1. Computer Vision TexPoint fonts used in EMF: AAA Niels Chr Overgaard 2010 Lecture 8: Structure from Motion RANSAC Structure from motion problem Structure estimation Motion estimation Structure and motion estimation Goal: To understand the general ideas and Some of the methods. Read: Forsyth & Ponce Chapter: 12 - 13

2. Datorseende vt-10Föreläsning 8 RANSAC Random sampling concensus RANSAC - is a general probabilistic method for model estimation given noisy and contaminated data. Example: Line fitting (15 noisy + 5 outliers) TheoryPractice

3. Datorseende vt-10Föreläsning 8 RANSAC – algorithm (outline) 1.Input: S = data points n = sample size k = number of iterations t = threshold for godness of fit ( d = sufficient number of inliers (optional) ) 2.Loop: repeat k times Pick n-sample at random from S Fit model to sample Count #inliers (i.e. points in S fitting the model within threshold t) Store sample and inliers if better than the previous one. ( Stop if #inliers > d (optional) ) 3.Finalization: Fit model to the inliers of the best sample obtained.

4. Datorseende vt-10Föreläsning 8 Example: line fitting (again) Recall our situation: 20 points given, 5 outliers: Sample size: n = 2. Number of iterations: k>6 (we use k=7) Threshold for goodness of fit: d=0.5 (wrt. scale in figure)

5. Datorseende vt-10Föreläsning 8 The first iteration:

6. Datorseende vt-10Föreläsning 8 The following 6 iterations:

7. Datorseende vt-10Föreläsning 8 The final line estimation: Notice: Exhaustive search for the line with most inliers requires 190 iterations!

8. Datorseende vt-10Föreläsning 8

9. RANSAC : How many iterations? Let w denote (#inliers)/(#data points). n = the sample size (n=2 for lines, n=4 for plane homographies) k iterations. The probability that a random n-sample is correct: The probability that k random n-sample contains at least one outlier each: Choose k so large that the fraction of failures is smaller than a given tolerance z.

10. Sampel storlek Andelen outliers N 5%10%20%25%30%40%50% 2235671117 33479111935 435913173472 54612172657146 64716243797293 748203354163588 8592644782721177 från Hartley & Zisserman RANSAC: k for p=1-z=0.99

11. Datorseende vt-10Föreläsning 8 x Bildplan Kamera- centrum X

12. Datorseende vt-10Föreläsning 8 The Structure from Motion Problem Many cameras (images) Many scene points Estimate all of them! Let us see how this is done in principle

13. Datorseende vt-10Föreläsning 8

14. 3D-modell Exempel: Punkter Bilder Följda punkter

15. Datorseende vt-10Föreläsning 8 Exempel: Linjer och kägelsnitt Bilder 3D-modell

16. Datorseende vt-10Föreläsning 8

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