of finding model better
  than S falls under h0.

8
Six Point EG estimator

• Stewénius CVPR2005 has shown that EG can be
  estimated from six points under following
  assumptions on the cameras
• Unit aspect ratio, zero skew, principal point in
  the middle of the image.
• Unknown but the same focal length.
• We take Stewénius assumptions.
• Is it possible to estimate EG from two Local
  Affine Frame correspondences?

9
Planar Degeneracy

• Chum CVPR2005 has that planar degeneracy causes
  suboptimal EG estimates.
• Six points on a plane do not provide enough
  constraints for estimation of the fundamental
  matrix.
• Planar degeneracy test - take sample points and
  check if they lie on a plane.
• When a plane is detected do plane and parallax
  search
• Local optimisation that takes into account plane
  found.
• Sample another pair not consistent with the plane
  and compute fundamental matrix

10
2LAF-LO-RANSAC Algorithm

• Input set of data points T, confidence h0,
  sample size m 2.
• Output model M of EG with largest support S.
• Repeat
• Draw a random sample of minimal size m from data
  points.
• Compute model parameters Mi and its support Si.
• If new maximum has occurred (i.e. Si gt Sj for
  (j lt i)
• If a degenerated sample configuration is
  detected
• Perform plane-and-parallax search for EG,
• otherwise
• Apply local optimisation.
• Store the best model with support S.
• Until the probability of finding model better
  than S falls under h0.

Performed at most log(k) times.
11
Experiments Scenes
Corner e 0.26
The China Wall e 0.28
Wash e 0.23
12
Experiment 1 Efficiency

• What is the efficiency of six point estimator?
  Is the planar degeneracy test necessary?
• Comparison of 2LAF, 3LAF with and without planar
  degeneracy test and PTS7 algorithm against
  reference EG.
• Number of good EG models generated from 1000
  all-inlier samples (quality is measured by number
  of inliers, good at least 90).

ND - without degeneracy test
13
Experiment 2 Performance

• Average number of inliers and samples measured in
  hundred runs of 7PTS, 3LAF, 2LAF-LO-RANSAC
  algorithms.
• Small drop (two to four percent) in average
  number of inliers on difficult and noisy scenes
  for both 3LAF and 2LAF algorithm.
• Note the significant speedup (measured by number
  of samples) of 2LAF algorithm.

14
Experiment 3 Assumptions violation

• The six point solver assumes equal focal length
  in both images.
• The focal length ratio on the scene Corner is 13
  and off-plane rotation about 30 degrees.
• No performance drop was observed.

15
Experiment 3 Assumptions violation

• Robustness to change of pixel aspect ratio was
  measured on scene China Wall (inliers in 100
  runs).
• Width of pixel scaled 0.5, 1.0, 2.0 times from
  top to bottom.

16
Conclusions

• Contribution
• Novel wide-baseline stereo algorithm using two
  LAF correspondences was proposed.
• 6 point ( 2 LAFs) EG estimator can be used in
  LO-RANSAC framework with the planar degeneracy
  test.
• Performance was experimentally tested and
  compared to 7PTS and 3LAF algorithms in
  wide-baseline stereo setup.
• Future work
• Detect other degenerated configurations.
• Compare the number of verifications instead of
  the number of samples.

Thank you for your attention.