1. (RANdom SAmple Consensus)
RANSAC
(RANdom SAmple Consensus)
Prof. Mariolino De Cecco, Ing. Luca Baglivo, Ing. Nicolò Blasi, Ilya Afanasyev
Department of Structural Mechanical Engineering, University of Trento

2. Object detection and localization
The block diagram of recognition task:
A-priori know:
- Camera, laser
Measurement system
- Dimensions;
Object
- Kinect
- Features (shape, curvature, etc.)
- Multicamera system
Data
- Point clouds
Math. model
- Images, video
- Superquadrics, etc.
- Eliminating the background;
Pre-processing
- Changing the contrast, size, etc.
- Optimization, iterations, etc.;
- RANSAC;
Fitting Algorithm
- Criteria of quality.
- Transformation matrix, angles, distance, accuracy, etc.
Results
02/04/2012

3. Robust parameter estimation methods
Robust parameter estimation methods are the general tool in computer vision, widely used for such tasks as multiple view relation estimation and camera calibration.
There are a lot of variety of Robust parameter estimation methods: pure RANSAC [1], LMEDS (Least Median of Squares) [2], MSAC, MAPSAC (MAximum aPosteriory Samling Consensus), MLESAC, etc. [3].
Analisi di regressione Fondamenti

4. Robust parameter estimation methods
One of the key tasks in the field of computer vision is to establish a relation between measured data and some mathematical parametric model. Assume that we need to estimate the parameters vector θ, so that F(X, θ) = 0.
where X{xi}, i = 1..N – is the measured data vector, F(X, θ) is the target mathematical model.
A classical example of parameter estimation task is fitting straight line to a set of points. In this case, x is the point set F(X, θ) is line equation, θ - line parameters to be estimated.
Analisi di regressione Fondamenti

5. A simple example is fitting of a line in two dimensions to a set of observations
Analisi di regressione Scopo
A data set with many outliers for which a line has to be fitted.
Fitted line with RANSAC; outliers have no influence on the result.

6. Estimating 2D lines [4]
Analisi di regressione Fondamenti

7. Robust parameter estimation methods
The data points generated by the target model (I the framework of set distance) are called inliers with respect to the model F, the other points are called outliers.
RANSAC is an abbreviation for "RANdom SAmple Consensus". It is an iterative method to estimate parameters of a mathematical model from a set of observed data which contains outliers.
The algorithm was first published by Fischler and Bolles in 1981 [1].
Analisi di regressione Fondamenti

8. RANSAC algorithm [3]
Analisi di regressione Fondamenti

9. Estimating 2D lines for SAC-algorithms [5]
Analisi di regressione Fondamenti

10. Scopo
A basic assumption is that the data consists of "inliers", i.e., data whose distribution can be explained by some set of model parameters, and "outliers" which are data that do not fit the model.
Analisi di regressione Scopo
3. regressione del modello

11. L’insieme dei parametri si determina: metodo dei minimi quadrati
Nel metodo dei minimi quadrati l’indice di prestazione è costituito dalla somma dei quadrati degli scarti (anche detti residui) [6]:
Essendo:
L’insieme dei parametri si determina:
metodo dei minimi quadrati
3. regressione del modello

12. Caso lineare - Calcolo retta ai minimi quadrati:
Si noti che i residui risultano essere lineari in funzione dei parametri da determinare
Dunque anche il fitting con un polinomio qualsiasi risulta risolvibile in maniera analoga y ei
parametri (a,b)
Il metodo dei minimi quadrati Esempio: retta ai minimi quadrati
N Dati sperimentali x
3. regressione del modello

13. 3. regressione del modello
Posizione del problema
(La soluzione ha un solo minimo)
Il metodo dei minimi quadrati Segue (retta ai minimi quadrati)
Posizione del problema
3. regressione del modello

14. Soluzione Dove: 3. regressione del modello
Il metodo dei minimi quadrati Segue (retta ai minimi quadrati)
soluzione
3. regressione del modello

15. What happens if we have outliers ?
Fitted line
outlier
Fitting with outliers

16. Let’s try to formalize a more general way of fitting data with outliers.
The problem can be stated in the following way: given a set of 2D data points, find the line which minimizes the sum of squared perpendicular distances (orthogonal regression), subject to the condition that none of the valid points deviates from this line by more than a threshold t
This is actually two problems:
classification of the data into inliers (valid points) and outliers. The threshold t is set according to the measurement noise (for example t = 3),
line fit to the data
Fitting with outliers

