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RANSAC

RANSAC. Robust model estimation from data contaminated by outliers. Ondřej Chum. Fitting a Line. Least squares fit. RANSAC. Select sample of m points at random. RANSAC. Select sample of m points at random Calculate model parameters that fit the data in the sample. RANSAC.

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RANSAC

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  1. RANSAC Robust model estimation from data contaminated by outliers Ondřej Chum

  2. Fitting a Line Least squares fit

  3. RANSAC • Select sample of m points at random

  4. RANSAC • Select sample of m points at random • Calculate model parameters that fit the data in the sample

  5. RANSAC • Select sample of m points at random • Calculate model parameters that fit the data in the sample • Calculate error function for each data point

  6. RANSAC • Select sample of m points at random • Calculate model parameters that fit the data in the sample • Calculate error function for each data point • Select data that support current hypothesis

  7. RANSAC • Select sample of m points at random • Calculate model parameters that fit the data in the sample • Calculate error function for each data point • Select data that support current hypothesis • Repeat sampling

  8. RANSAC • Select sample of m points at random • Calculate model parameters that fit the data in the sample • Calculate error function for each data point • Select data that support current hypothesis • Repeat sampling

  9. How Many Samples? P(good) = On average N … number of point I … number of inliers m … size of the sample mean time before the success E(k) = 1 / P(good)

  10. How Many Samples? With confidence p

  11. How Many Samples? P(good) = With confidence p N … number of point I … number of inliers m … size of the sample P(bad) = 1 – P(good) P(bad k times) = (1 – P(good))k

  12. How Many Samples? With confidence p P(bad k times) = (1 – P(good))k≤1 - p k log (1 – P(good))≤ log(1 – p) k≥ log(1 – p) / log (1 – P(good))

  13. How Many Samples I / N [%] Size of the sample m

  14. RANSAC log(1 – p) log (1- ) I I-1 N N-1 k = k … number of samples drawn N … number of data points I … time to compute a single model p … confidence in the solution (.95)

  15. RANSAC [Fischler, Bolles ’81] In: U = {xi} set of data points, |U| = N function f computes model parameters p given a sample S from U the cost function for a single data point x Out: p* p*, parameters of the model maximizing the cost function k := 0 Repeat until P{better solution exists} < h (a function of C* and no. of steps k) k := k + 1 I. Hypothesis (1) select randomly set , sample size (2) compute parameters II. Verification (3) compute cost (4) if C* < Ck then C* := Ck, p* := pk end

  16. PROSAC – PROgressive SAmple Consensus • Not all correspondences are created equally • Some are better than others • Sample from the best candidates first 4 1 2 3 5 … N-2 N-1 N Sample from here

  17. PROSAC Samples l+2 l-1 l l+1 … … Draw Tlsamples from (1 … l) Draw Tl+1samples from (1 … l+1) l+1 Samples from (1 … l) that are not from (1 … l+1) contain l+1 Draw Tl+1 - Tlsamples of size m-1 and add

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