1. 1 Imaging Techniques for Flow and Motion Measurement Lecture 7 Lichuan Gui University of Mississippi 2011 Correlation Interrogation & FFT Acceleration

2. 2 Correlation Interrogation Correlation interrogation basics Interrogation grid & interrogation window PIV recording Interrogation grid (M g  N g ) MgMg NgNg Interrogation window (M  N) M N

3. 3 Correlation Interrogation Correlation interrogation basics Evaluation sample PIV recording

4. 4 Correlation Interrogation Correlation interrogation basics Coordinate systems G(x,y) g(i,j) o y x o j i i=1,2,,M j=1,2,,N x=1,2,,nx y=1,2,,ny PIV recording (nx  ny pixels)Evaluation sample (M  N pixels)

5. 5 Correlation Interrogation Correlation interrogation basics Interrogation window overlap -Grid distance smaller than window side length -Enable reuse of particle images -Over sampling requires too much evaluation time o x =o y = 75%o x =o y =50%

6. 6 Correlation Interrogation Evaluation function g(i,j) (m*=10, n*=-5) (m*=-10, n*=5) Auto-correlation function -One double exposed evaluation sample, i.e. g(i,j)=g 2 (i,j)=g 1 (i,j) -Displacement determined by positions of the secondary maxima -Two possible velocity directions

7. 7 Correlation Interrogation Evaluation function (m*=3, n*=5) Cross-correlation function -Two single exposed evaluation samples, i.e. g 2 (i,j) & g 1 (i,j) -Displacement determined by position of the maximum -Velocity direction clear g 1 (i,j) g 2 (i,j)

8. 8 Correlation Interrogation Acceleration with radix-2 based FFT algorithm - Side length of interrogation window selected to be powers of 2 - Evaluation sample assumed to be periodically distributed FFT -1 FFT Changing the sign of the image part FFT Complex conjugate Fast computation of evaluation function

9. 9 Correlation Interrogation Acceleration with radix-2 based FFT algorithm - Periodical padding when using FFT Fast computation of evaluation function A : Effective correlation region B,C,D: Periodically Padded regions

10. 10 Correlation Interrogation Performance of FFT-based correlation Reliability: RMS error: S : measured value S’: real value Some test results with synthetic images f: interrogation window S S  - Reliability & RMS error dependent on particle image displacement - Reliability & RMS error dependent on interrogation window size - Impossible to determine displacement components not less than half of window side length

11. 11 Correlation Interrogation Arbitrarily sized interrogation window Test results with synthetic images (f=32x32) 1  original 2  padded to f=64x64  SS

12. 12 Correlation Interrogation Acceleration with radix-2 based FFT algorithm - Computation time test Fast computation of evaluation function

13. 13 Correlation Interrogation Effect of linear transformation of evaluation samples

14. 14 –Reading Willert and Gharib 1991, Digital particle image velocimetry. Exp. Fluids 10: 181-193 Gui and Merzkirch 1998, Generating arbitrarily size interrogation windows for correlation-based analysis of particle image recordings, Exp. Fluids 24: 66-69 –Programming Compute auto-correlation function with double exposed image ac01.bmp Compute cross-correlation function with single exposed image pair cc01.bmp and cc02.bmp Determine particle image displacement in both cases Homework