Instructor: Lichuan Gui lichuan-gui@uiowa
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Transcript of Instructor: Lichuan Gui lichuan-gui@uiowa
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 [email protected]
Phone: 319-384-0594 (Lab), 319-400-5985 (Cell) http://lcgui.net
2
Lecture 33. Peak-locking effect
3
Evaluation Errors Bias & random error for replicated measurement
Measuring variable X for N times
N
ii
N
ii N
XXN 1
2
1
2 11
RMS fluctuation (random error)
22
1
21
N
ioi XX
NRMS error
error)ramdon:εerror,bias:β value,true :( ioioi XXX
Individuale reading of X:
o
N
iio
N
ii X
NXX
NX
11
11
Mean value0
4
Peak-locking Effect Example: PIV test in a thermal convection flow
One of PIV recordings 3232-pixel window
5
Peak-locking Effect Example: PIV test in a thermal convection flow
One of vector maps Histogram of U & V
U component [pixel]
Num
ber
-4 -3 -2 -1 0 1 2 3 40
200
400
600
800
V component [pixel]
Num
ber
-4 -3 -2 -1 0 1 2 3 40
200
400
600
800
6
Peak-locking Effect Example: PIV test in a thermal convection flow
U component [pixel]
Num
ber
-4 -3 -2 -1 0 1 2 3 40
200
400
600
800
V component [pixel]
Num
ber
-4 -3 -2 -1 0 1 2 3 40
200
400
600
800
Correlation-basedinterrogation
U component [pixel]
Num
ber
-4 -3 -2 -1 0 1 2 3 40
200
400
600
800
V component [pixel]
Num
ber
-4 -3 -2 -1 0 1 2 3 40
200
400
600
800
Correlation-basedtracking
U component [pixel]
Num
ber
-4 -3 -2 -1 0 1 2 3 40
200
400
600
800
V component [pixel]
Num
ber
-4 -3 -2 -1 0 1 2 3 40
200
400
600
800
MQD-tracking
Histograms resulting from different algorithms
Peak-locking
Is the peak-locking an error?
Why does the peak-locking exist?
How to reduce the peak-locking effect?
7
Histogram for measuring 0.5 pixels
2
2
2
2
1
oXX
eXp
Probability density function (PDF)
o
oo
X
XXX
o
o eX
XXp2
2
2
2
1,
Source of Peak-locking
Probability to get X when measuring Xo
8
Distribution density function (DDF)
Source of Peak-locking
Distribution density function of true value Xo in region [a,b]:
b
aooo dXX
abforX 1
1
- (Xo)/(b-a): probability to find true value Xo in region [a,b] - Physical truth to be investigated
b
aooo dXXXpXX ,
Distribution density function of measured value X:
- (X)/(b-a): probability to get value X when measuring Xo in region [a,b]- Investigated phenomenon - Defined in region [-,+]:
2
2
X
X
dXX
MXHHistogram of measured variable X:
- Number of samples in [X-/2,X +/2]- M: average number in
9
Source of Peak-locking
b
ao
X
XXX
o
oo
b
aoo dXe
XXdXXXpXX o
oo2
2
2
2
1,
Distribution density function (DDF)
dXdXe
XX
MdXX
MXH
X
X
b
ao
X
XXX
o
o
X
X
o
oo
2
2
22
2
2
2
2
1
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
10
Bias & Random Error Distribution Simulation of Gaussian particle images
Test results with simulated PIV recording pairs- particle image diameter: 2 5 pixels- particle image brightness: 130 150- particle image number density: 20 particles in 3232-pixel window- vector number used for statistics:15,000
Displacement [pixel]
[p
ixel
]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.00
0.05
0.10
0.15CDWS
CCWS
FCTR
(a)
Displacement [pixel]
[p
ixel
]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
(b) Displacement [pixel]
[p
ixel
]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.15
-0.10
-0.05
0.00
0.05
0.10
0.15
(b)
Displacement [pixel]
[p
ixel
]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.00
0.05
0.10
0.15CDWS & random noiseCCWS & random noiseFCTR & random noise
(a)
w/o single pixel random noise with single pixel random noise
(CDWS=DWS, CCWS=CWS, FCTR=correlation-base tracking)
