Instructor: Lichuan Gui lichuan-gui@uiowa lcgui

Measurements in Fluid Mechanics058:180:001 (ME:5180:0001)

Time & Location: 2:30P - 3:20P MWF 218 MLH

Office Hours: 4:00P – 5:00P MWF 223B-5 HL

Instructor: Lichuan [email protected]

http://lcgui.net

Students are encouraged to attend the class. You may not be able to understand by just reading the lecture notes.

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Lecture 32. Erroneous Vector Identification& Correction

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Erroneous Vectors

Example: Air flow near a pipe wallDouble exposed PIV recording Evaluation results with 6464-pixel window

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Identification Erroneous Vectors

Range limit validation Identify vectors larger than a threshold Remove vectors larger than the threshold

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Median filter (3x3)

1 2 3

4 5 6

7 8 9

5S

123456789

xiS ,

396572418

xiS ,

xmS ,

1 2 3 4 5 6 7 8 9

yiS , yiS ,

3 2 8 7 5 6 9 4 1

ymS ,mS

xmS ,

ymS ,

Validation criterion: gm SSS

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Identification Erroneous Vectors

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Identification Erroneous Vectors

Effect of median filter Original vector map Filtered vector map

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i-1

i

i+1

j+1

j

j-1

t+t

t

t-t

Vm

Vc

|Vc-Vm| Vg

3D Median filter - Successively recorded PIV image pairs

- Time interval between recording pairs t much smaller than flow variation period

- Median of 27 vectors

- Good for non-uniformly seeded flow

Identification Erroneous Vectors

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Reversibility of particle image pattern tracking

gyyxx SSSSS 2*2**SSValidation criterion:

Identification Erroneous Vectors

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Correction of Erroneous Vectors

Possible particle image displacements

Evaluation function with strong noises

Local maxima

Possible displacements

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Target vector method

gkikkp Si ||and,3,2,1|,|min||if mmm SSSSSSSS

Correction of Erroneous Vectors

Possible vectors

Sm

Target vector

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Maximal reversibility method

gkkiikkkp Si ||and,3,2,1|,|min||if *** SSSSSSSS

Correction of Erroneous Vectors

Possible vectors Reversed vectors

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Filtered vector map

Correction of Erroneous Vectors

Effect of target vector error correction Corrected vector map

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Further Error Reduction

Use small search radius=31 pixels, 873 valid vectors =17 pixels, 897 valid vectors

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Further Error Reduction

Use window shift=17 pixels, no window shift =12 pixels, xs=13 pixels, ys=0

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Further Error Reduction

Use iterated evaluation (w. window shift)W/o iteration, 947 valid vectors Iterated evaluation, 953 valid vectors

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Further Error Reduction

Process digital image before evaluationW/o iteration, unsharp mask (r=11) Iterated evaluation, unsharp mask (r=11)

Matlab function for erroneous vector identification File name: errordetection.m

function[valid]=errordetection(U,V,dU,dV)% U - horizontal particle image displacement% V - vertical particle image displacement% dU(1) - minimal horizontal particle image displacement% dU(2) - maximal horizontal particle image displacement% dU(3) - error tolerance of horizontal particle image displacement% dV(1) - minimal vertical particle image displacement% dV(2) - maximal vertical particle image displacement% dV(3) - error tolerance of vertical particle image displacement% Valid - validation index of the particle image displacement% 1 for valid vector, 0 for evaluation error

[nx ny]=size(U); % determine number of rows & collumns of interrogation grid

for i=1:nx % range filtering begin for j=1:ny if U(i,j)>=dU(1) & U(i,j)<=dU(2) & V(i,j)>=dV(1) & V(i,j)<=dV(2) valid1(i,j)=1; else valid1(i,j)=0; end endend % range filtering end

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File name: errordetection.m

for i=1:nx % median filter begin for j=1:ny if valid1(i,j)==0 % skip detected errors valid(i,j)=0; continue; end % skip detected errors nr=0; % search for valid vectors in 3x3 neighborhood for p=-1:1 for q=-1:1 if i+p>=1 & i+p<=nx & j+q>=1 & j+q<=ny & valid1(i+p,j+q)==1 nr=nr+1; u(nr)=U(i+p,j+q); v(nr)=V(i+p,j+q); end end end % search for valid vectors in 3x3 neighborhood if nr<3 % skip vector without enough valid neighbors valid(i,j)=0; continue; end % skip vector without enough valid neighbors

Matlab function for erroneous vector identification

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File name: errordetection.m

for p=1:nr-1 % determine median vector for q=p+1:nr if u(q)>u(p) um=u(p); u(p)=u(q); u(q)=um; end if v(q)>v(p) vm=v(p); v(p)=v(q); v(q)=vm; end end end nm=int16(nr/2); um=u(nm); vm=v(nm); % determine median vector if abs(U(i,j)-um)<=dU(3) & abs(V(i,j)-vm)<=dV(3) % detect wrong vector valid(i,j)=1; else valid(i,j)=0; end % detect wrong vector end

end % median filter end

Matlab function for erroneous vector identification

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Matlab function for local vector interpolationFile name: interpolation.mfunction[U V valid2]=interpolation(U,V, valid1)[nx ny]=size(U);for i=1:nx for j=1:ny if valid1(i,j)==1 valid2(i,j)=1; continue; end nr=0; um=0; vm=0; for p=-1:1 for q=-1:1 if i+p>=1 & i+p<=nx & j+q>=1 & j+q<=ny & valid1(i+p,j+q)==1 nr=nr+1; um=um+U(i+p,j+q); vm=vm+V(i+p,j+q); end end end if nr>0 valid2(i,j)=1; U(i,j)=um/nr; V(i,j)=vm/nr; else valid2(i,j)=0; end end end

