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Measurements in Fluid Mechanics 058:180:001 (ME:5180:0001) Time & Location: 2:30P - 3:20P MWF 218 MLH Office Hours: 4:00P – 5:00P MWF 223B-5 HL Instructor: Lichuan Gui lichuan-gui@uiowa.edu 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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2 Lecture 29. Interrogation Window Shift
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3 Reduced working region for correlation interrogation Interrogation Window Shift Evaluation error for ideal PIV recordings by using different algorithms with a 64x64-pixel interrogation window Original peak search radius usually M/3 or N/3 Reduced peak search radius oo rr r < o
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4 Discrete window shift (DWS) Interrogation Window Shift G 2 (x,y) o y x xmxm ymym o j i g 1 (i,j) S WS x m +x s y m +y s f 2 (i,j) o j i S’ S Cross-correlation of single exposed PIV recording pair
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5 Discrete window shift (DWS) Interrogation Window Shift Auto correlation of double exposed PIV recording No secondary maximum detected because of noises g(i,j)
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6 Cross-correlation of double exposed PIV recording Secondary maximum appears S ws Limited search area: <x s & <y s Discrete window shift (DWS) Interrogation Window Shift g(i,j)f(i,j)
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7 Discrete window shift (DWS) Interrogation Window Shift Determine initial window shift S WS Determine g 1 (i,j) Determine f 2 (i,j) Compute (m,n) from g 1 (i,j) and f 2 (i,j) Maximum search to determine S’, S=S WS +S’ S’ small enough? S WS =S YES NO Too many iterations? NO YES Begin By previous knowledge or set to zero Usually 3 or 4 iterations Accelerated with FFT Sub-pixel fit End S: integer number of pixels DWS Flow Chart
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8 Continuous window shift (CWS) Interrogation Window Shift S ws ={I+x, J+y} – I,J: integer numbers – x and y: decimal numbers and 0 x<1;0 y< 1 BA DC ab cd g(i,j) I+i J+j i j f(i,j) x y Binlear interpolation Other interpolation methods available
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9 Continuous window shift (CWS) Interrogation Window Shift Determine initial window shift S WS Determine g 1 (i,j) Determine f 2 (i,j) Compute (m,n) from g 1 (i,j) and f 2 (i,j) Maximum search and sub-pixel fit to determine S’, S=S WS +S’ S’ small enough? S WS =S YES NO Too many iterations? NO YES Begin By previous knowledge or set to zero Usually 4 to 6 iterations Accelerated with FFT End Bilinear interpolation or other CWS Flow Chart
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10 Evaluation error distribution of DWS and CWS Interrogation Window Shift Test of three different algorithms with synthetic PIV images Periodical functions of particle image displacement of 1-pixel period; DWS better than correlation tracking around integer-pixel displacements but worse around mid-pixel displacements CWS has much lower error level than DWS and correlation-based tracking
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11 Matlab function for CWS File name: sample3.m function[g]=sample3(G,M,N,X,Y) I=int16(X); J=int16(Y); x=double(X)-double(I); y=double(Y)-double(J); if x<0 I=I-1; x=x+1; end if y<0 J=J-1; y=y+1; end for i=1:M for j=1:N ii=i+I-int16(M/2); jj=j+J-int16(N/2); 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 end %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
![11 Matlab function for CWS File name: sample3.m function[g]=sample3(G,M,N,X,Y) I=int16(X); J=int16(Y); x=double(X)-double(I); y=double(Y)-double(J); if x<0 I=I-1; x=x+1; end if y<0 J=J-1; y=y+1; end for i=1:M for j=1:N ii=i+I-int16(M/2); jj=j+J-int16(N/2); 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 end %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](http://images.slideplayer.com./26/8823268/slides/slide_11.jpg "11 Matlab function for CWS File name: sample3.m function[g]=sample3(G,M,N,X,Y) I=int16(X); J=int16(Y); x=double(X)-double(I); y=double(Y)-double(J); if x<0 I=I-1; x=x+1; end if y<0 J=J-1; y=y+1; end for i=1:M for j=1:N ii=i+I-int16(M/2); jj=j+J-int16(N/2); 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 end %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")
