Computer Vision TexPoint fonts used in EMF: AAA Niels Chr Overgaard 2010 Lecture 8: Structure from...

Computer Vision

Niels Chr Overgaard2010

Lecture 8: Structure from Motion

•RANSAC•Structure from motion problem•Structure estimation•Motion estimation•Structure and motion estimation

Goal: To understand the general ideas andSome of the methods.

Read: Forsyth & PonceChapter: 12 - 13

Datorseende vt-10 Föreläsning 8

RANSACRandom sampling concensus

RANSAC - is a general probabilistic method for model estimation given noisy and contaminated data.

Example: Line fitting (15 noisy + 5 outliers)

Theory Practice


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.


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)


The first iteration:


The following 6 iterations:


The final line estimation:

Notice: Exhaustive search for the line with most inliers requires 190 iterations!

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.

nw

knw )1(

Sampel storlek Andelen outliers

N 5% 10% 20% 25% 30% 40% 50%

2 2 3 5 6 7 11 17

3 3 4 7 9 11 19 35

4 3 5 9 13 17 34 72

5 4 6 12 17 26 57 146

6 4 7 16 24 37 97 293

7 4 8 20 33 54 163 588

8 5 9 26 44 78 272 1177från Hartley & Zisserman

RANSAC: k for p=1-z=0.99


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BildplanKamera-centrum

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The Structure from Motion Problem

•Many cameras (images)•Many scene points•Estimate all of them!

Let us see how this is done in principle


3D-modell

Exempel: Punkter

Bilder Följda punkter


Exempel: Linjer och kägelsnitt

Bilder3D-modell


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Computer Vision TexPoint fonts used in EMF: AAA Niels Chr Overgaard 2010 Lecture 8: Structure from...

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