% U - horizontal particle image displacement% V - vertical particle image displacement% valid - validation index of the particle image displacement

Example of Matlab program for evaluating double exposed recording D001_1.bmp Main programclear;A=imread('D001_1.bmp'); % input image file G=img2xy(A); % convert image to gray value distribution

Mg=32; % interrogation grid width Ng=32; % interrogation grid height M1=64; % initial interrogation window width N1=64; % initial interrogation window heightM2=48; % final interrogation window width N2=48; % final interrogation window height sr1=15; % initial search radiussr2=6; % final search radiusNN=6; % iteration numberdU=[0 20 3]; % parameters for error detectiondV=[-5 5 3]; % parameters for error detection

[nx ny]=size(G); % determine size of the imagerow=ny/Mg-1; % number of grid rows col=nx/Mg-1; % number of grid columns

for nn=1:NN % iteration begin M=(nn-1)*(M2-M1)/(NN-1)+M1; % determine window width N=(nn-1)*(N2-N1)/(NN-1)+N1; % determine window height sr=(nn-1)*(sr2-sr1)/(NN-1)+sr1; % determine search radius

if nn>1 [U V valid]=interpolation(U,V, valid); % interpolation for window shift [U V valid]=interpolation(U,V, valid); % interpolation for window shift end

Example of Matlab program for evaluating double exposed recording D001_1.bmp for i=1:col for j=1:row if nn==1 % determine window shift wsx=sr; wsy=0; else if valid(i,j)>0 wsx=U(i,j); wsy=V(i,j); if sr>int16(wsx) % adjust search radius for double exposed recording sr=int16(wsx); end end end x=i*Mg; % determine horizontal coordinate of interrogation point y=j*Ng; % determine vertical coordinate of interrogation point g1=sample3(G,M,N,x-wsx/2,y-wsy/2); % evaluation sample with backward wondow shift g2=sample3(G,M,N,x+wsx/2,y+wsy/2); % evaluation sample with forward wondow shift [C m n]=correlation(g1,g2); % claculating corraltion function [cm vx vy]=peaksearch(C,m,n,sr,0,0); % determine particle image displacement U(i,j)=vx+wsx; % adjust particle image displacement with window shift V(i,j)=vy+wsy; % adjust particle image displacement with window shift X(i,j)=x; % record evaluation coordinate Y(i,j)=y; % record evaluation coordinate end end

valid=errordetection(U,V,dU,dV); % detect evaluation errorsend % iteration end

Example of Matlab program for evaluating double exposed recording D001_1.bmp for i=1:col % remove evaluation errors for j=1:row if valid(i,j)==0 U(i,j)=0; V(i,j)=0; end endend quiver(X,Y,U,V);

Unsharp mask filter may be used for further improvement

Example of Matlab program for evaluating single exposed recording pair Main programclear;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 heightsr1=12; % initial search radiussr2=6; % final search radiusNN=6; % iteration numberdU=[-12 12 3]; % parameters for error detectiondV=[-12 12 3]; % parameters for error detection

[nx ny]=size(G1); % determine size of the imagerow=ny/Mg-1; % number of grid rows col=nx/Mg-1; % number of grid columns

for nn=1:NN % iteration begin sr=int16((nn-1)*(sr2-sr1)/(NN-1)+sr1); % determine search radius

if nn>1 [U V valid]=interpolation(U,V, valid); % interpolation for window shift [U V valid]=interpolation(U,V, valid); % interpolation for window shift end

Example of Matlab program for evaluating single exposed recording pair

for i=1:col for j=1:row if nn==1 % determine window shift wsx=0; wsy=0; else if valid(i,j)>0 wsx=U(i,j); wsy=V(i,j); end end x=i*Mg; % determine horizontal coordinate of interrogation point y=j*Ng; % determine vertical coordinate of interrogation point g1=sample3(G1,M,N,x-wsx/2,y-wsy/2); % evaluation sample with backward window shift g2=sample3(G2,M,N,x+wsx/2,y+wsy/2); % evaluation sample with forward window shift [C m n]=correlation(g1,g2); % calculating correlation function [cm vx vy]=peaksearch(C,m,n,sr,0,0); % determine particle image displacement U(i,j)=vx+wsx; % adjust particle image displacement with window shift V(i,j)=vy+wsy; % adjust particle image displacement with window shift X(i,j)=x; % record evaluation coordinate Y(i,j)=y; % record evaluation coordinate end end

valid=errordetection(U,V,dU,dV); % detect evaluation errors

end % iteration end

for i=1:col % remove evaluation errors for j=1:row if valid(i,j)==0 U(i,j)=0; V(i,j)=0; end endend quiver(X,Y,U,V); % plot vector map

CDIC may be used for further improvement