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 Guilichuan-gui@uiowa.edu

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