J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
Center for
Machine
Perception
Prague
WaldSAC – Optimal Randomised RANSAC
PROSAC - Progressive Sampling and Consensus
Jiří Matas and Onřej Chum
Centre for Machine Perception
Czech Technical University, Prague
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
Center for
Machine
Perception
Prague
WaldSAC – Optimal Randomised
RANSAC
References:
Matas, Chum: Optimal
RANSAC,
Randomised
International Conference on
Computer Vision, Beijing,
2005.
Chum, Matas: Randomized
with Td,d test,
RANSAC
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
RANSAC

Time
Complexity
Center for
Machine
Perception
Prague
Time
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
Total running
set) by
time:

verifying all N data
points
I- the number of
inliers
N - the number of
points
New York, RANSAC 25 @ CVPR06
data
WaldSAC - Optimal Randomised RANSAC
3
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
RANSAC time complexity
where
P is
a
where
New York, RANSAC 25 @ CVPR06
probability
m is
size
of
WaldSAC - Optimal Randomised RANSAC
of
Center for
Machine
Perception
Prague
drawi
the
samp
4
Randomised RANSAC
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
[Matas,
Chum 02]
Center for
Machine
Perception
Prague
Time
Repeat k/(1-a) times
1. Hypothesis
generation
2. Model preverification Td,d test

Verify d << N data
points, reject

the model if not all
d data points
V - are
average
number
of data points
consistent
with
the model
a - probability
that a good model
3. Model verification
Verify the rest of
the data points
New York, RANSAC 25 @ CVPR06
WaldSAC - Optimal Randomised RANSAC
5
Optimal Randomised
Strategy
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
Center for
Machine
Perception
Prague
Model Verification is
Sequential Decision Making
where
Hg - hypothesis of a `good` model (H from
an uncontaminated sample)
Hb - hypothesis of a `bad` model, (H from
a contaminated sample)
 - probability of a data point being
Optimal
(the fastest) test that
consistent with an arbitrary model
ensures with probability a that
that Hg is not incorrectly
rejected
is
the
probability
ratio 6
New York, Sequential
RANSAC 25 @ CVPR06
WaldSAC
- Optimal Randomised RANSAC
SPRT
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
[simplified from Wald
47]
Compute the
likelihood
Two important properties
Center for
Machine
Perception
Prague
ratio
of
SPRT:
1.probability of rejecting
\good\ model a < 1/A
a
2.average number of
verifications V=C log(A)
New York, RANSAC 25 @ CVPR06
WaldSAC - Optimal Randomised RANSAC
7
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
SPRT properties
Center for
Machine
Perception
Prague
1. Probability of rejecting
a \good\ model a=1/A
New York, RANSAC 25 @ CVPR06
WaldSAC - Optimal Randomised RANSAC
8
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
WaldSAC
Repeat k/(11/A) times
1. Hypothesis
generation
2. Model
verification
use SPRT
Center for
Machine
Perception
Prague
Time
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
New York, RANSAC 25 @ CVPR06
WaldSAC - Optimal Randomised RANSAC
9
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
Optimal test (optimal A)
given e and 
Center for
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Perception
Prague
Optimal A*
Optimal A*
New York, RANSAC 25 @ CVPR06
found by
solving
WaldSAC - Optimal Randomised RANSAC
10
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
SPRT
Center for
Machine
Perception
Prague
decision
bad
model
New York, RANSAC 25 @ CVPR06
good
WaldSAC - Optimal Randomised RANSAC
mode
11
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
Exp. 1: Widebaseline matching
New York, RANSAC 25 @ CVPR06
WaldSAC - Optimal Randomised RANSAC
Center for
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Perception
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12
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
Exp. 2 Narrowbaseline stereo
New York, RANSAC 25 @ CVPR06
WaldSAC - Optimal Randomised RANSAC
Center for
Machine
Perception
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13
Randomised Verification
in RANSAC: Conclusions
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
Center for
Machine
Perception
Prague
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 .
 Results with e and 
estimated
during the course of RANSAC are not
significantly different.
Performance of SPRT is insensitive
to errors in the estimate.
•
 can be learnt, an initial
estimate can be obtained by
geometric
consideration
• Lower bound on e is given by the
support
Newbest-so-far
York, RANSAC 25 @ CVPR06
WaldSAC - Optimal Randomised RANSAC

14
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
Center for
Machine
Perception
Prague
PROSAC - Progressive Sampling and Consensus
O. Chum and J. Matas.
Matching with PROSACprogressive sampling consensus.
CVPR, 2005.
New York, RANSAC 25 @ CVPR06
WaldSAC - Optimal Randomised RANSAC
15
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
PROSAC: Progressive Sampling
and Consensus
Center for
Machine
Perception
Prague
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
New York, RANSAC 25 @ CVPR06
WaldSAC - Optimal Randomised RANSAC
16
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
The PROSAC Algorithm
New York, RANSAC 25 @ CVPR06
WaldSAC - Optimal Randomised RANSAC
Center for
Machine
Perception
Prague
17
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
EG estimation
experiment
Center for
Machine
Perception
Prague
 The fraction
of inliers in
top n
correspondences
:
New York, RANSAC 25 @ CVPR06
WaldSAC - Optimal Randomised RANSAC
18
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
Multiple motion
experiment
Note: What is an inlier does
depend on the on the
similarity of appearance!
New York, RANSAC 25 @ CVPR06
WaldSAC - Optimal Randomised RANSAC
Center for
Machine
Perception
Prague
not
19
Can the probability
of a correspondence
by established?
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
New York, RANSAC 25 @ CVPR06
WaldSAC - Optimal Randomised RANSAC
Center for
Machine
Perception
Prague
20
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
PROSAC: Conclusions
Center for
Machine
Perception
Prague
 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)
York, RANSAC 25 @
CVPR06
WaldSAC - Optimalwere
Randomised RANSAC
21
 New
PROSAC
experiment
a
J. Matas, O.
Chum, 25RANSAC
workshop CVPR
2006 
Center for
Machine
Perception
Prague
New York, RANSAC 25 @ CVPR06
WaldSAC - Optimal Randomised RANSAC
22