CDWS – Correlation-based discrete window shift (=DWS)
CCWS – Correlation-based continuous window shift (=CWS)
FCTR – FFT accelerated correlation-based tracking
11
Peak-locking Factor DDFs and histograms for the test results
flow)Millroll4androtationobjectsolid(e.g.,in1XωforXΩDefine oo
Displacement [pixel]
H
0 1 2 3 40
1000
2000
3000
4000
5000
6000
7000
8000
9000CDWS & ideal image
(d)
Displacement [pixel]
H
0 1 2 3 40
1000
2000
3000
4000
5000
6000
7000
8000
9000
(e)
CCWS & ideal image
Displacement [pixel]
H
0 1 2 3 40
1000
2000
3000
4000
5000
6000
7000
8000
9000
(f)
FCTR & ideal image
Displacement [pixel]
o
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8CDWS & ideal image
(a)
Displacement [pixel]
o
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(b)
CCWS & ideal image
Displacement [pixel]
o
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8FCTR & ideal image
(c)
Displacement [pixel]
H
0 1 2 3 40
1000
2000
3000
4000
5000
6000
7000
8000
9000
(d)
CDWS & random noise
Displacement [pixel]
H
0 1 2 3 40
1000
2000
3000
4000
5000
6000
7000
8000
9000
(e)
CCWS & random noise
Displacement [pixel]
H
0 1 2 3 40
1000
2000
3000
4000
5000
6000
7000
8000
9000
(f)
FCTR & random noise
Displacement [pixel]
o
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(b)
CCWS & random noise
Displacement [pixel]
o
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(c)
FCTR & random noise
Displacement [pixel]
o
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8CDWS & random noise
(a)
1
0o 1XΩ:factor locking-peack Define dX
12
Response of to bias and random error distribution
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
Peak-locking Factor
oo XAX 2sin 02cos1 oo XAXSimulation of error distributions:
Particle image displacement [pixel]
[p
ixel
]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.07
-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
(b)
A= -0.050.05
Particle image displacement [pixel]
[p
ixel
]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
0.13
0.14
(a)
A= -0.010.050= 0.025
Simulated error distributions
A
-0.04 -0.02 0 0.02 0.040
0.05
0.1
0.15
0.2
A= 0, 0= 0.025
(a)
0
0 0.02 0.04 0.06 0.080
0.05
0.1
0.15
0.2
A= 0.01,A= 0.01
A= 0.01,A= -0.01
(c)
A
-0.02 0 0.02 0.04 0.060
0.05
0.1
0.15
0.2
A= 0, 0= 0.025
(b)
0
0 0.02 0.04 0.06 0.080
0.05
0.1
0.15
0.2
A= -0.01,A= -0.01
A= -0.01,A= 0.01
(d)
Response of peak-locking factor
13
0.02
0.02
0.02
0.04
0.04
0.04
0.04
0.04
0. 0
6
0.06
0.06
0.06
0.06
0.08
0.08
0.08
0.08
0.1
0.1
0.1
0.1
0.12
0.12
0.12
0.12
0.1
4
0.14
0.14
0.14
0.14
0.16
0.16
0.16
0.18
0.18
0.1
8
0.2
0.2
0.22
0.24
A [pixel]
A
[pix
el]
-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05-0.01
0
0.01
0.02
0.03
0.04
0.05Contours of peak-locking factor for o=0.025
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
Peak-locking Factor Response of to bias and random error distribution
14
Influence of particle size on Test results
Particle image diameter [pixel]
1 2 3 4 50
0.1
0.2
0.3
0.4
0.5FCTRCDWSCCWS
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)
Increasing A when A>0 for CCWS
Peak-locking Factor
15
Influence of particle number density on Test results
Particle number in the 32x32-pixel window
10 15 20 25 30 35 400
0.1
0.2
0.3
0.4FCTRCDWSCCWS
not sensitive to particle image number density
generally smallest by FCTR (for Gaussian image profile)
Peak-locking Factor
16
Influence of window size on Test results
Side length of the interrog. window [pixel]
16 24 32 40 48 56 640
0.1
0.2
0.3
0.4FCTRCDWSCCWS
