Instructor: Lichuan Gui lichuan-gui@uiowa

Measurements in Fluid Mechanics058:180 (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

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Lecture 26. Sub-pixel Displacement

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Correlation Interrogation & FFT Acceleration

Evaluation function

Auto-correlation function

g(i,j)

0

0.2

0.4

0.6

0.8

1

(m,n)

-30-20

-100

1020

30

m

-30-20

-100

1020

30

n

M

i

N

j

njmigjignm1 1

,,,

(m*=10, n*=-5) (m*=-10, n*=5)

- One double exposed evaluation sample, i.e. g(i,j)=g2(i,j)=g1(i,j)

- Displacement determined by positions of the secondary maxima

- Two possible velocity directions

4

Correlation Interrogation & FFT Acceleration

Evaluation function

Cross-correlation function

- Two single exposed evaluation samples, i.e. g2(i,j) & g1(i,j)

0

0.2

0.4

0.6

0.8

1

(m,n)

-30-20

-100

1020

30

m

-30-20

-100

1020

30

n

M

i

N

j

njmigjignm1 1

21 ,,,

(m*=3, n*=5)

g1(i,j)

g2(i,j)

- Velocity direction clear

- Displacement determined by position of the maximum

5

Sub-pixel Displacement

Local surface fit of evaluation function

Discrete correlation function around point of maximum (m*,n*)

-1

-1

6



Discrete distribution fitted with continuous surface z(x,y)

7



Sub-pixel displacement (x*,y*) determined with position of the maximum of z(x,y)

8



5-point fit most commonly used

9



5-point parabolic fit

Fit function 2

42

3210),( xcycxcyccyxz

Linear equation system for determining constants

0,1

0,1

1,0

1,0

0,0

10101

10101

01011

01011

00001

**

**

**

**

**

4

3

2

1

0

nm

nm

nm

nm

nm

c

c

c

c

c

02

02*

31

*42

ycc

xcc

1,21,2,4

1,1,

,12,12,4

,1,1

******

*****

******

*****

nmnmnm

nmnmy

nmnmnm

nmnmx

0,

,

0,

,

**

**

yxy

yxz

yxx

yxz

Position of the maximum

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5-point exponential fit (3-point Guassian curve fit)

0,

,

0,

,

**

**

yxy

yxz

yxx

yxz

Position of the maximum

Fit function 2

42

3210),( xcycxcycceyxz

Linear equation system for determining constants

0,1ln

0,1ln

1,0ln

1,0ln

0,0ln

10101

10101

01011

01011

00001

**

**

**

**

**

4

3

2

1

0

nm

nm

nm

nm

nm

c

c

c

c

c

1,ln21,ln2,ln4

1,ln1,ln

,1ln2,1ln2,ln4

,1ln,1ln

******

*****

******

*****

nmnmnm

nmnmy

nmnmnm

nmnmx

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Arbitrarily sized interrogation window

Performance of FFT-based correlation

- Effective correlation region

A : Effective correlation region

B,C,D: Periodically Padded regions

DCBA

M

i

N

j

njmigjignm

1 1

21 ,,,

12


Performance of FFT-based correlation- Reliability & RMS error dependent on particle image displacement

- Reliability & RMS error dependent on interrogation window size

- Impossible to determine displacement components not less than half of

window side length

total

valid

N

NReliability:

RMS error:

validN

kkk

valid

SSN 1

21

S : measured valueS’: real value

Some test results with synthetic images

f: interrogation window size

0 4 8 12 16 20 24 28 320

0.2

0.4

0.6

0.8

1

0 4 8 12 16 20 24 28 320

0.1

0.2

0.3

0.4

0.5

f =64X64

f =32X32

16X16

f =64X64

f =16X16

f =32X32

S

S

13


Average padding to next power of 2

22

22

2

21

1

*

*

N

Mfor

N

M

* *

1 1

*2

*1

* ,,,M

i

N

j

njmigjignm

NjMijigjig 0,1,,*

**

**

1 1

*

,1

1,

,1

,

NjNMi

or

NjMiM

qpgMN

jigM

p

N

q

3,2,1,0,for,, **** lkjiglNjkMig

Test results with synthetic

images (f=32x32)

