Lecture33 - Lcgui.net
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Transcript Lecture33 - Lcgui.net
Measurements in Fluid Mechanics
058:180 (ME:5180)
Time & Location: 2:30P - 3:20P MWF 3315 SC
Office Hours:
4:00P – 5:00P MWF 223B-5 HL
Instructor: Lichuan Gui
[email protected]
Phone: 319-384-0594 (Lab), 319-400-5985 (Cell)
http://lcgui.net
Lecture 33. Peak-locking effect
2
Evaluation Errors
Bias & random error for replicated measurement
Measuring variable X for N times
Individuale reading of X:
X i X o i ( X o : true value,β : bias error, ε i : ramdon error)
Mean value
1
X
N
N
1
X
X
i o
N
i 1
N
0
i X o
i 1
RMS fluctuation (random error)
RMS error
1
N
1
N
X i X
N
2
i 1
N
X i X o
2
1
N
N
i2
i 1
2 2
i 1
3
Peak-locking Effect
Example: PIV test in a thermal convection flow
One of PIV recordings
3232-pixel window
4
Peak-locking Effect
Example: PIV test in a thermal convection flow
One of vector maps
Histogram of U & V
Number
800
600
400
200
0
-4
-3
-2
-3
-2
-1
0
1
2
3
4
-1
0
1
2
3
4
U component [pixel]
800
Number
600
400
200
0
-4
V component [pixel]
5
Peak-locking Effect
Example: PIV test in a thermal convection flow
800
800
600
600
Number
Correlation-based
interrogation
Number
Histograms resulting from different algorithms
400
200
400
200
-3
-2
-1
0
1
2
3
0
-4
4
800
800
600
600
-2
-1
0
1
2
3
4
1
2
3
4
1
2
3
4
400
Why does the peak-locking
exist?
400
200
0
-4
-3
V component [pixel]
Is the peak-locking an error?
U component [pixel]
Number
Correlation-based
tracking
Number
0
-4
200
-3
-2
-1
0
1
U component [pixel]
2
3
0
-4
4
-3
-2
-1
0
V component [pixel]
How to reduce the peak-locking effect?
800
800
600
600
Number
MQD-tracking
Number
400
200
Peak-locking
0
-4
400
200
-3
-2
-1
0
1
U component [pixel]
2
3
4
0
-4
-3
-2
-1
0
V component [pixel]
6
Source of Peak-locking
Probability density function (PDF)
Probability to get X when measuring Xo
p X
1
2
e
X X o 2
Histogram for measuring 0.5 pixels
2 2
p X , X o
1
2 X o
e
X X o X o 2
2 2 X o
7
Source of Peak-locking
Distribution density function (DDF)
Distribution density function of true value Xo in region [a,b]:
X o
for
1 b
X o dX o 1
ba a
- (Xo)/(b-a): probability to find true value Xo in region [a,b]
- Physical truth to be investigated
Distribution density function of measured value X:
b
X X o p X , X o dX o
a
- (X)/(b-a): probability to get value X when measuring Xo in region [a,b]
- Investigated phenomenon
- Defined in region [-,+]:
Histogram of measured variable X:
- Number of samples in [X-/2,X +/2]
- M: average number in
X
H X M
2
X
X
dX
2
8
Source of Peak-locking
Distribution density function (DDF)
X X o X o 2
1
2 2 X o
X X o p X , X o dX o X o
e
dXo
2 X o
a
a
b
b
X
H X
M
2
X dX
X
2
X
M
2
X
2
X X o X o 2
1
2 2 X o
X
e
dX o dX
o 2 X
a
o
b
Histogram determined by
1)
Sample number M
2)
Sub region size
3)
Physical truth
(Xo)
4)
Bias error
(Xo)
5)
Random error
(Xo)
Possible sources of peak-locking
9
Bias & Random Error Distribution
Simulation of Gaussian particle images
Test results with simulated PIV recording pairs
- particle image diameter:
- particle image brightness:
- particle image number density:
- vector number used for statistics:
2 5 pixels
130 150
20 particles in 3232-pixel window
15,000
0.15
CDWS
CCWS
FCTR
0.10
[pixel]
[pixel]
0.15
0.05
0.00
(a)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(a)