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)
Peak-locking Factor
17
Image samples of different quality
Displacement [pixel]
[p
ixel
]
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
(a)
FCTR
Displacement [pixel]
[p
ixel
]
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15GaussianOverexposedBinariy
(d)
CCWS
Displacement [pixel]
[p
ixel
]
0 0.2 0.4 0.6 0.8 1-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
(b)
FCTR
Displacement [pixel]
[p
ixel
]
0 0.2 0.4 0.6 0.8 1-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
(e)
CCWS
Displacement [pixel]
[p
ixel
]
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
(c)
FCTR
Displacement [pixel]
[p
ixel
]
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
(f)
CCWS
Displacement [pixel]
o
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(b)
FCTR & overexposed image
Displacement [pixel]
o
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(c)
FCTR & binary image
Displacement [pixel]
o
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(d)
CCWS & Gaussian image
Displacement [pixel]
o
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(e)
CCWS & overexposed image
Displacement [pixel]
o
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(f)
CCWS & binary image
Displacement [pixel]
o
-2 -1 0 1 20
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(a)
FCTR & Gaussian image
Non-Gaussian Particle Images Influence of particle image profile
18
Application Examples
PIV measurement in a thermal convection flowGray value histogram & evaluation sample
Particle image displacement [pixel]
Num
ber
-3 -2 -1 0 1 2 3 4 50
2000
4000
6000
8000
10000
12000
(a)
CDWS
Particle image displacement [pixel]
Num
ber
-3 -2 -1 0 1 2 3 4 50
2000
4000
6000
8000
10000
12000
(c)
FCTR
Particle image displacement [pixel]
Num
ber
-3 -2 -1 0 1 2 3 4 50
2000
4000
6000
8000
10000
12000
(d)
CCWS
Histogram of particle image displacement
- Overexposed particle images
- Particle image diameter 3 4 pixels
- No peak-locking for CCWS
19
Application Examples PIV measurement in a wake vortex flow
Gray value histogram & evaluation sample
Particle image displacement [pixel]
Num
ber
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 90
200
400
600
800
1000
1200
(a)
CDWS
Particle image displacement [pixel]
Num
ber
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 90
200
400
600
800
1000
1200
(b)
FCTR
Particle image displacement [pixel]
Num
ber
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 90
200
400
600
800
1000
1200
(c)
CCWS
Histogram of particle image displacement
- Particle image diameter 1 pixels
- Least peak-locking for CCWS
20
Application Examples PIV measurement in a micro channel flow
Gray value histogram & evaluation sample
Particle image displacement [pixel]
Num
ber
1 2 3 4 5 6 7 8 9 10 11 120
100
200
300
400
500
(c)
FCTR
Particle image displacement [pixel]
Num
ber
1 2 3 4 5 6 7 8 9 10 11 120
100
200
300
400
500
(d)
CCWS
Particle image displacement [pixel]
Num
ber
1 2 3 4 5 6 7 8 9 10 11 120
100
200
300
400
500
(a)
CDWS
Histogram of particle image displacement
- Mid-pixel peak-locking for CCWS
- Particle image diameter 4 6 pixels
21
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
References
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 distributionG2=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 numbercol=nx/Mg-1; % grid column numbersr=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; endend % 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 endend
plot(D,Px,'r*-') % make plotshold onplot(D,Py,'b*-')hold off