1 original

2 padded to f=64x64

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1

2

1 2

S S

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More about Correlation Interrogation

Fast computation of evaluation functionAcceleration with radix-2 based FFT algorithm- Computation time test

15

Effect of linear transformation of evaluation samples

211 1

2121 1

12121 ,,,, BBnjmigBAjigBAnmAAnmM

i

N

j

M

i

N

j

jigjig ,,, 21

2222

1111

,,

,,

BjigAjig

BjigAjig

jigjig ,,, 21

M

i

N

j

njmigjignm1 1

21 ,,,

M

i

N

j

njmigjignm1 1

21 ,,,

DnmCnm ,,


on the correlation function

- no influence

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on the sub-pixel displacement determination with Parabolic surface/curve fit

- no influence

DnmCnm ,,

nmx

mnmnm

nmnm

DmCDnmCDnmC

DnmCDnmC

mnmnm

nmnmnmx

,

0,12,12,4

,1,1

0,12,12,4

,1,1

0,12,12,4

,1,1,

*

*****

****

*****

****

*****

*****

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on the sub-pixel displacement determination with exponential surface/curve fit

DnmCnm ,,

0,1ln2,1ln2,ln4

,1ln,1ln

0,1ln2,1ln2,ln4

,1ln,1ln*****

****

*****

****

mnmnm

nmnm

mnmnm

nmnm

nmxnmx ,, **

- obvious influence

- no zero and negative values allowed

- normalization suggested before sub-pixel fit

Class project: example of Matlab functions for PIV recording evaluation File name: correlation.m

function[c m n]=correlation(g1,g2)% INPUT PARAMETERS% g1 - 1st evaluation sample, g2 - 2nd evaluation sample% OUTPUT PARAMETERS% c - correlation function, % m - horizontal coordinate, n - vertical coordinate

[M N]=size(g1); % determine size of evaluation sample for k=3:12 % determine window size for FFT - begin if 2^k>=M & 2^k>=N break; endendM2=2^k;N2=2^k; % determine window size for FFT - end

gm1=mean(mean(g1)); % average gray value padding - begin gm2=mean(mean(g2));for i=1:M2 for j=1:N2 if i<=M & j<=N gg1(i,j)=g1(i,j); gg2(i,j)=g2(i,j); else gg1(i,j)=gm1; gg2(i,j)=gm2; end endend % average gray value padding - end

gfft1=fft2(gg1); % compute correlation function - begingfft2=conj(fft2(gg2));C=real(ifft2(gfft1.*gfft2)); % compute correlation function - end

for i=1:M2 % determine coordinates in correlation plane - begin for j=1:N2 m(i,j)=i-M2/2-1; n(i,j)=j-N2/2-1; endend % determine coordinates in correlation plane - end

for i=1:M2 % periodical reconstruction - begin for j=1:N2/2 t=C(i,j); C(i,j)=C(i,N2-j+1); C(i,N2-j+1)=t; endendfor j=1:N2 for i=1:M2/2 t=C(i,j); C(i,j)=C(M2-i+1,j); C(M2-i+1,j)=t; endendfor i=1:M2/2 C1(i,1:N2)=C(i+M2/2,1:N2);endfor i=M2/2+1:M2 C1(i,1:N2)=C(i-M2/2,1:N2);endfor j=1:N2/2 c(1:M2,j)=C1(1:M2,j+N2/2);endfor j=N2/2+1:N2 c(1:M2,j)=C1(1:M2,j-N2/2);end % periodical reconstruction of correlation function - end

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Class project: example of Matlab functions for PIV recording evaluation File name: peaksearch.mfunction [cm vx vy]=peaksearch(C,m,n,sr,mode,direction)% INPUT PARAMETERS% C - correlation function% m - horizontal coordinate% n - vertical coordinate% sr - search radius % mode - (=0) cross-correlation, (=1) auto-correlation% direction - 1. to right, 2. to left, 3. upwards, 4. downwards% OUTPUT PARAMETERS% cm - high correlation peak value% (vx,vy) – particle image displacement [M N]=size(C); % high peak search - beginix=0;jy=0;cm=0;for i=M/2+1-sr:M/2+1+sr for j=N/2+1-sr:N/2+1+sr if mode==1 & direction==1 & i<=M/2+1 continue; end if mode==1 & direction==2 & i>=M/2+1 continue; end if mode==1 & direction==3 & j>=N/2+1 continue; end if mode==1 & direction==4 & j<=N/2+1 continue; end

max=0; for i1=i-1:i+1 for j1=j-1:j+1 if max<C(i1,j1) max=C(i1,j1); end end end if C(i,j)>=max & C(i,j)>cm ix=i; jy=j; cm=C(i,j); end endendmm=m(ix,jy);nn=n(ix,jy);% high peak search -end

