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Moment_of_Correlation_2DPIV_uncertainty/

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Moment_of_Correlation_2DPIV_uncertainty/**Moment_of_correlation.m**

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function[Ixx,Iyy,biasx,biasy,Neff,Autod]=Moment_of_correlation(P21,f1,f2,Sx,Sy,cnorm,D,fftindx,fftindy,G,DXtemp,DYtemp,region1,region2,MIest) | |

%% This function calculates the standard uncertainty from the cross-correlation plane using Moment of Correlation method. | |

% written by Sayantan Bhattacharya | |

% Inputs | |

%P21= cross-correlation plane in fourier domain | |

%f1= fft of window1/image1 | |

%f2= fft of window2/image2 | |

%Sx= window size Y | |

%Sy= window size X | |

%D= Approx Diameter of particle image (e.g. [2.8 2.8]), this is used to | |

%initialize the autocorrelation diameter estimation using subpixel fit | |

%cnorm= Matrix of ones of size Sx,Sy this is non unity if you want to | |

%correct for spatial winfow filtering | |

% fftindx,fftindy= fftshift indices I guess a simple fftshift can be used | |

% instead, but just to be consistent with prana functions | |

%G= Cross correlation plane of two windows | |

%region1, region2= spatially windowed image | |

%DXtemp,DYtemp= Cross correlation peak diameter estimated from 3pt subpixel | |

%fit on cross-correlation plane G | |

%MIest = If MI is already estimated then use that otherwise calculate MI | |

%Output | |

%Ixx=PDF diameter in X direction | |

%Iyy=PDF diameter in X direction | |

%biasx=bias uncertainty in X direction | |

%biasy=bias uncertainty in Y direction | |

%Neff=Effective number of particles | |

%Autod= Average Autocorrelation diameter | |

if MIest==-1 | |

% If Mi has not been estimated | |

%Autocorrelations | |

P11 = f1.*conj(f1); | |

P22 = f2.*conj(f2); | |

Auto1 = ifftn(P11,'symmetric'); | |

Auto2 = ifftn(P22,'symmetric'); | |

Auto1 = Auto1(fftindy,fftindx); | |

Auto2 = Auto2(fftindy,fftindx); | |

Auto1=abs(Auto1); | |

Auto2=abs(Auto2); | |

nAuto1 = Auto1-min(Auto1(:)); % Autocorrelation plane of image 1 | |

nAuto2 = Auto2-min(Auto2(:)); % Autocorrelation plane of image 2 | |

% 3 pt Gaussian fit to Autocorrelation Diameter | |

[~,~,~,~,Dauto1x3,Dauto1y3,~]=subpixel(nAuto1,Sx,Sy,cnorm,1,0,D); | |

[~,~,~,~,Dauto2x3,Dauto2y3,~]=subpixel(nAuto2,Sx,Sy,cnorm,1,0,D); | |

Diap1=sqrt(Dauto1x3*Dauto1y3/2); | |

Diap2=sqrt(Dauto2x3*Dauto2y3/2); | |

%Average Autocorrelation Diameter | |

Autod=mean([Diap1 Diap2]); | |

%MI Calculation | |

INTS1 = max(region1(:)); | |

INTS2 = max(region2(:)); | |

[MI,~,~,~,~,~] = MI_Cal_SCC(G,nAuto1,nAuto2,INTS1,INTS2,Dauto1x3,Dauto1y3,Dauto2x3,Dauto2y3,Sx,Sy,fftindx,fftindy); | |

else | |

% Use estimated MI | |

MI=MIest; | |

Autod=0; | |

end | |

%Cross-correlation subpixel fit | |

% [U,V,~,~,DXtemp,DYtemp,~]=subpixel(G,Sx,Sy,cnorm,1,0,D); | |

if isnan(DXtemp) | |

DXtemp=sqrt(2)*2.8; | |

end | |

if isnan(DYtemp) | |

DYtemp=sqrt(2)*2.8; | |

end | |

% Estimate the diameter of the cross-correlation peak (with initiation of | |

% 3pt fit estimates of correlation diameter) | |

[~,~,~,~,dxc1,dyc1,alpha2]=subpixel(G,Sx,Sy,cnorm,3,0,(1/sqrt(2))*[DXtemp DYtemp]); | |

