Lecture 3.3 Robust estimation with RANSAC. Thomas Opsahl

Transcription

1 Lecture 3.3 Robust estimation with RANSAC Thomas Opsahl

2 Motivation If two perspective cameras captures an image of a planar scene, their images are related by a homography HH 2

3 Motivation If two perspective cameras captures an image of a planar scene, their images are related by a homography HH It can be estimated if we know at least 4 pointcorrespondences uu ii uuu ii 3

4 Motivation If two perspective cameras captures an image of a planar scene, their images are related by a homography HH It can be estimated if we know at least 4 pointcorrespondences uu ii uuu ii Correspondences can be found automatically, but typically some of them will be wrong 4

5 Motivation If two perspective cameras captures an image of a planar scene, their images are related by a homography HH It can be estimated if we know at least 4 pointcorrespondences uu ii uuu ii Correspondences can be found automatically, but typically some of them will be wrong A robust estimation method provides a good estimate of HH despite the presence of these wrong correspondences 5

6 RANdom SAmple Consensus - RANSAC SS xx ii, yy ii Observed data Estimation method yy = ff xx; αα Mathematical model with parameters αα = αα 1,, αα nn RANSAC is an iterative method for estimating the parameters of a mathematical model from a set of observed data containing outliers 6

7 RANdom SAmple Consensus - RANSAC SS yy = aaaa + bb aaaa + bbbb + cc = 0 xx ii, yy ii Observed data Estimation method yy = ff xx; αα Mathematical model with parameters αα = αα 1,, αα nn yy = aaxx 2 + bbbb + cc xx 2 aa 2 + yy2 bb 2 = 1 RANSAC is an iterative method for estimating the parameters of a mathematical model from a set of observed data containing outliers 7

8 RANdom SAmple Consensus - RANSAC SS xx ii, yy ii Observed data Estimation method yy = ff xx; αα Mathematical model with parameters αα = αα 1,, αα nn RANSAC is an iterative method for estimating the parameters of a mathematical model from a set of observed data containing outliers Robust method (handles up to 50% outliers) 8

9 RANdom SAmple Consensus - RANSAC SS xx ii, yy ii Observed data Estimation method yy = ff xx; αα Mathematical model with parameters αα = αα 1,, αα nn RANSAC is an iterative method for estimating the parameters of a mathematical model from a set of observed data containing outliers Robust method (handles up to 50% outliers) The estimated model is random but reasonable 9

10 RANdom SAmple Consensus - RANSAC SS Inliers Outliers xx ii, yy ii Observed data Estimation method yy = ff xx; αα Mathematical model with parameters αα = αα 1,, αα nn RANSAC is an iterative method for estimating the parameters of a mathematical model from a set of observed data containing outliers Robust method (handles up to 50% outliers) The estimated model is random but reasonable The estimation process divides the observed data into inliers and outliers 10

11 RANdom SAmple Consensus - RANSAC SS Inliers Outliers xx ii, yy ii Observed data Estimation method yy = ff xx; αα Mathematical model with parameters αα = αα 1,, αα nn RANSAC is an iterative method for estimating the parameters of a mathematical model from a set of observed data containing outliers Robust method (handles up to 50% outliers) The estimated model is random but reasonable The estimation process divides the observed data into inliers and outliers Usually an improved estimate of the model is determined based on the inliers using a less robust estimation method, e.g. least squares 11

12 Basic RANSAC Objective Robustly fit a model yy = ff xx; αα to a data set SS = xx ii Algorithm 1. Determine a test model yy = ff xx; αα tttttt from nn random data points xx 1, xx 2,, xx nn 2. Check how well each individual data point in SS fits with the test model Data points within a distance tt of the model constitute a set of inliers SS tttttt SS Data points outside a distance tt of the model are outliers 3. If SS tttttt is the largest set of inliers encountered so far, we keep this model Set SS IIII = SS tttttt and αα = αα tttttt 4. Repeat steps 1-3 until NN models have been tested 12

13 Basic RANSAC Comments The number of random samples, nn, is typically the smallest number of data points required to estimate the model Assuming Gaussian noise in the data, the threshold value tt should be in the region of 2σσ were σσ is the expected noise in the data set The maximal number of tests, NN, can be chosen according to how certain we want to be of sampling at least one data set xx 1, xx 2,, xx nn with no outliers If pp is the desired probability of sampling at least one nn-tuple with no outliers and ωω is the probability of a random data point to be an inlier, then NN = llllll 1 pp llllll 1 ωω nn 13

14 Basic RANSAC Comments Standard value pp = 0.99 We rarely know the ratio of inliers in our set of data points, so in most situations, ωω is unknown Instead of maximizing ωω, leading to a larger than necessary NN, we can modify RANAC to adaptively estimate NN as we perform the iterations nn ωω NN NN = llllll 1 pp llllll 1 ωω nn pp =