% SUB-PIXEL FITdx=(log(C(ix+1,jy))-log(C(ix-1,jy)))/(4*log(C(ix,jy))-2*log(C(ix+1,jy))-2*log(C(ix-1,jy)));dy=(log(C(ix,jy+1))-log(C(ix,jy-1)))/(4*log(C(ix,jy))-2*log(C(ix,jy+1))-2*log(C(ix,jy-1)));

%Displacementvx=mm+dx;vy=nn+dy;

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Class project: example of Matlab functions for PIV recording evaluation File name: sample01.m

function[g]=sample01(G,M,N,X,Y)%INPUT PARAMETERS% G - gray value distribution of the PIV recording% M - interrogation sample width% N - interrogation sample height% X, Y - interrogation sample coordinates% ----------------------------------------------------% OUTPUT PARAMETERS% g - gray value distribution of the evaluation sample% ----------------------------------------------------% I, J – integer part of evaluation sample coordinates% x,y – decimal part of evaluation sample coordinates

[nx ny]=size(G);

I=int16(X); % integer part of coordinatesJ=int16(Y);

x=double(X)-double(I); %decimal part of coordinates

y=double(Y)-double(J);

if x<0 I=I-1; x=x+1; % x should be positiveend

if y<0 J=J-1; y=y+1; % y should be positive end

gm=0; % initial value of mean gray valuenr=0; % initial value of number of effective pixelsfor i=1:M for j=1:N ii=i+I-int16(M/2); jj=j+J-int16(N/2); if ii>=1 & ii<nx & jj>=1 & jj<ny % pixels in the image frame Ga=double(G(ii,jj)); Gb=double(G(ii+1,jj)); Gc=double(G(ii,jj+1)); Gd=double(G(ii+1,jj+1)); A=(1-x)*(1-y); B=x*(1-y); C=(1-x)*y; D=x*y; g(i,j)=A*Ga+B*Gb+C*Gc+D*Gd; % bilinear interpolation gm=gm+g(i,j); nr=nr+1; % count effective pixels else g(i,j)=-1; % pixels out of image frame end endendgm=gm/double(nr); % determine mean gray value

for i=1:M for j=1:N if g(i,j)<0 g(i,j)=gm; % fill with mean value end endend

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Class project: example of Matlab functions for PIV recording evaluation File name: img2xy.m

function [G]=img2xy(A)% A - image% G - gray value distribution[ny nx]=size(A);for x=1:nx for y=1:ny G(x,y)=A(ny-y+1,x); endend

File name: xy2img.m

function [A]=xy2img(G)% A - image% G - gray value distribution[nx ny]=size(G);Gmax=max(max(G));Gmin=min(min(G));for x=1:nx for y=1:ny A(ny-y+1,x)=int8(double(G(x,y)-Gmin)/double(Gmax-Gmin)*245+5); endend

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Class project: example of Matlab functions for PIV recording evaluation File name: main.m

clear;A=imread('D001_1.bmp'); % input image file G=img2xy(A); % conver image to gray value distribution

% evaluation pointx=100;y=200;

% interrogation window size M=64;N=64;

% evaluation samples g1=sample(G,M,N,x,y);g2=g1;

% correlation function[C m n]=correlation(g1,g2);

sr=20; % search radiusmode=1; % PIV recording modedirection=1; % main flow direction[cm vx vy]=peaksearch(C,m,n,sr,mode,direction) % particle image displacement

D=xy2img(C); % convert correlation function to imageimshow(D); % display correlation function

Instructor: Lichuan Gui lichuan-gui@uiowa

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