0.6
Displacement [pixel]
0.7
0.8
0.9
1
0.7
0.8
0.9
1
CCWS – Correlation-based continuous window shift (=CWS)
0.15
[pixel]
[pixel]
0.05
FCTR – FFT accelerated correlation-based
tracking
0.10
0.10
0.05
0.00
-0.05
0.05
0.00
-0.05
-0.10
-0.10
(b)
0.10
CDWS – Correlation-based discrete
0.00window shift (=DWS)
0
0.1
0.2
0.3
0.4
0.5
Displacement [pixel]
0.15
-0.15
CDWS & random noise
CCWS & random noise
FCTR & random noise
0
0.1
0.2
0.3
0.4
0.5
0.6
Displacement [pixel]
0.7
0.8
w/o single pixel random noise
0.9
1
-0.15
(b)
0
0.1
0.2
0.3
0.4
0.5
0.6
Displacement [pixel]
with single pixel random noise
(CDWS=DWS, CCWS=CWS, FCTR=correlation-base tracking)
10
Peak-locking Factor
DDFs and histograms for the test results
Define Ωo X for ωX o 1 in , (e.g. solidobject rotationand 4 roll Mill flow)
(b)
CCWS & ideal image
H
-1
0
1
2
Displacement [pixel]
(c)
H
-1
0
1
Displacement [pixel]
2
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
0
(e)
FCTR & ideal image
o
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2
(d)
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
0
(f)
1
2
3
4
Displacement [pixel]
CCWS & ideal image
(a)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2
H
-1
0
1
2
Displacement [pixel]
1
2
3
4
Displacement [pixel]
FCTR & ideal image
(b)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2
CCWS & random noise
-1
0
1
2
Displacement [pixel]
1
2
3
Displacement [pixel]
4
(c)
-1
0
1
2
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
0
(e)
FCTR & random noise
Displacement [pixel]
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
0
(d)
H
2
CDWS & random noise
H
1
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2
o
H
0
o
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2
-1
Displacement [pixel]
CDWS & ideal image
o
(a)
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
0
o
CDWS & ideal image
o
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2
9000
8000
7000
6000
5000
4000
3000
2000
1000
0
0
(f)
CDWS & random noise
1
2
3
4
Displacement [pixel]
CCWS & random noise
1
2
3
4
Displacement [pixel]
FCTR & random noise
1
2
3
4
Displacement [pixel]
1
Define peack - lockingfactor : Ω o X 1 dX
0
11
Peak-locking Factor
Response of to bias and random error distribution
Simulation of error distributions: X o A 1 cos2X o 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Particle image displacement [pixel]
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
-0.07
A= 0, 0= 0.025
0.15
0.1
(a)
0
-0.04
-0.02
0
A
0.02
0.04
0.2
(b)
0.3
0.4
0.5
0.6
0.7
Particle image displacement [pixel]
0.8
0.9
1
0.1
(c)
0
0
0
-0.02
0
0.02
A
0.04
0.06
A= -0.01,A= -0.01
A= -0.01,A= 0.01
0.15
0.05
0.1
0.1
0.2
A= 0.01,A= 0.01
A= 0.01,A= -0.01
0.15
0
A= 0, 0= 0.025
0.05
0.2
A= -0.050.05
[pixel]
0.15
0.2
0.05
(a)
(b)
0.2
A= -0.010.05
0= 0.025
0.14
0.13
0.12
0.11
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
Response of peak-locking factor
[pixel]
Simulated error distributions
X o A sin2X o
0.1
0.05
0.02
0.04
0
0.06
0.08
(d)
0
0
0.02
0.04
0
0.06
0.08
very sensitive to bias error amplitude A
sensitive to random error amplitude A when >0.02
not sensitive to constant portion of random error 0
12
Peak-locking Factor
Response of to bias and random error distribution
Contours of peak-locking factor for o=0.025
6
0.0
6
0.1
4
0.1
0.
