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
