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Robust Estimator

Robust Estimator. 學生 : 范育瑋 老 師 : 王聖智. Outline. Introduction LS-Least Squares LMS-Least Median Squares RANSAC- Random Sample Consequence MLESAC-Maximum likelihood Sample consensus MINPRAN-Minimize the Probability of Randomness. Outline. Introduction LS-Least Squares

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Robust Estimator

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  1. Robust Estimator 學生:范育瑋 老師:王聖智

  2. Outline • Introduction • LS-Least Squares • LMS-Least Median Squares • RANSAC- Random Sample Consequence • MLESAC-Maximum likelihood Sample consensus • MINPRAN-Minimize the Probability of Randomness

  3. Outline • Introduction • LS-Least Squares • LMS-Least Median Squares • RANSAC- Random Sample Consequence • MLESAC-Maximum likelihood Sample consensus • MINPRAN-Minimize the Probability of Randomness

  4. Introduction • Objective: Robust fit of a model to a data set S which contains outliers.

  5. Outline • Introduction • LS-Least Squares • LMS-Least Median Squares • RANSAC- Random Sample Consequence • MLESAC-Maximum likelihood Sample consensus • MINPRAN-Minimize the Probability of Randomness

  6. LS • Consider the data generating process Yi = b0 + b1Xi + ei , where ei is independently and identically distributed N(0, σ). • If any outlier exists in the data. Least squares performs poorly.

  7. Outline • Introduction • LS-Least Squares • LMS-Least Median Squares • RANSAC- Random Sample Consequence • MLESAC-Maximum likelihood Sample consensus • MINPRAN-Minimize the Probability of Randomness

  8. LMS • The method can tolerates the highest possible breakdown point of 50% .

  9. Outline • Introduction • LS-Least Squares • LMS-Least Median Squares • RANSAC- Random Sample Consequence • MLESAC-Maximum likelihood Sample consensus • MINPRAN-Minimize the Probability of Randomness

  10. Main idea

  11. 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. • If the size of Si (number of inliers) is greater than threshold T, re-estimate the model using all points in Si and terminate. • If the size of Si is less than T , select a new subset and repeat above. • After N trials the largest consequence ser Si is selected, and the model is re-estimate the model using all points in the subset Si.

  12. Algorithm • What is the distance threshold? • How many sample? • How large is an acceptable consensus set?

  13. What is the distance threshold? • We would like to choose the distance threshold, t, such that with a probability α the point is an inlier. • Assume the measurement error is Gaussian with zero mean and standard deviation σ. • In this case the square of point distance, d2⊥ , is sum of squared Gaussian variables and follows a χ2m distribution with m degrees of freedom.

  14. What is the distance threshold? Note: The probability that the value of a random variable is less than k2 is given by the cumulative chi-squared distribution. Choose αas 0.95

  15. How many sample? • If w = proportion of inliers = 1-ε • Prob(sample with all inliers)=ws • Prob(sample with an outlier)=1-ws • Prob(N samples an outlier)=(1-ws)N We want Prob (N samples an outlier)<1-p Usually p is chosen at 0.99. • (1-ws)N<1-p • N>log(1-p)/log(1-ws)

  16. How large is an acceptable consensus set? • If we know the fraction of data consisting of outliers . Use a rule of thumb : T=(1-ε)n • For example: Data points : n=12 The probability of outlier ε=0.2 T=(1-0.2)12=10

  17. Outline • Introduction • LS-Least Squares • LMS-Least Median Squares • RANSAC- Random Sample Consequence • MLESAC-Maximum likelihood Sample consensus • MINPRAN-Minimize the Probability of Randomness

  18. THE TWO-VIEW RELATIONS

  19. Maximum Likelihood Estimation • Assume the measurement error is Gaussian with zero mean and standard deviation σ n: The number of correspondences M: The appropriate two-view relation, D: The set of matches.

  20. Maximum Likelihood Estimation

  21. Maximum Likelihood Estimation Modify the mode: Where γ is the mixing parameter, v is just a constant. Here it is assumed that the outlier distribution is uniform, with - v/2,…,v/2 being the pixel range within which outliers are expected to fall.

  22. How to estimate γ • The initial estimate of γis ½ • Estimate the expectation of the ηifrom the current estimate of γ Where ηi =1 if the ith correspondence is an inlier, and ηi =0 if the ith correspondence is an outlier.

  23. How to estimate γ piis the likelihood of a datum given that it is an inlier and pois the likelihood of a datum given that it is an outlier: • Make a new estimate of γ • Repeat step2 and step3 until convergence.

  24. Outline • Introduction • LS-Least Squared • LMS-Least Median Squared • RANSAC- Random Sample Consequence • MLESAC-Maximum likelihood Sample consensus • MINPRAN-Minimize the Probability of Randomness

  25. MINPRAN • The first technique that reliably tolerates more than 50% outliers without assuming a know inlier bound. • It only assumes the outliers are uniformly distributed within the dynamic range of the sensor.

  26. MINPRAN

  27. MINPRAN N: The points are drawn from a uniform distribution of z value in range Zmin to Zmax. r: A distance between a curve φ k: Least points randomly full with in the range φ± r Z0:(Zmax-Zmin)/2

  28. MINPRAN

  29. Algorithm • Assuming p points are required to completely instantiate a fit. • Chooses S distinct, but not necessarily disjoint subsets of p points from data, and finds the fit to each random subsets to form the data, and finds the fit to each random subset to form S hypothesized fits φ1,…., φS • Select the minimizing as the “best fit”. • If , then φ* is accepted as a correct fit. • A final least-squares fit involving the p + i* inliers to φ* produces a accurate estimate of the model parameters.

  30. How many sample? • If w = proportion of inliers = 1-ε • Prob(sample with all inliers)=wp • Prob(sample with an outlier)=1-wp • Prob(S samples an outlier)=(1-wp)S We want Prob (S samples an outlier)<1-p Usually p is chosen at 0.99. • (1-wp)S<1-p • S>log(1-p)/log(1-wp)

  31. Randomness Threshold F0 Choose probability P0, and solve the equation. We can get Randomness Threshold F0.

  32. Randomness Threshold F0 • Define the equation • There is a unique value, ,which can found by bisection search, such that • By Lemma 2, if and only if • In order to for all i, we get the constraints • ci, 0≦i＜N, denote the number of the residuals in the range (fi…fi+1]

  33. Randomness Threshold F0 • Because the residuals are uniformly distributed, the probability any particular residuals are in the range fi to fi+1 is , where • The probability ci particular residuals are in the range fi to fi+1 is • Based on this, we can calculate the probability

  34. Randomness Threshold F0 • To make the analysis feasible, we assume the S fits and their residuals are independent. • So the probability each of the S samples has is just • The probability at least one sample has a minimum less than is • We can get F0 , since we know N, S , P0.

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