17. Fitting with outliers The key idea is very simple:
repeat N times the following:
select randomly two points
compute the connecting line
the support for this line is measured by the number of points that lie within the distance threshold t
the line with most support is deemed the robust fit
The points within the threshold distance are the inliers (and constitute the consensus set).
Fitting with outliers

18. The intuition is that if one of the points is an outlier then the line will not gain much support
Fitting with outliers

19. Scoring a line by its support has the additional advantage of favouring better fits. The line (a, b) has a support of 10, whereas the line (a, d), where the sample points are neighbours, has a support of only 4.
Consequently, even though both samples contain no outliers, the line (a. b) will be selected
Fitting with outliers

20. As stated by Fischler and Bolles [Fischler, 1981]:
"The RANSAC procedure is opposite to that of conventional smoothing techniques: Rather than using as much of the data as possible to obtain an initial solution and then attempting to eliminate the invalid data points (like for example using the Chauvenet criteria), RANSAC uses as small an initial data set as feasible and enlarges this set with consistent data when possible".
Fitting with outliers

21. Objective
Robust fit of a model to a data set S which contains outliers.
Algorithm
Randomly select a sample of s data points from S 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 the sample and defines the inliers of S
If the size of Si (the number of inliers) is greater than some threshold T, re-estimate the model using all the points in Si and terminate
(iv) If the size of Si is less than T, select a new subset and repeat the above,
(v) After N trials the largest consensus set Si is selected, and the model is re-estimated using all the points in the subset Si
Algorithm

22. Select a sample of s data{
This to verify if the sample is able to provide a reliable fitting
Verify the sample data
Use the s data to estimate the model
Determine the consensus set Si and its cardinality ns no
ns >T {
yes
Use Si to estimate the model
This to ensure a Maximum Likelikood solution
Determine the new consensus set Si and the cardinality nsnew no
nnew = nold
yes
Stop
Algorithm revised

23. It is often computationally infeasible and unnecessary to try every possible sample. Instead the number of samples N is chosen sufficiently high to ensure with a probability, p, that at least one of the random samples of s points is free from outliers.
Usually p is chosen at 0.99
The number N of samples required to ensure, with a probability p = 0.99, that at least one sample has no outliers for a given size of sample S, and proportion of outliers 
Number of samples

24. Number of samples - consensus set
For a given  and p the number of samples increases with the size of the minimal subset
It might be thought that it would be advantageous to use more than the minimal subset, three or more points in the case of a line, because then a better estimate of the line would be obtained, and the measured support would more accurately reflect the true support. However, this possible advantage in measuring support is generally outweighed by the severe increase in computational cost incurred by the increase in the number of samples.
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, i.e. for n data points T < (1 - ) n
Number of samples - consensus set

25. An advantage of RANSAC is its ability to achieve robust estimation of the model parameters, i.e., it can estimate the parameters with a high degree of accuracy even when significant amount of outliers are present in the data set.
A disadvantage of RANSAC is that there is no upper bound on the time it takes to compute these parameters. When an upper time bound is used (a maximum number of iterations) the solution obtained may not be the optimal one, it may not even be one that fits the data in a good way. A reasonable model can be produced by RANSAC only with a certain probability, a probability that becomes larger the more iterations that are used
Another disadvantage of RANSAC is that it requires the setting of problem-specific thresholds.
RANSAC - conclusions

26. Suppose we acquired the 3D cloud of points of a cube:
Example - plane fitting

27. Example - main function
close all
clear all
%% CARICHIAMO E PLOTTIAMO IL SET DI DATI
load('Points.mat');
all_point = Points(: , 1:3);
all_point(: , 1) = -all_point(: , 1); % convenzione Kinect ?
% Plot all points
figure('Name','All points')
hold on
axis equal
plot3(Points(:,1), Points(:,2), Points(:,3), '.' )
%% FITTIAMO CON RANSAC IL PIANO
% decimiamo i punti
ss = length(all_point);
ii = 1 : 10 : ss;
all_pointDec = all_point(ii , :);
% FIT RANSAC
[B, P, inliers] = ransacfitplane(all_pointDec', 0.01, 1);
% Plot risultato
plotPlane3D( all_point' , B )
Example - main function

