1. Jiří Matas and Onřej Chum Centre for Machine Perception
WaldSAC – Optimal Randomised RANSAC PROSAC - Progressive Sampling and Consensus
Jiří Matas and Onřej Chum
Centre for Machine Perception
Czech Technical University, Prague

2. WaldSAC – Optimal Randomised RANSAC
References:
Matas, Chum: Optimal Randomised RANSAC,
International Conference on Computer Vision, Beijing,
2005.
Chum, Matas: Randomized RANSAC with Td,d test,
British Machine Vision Conference, Cardiff,
2002.

3. RANSAC – Time Complexity
Repeat k times (k is a function of h, I, N)
1. Hypothesis generation
Select a sample of m data points
Calculate parameters of the model(s)
2. Model verification
Find the support (consensus set) by
verifying all N data points
Time
Total running time:
–- the number of inliers
N - the number of data points
h - confidence in the solution
New York, RANSAC CVPR WaldSAC - Optimal Randomised RANSAC

4. RANSAC time complexity
where P is a probability of drawing an all-inlier sample
where m is size of the sample
New York, RANSAC CVPR WaldSAC - Optimal Randomised RANSAC

5. Randomised RANSAC [Matas, Chum 02]
Repeat k/(1-a) times
1. Hypothesis generation
2. Model pre-verification Td,d test
Verify d << N data points, reject
the model if not all d data points
are consistent with the model
3. Model verification
Verify the rest of the data points
Time
V - average number of data points verified
a - probability that a good model is rejected by Td,d test
New York, RANSAC CVPR WaldSAC - Optimal Randomised RANSAC

6. Optimal Randomised Strategy
Model Verification is Sequential Decision Making
where
Hg - hypothesis of a `good` model (≈ from an uncontaminated sample)
Hb - hypothesis of a `bad` model, (≈ from a contaminated sample)
- probability of a data point being consistent with an arbitrary model
Optimal (the fastest) test that ensures with probability a that that Hg is not incorrectly rejected
is the
Sequential probability ratio test (SPRT) [Wald47]
New York, RANSAC CVPR WaldSAC - Optimal Randomised RANSAC

7. SPRT [simplified from Wald 47]
Compute the likelihood ratio
Two important properties of SPRT:
probability of rejecting a \good\ model a < 1/A
average number of verifications V=C log(A)
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SPRT properties
1. Probability of rejecting a \good\ model a=1/A
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Time
Repeat k/(1-1/A) times
1. Hypothesis generation
2. Model verification
use SPRT
In sequential statistical decision problem decision errors are traded off for time. These are two incomparable quantities, hence the constrained optimization.
In WaldSAC, decision errors cost time (more samples) and there is a single minimised quantity, time t(A), a function of a single parameter A.
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10. Optimal test (optimal A) given e and d
Optimal A* found by solving
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SPRT
decision
good model
bad model
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12. Exp. 1: Wide-baseline matching
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13. Exp. 2 Narrow-baseline stereo
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14. Randomised Verification in RANSAC: Conclusions
The same confidence h in the solution reached faster (data dependent, ¼ 10x)
No change in the character of the algorithm, it was randomised anyway.
Optimal strategy derived using Wald`s theory for known e and d.
Results with e and d estimated during the course of RANSAC are not significantly different. Performance of SPRT is insensitive to errors in the estimate.
d can be learnt, an initial estimate can be obtained by geometric consideration
Lower bound on e is given by the best-so-far support
Note that the properties of WaldSAC are quite different from preemptive RANSAC!
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PROSAC - Progressive Sampling and Consensus
O. Chum and J. Matas. Matching with PROSAC-progressive sampling consensus. CVPR, 2005.
New York, RANSAC CVPR WaldSAC - Optimal Randomised RANSAC

16. PROSAC: Progressive Sampling and Consensus
Idea: exploit the fact that all correspondences are not created equal.
In all (?) multi-view problems, the correspondences are from a totally ordered set (they are selected by thresholding of a similarity function)
Challenge:
organize computation in a way that exploits ordering, and yet is equally robust as RANSAC
PROSAC properties:
selection of correspondences and RANSAC integrated (no need for threshold on local similarity).
Same guarantees about the solution as RANSAC
Depending on the data, can be orders of magnitude faster
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The PROSAC Algorithm
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18. EG estimation experiment
The fraction of inliers in top n correspondences:
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19. Multiple motion experiment
Note: What is an inlier does not depend on the on the similarity of appearance!
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20. Can the probability of a correspondence by established?
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PROSAC: Conclusions
In many cases, PROSAC draws only a few samples.
The information needed to switch from RANSAC to PROSAC - ordering of features - is typically available
Note that the set of data points that is sampled to hypothesise models and the set used in verification need not be the same. This suggest a combination with WaldSAC, but this is not straightforward (e is no more a constant)
PROSAC experiment were a carried a version of RANSAC with protection against degenerate configurations and Local Optimisation (effectively we ran LO-DEGEN-PROSAC). This is important, since often top correspondences lie on the same plane.
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