% Taking care of peak rotation | |

DCCx = sqrt( (cos(alpha2)^2*dxc1^2 + sin(alpha2)^2*dyc1^2) ); | |

DCCy = sqrt( (sin(alpha2)^2*dxc1^2 + cos(alpha2)^2*dyc1^2) ); | |

% If estimates are too big than the 3pt fit estimate then default to 3 pt | |

% fit estimate of the correlation diameter | |

if DCCx >sqrt(2)*DXtemp | |

DCCx=DXtemp; | |

end | |

if DCCy >sqrt(2)*DYtemp | |

DCCy=DYtemp; | |

end | |

%Use Cross-correlation diameter estimate to define convolving Gaussian Diameter | |

Dconv=(1/sqrt(2))*[DCCx DCCy]; | |

% Finding the Phase correlation | |

W = ones(Sy,Sx); | |

Wden = sqrt(P21.*conj(P21)); | |

W(Wden~=0) = Wden(Wden~=0); | |

R = P21./W; % This is effectively the fourier transform of the PDF | |

% constructing the gaussian which will be convolved with the pdf | |

[Xt,Yt]=meshgrid(1:Sx,1:Sy); | |

gfilt=exp(-(4/Dconv(1)^2).*(Xt-Sx/2-1).^2-(4/Dconv(2)^2).*(Yt-Sy/2-1).^2); | |

Gfilt=abs(fftn(gfilt,[Sy Sx])); | |

%Convolving the PDF and the Gaussian | |

G1 = ifftn(R.*Gfilt,'symmetric'); | |

G1 = G1(fftindy,fftindx); | |

G1 = abs(G1); | |

G1=G1-min(G1(:)); | |

%subpixel estimation using least squres guassfit for the convolved plane G1 | |

[gpx,gpy,~,~,dx1,dy1,alpha1]=subpixel(G1,Sx,Sy,cnorm,3,0,Dconv); | |

%find the PDF major and minor axis | |

Px=real(((dx1(1)^2 - 2*(Dconv(1))^2)).^0.5); | |

Py=real(((dy1(1)^2 - 2*(Dconv(2))^2)).^0.5); | |

%If the pdf diameter comes out to be zero or imaginary try 3point fit for | |

%the convolved plane G1 | |

if Px==0 || Py==0 | |

% if zero try 3point fit | |

[gpx,gpy,~,~,dx1,dy1,~]=subpixel(G1,Sx,Sy,cnorm,1,0,Dconv); | |

Px=real(((dx1(1)^2 - 2*(Dconv(1))^2)).^0.5); | |

Py=real(((dy1(1)^2 - 2*(Dconv(2))^2)).^0.5); | |

alpha1=0; | |

%project major and minor axis to X and Y axes | |

Ixx = sqrt( 1/16 * (cos(alpha1)^2*Px^2 + sin(alpha1)^2*Py^2) ); | |

Iyy = sqrt( 1/16 * (sin(alpha1)^2*Px^2 + cos(alpha1)^2*Py^2) ); | |

else | |

% If not zero then continue | |

%project to axes | |

Ixx = sqrt( 1/16 * (cos(alpha1)^2*Px^2 + sin(alpha1)^2*Py^2) ); | |

Iyy = sqrt( 1/16 * (sin(alpha1)^2*Px^2 + cos(alpha1)^2*Py^2) ); | |

end | |

% The mean particle diameter is estimated from the mean cross-correlation | |

% peak diameter divided by sqrt(2) | |

part_dia=(1/sqrt(2))*mean([DCCx DCCy]); | |

% Effective number of pixels is MI times a circular blob of pixels with | |

% mean particle diameter | |

Neff=MI*((pi/4)*(part_dia)^2); | |

%Bias: This is the peak location of the convolved gaussian plane, typically | |

%this is done for multipass converged correlation plane in which case the | |

%peak should be at zero and any deviation is the bias. However this will | |

%not work for first pass. | |

biasx=gpx; | |

biasy=gpy; | |

% The gradient correction and scaling is done outside PIVwindowed as the | |

% gradient estimation requires the full velcoity field | |

% %Gradient correction using velocity gradient field Udiff and Vdiff | |

% Ixxt= real(sqrt(Ixx.^2 - (Autod.^2/16).*(Udiff).^2 )); | |

% Iyyt= real(sqrt(Iyy.^2 - (Autod.^2/16).*(Vdiff).^2 )); | |

% | |

% | |

% %MC uncertainty after scaling and bias correction | |

% MCx=sqrt(biasx^2+(Ixxt^2)/Neff); | |

% MCy=sqrt(biasy^2+(Iyyt^2)/Neff); | |

end | |

% JJC: this function is an exact duplicate of the external function. | |

% Is it necessary? | |

function [MI,Nim1,Nim2,INTS,Diapx,Diapy] = MI_Cal_SCC(G,nAuto1,nAuto2,INTS1,INTS2,Diap1x,Diap1y,Diap2x,Diap2y,Sx,Sy,fftindx,fftindy) | |

