EM and RANSAC
Jana Kosecka, CS 223b
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EM (Expectation Maximization)
Brief tutorial by example:
EM well known statistical technique for estimation of models from data
Set up: Given set of datapoints which were generated by multiple models
estimate the parameters of the models and assignment of the
data points to the models
Here: set of points in the plane with coordinates (x,y), two lines with parameters
(a1,b1) and (a2,b2)
1. Guess the line parameters and estimate error of each point wrt
to current model
2. Estimate Expectation (weight for each point)
Jana Kosecka, CS 223b
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EM
Maximization step:
Traditional least squares:
Here weighted least squares:
Iterate until no change
Problems: local minima, how many models ?
Jana Kosecka, CS 223b
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EM - example
Jana Kosecka, CS 223b
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Difficulty in motion estimation using widebaseline matching
Jana Kosecka, CS 223b
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Least square estimator can’t tolerate any
outlier
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inliers
Outlier
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Inliers
Inliers
Outlier
Outlier
Erroneous line
Correct line
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Robust techniques is needed to solve the problem.
Jana Kosecka, CS 223b
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Robust estimators for dealing with outliers
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Use robust objective functions
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The M-estimator and Least Median of Squares (LMedS) Estimator
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Neither of them can tolerate more than 50% outliers
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The RANSAC (RANdom SAmple Consensus) algorithm
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Proposed by Fischler and Bolles
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The most popular technique used in Computer Vision community
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It can tolerate more than 50% outliers
Jana Kosecka, CS 223b
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The RANSAC algorithm
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Generate M (a predetermined number) model hypotheses, each of
them is computed using a minimal subset of points
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Evaluate each hypothesis
•
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Compute its residuals with respect to all data points.
Points with residuals less than some threshold are classified as its
inliers
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The hypothesis with the maximal number of inliers is chosen. Then
re-estimate the model parameter using its identified inliers.
Jana Kosecka, CS 223b
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RANSAC – Practice

The theoretical number of samples needed to ensure 95%
confidence that at least one outlier free sample could be obtained.
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It has been noticed that the theoretical estimates are wildly
optimistic
Usually the actual number of required samples is almost an
magnitude more than the theoretical estimate.
Jana Kosecka, CS 223b
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More correspondences and robust matching
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
Select set of putative correspondences
Repeat
1. Select at random a set of 8 successful matches
2. Compute fundamental matrix
3. Determine the subset of inliers, compute distance to
epipolar line
4. Count the number of points in the consensus set
Jana Kosecka, CS 223b
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RANSAC in action
Inliers
Outliers
Jana Kosecka, CS 223b
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Epipolar Geometry
• Epipolar geometry in two views
• Refined epipolar geometry using nonlinear estimation of F
Jana Kosecka, CS 223b
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The difficulty in applying RANSAC

Drawbacks of the standard RANSAC algorithm
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Requires a large number of samples for data with many outliers
(exactly the data that we are dealing with)
Needs to know the outlier ratio to estimate the number of samples
Requires a threshold for determining whether points are inliers
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Various improvements to standard approaches
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Still rely on finding outlier-free samples.
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[Torr’99, Murray’02, Nister’04, Matas’05, Sutter’05 and many
others]
Jana Kosecka, CS 223b
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Robust technique
– result
Jana Kosecka, CS 223b
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