02
0.0
0.1
2
0.0
4
0.16
0.12
0.04
-0.03
-0.02
0.18
0.06
0.14
-0.04
0.14
0.08
0.1
0.02
0.04
0.06
0.02
0.06
0.08
0.1
0
-0.01
-0.05
0.0
8
0.0
4
8
0.0
0 .1
2
6
0.12
0.16
0.18
0.2
0.14
0.02
0.01
0.1
0 .2
0.22
0.03
0.1
8
0.1
0.1
0.2
0.04
4
4
0.0
4
0.05
A [pixel]
-0.01
0
0.01
0.02
0.03
0.04
0.05
A [pixel]
Peaks locked at integer pixels in bright area and at midpixels in dark area
Peak-locking minimum around A=0
Increasing A increaes for A<0 but reduces for A>0
13
Peak-locking Factor
Influence of particle size on
Test results
Increasing A when A>0 for CCWS
0.5
0.4
0.3
FCTR
CDWS
CCWS
0.2
0.1
0 1
2
3
4
5
Particle image diameter [pixel]
increases with incresing particle size by CDWS
descreses with incresing particle size by FCTR & CCWS
increases when particle szie too small by FCTR & CDWS
smallest when particle szie too small by CCWS
generally smallest by FCTR (for Gaussian image profile)
14
Peak-locking Factor
Influence of particle number density on
Test results
0.4
FCTR
CDWS
CCWS
0.3
0.2
0.1
0 10
15
20
25
30
35
40
Particle number in the 32x32-pixel window
not sensitive to particle image number density
generally smallest by FCTR (for Gaussian image profile)
15
Peak-locking Factor
Influence of window size on
Test results
0.4
FCTR
CDWS
CCWS
0.3
0.2
0.1
0 16
24
32
40
48
56
64
Side length of the interrog. window [pixel]
decreases with incresing window size by CDWS
slightly increses with incresing window size by CCWS
slightly decrease with incresing window size by FCTR
generally smallest by FCTR (Gaussian image profile)
16
Non-Gaussian Particle Images
Influence of particle image profile
Gaussian
Overexposed
Binariy
0.05
(a)
0.2
0.4
0.6
0.8
1
Displacement [pixel]
(d)
0.08
0
0
0.4
0.6
[pixel]
0
-0.02
CCWS
0.02
0
-0.02
-0.04
-0.04
-0.06
-0.06
-0.08
0
-0.08
0
0.2
0.4
0.6
0.8
1
Displacement [pixel]
(e)
0.15
0.2
0.4
0.6
1
0.15
CCWS
0.05
0.2
0.4
0.6
0.8
Displacement [pixel]
1
(a)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2
FCTR & Gaussian image
-1
0
1
2
Displacement [pixel]
0.1
(b)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2
FCTR & overexposed image
-1
0
1
2
Displacement [pixel]
o
0.1
[pixel]
[pixel]
0.8
Displacement [pixel]
FCTR
(c)
1
0.04
0.02
0
0
0.8
Displacement [pixel]
0.06
0.04
(b)
0.2
0.08
FCTR
0.06
[pixel]
Image samples of different quality
0.05
o
0
0
0.1
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2
o
[pixel]
[pixel]
0.1
CCWS
0.05
(f)
0
0
0.2
0.4
0.6
0.8
Displacement [pixel]
1
(c)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2
o
0.15
FCTR
(d)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2
CCWS & Gaussian image
-1
0
1
2
Displacement [pixel]
o
0.15
(e)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
-2
CCWS & overexposed image
-1
0
1
2
Displacement [pixel]
o
FCTR & binary image
-1
0
1
Displacement [pixel]
2
(f)
CCWS & binary image
-1
0
1
Displacement [pixel]
17
2
Application Examples
PIV measurement in a thermal convection flow