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12 Matlab function for CWS Test with ideal window shift clear; A1=imread('lecture27-image01.bmp'); % input image files A2=imread('lecture27-image02.bmp'); G1=img2xy(A1);G2=img2xy(A2); % convert image to gray value distribution Mg=64; Ng=64; % interrogation grid width and height M=64; N=64; % interrogation window width and height [nx ny]=size(G1); row=ny/Mg-1; % grid row number col=nx/Mg-1; % grid column number sr=10; % search radius wsx=3.5; wsy=5.5;% window shift for i=1:col for j=1:row x=i*Mg; y=j*Mg; g1=sample3(G1,M,N,x,y); % evaluation samples g2=sample3(G2,M,N,x+wsx,y+wsy); [C m n]=correlation(g1,g2); [cm vx vy]=peaksearch(C,m,n,sr,0,0); U(i,j)=vx+wsx; V(i,j)=vy+wsy; X(i,j)=x; Y(i,j)=y; end quiver(X,Y,U,V); % plot vector map mean(mean(U))-3.5 % bias of x-component mean(mean(V))-5.5 % bias of y-component
![12 Matlab function for CWS Test with ideal window shift clear; A1=imread( lecture27-image01.bmp ); % input image files A2=imread( lecture27-image02.bmp ); G1=img2xy(A1);G2=img2xy(A2); % convert image to gray value distribution Mg=64; Ng=64; % interrogation grid width and height M=64; N=64; % interrogation window width and height [nx ny]=size(G1); row=ny/Mg-1; % grid row number col=nx/Mg-1; % grid column number sr=10; % search radius wsx=3.5; wsy=5.5;% window shift for i=1:col for j=1:row x=i*Mg; y=j*Mg; g1=sample3(G1,M,N,x,y); % evaluation samples g2=sample3(G2,M,N,x+wsx,y+wsy); [C m n]=correlation(g1,g2); [cm vx vy]=peaksearch(C,m,n,sr,0,0); U(i,j)=vx+wsx; V(i,j)=vy+wsy; X(i,j)=x; Y(i,j)=y; end quiver(X,Y,U,V); % plot vector map mean(mean(U))-3.5 % bias of x-component mean(mean(V))-5.5 % bias of y-component](http://images.slideplayer.com./26/8823268/slides/slide_12.jpg "12 Matlab function for CWS Test with ideal window shift clear; A1=imread( lecture27-image01.bmp ); % input image files A2=imread( lecture27-image02.bmp ); G1=img2xy(A1);G2=img2xy(A2); % convert image to gray value distribution Mg=64; Ng=64; % interrogation grid width and height M=64; N=64; % interrogation window width and height [nx ny]=size(G1); row=ny/Mg-1; % grid row number col=nx/Mg-1; % grid column number sr=10; % search radius wsx=3.5; wsy=5.5;% window shift for i=1:col for j=1:row x=i*Mg; y=j*Mg; g1=sample3(G1,M,N,x,y); % evaluation samples g2=sample3(G2,M,N,x+wsx,y+wsy); [C m n]=correlation(g1,g2); [cm vx vy]=peaksearch(C,m,n,sr,0,0); U(i,j)=vx+wsx; V(i,j)=vy+wsy; X(i,j)=x; Y(i,j)=y; end quiver(X,Y,U,V); % plot vector map mean(mean(U))-3.5 % bias of x-component mean(mean(V))-5.5 % bias of y-component")
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13 Matlab function for CWS Test with iterated window shift clear; A1=imread('lecture27-image01.bmp'); A2=imread('lecture27-image02.bmp'); G1=img2xy(A1); G2=img2xy(A2); Mg=64; Ng=64; M=64; N=64; [nx ny]=size(G1); row=ny/Mg-1; col=nx/Mg-1; sr=10; for i=1:col for j=1:row U(i,j)=0; V(i,j)=0; end for iteration=1:6 % iterated 6 times for i=1:col for j=1:row wsx=U(i,j); wsy=V(i,j); x=i*Mg; y=j*Mg; g1=sample3(G1,M,N,x,y); g2=sample3(G2,M,N,x+wsx,y+wsy); [C m n]=correlation(g1,g2); [cm vx vy]=peaksearch(C,m,n,sr,0,0); U(i,j)=vx+wsx; V(i,j)=vy+wsy; X(i,j)=x; Y(i,j)=y; end End quiver(X,Y,U,V); % plot vector map mean(mean(U))-3.5 % bias of x-component mean(mean(V))-5.5 % bias of y-component
![13 Matlab function for CWS Test with iterated window shift clear; A1=imread( lecture27-image01.bmp ); A2=imread( lecture27-image02.bmp ); G1=img2xy(A1); G2=img2xy(A2); Mg=64; Ng=64; M=64; N=64; [nx ny]=size(G1); row=ny/Mg-1; col=nx/Mg-1; sr=10; for i=1:col for j=1:row U(i,j)=0; V(i,j)=0; end for iteration=1:6 % iterated 6 times for i=1:col for j=1:row wsx=U(i,j); wsy=V(i,j); x=i*Mg; y=j*Mg; g1=sample3(G1,M,N,x,y); g2=sample3(G2,M,N,x+wsx,y+wsy); [C m n]=correlation(g1,g2); [cm vx vy]=peaksearch(C,m,n,sr,0,0); U(i,j)=vx+wsx; V(i,j)=vy+wsy; X(i,j)=x; Y(i,j)=y; end End quiver(X,Y,U,V); % plot vector map mean(mean(U))-3.5 % bias of x-component mean(mean(V))-5.5 % bias of y-component](http://images.slideplayer.com./26/8823268/slides/slide_13.jpg "13 Matlab function for CWS Test with iterated window shift clear; A1=imread( lecture27-image01.bmp ); A2=imread( lecture27-image02.bmp ); G1=img2xy(A1); G2=img2xy(A2); Mg=64; Ng=64; M=64; N=64; [nx ny]=size(G1); row=ny/Mg-1; col=nx/Mg-1; sr=10; for i=1:col for j=1:row U(i,j)=0; V(i,j)=0; end for iteration=1:6 % iterated 6 times for i=1:col for j=1:row wsx=U(i,j); wsy=V(i,j); x=i*Mg; y=j*Mg; g1=sample3(G1,M,N,x,y); g2=sample3(G2,M,N,x+wsx,y+wsy); [C m n]=correlation(g1,g2); [cm vx vy]=peaksearch(C,m,n,sr,0,0); U(i,j)=vx+wsx; V(i,j)=vy+wsy; X(i,j)=x; Y(i,j)=y; end End quiver(X,Y,U,V); % plot vector map mean(mean(U))-3.5 % bias of x-component mean(mean(V))-5.5 % bias of y-component")