%[MI,INTS,DiaP] = MI_Cal_SCC(G,nAuto1,nAuto2,INTS1,INTS2,Sx,Sy,fftindx,fftindy) | |

%[MI,INTS,DiaP] = MI_Cal_SCC( region1,region2,Sx,Sy,fftindx,fftindy) | |

%This function is used to calculate the mutual information between two | |

%consecutive frames using SCC method | |

% region 1 and 2 are the particle image within the interrogation window. | |

% Sx and Sy are the correlation window size | |

% fftindx and fftindy are fftshift indicies | |

% We will first find the maximum intensity from the image, and average | |

% particle diameter from particle image autocorrelation. A standard | |

% gaussian particle is built based on these two parameter. Then the | |

% contritbution of one particle is caluclate by take the auotcorrelatio | |

% peak of stardard gaussian particle. The primary peak on correlation | |

% plane is a summation of the contribution of all correlated particles. | |

% MI is the ratio between contribution of all correlated particles and | |

% one particle. | |

% Diapx=(1/sqrt(2))*mean([Diap1x Diap2x]); | |

% Diapy=(1/sqrt(2))*mean([Diap1y Diap2y]); | |

Diapx=(1/sqrt(2))*(Diap1x*Diap2x)^0.5; | |

Diapy=(1/sqrt(2))*(Diap1y*Diap2y)^0.5; | |

%make analytical gaussian particle and calculate | |

%autocorrelaiton of the standard particle | |

xco = 1:Sx;xco = repmat(xco,[Sx,1]); % build X axis | |

yco = 1:Sy;yco = repmat(yco',[1,Sy]); % build Y axis | |

% INTS = (INTS1+INTS2)/2; % intensity of standard Gaussian particle is the mean (maximum) intensity of two frames | |

INTS = sqrt(INTS1*INTS2); % intensity of standard Gaussian particle is the mean (maximum) intensity of two frames | |

% % DiaP = (DiaP1+DiaP2)/2; % diameter of standard Gaussian particle is the mean diameter of two frames | |

% % fg = INTS*exp(-8*((xco-round(Sx/2)).^2+(yco-round(Sy/2)).^2)/DiaP^2); % build the particle intensity distribution based on 2D Gaussian distribution | |

fg = INTS*exp(-8*((xco-round(Sx/2)).^2/(Diapx^2)+(yco-round(Sy/2)).^2/(Diapy^2))); % build the particle intensity distribution based on 2D Gaussian distribution | |

fg = fftn(fg,[Sy,Sx]); %FFT | |

Pg = fg.*conj(fg); %FFT based correlation | |

Sp = ifftn(Pg,'symmetric'); % convert to time domain | |

Sp = Sp(fftindy,fftindx); % Sp is the autocorrelation of the standadrd Gaussian particle | |

%use cross correaltion plane to get MI | |

Gnorm = G-min(G(:)); %minimum correlation subtraction to elminate background noise effect | |

[~,Gind] = max(Gnorm(:)); % find the primary peak location | |

shift_locyg = 1+mod(Gind-1,Sy); % find the Y coordinate of the peak | |

shift_locxg = ceil(Gind/Sy); % find the X coordinate of the peak | |

GXshift = Sy/2+1-shift_locxg; % find the distance between the peak location to the center of correlation plane in X direction | |

GYshift = Sx/2+1-shift_locyg; % find the distance between the peak location to the center of correlation plane in Y direction | |

SGnorm = circshift(Gnorm,[GYshift,GXshift]); % shift the peak to the center of the plane | |

%keyboard; | |

MI = max(SGnorm(:))/max(Sp(:)); % MI is the ratio between the contribution of all correlated particle and the contribution of one particle | |

Nim1=max(nAuto1(:))/max(Sp(:)); | |

Nim2=max(nAuto2(:))/max(Sp(:)); | |

end | |