1. Robot & Vision Lab. Wanjoo Park (revised by Dong-eun Seo)
RANSAC
Robot & Vision Lab.
Wanjoo Park
(revised by Dong-eun Seo)

2. What is RANSAC ?
A new paradigm, Random Sample Consensus (RANSAC), for fitting a model to experimental data.
Authors are Martin A. Fischler and Robert C. Bolles
Communications of the ACM, June 1981

3. Interpretation involves two distinct activities
First
There is the problem of finding the best match between the data and one of the available models (the classification problem).
Second
there is the problem of computing the best values for the free parameters of the selected model (the parameter estimation problem).

4. Two types of Errors Measurement Error Classification Error
Measurement errors occur when the feature detector correctly identifies the feature, but slightly miscalculates one of its parameters
Measurement errors generally follow a normal distribution, and therefore the smoothing assumption is applicable to them.
Classification Error
Classification errors occur when a feature detector incorrectly identifies a portion of an image as an occurrence of a feature.
Classification errors, however, are gross errors, having a significantly larger effect than measurement errors, and do not average out.

5. Gross Error

6. Data set of Experimentation
Original Data set : 12 elements (y=x)
Add Gaussian noise with zero mean, ½ variance
Add 2 gross error

7. Conventional Least Square
Eq.1
Eq.2
Eq.3

8. RANSAC – Algorithm
Randomly select a sample of s data points from S (data set) and instantiate the model from this subset.
Determine the set of data points Si which are within a distance threshold t of the model. The set Si is the consensus set of sample and defines the inliers of S.
If the size of Si (the number of inliers) is greater than some threshold T (the size of consensus set), re-estimate the model using all the points in Si and terminate.
If the size of Si is less than T, select a new subset and repeat the above.
After N trials the largest consensus set Si is selected, and the model is re-estimated using all the points in the subset Si.

9. Randomly select sample set
Randomly selected 2 points
Determine line equation
Number of pair is 66

10. Inlier and Outlier
Threshold: , α is probability that the point is an inlier
For example α=0.95 t
Inlier d2 < t2
Outlier d2 ≥ t2 d
Threshold 결정은 일반적으로 경험적으로
결정한다. 여기서는 noise가 zero mean Gaussian
이라는 가정하에 d^2은 Gaussian variable의 sum
이라 생각해서 cha-square distribution을 따른다고
modeling 하였다.

11. How many Samples ?
1) W is the probability that any selected data point is an inlier, and thus ε=1-w is the probability that it is an outlier. K is number of random selection.

12. SD(k) will be approximately equal to E(k) :
In general, we would probably want to exceed E(k) trials by one or two standard deviations before we give up.
Standard deviation
SD(k) will be approximately equal to E(k) :
This means that one might want to try two or three times the expected number of random selections

13. How many Samples ?
2) if we want to ensure with probability z that at least one of our random selections is an error-free set of n data points
At least z selections are required, where
so that
Examples of k for z = 0.99

14. How large is an acceptable consensus set?
A rule of thumb is to terminate if the size of the consensus set is similar to the number of inliers believed to be in the data set, given the assumed proportion of outliers.
T = (1-ε)n = ( )12 = 10

15. Result of Consensus set

16. The Last Result
RANSAC
Conventional Least square

17. The Last Result 2nd order curve y = ax^2+bx+c : a=1 ,b =-20, c= 110
Datum are corrupted by zero mean, 0.9 SD Gaussian noise and three gross errors.
Number of data : 20, Gross error : 3
3/20 = 15/100 = 0.15

18. final estimation of a,b,c
E(k)=2.0 -> k = 2.0*2 = 4
Estimated a,b,c
Number of inlier
final estimation of a,b,c
Maximum

19. plot
: Real curve
: Estemated curve
: Measured data

20. Thank you for your attention
Practice makes Perfect
Only One Step at a time
Robot & Vision Lab.
Wanjoo Park –
(revised by Dong-eun Seo)