Gray value histogram & evaluation sample
Histogram of particle image displacement
12000
10000
CDWS
Number
8000
6000
4000
2000
(a)
0-3
-2
-1
0
1
2
3
4
5
4
5
4
5
Particle image displacement [pixel]
12000
10000
Number
6000
4000
2000
- Overexposed particle images
- Particle image diameter 3 4 pixels
FCTR
8000
(c)
0-3
-2
-1
0
1
2
3
Particle image displacement [pixel]
12000
- No peak-locking for CCWS
10000
Number
CCWS
8000
6000
4000
2000
(d)
0-3
-2
-1
0
1
2
3
Particle image displacement [pixel]
18
Application Examples
PIV measurement in a wake vortex flow
Gray value histogram & evaluation sample
Histogram of particle image displacement
1200
Number
1000
CDWS
800
600
400
200
(a)
0-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
7
8
9
7
8
9
Particle image displacement [pixel]
1200
Number
1000
- Least peak-locking for CCWS
FCTR
800
600
400
200
- Particle image diameter 1 pixels
(b)
0-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
Particle image displacement [pixel]
1200
1000
Number
CCWS
800
600
400
200
(c)
0-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
Particle image displacement [pixel]
19
Application Examples
PIV measurement in a micro channel flow
Gray value histogram & evaluation sample
Histogram of particle image displacement
500
CDWS
Number
400
300
200
100
(a)
01
2
3
4
5
6
7
8
9
10
11
12
10
11
12
10
11
12
Particle image displacement [pixel]
500
FCTR
Number
400
- Particle image diameter 4 6 pixels
300
200
100
(c)
- Mid-pixel peak-locking for CCWS
01
2
3
4
5
6
7
8
9
Particle image displacement [pixel]
500
CCWS
400
Number
300
200
100
(d)
01
2
3
4
5
6
7
8
9
Particle image displacement [pixel]
20
References
Gui and Wereley (2002) A correlation-based continues window shift technique for reducing the
peak locking effect in digital PIV image evaluation. Exp Fluids 32: 506-517
21
Matlab program for showing peak-locking effect
A1=imread('A001_1.bmp'); % input image file
A2=imread('A001_2.bmp'); % input image file
G1=img2xy(A1); % convert image to gray value distribution
G2=img2xy(A2); % convert image to gray value distribution
Mg=16; % interrogation grid width
Ng=16; % interrogation grid height
M=32; % interrogation window width
N=32; % interrogation window height
[nx ny]=size(G1);
row=ny/Mg-1; % grid row number
col=nx/Mg-1; % grid column number
sr=12; % search radius
for i=1:col % correlation interrogation begin
for j=1:row
x=i*Mg;
y=j*Ng;
g1=sample01(G1,M,N,x,y);
g2=sample01(G2,M,N,x,y);
[C m n]=correlation(g1,g2);
[cm vx vy]=peaksearch(C,m,n,sr,0,0); U(i,j)=vx;
V(i,j)=vy;
X(i,j)=x;
Y(i,j)=y;
end
end % correlation interrogation end
nn=0; % count number of displacements with 0.1 pixel steps
for k=-120:120
nn=nn+1;
D(nn)=double(k/10);
Px(nn)=0;
Py(nn)=0;
for i=1:col
for j=1:row
if U(i,j)>= D(nn)-0.05 & U(i,j) < D(nn)+0.05
Px(nn)=Px(nn)+1;
end
if V(i,j)>= D(nn)-0.05 & V(i,j) < D(nn)+0.05
Py(nn)=Py(nn)+1;
end
end
end
end
plot(D,Px,'r*-') % make plots
hold on
plot(D,Py,'b*-')
hold off