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Motivation - I
Estimation of model parameters in presence of noise andoutliers is a crucial task in image processing, computervision, patter recognition. . .
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Inliers
Outliers
Unbiased estimate(RANSAC)
Biased estimate(least squares) Least squares produce
biased estimates in
presence of outliers
Robust statistic can toler-
ate up to 50% outliers
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Motivation - I
Estimation of model parameters in presence of noise andoutliers is a crucial task in image processing, computervision, patter recognition. . .
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Affine model
Outliers
Inliers
RANSAC algorithm
[Fischler 1981]
RANdom SAmple Con-
sensus
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Motivation - II
What happens if there are multiple models?
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(a) (b)(a) Multiple segments, (b) Multiple planes
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Motivation - II
Traditional RANSAC-based approaches:
1. apply standard RANSAC
2. remove the detected set of inliers
3. go back to 1
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Motivation - II
Traditional RANSAC-based approaches:
1. apply standard RANSAC
2. remove the detected set of inliers
3. go back to 1
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Motivation - II
Traditional RANSAC-based approaches:
1. apply standard RANSAC
2. remove the detected set of inliers
3. go back to 1
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Motivation - II
Traditional RANSAC-based approaches:
1. apply standard RANSAC
2. remove the detected set of inliers
3. go back to 1
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Motivation - II
Traditional RANSAC-based approaches:
1. apply standard RANSAC
2. remove the detected set of inliers
3. go back to 1
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Motivation - II
Traditional RANSAC-based approaches:
1. apply standard RANSAC
2. remove the detected set of inliers
3. go back to 1
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Motivation - III
Inaccurate inlier detection for the initial (or subsequent) pa-rameter estimation contributes heavily to the instability of theestimates of the parameters for the remaining models.
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Goals
Extend RANSAC to multiRANSAC
Present a theoretical analysis of multiRANSAC
Show its effectiveness on synthetic and real data sets
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Presentation Overview
Description of multiRANSAC algorithm
Experimental Results
Conclusions and Future Work
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Presentation Overview
Description of multiRANSAC algorithm
Experimental Results
Conclusions and Future Work
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From RANSAC to multiRANSAC - I
RANSAC key ideas:Estimate the model parameters using the minimum number of data possible
Check which of the remaining data points fit the model instantiated with the
estimated parameters.
Do this a sufficiently large number of times (more to come. . . )
RANSAC iteration:
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From RANSAC to multiRANSAC - II
multiRANSAC key ideas:At each iteration instantiate
models and find the corresponding CSs
Fuse the new CSs with the previously detected ones in a sensible way.
multiRANSAC iteration:
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From RANSAC to multiRANSAC - II
multiRANSAC key ideas:At each iteration instantiate
models and find the corresponding CSs
Fuse the new CSs with the previously detected ones in a sensible way.
multiRANSAC iteration:
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H M It ti ? I
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How Many Iterations? - I
What is the probability that we draw MSSs onlycomposed by inliers?
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H M It ti ? I
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How Many Iterations? - I
What is the probability that we draw MSSs onlycomposed by inliers?
It can be shown that:
where:
is the number of models
is the total number of data points
is the number of inliers for the
model
is the cardinality of the MSS for the
model
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H M It ti ? II
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How Many Iterations? - II
The probability that MSSs are not entirely composedby inliers is
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Ho Man Iterations? II
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How Many Iterations? - II
The probability that MSSs are not entirely composedby inliers is
The probability that this happens for trials is
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How Many Iterations? II
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How Many Iterations? - II
The probability that MSSs are not entirely composedby inliers is
The probability that this happens for trials is
Goal: make sure that
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How Many Iterations? II
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How Many Iterations? - II
The probability that MSSs are not entirely composedby inliers is
The probability that this happens for trials is
Goal: make sure that
Then:
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How Many Iterations? II
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How Many Iterations? - II
The probability that MSSs are not entirely composedby inliers is
The probability that this happens for trials is
Goal: make sure that
Then:
Problem: a priori we dont know
!
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How Many Iterations? II
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How Many Iterations? - II
The probability that MSSs are not entirely composedby inliers is
The probability that this happens for trials is
Goal: make sure that
Then:
Problem: a priori we dont know
!
Solution: the estimate for is the cardinality oflargest CS found so far.