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Class project: corrected Matlab functions File name: correlation.m function[c m n]=correlation(g1,g2) [M N]=size(g1); % determine size of evaluation sample % determine window size for FFT - begin for k=3:12 if 2^k>=M & 2^k>=N break; end M2=2^k; N2=2^k; % determine window size for FFT - end % average gray value padding - begin gm1=mean(mean(g1)); 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 % average gray value padding - end % compute correlation function - begin gfft1=fft2(gg1); gfft2=conj(fft2(gg2)); C=real(ifft2(gfft1.*gfft2)); % compute correlation function - end % determine coordinates in correlation plane - begin for i=1:M2 for j=1:N2 m(i,j)=i-M2/2; n(i,j)=j-N2/2; end % determine coordinates in correlation plane - end % periodical reconstruction of correlation function – begin for i=1:M2 for j=1:N2/2 t=C(i,j); C(i,j)=C(i,N2-j+1); C(i,N2-j+1)=t; end for 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; end for i=1:M2/2 C1(i,1:N2)=C(i+M2/2,1:N2); end for i=M2/2+1:M2 C1(i,1:N2)=C(i-M2/2,1:N2); end for j=1:N2/2 c(1:M2,j)=C1(1:M2,j+N2/2); end for j=N2/2+1:N2 c(1:M2,j)=C1(1:M2,j-N2/2); end % periodical reconstruction of correlation function - end
![Class project: corrected Matlab functions File name: correlation.m function[c m n]=correlation(g1,g2) [M N]=size(g1); % determine size of evaluation sample % determine window size for FFT - begin for k=3:12 if 2^k>=M & 2^k>=N break; end M2=2^k; N2=2^k; % determine window size for FFT - end % average gray value padding - begin gm1=mean(mean(g1)); 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 % average gray value padding - end % compute correlation function - begin gfft1=fft2(gg1); gfft2=conj(fft2(gg2)); C=real(ifft2(gfft1.*gfft2)); % compute correlation function - end % determine coordinates in correlation plane - begin for i=1:M2 for j=1:N2 m(i,j)=i-M2/2; n(i,j)=j-N2/2; end % determine coordinates in correlation plane - end % periodical reconstruction of correlation function – begin for i=1:M2 for j=1:N2/2 t=C(i,j); C(i,j)=C(i,N2-j+1); C(i,N2-j+1)=t; end for 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; end for i=1:M2/2 C1(i,1:N2)=C(i+M2/2,1:N2); end for i=M2/2+1:M2 C1(i,1:N2)=C(i-M2/2,1:N2); end for j=1:N2/2 c(1:M2,j)=C1(1:M2,j+N2/2); end for j=N2/2+1:N2 c(1:M2,j)=C1(1:M2,j-N2/2); end % periodical reconstruction of correlation function - end](http://images.slideplayer.com./26/8823268/slides/slide_14.jpg "Class project: corrected Matlab functions File name: correlation.m function[c m n]=correlation(g1,g2) [M N]=size(g1); % determine size of evaluation sample % determine window size for FFT - begin for k=3:12 if 2^k>=M & 2^k>=N break; end M2=2^k; N2=2^k; % determine window size for FFT - end % average gray value padding - begin gm1=mean(mean(g1)); 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 % average gray value padding - end % compute correlation function - begin gfft1=fft2(gg1); gfft2=conj(fft2(gg2)); C=real(ifft2(gfft1.*gfft2)); % compute correlation function - end % determine coordinates in correlation plane - begin for i=1:M2 for j=1:N2 m(i,j)=i-M2/2; n(i,j)=j-N2/2; end % determine coordinates in correlation plane - end % periodical reconstruction of correlation function – begin for i=1:M2 for j=1:N2/2 t=C(i,j); C(i,j)=C(i,N2-j+1); C(i,N2-j+1)=t; end for 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; end for i=1:M2/2 C1(i,1:N2)=C(i+M2/2,1:N2); end for i=M2/2+1:M2 C1(i,1:N2)=C(i-M2/2,1:N2); end for j=1:N2/2 c(1:M2,j)=C1(1:M2,j+N2/2); end for 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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