15 Adaptive RANSAC Objective Robustly fit a model yy = ff xx; αα to a data set SS = xx ii Algorithm 1. Let NN =, SS IIII = 2. As long as the number of iterations are smaller than NN repeat steps Determine a test model yy = ff xx; αα tttttt from nn random data points xx 1, xx 2,, xx nn 4. Check how well each individual data point in SS fits with the test model Data points within a distance tt of the model constitute a set of inliers SS tttttt SS 5. If SS tttttt is the largest set of inliers encountered so far, we keep this model Set SS IIII = SS tttttt and αα = αα tttttt llllll 1 pp Compute NN = using that ωω = SS IIII and pp = 0.99 llllll 1 ωω nn SS 16

16 Example Fit a circle xx xx yy yy 0 2 = rr 2 to these data points by estimating the 3 parameters xx 0, yy 0 and rr

17 Example Circle + Gaussian noise Random points Fit a circle xx xx yy yy 0 2 = rr 2 to these data points by estimating the 3 parameters xx 0, yy 0 and rr The data consists of some points on a circle with Gaussian noise and some random points

18 Example Least-squares approach Separate observables from parameters: ( ) ( ) [ ] x x + y y = r x 2xx + x + y 2yy + y = r xx0 + 2yy0 + r x0 y0 = x + y 2x0 x y 1 2y 0 = x + y r x0 y0 p1 [ x y 1 ] p 2 = x + y p So for each observation xx ii, yy ii we get one equation p1 2 2 [ xi yi 1 ] p 2 = xi + y i p 3 From all our NN observations we get a system of linear equations 2 2 x1 y1 1 x1 + y 1 p1 2 2 x2 y2 1 x2 y2 p + 2 = p3 2 2 xn yn 1 xn + yn Ap= b

19 Example One way of solving the equation AApp = bb is to take the pseudo inverse pp = AA TT AA 1 AA TT bb This give us the solution that minimizes AApp bb

20 Example One way of solving the equation AApp = bb is to take the pseudo inverse pp = AA TT AA 1 AA TT bb This give us the solution that minimizes AApp bb NOT GOOD! All points are treated equally, so the random points shifts the estimated circle away from the desired solution

21 Example To estimate the circle using RANSAC, we need two things 1. A way to estimate a circle from nn points, where nn is as small as possible 2. A way to determine which of the points are inliers for an estimated circle

22 Example To estimate the circle using RANSAC, we need two things 1. A way to estimate a circle from nn points, where nn is as small as possible 2. A way to determine which of the points are inliers for an estimated circle The smallest number of points required to determine a circle is 3, i.e. nn = 3, and the algorithm for computing the circle is quite simple

23 Example To estimate the circle using RANSAC, we need two things 1. A way to estimate a circle from nn points, where nn is as small as possible 2. A way to determine which of the points are inliers for an estimated circle The distance from a point xx ii, yy ii to a circle xx xx yy yy 0 2 = rr 2 is given by xx ii xx yy ii yy 0 2 rr xx ii, yy ii xx ii xx yy ii yy 0 2 xx 0, yy 0 rr

24 Example To estimate the circle using RANSAC, we need two things 1. A way to estimate a circle from nn points, where nn is as small as possible 2. A way to determine which of the points are inliers for an estimated circle The distance from a point xx ii, yy ii to a circle xx xx yy yy 0 2 = rr 2 is given by xx ii xx yy ii yy 0 2 rr So for a threshold value tt, we say that xx ii, yy ii is an inlier if xx ii xx yy ii yy 0 2 rr < tt

25 Example Objective To robustly fit the model xx xx yy yy 0 2 = rr 2 to our data set SS = xx ii, yy ii Algorithm 1. Let NN =, SS IIII =, pp = 0.99, tt = 2 eeeeeeeeeeeeeeee nnnnnnnnnn 2. As long as the number of iterations are smaller than NN repeat steps Determine parameters xx tttttt, yy tttttt, rr tttttt from three random points from SS 4. Check how well each individual data point in SS fits with the test model SS tttttt = xx ii, yy ii SS such that xx ii xx 2 tttttt + yy ii yy 2 tttttt rr tttttt < tt 5. If SS tttttt is the largest set of inliers encountered so far, we keep this model Set SS IIII = SS tttttt and xx 0, yy 0, rr = xx tttttt, yy tttttt, rr tttttt llllll 1 pp Recompute NN = using that ωω = SS IIII llllll 1 ωω nn SS

26 Example The RANSAC algorithm evaluates many different circles and returns the circle with the largest inlier set Inliers

27 Example The RANSAC algorithm evaluates many different circles and returns the circle with the largest inlier set Inliers An improved estimate for the circle can found from the set of inliers using a less robust algorithm e.g. least squares

28 Robust estimation But RANSAC is not perfect Several other robust estimation methods exist Least Median Squares (LMS) Preemptive RANSAC PROgressive Sample and Consensus (PROSAC) M-estimator Sample and Consensus (MSAC) Maximum Likelihood Estimation Sample and Consensus (MLESAC) Randomized RANSAC (R-RANSAC) KALMANSAC +++

29 Summary RANSAC A robust iterative method for estimating the parameters of a mathematical model from a set of observed data containing outliers Separates the observed data into inliers and outliers which is very useful if we want to use better, but less robust, estimation methods Additional reading Szeliski: Homework? Implement a RANSAC algorithm for estimating a line 30