Properties of 2D discrete FFT

Fixed sample size Size of window has to be a base-2 dimension, 32x32, or

64x64

Periodicity assumption Particle image is assumed to be periodic

Aliasing Correlation data is periodic, peak located at area outside

of IC window will be found on the opposite side

Bias error Large shift (less overlapping pattern) leads to smaller

and biased peak

Periodicity assumption

Applying FFT to a domain assumes the domain is periodic.

Periodicity assumption

Applying FFT to a domain assumes the domain is periodic.

Periodic condition for the domain

Periodicity assumption

Applying FFT to a domain assumes the domain is periodic.

Periodic condition for the domain

Particle image does not satisfy this condition

Periodicity assumption

Applying FFT to a domain assumes the domain is periodic.

Periodic condition for the domain

Particle image does not satisfy this condition

Bias error

Shift (s)

Cor

rela

tion

Bias error

True peakMeasured peak

1

Solution for the bias error

1D example:

f0

1

0 Ng

0

1

0 N

convolving

0

1

-N/2 N/20Center 0: 1; ±N/2: 1/2

Reduce bias error

Multiplying weighting factors

Adjusting correlation coefficients

Re-scale to [1/2, 1]

Image is convolved with itself

Subpixel interpolation

Reason

shift

correlationCalculating in digital form

shift

correlation

Solution: subpixel interpolation-curve fitting from 3 points near peak- finding the peak in the curve

shift

correlationCurve fitting

Schemes for subpixel interpolation

Remark:

-Gaussian fitting is more commonly used-Parabolic fitting has a divided-by-zero problem when Ri-1 = Ri = Ri+1

Advanced techniques- multiple pass interrogation algorithm

Principle Crosscorrelating one IC window with another IC window

with known shift Determination of known shift – normal crosscorrelation

Implementation Using normal crosscorrelation method to get the velocity

map; Detecting bad vectors and replacing with interpolated

vectors; Computing crosscorrelation coefficients by shifting

another IC window with the displacement determined by the obtained velocity;

Advanced techniques- multiple pass interrogation algorithm

P-I

P-II

Normal crosscorrelation

P-I

P-II

multiple pass interrogation

IC

Advanced techniques- multiple pass interrogation algorithm

P-I

P-II

Normal crosscorrelation

P-I

P-II

multiple pass interrogation

IC

Advanced techniques- multiple pass interrogation algorithm

P-I

P-II

Normal crosscorrelation

P-IP-II

multiple pass interrogation

IC

Advanced techniques – hierarchical approach

Principle Similar to the multiple pass algorithm but the sampling

grid system is gradually refined with reducing IC size simultaneously

Advantage Able to capture complex structure;

Implementation Run multiple pass algorithms under adaptive grid

systems and correspondingly changed IC window

Example of hierarchical approach

Multiple peak detection

Reason The strongest peak is not always associated with the

correct shift, especially in the area of complex structure, e.g., strong vorticity, strong shear

Implementation Detecting local maximums and keeping them as “peak

candidates”; Compare velocity with surrounding velocities, ruling out

the unreasonable “peak candidates” until the searching status is not changed

Example of multiple peak detection

Multiple peak detectionSingle peak detection

Data validation

Reason of the occurrence of bad vectors Inhomogeneous seeding Noise Complex flow structure

Validation method Direct comparison Median Filter Mean Filter

Data validation (cont’d)

U1=Ui+1,j

U2=Ui+1,j-1

U8=Ui+1,j+1

U9=Ui,j

U3=Ui,j-1

U7=Ui,j+1

U5=Ui-1,j

U4=Ui-1,j-1

U6=Ui-1,j+1Direct comparison

00;1,1,,,

jiji UU

Median Filter

jiji UUMedian ,, )(

Mean Filter

jiji UUMean ,, )(

Interpolation

Linear interpolation

4

10

43

21

00

4/)1)(1(4/)1)(1(

4/)1)(1(4/)1)(1(

2/)(2/)(

iii

cc

UU

yyyxxx

x

y

x1,y1x1,y1

x3,y3 x4,y4

x0,y0xc,yc

Read Tecplot data file

Format of head before data

TITLE = "Import0 in Mae513 Velocity vectors [positions in cm] [velocities in cm/s]"VARIABLES = " Position x "," Position y "," Velocity u "," Velocity v "ZONE T="Data", I=34, J=34, F=POINTDT=( SINGLE, SINGLE, SINGLE, SINGLE )-3.200000000e+001 -3.200000000e+001 0.000000000e+000 0.000000000e+0000.000000000e+000 -3.200000000e+001 0.000000000e+000 0.000000000e+000 x y u v

Program list (Matlab)

FileName = 'Take_01000.dat';%% TecPlot data file which saves velocity datafid = fopen(FileName,'r');Line = fgetl( fid ); Line = fgetl( fid ); Line = fgetl( fid );Dim = sscanf( Line, 'ZONE T="Data", I=%d, J=%d, F=POINT');Line = fgetl( fid );nx = Dim(1); %% Grid number in x directionny = Dim(2); %% Grid number in y directionu = zeros(nx, ny); v = zeros(nx, ny); x = zeros(nx, 1); y = zeros(1, ny);temp = fscanf(fid, '%e');s = 1;for j = 1:ny for i = 1:nx x(i, 1) = temp(s); y(1, j) = temp(s+1); u(i,j) = temp(s+2); v(i,j) = temp(s+3); s = s + 4; endendfclose(fid);%% now you have x(1:nx, 1), y(1, 1:ny), u(1:nx, 1:ny), v(1:nx,1:ny)%% you can do further processing of x, y, u, v

Program list (Fortran)

!! maximum dimension for u and vparameter (mx = 101, my = 101)real x(mx), y(my), u(mx,my), v(mx,my)character*32 FileName /'Take_01000.dat'/character*80 Lineopen(100, file=FileName, status = 'old')read(100, *); read(100, *); read(100, '(A80)') Line; read(100, *)!! The line contains the string of!! 'ZONE T="Data", I=%d, J=%d, F=POINT'ie1 = index(Line, 'I=') + 2ie2 = index(Line, ', J=') - 1ie3 = index(Line, 'J=') + 2ie4 = index(Line, ', F=POINT') - 1read(Line(ie1:ie2), '(I)') nxread(Line(ie3:ie4), '(I)') nydo j=1, nydo i=1, nx

read(100, *) x(i), y(j), u(i,j), v(i,j)enddoenddoclose(100)

Program list (C/C++)#include <stdio.h>#include <stdlib.h>// maximum dimension for x and y directions#define MX 101#define MY 101int main() {

char FileName[32]; int nx, ny; char Line[81];float x[MX], y[MY], u[MY][MX], v[MY][MX];strcpy(FileName, "Take_01000.dat");FILE *fp = fopen(FileName, "r"); if ( !fp ) return -1;fgets(Line, 80, fp); fgets(Line, 80, fp); fgets(Line, 80, fp);fgets(Line, 80, fp); // printf("%s", Line);sscanf(Line, "ZONE T=\"Data\", I=%d, J=%d, F=POINT\n", &nx, &ny);printf("nx=%d ny=%d\n", nx, ny);fgets(Line, 80, fp);for (int j=0; j<ny; j++)for (int i=0; i<nx; i++) { fscanf(fp, "%e %e %e %e", &x[i], &y[j], &u[j][i], &v[j][i]);}fclose(fp);

}