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Fusing the CSs
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Fusing the CSs
Observation: the CSs obtained in current iteration canbe fused with the CSs obtained in the previousiterations to produce better CSs.
More specifically:Better is quantified in terms of a fitness function
Cardinality
MSE
Simplification: just look at the previous iteration
Greedy algorithm
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Fusing the CSs
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Fusing the CSs
Observation: the CSs obtained in current iteration canbe fused with the CSs obtained in the previousiterations to produce better CSs.
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Presentation Overview
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Presentation Overview
Description of multiRANSAC algorithm
Experimental Results
Conclusions and Future Work
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Presentation Overview
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Presentation Overview
Description of multiRANSAC algorithm
Experimental Results
Conclusions and Future Work
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Experiments: Detecting Lines - I
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Experiments: Detecting Lines I
Toy problem: Identify the lines containing the segments.
The stair data set
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Wrong estimate
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Experiments: Detecting Lines - I
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Experiments: Detecting Lines I
Toy problem: Identify the lines containing the segments.
The stair data set
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Correct estimate
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Experiments: Detecting Lines - II
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Experiments: Detecting Lines II
Definitions:Set of correctly detected inliers for the model
:
where
is
the detected set of inliers
Percentage of correctly detected inliers:
is the averaged value of
over 50 trials and
models
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Experiments: Detecting Lines - II
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Experiments: Detecting Lines II
Definitions:Set of correctly detected inliers for the model
:
where
is
the detected set of inliers
Percentage of correctly detected inliers:
is the averaged value of
over 50 trials and
models
Noise std multiRANSAC sequential RANSAC
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95.96% 7499 93.80% 2016
6.0
95.22% 7626 86.00% 2049
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90.08% 8194 67.14% 20807.0
90.13% 8699 47.92% 2103
7.5
86.01% 8110 37.29% 2103
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Experiments: Detecting Homographies
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Experiments: Detecting Homographies
,
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Experiments: Detecting Homographies
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,
multiRANSAC
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Experiments: Detecting Homographies
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p g g p
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sequential RANSAC
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Experiments: Detecting Homographies
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p g g p
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Experiments: Detecting Homographies
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p g g p
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multiRANSAC
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Experiments: Detecting Homographies
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p g g p
, ,
,
sequential RANSAC
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Experiments: Detecting Homographies
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p g g p
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Experiments: Detecting Homographies
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multiRANSAC
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Experiments: Detecting Homographies
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, ,
,
sequential RANSAC
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Presentation Overview
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Description of multiRANSAC algorithm
Experimental Results
Conclusions and Future Work
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Presentation Overview
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Description of multiRANSAC algorithm
Experimental Results
Conclusions and Future Work
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Conclusion & Future Work
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We presented the multiRANSAC algorithm
On average, better performance than sequentialRANSAC
Synthetic data (quantitative results)Real data
Future Work
Quantitative experiments on real imagery
Improved CSs fusion strategy
Can we detect the number of models?
Explore waiting time between updates of the CSs
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The End
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Thanks for your attention.
Special thanks to Dr. M. Bober, E. Drelie, D. Fedorov and prof. K. Rose for the helpful
discussion
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How Many Iterations? - I
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What is the probability that we draw MSSs onlycomposed by inliers?
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How Many Iterations? - I
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What is the probability that we draw MSSs onlycomposed by inliers?
Think in terms of ratio of number of favorable to number
of possible
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How Many Iterations? - I
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What is the probability that we draw MSSs onlycomposed by inliers?
The chance of drawing only inliers is given by:
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What is the probability that we draw MSSs onlycomposed by inliers?
The chance of having the inliers belonging to the
correct model:
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What is the probability that we draw MSSs onlycomposed by inliers?
Putting things together we have:
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How Many Iterations? - I
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What is the probability that we draw MSSs onlycomposed by inliers?
Putting things together we have:
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RANSAC vs. multiRANSAC iterations
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How big is
in a typical problem?
Number of data
Number of models
Cardinality of a MSS
Probability threshold
Total percentage of inliers
Inliers for each class
,
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RANSAC vs. multiRANSAC iterations
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How big is
in a typical problem?
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104
106
rI
Titer
N=250
W = 4
W = 3
W = 2
W = 1
W = 3
W = 2
W = 1
RANSAC
multiRANSAC
W = 4
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