28. Example – RANSACFITPLANE [7]
% RANSACFITPLANE - fits plane to 3D array of points using RANSAC
% Usage [B, P, inliers] = ransacfitplane(XYZ, t, feedback)
% This function uses the RANSAC algorithm to robustly fit a plane
% to a set of 3D data points. %
% Arguments:
% XYZ - 3xNpts array of xyz coordinates to fit plane to.
% t - The distance threshold between data point and the plane
% used to decide whether a point is an inlier or not.
% feedback - Optional flag 0 or 1 to turn on RANSAC feedback
% information.
% Returns:
% B - 4x1 array of plane coefficients in the form
% b(1)*X + b(2)*Y +b(3)*Z + b(4) = 0
% The magnitude of B is 1.
% This plane is obtained by a least squares fit to all the
% points that were considered to be inliers, hence this
% plane will be slightly different to that defined by P below.
% P - The three points in the data set that were found to
% define a plane having the most number of inliers.
% The three columns of P defining the three points.
% inliers - The indices of the points that were considered
% inliers to the fitted plane.
% Copyright (c) Peter Kovesi
Example – RANSACFITPLANE [7]

29. Example - RANSACFITPLANE
s = 3; % Minimum No of points needed to fit a plane.
fittingfn
distfn
degenfn
[P, inliers] = ransac(XYZ, fittingfn, distfn, degenfn, s, t, feedback);
% Perform least squares fit by means of the inlying points
B = fitplane(XYZ(:,inliers)); %
% Function to define a plane given 3 data points as required by
% RANSAC. In our case we use the 3 points directly to define the plane.
function P = defineplane(X);
P = X;
Example - RANSACFITPLANE

30. Example - RANSACFITPLANE
% Function to calculate distances between a plane and a an array of points in the X matrix where
% each column represents the x, y, z coordinates (X: 3 x Npts array of xyz coordinates)
% The plane is defined by a 3x3 matrix, P. The three columns of P defining
% three points that are within the plane.
function [inliers, P] = planeptdist(P, X, t)
n = cross(P(:,2)-P(:,1), P(:,3)-P(:,1)); % Plane normal.
n = n/norm(n); % Make it a unit vector.
npts = length(X);
d = zeros(npts,1); % d will be an array of distance values.
% The perpendicular distance from the plane to each point:
% d = n’ * ( X - P(:,1) ) [1 x Npts]
for i=1:3
d = d + ( X(i,:)’ - P(i,1) ) * n(i) ;
end
inliers = find(abs(d) < t);
% Function to determine whether a set of 3 points are in a degenerate
% configuration for fitting a plane as required by RANSAC. In this case
% they are degenerate if they are colinear.
function r = isdegenerate(X)
% The three columns of X are the coords of the 3 points.
r = iscolinear(X(:,1),X(:,2),X(:,3));
Example - RANSACFITPLANE

31. At the end we obtain the following fitted ‘floor’ plane
Example - OUTCOME

32. Ideas for homework Ideas for homework
To use the camera view to assert consensus. Colour or curvature information could be used
To use the 3D and colour information to assert consensus
To modify the consensus criteria simulating the laser view of the fitted object
Ideas for homework

33. Links Grazie per attenzione!!
M.A.Fischler, R.C.Bolles. “Random Sample Consensus: A paradigm for model fitting with applications to image analysis and automated cartography”. // In Communications of the ACM, 24 (6), pp ,1981.
P.J.Rousseeuw. “Least median of squares regression”. // In Journal of the American Statistical Association, 79(388), pp , 1984.
Konouchine A., Gaganov V., Veznevets V. “AMLESAC: A New Maximum Likelihood Robust Estimator”. // In proceedings of conf. “Graphicon 2005” (Novosibirsk, Russia). P.8.
Marco Zuliani. RANSAC for Dummies. // January 31, vision.ece.ucsb.edu/~zuliani/Research/RANSAC/docs/RANSAC4Dummies.pdf
GML RANSAC Matlab Toolbox v 0.1:
R. Hartley and A. Zisserman, “Multiple View Geometry in Computer Vision”. // Cambridge University Press, 2003.
Kovesi Peter. RANSAC software in MATLAB
Grazie per attenzione!!