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 Gui Phone: (Lab), (Cell)

2 Lecture 27. Bias Error of Correlation Interrogation

3 Bias Error of Correlation Interrogation Performance of FFT-based correlation Reduced effective correlation region with increasing magnitude of (m, n) A : Effective correlation region B,C,D: Periodically Padded regions - The reliability & accuracy decrease with increasing particle image displacement (m*,n*) - The evaluated particle image displacement has smaller magnitude than that of the true value

4 Bias Error of Correlation Interrogation Bias & random errors for measuring variable X N times RMS fluctuation (random error) RMS error Individuale reading of X: Mean value 0

5 Probability density function (PDF) Bias Error of Correlation Interrogation Bias & random errors for measuring variable X N times PDF for measuring displacement 0.5 pixels   

6 Bias Error of Correlation Interrogation Correlation interrogation with Gaussian window mask f=32x32 Gui et al 2000, A digital mask technique for reducing the bias error of the correlation-based PIV interrogation algorithm. Exp. Fluids 29, 30-35

7 Bias Error of Correlation Interrogation Correlation interrogation with Gaussian window mask function [gm]=wmask(go) % INPUT: go - original evaluation sample % OUTPUT: gm - masked evaluation sample [M N]=size(go); for i=1:M for j=1:N gm(i,j)=exp(-4*((i-M/2)*(i-M/2)/(M*M/4)+(j-N/2)*(j-N/2)/(N*N/4)))*go(i,j); end Matlab function for Gaussian window mask File name: wmask.m Test image pair: - simulated single-exposed PIV recording pair of 1024  1024 pixels - given uniform particle image displacement:  x=3.5,  y=5.5 pixels

8 Bias Error of Correlation Interrogation Main program to test Gaussian window mask A1=imread('lecture27-image01.bmp'); % input image file A2=imread('lecture27-image02.bmp'); % input image file G1=img2xy(A1); % convert image to gray value distribution G2=img2xy(A2); % convert image to gray value distribution Mg=64; % interrogation grid width Ng=64; % interrogation grid height M=64; % interrogation window width N=64; % interrogation window height [nx ny]=size(G1); row=ny/Mg-1; % grid row number col=nx/Mg-1; % grid column number sr=10; % search radius for i=1:col for j=1:row x=i*Mg; y=j*Mg; g1=sample01(G1,M,N,x,y); % evaluation samples g2=sample01(G2,M,N,x,y); g1=wmask(g1); % apply Gaussian window mask [C m n]=correlation(g1,g2); [cm vx vy]=peaksearch(C,m,n,sr,0,0); % particle image displacement U(i,j)=vx; V(i,j)=vy; 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 Unmasked: Masked: