# 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
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
3232-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

ba 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 3232-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 ωX 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  cos2X 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= -0.050.05

 [pixel]
0.15
0.2
0.05
(a)
(b)
0.2
A= -0.010.05
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 sin2X 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
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
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
```

– Cards

– Cards

– Cards

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– Cards