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OPTIPRO
Computational Methods forOn-Line Shape Inspection
September 18, 2009
Per Bergström
Department of Mathematics,Luleå University of Technology
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Project Management
Svensk Verktygsteknik (Swedish tool & die technology)
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Licentiate Thesis in Scientific Computing
Computational Methods forOn-Line Shape Inspection
ISSN: 1402-1757ISBN 978-91-86233-09-9
http://pure.ltu.se/ws/fbspretrieve/2505373
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Abstract
Computer vision problem
Industrial production line
Aimed for on-line shape inspection
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Introduction
Measurement of the shape of the surface
Transformation of the measured surface
Fitting the surface of the CAD object
Comparison with the CAD object
Measured surface is inspected
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The Shape Measurement Method
Projected fringes
Optical non-touching measurement method
One projector, two cameras
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The Shape Measurement Method
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The Shape Measurement Method
The measurement results in a huge number of data pointsrepresenting the measured surface.
Notation of data points: P = pkk=0,...,NP−1, NP ∼ 105
The background will be measured
Troubles around sharp edges
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About Surfaces in CAD
Example of NURBS surface with bidirectional control net.
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About Surfaces in CAD
Trimmed NURBS Surfaces
Concept of trimming
Arbitrarily shaped domainDomain is a connected spaceBounded by an outer curve and possible inner curve(s)
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Transformation of the Measured Surface
CAD surface: model points, set of points X
measured surface: data points, set of pointsP = pkk=0,...,NP−1
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Transformation of the Measured Surface
Rotation matrix R ∈ Ω,
Ω = R ∈ R3×3|RTR = I3,det (R) = +1
Translation vector T ∈ R3×1
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Transformation of the Measured Surface
Surface matching problem
minR∈Ω,T
NP−1∑k=0
(d(Rpk + T, X)
)2(1)
where d(p, X) = minx∈X‖x− p‖2
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Transformation of the Measured Surface
Problem (1) is solved using the ICP algorithm
ICP (iterative closest point) algorithm
Require: X,PR = I3, T = 0repeat
yk = C(Rpk + T, X), k = 0, . . . , NP − 1
[R,T] = arg minR∈Ω,T
NP−1∑k=0‖Rpk + T− yk‖2
2
until convergencereturn R , T
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Examples of Shape Inspection
p point on measured surface
p = Rp + T
p corresponding point on CAD object
x point on CAD object
x = R−1(x−T)
x corresponding point on measured surface
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Examples of Shape Inspection
Distance Measurement
Distance between two fix-points
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Examples of Shape Inspection
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Examples of Shape Inspection
x1 and x2 points on CAD object
Measured distance d between the points
d =∥∥R−1
2 (x2 −T2)−R−11 (x1 −T1)
∥∥2
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Using Best Fit: Finding Corners
Computer vision problem
The computer does not understand the shape
A minimization problem must be solved
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Using Best Fit: Finding Corners
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Using Best Fit: Finding Corners
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Using Best Fit: Finding Corners
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Brief Introduction of Robust Estimation
Decrease influence of strogly deviating data points
Apply methods from robust statistics tothe surface fitting problem
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Brief Introduction of Robust Estimation
x
y
Example of linear regression by minimizing a sum ofsquared residuals.
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Brief Introduction of Robust Estimation
−15 −10 −5 0 5 10 150
5
10
15
20
25
30
r
ρ
HuberTukeys Biweight
Huber (k > 0) Tukey’s Bi-weight (k > 0)
ρ(r) =
r2/2, |r| ≤ kk|r| − k2/2, |r| > k
ρ(r) =
(k2/6)[1− 1− (r/k)23], |r| ≤ kk2/6, |r| > k
Two examples of ρ-functions. Huber is convex,while Biweight is not.
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Brief Introduction of Robust Estimation
x
y
Regression examples: Linear robust regression using theHuber ρ-function, solid line.
Linear least squares regression, dotted line.
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Robust Problem Formulation
Instead of solving
minR∈Ω,T
NP−1∑k=0
(d(Rpk + T, X)
)2
the problem
minR∈Ω,T
NP−1∑k=0
ρ(d(Rpk + T, X)/σ
)(2)
is solved
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IRLS
IRLS (iteratively re-weighted least squares) method
weight function
w(r) =
ddrρ(r)r
if r 6= 0
d2
dr2ρ(r) if r = 0
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The ICP Algorithm with IRLS
Require: X,P,Rinit,Tinit
R = Rinit, T = Tinit
repeatpi = Rpi + T, i = 0, . . . , NP − 1yi = C(pi, X), i = 0, . . . , NP − 1ri = ‖pi − yi‖2 /σ, i = 0, . . . , NP − 1
[R,T] = arg minR∈Ω,T
NP−1∑i=0
wi(ri) ‖Rqi + T− yi‖22
until convergencereturn R , T
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Robust Estimators
−15 −10 −5 0 5 10 150
5
10
15
20
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r
ρ
Least−SquareHuberFairTukeys BiweightHampel
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Robust Estimators
−15 −10 −5 0 5 10 150
0.2
0.4
0.6
0.8
1
r
w
Least−SquareHuberFairTukeys BiweightHampel
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End of Presentation
Questions?
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