基于预检测的ransac

Conclusions • Benefits of RANSAC randomization of hypothesis verification studied. • A statistical preverification test was proposed. • The increased performance experimentally verified. 1 2 3 4 5 6 7 8 Remaining problems The parameter d is fixed at 1, not optimal Estimating ε, δ and consequently d during the sampling process 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
data points sample size model parameters cost function
Example II: Line detection by RANSAC
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Example II: Line detection by RANSAC
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• Randomly select two points
d
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+ (1 − ε )
m
1 − δd δ N+ 1−δ
d

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where δ is the probability that data point is consistent with a ”random” model α=ε β = δd Optimal d∗ satisfying
x∈U
ρ(pk , x)
(4) if C ∗ < Ck then C ∗ := Ck , p∗ := pk
Example I: Epipolar geometry estimation by RANSAC
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U : a set of correspondences, i.e. pairs of 2D points m=7 f : seven-point algorithm - gives 1 to 3 independent solutions ρ : thresholded Sampson’s error
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For each hypothesised model, all the data points are verified The more data points the longer
Randomized RANSAC Algorithm k := 0 Repeat until P{better solution exists} < η k := k + 1 I. Hypothesis (1) select randomly set Sk ⊂ U , |Sk | = m (2) compute parameters pk = f (Sk ) II. Preverification (3) perform test based on d N data points 9 10 11 12 13 14 15 16 17 18
∂J(Td,d ) ∂d ∗ d
=0
ln ε(tM + 1) d = ln /ln δ N (ln δ − ln ε) Randomized RANSAC is faster than the standard one, if J(T0,0) > J(T1,1) N > (tM 1−ε + 1) ε−δ
RANSAC
V´clav Hlav´ˇ a ac Czech Technical University, Faculty of Electrical Engineering Department of Cybernetics, Center for Machine Perception 121 35 Praha 2, Karlovo n´m. 13, Czech Republic a hlavac@fel.cvut.cz, http://cmp.felk.cvut.cz
Epipolar Geometry Estimation: Short Baseline Experiment 676 correspondences, approx. 60% of inliers tentative corr. by Harris operator and cross-correlation Leuven castle dataset 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 d 0 1 samples 480 960 models 1146 2301 tests 766875 83953 inliers 343 342 time 2.6 1.4 21 22 23 24 25
average R-RANSAC time: J= 1 εm α ¯ ¯ tM + PI (α N + (1 − α) tα) + PO (β N + (1 − β) tβ ) Standard RANSAC: α = 1, β=1 =⇒ 1 J = m (tM + N ) ε
Choosing the Preverification Test : The Td,d Test Td,d definition: test passed if all d randomly selected data points from U \ S are consistent with the model parameters p The time spent on the R-RANSAC with Td,d 1 J(Td,d) = m d tM + εm ε ε 1 − εd ε N+ 1−ε
RANSAC Algorithm
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k := 0 Repeat until P{better solution exists} < η (a function of C ∗ and no. of steps k) k := k + 1 I. Hypothesis (1) select randomly set Sk ⊂ U , |Sk | = m (2) compute parameters pk = f (Sk ) II. Verification (3) compute cost Ck = end
Example II: Line detection by RANSAC
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Randomly select two points • The hypothesised model is the line passing through the two
Example II: Line detection by RANSAC
x∈U
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(4) continue verification only if the test is passed III. Verification (5) compute cost Ck = ρ(pk , x)
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(6) if C ∗ < Ck then C ∗ := Ck , p∗ := pk end
Epipolar Geometry Estimation: Wide Baseline Experiment 413 correspondences, less than 40% of inliers tentative corr. by WBS algorithm [Matas, Chum, Urban, Pajdla ’01] bookshelf dataset 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 d 0 1 samples 3094 6366 models 7582 15583 tests 3078184 178217 inliers 161 164 time 12.9 8.7 21 22 23 24 25
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LECTURE PLAN Courtesy: O. Chum, J. Matas 1. 2. 3.
The RANSAC Algorithm [Fischler, Bolles ’81]
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In:
U = {xi} f (S) : S → p ρ(p, x) p∗
Out:
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 p∗, parameters of the model maximizing the cost function
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Randomly select two points The hypothesised model is the line passing through the two
Example II: Line detection by RANSAC
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Randomly select two points The hypothesised model is the line passing through the two
Example II: Line detection by RANSAC
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RANSAC Time Complexity