1. Imaging Techniques for Flow and Motion Measurement Lecture 13 Lichuan Gui University of Mississippi 2011 Central Difference Interrogation

2. Recording 1 Recording 2 g1(i,j) g2(i,j) Correlation-based Interrogation w/o Window Shift Gray value distribution in a PIV recording pair 2 nd window shifted by S ws =0

3. g 1 (i,j) g 2 (i,j) Correlation-based Interrogation w/o Window Shift Evaluation samples of M  N pixels at point (x m,y m ) Recording 1 Recording 2 2 nd window shifted by S ws =0

4. g 1 (i,j) g 2 (i,j) Correlation-based Interrogation w/o Window Shift Evaluation Function Recording 1 Recording 2 2 nd window shifted by S ws =0

5. g 1 (i,j) g 2 (i,j) Correlation-based Interrogation w/o Window Shift Particle image displacement determined by position of the maximal function value Recording 1 Recording 2 2 nd window shifted by S ws =0 (m*,n*)

6. Correlation-based Interrogation w/o Window Shift Compared to correlation tracking: – higher evaluation speed – insensitive to brightness Problems: – error dependent on displacement – evaluation bias g 1 (i,j) g 2 (i,j) S Recording 1 Recording 2 2 nd window shifted by S ws =0

7. Forward Difference Interrogation (FDI) g1(i,j) g2(i,j) S Recording 1 Recording 2 Gray value distribution in a PIV recording pair 2 nd window shifted by S ws =S Interrogation window shift

8. Forward Difference Interrogation (FDI) g 1 (i,j) f 2 (i,j) S Recording 1 Recording 2 2 nd window shifted by S ws =S Evaluation samples of M  N pixels at point (x m,y m )

9. Forward Difference Interrogation (FDI) g 1 (i,j) f 2 (i,j) S Recording 1 Recording 2 2 nd window shifted by S ws =S Evaluation Function

10. Forward Difference Interrogation (FDI) g 1 (i,j) f 2 (i,j) (m*,n*) Recording 1 Recording 2 2 nd window shifted by S ws =S Particle image displacement

11. Forward Difference Interrogation (FDI) g1(i,j) g2(i,j) Recording 1 Recording 2 2 nd window shifted by S ws =S S‘ Compared to correlation w/o shift: – multi-pass (iterated) – higher reliability – lower evaluation error – periodic error distribution (1 pixel) S = S ws +S ‘

12. Forward Difference Interrogation (FDI) Typical curvature flow

13. Forward Difference Interrogation (FDI) Interrogation point (x m,y m ) (x m,y m )

14. Forward Difference Interrogation (FDI) Flow direction at evaluation point (x m,y m )

15. Forward Difference Interrogation (FDI) True velocity to be determined (S o ) (x m,y m ) SoSo

16. Forward Difference Interrogation (FDI) (x m,y m ) SoSo Interrogation window in the first recording g 1 (i,j) g 1 (i,j)

17. Forward Difference Interrogation (FDI) (x m,y m ) SoSo Matched image pattern in the second recording f 2 (i,j) g 1 (i,j) f 2 (i,j)

18. Forward Difference Interrogation (FDI) (x m,y m ) SoSo g 1 (i,j) f 2 (i,j) S FDI interrogation result (S)

19. Forward Difference Interrogation (FDI) (x m,y m ) SoSo g 1 (i,j) f 2 (i,j) S FDI interrogation bias error (  ) SoSo S 

20. Forward Difference Interrogation (FDI) (x m,y m ) SoSo g 1 (i,j) g 2 (i,j) S SoSo S  Adjusted position (x’ m,y’ m ) FDI interrogation position deviation (x m -x’ m,y m -y’ m )

21. f 1 (i,j) f 2 (i,j) Recording 1 Recording 2 S‘ S ws2 = S/2 S ws1 = -S/2 S = S ws +S ‘ Central Difference Interrogation (CDI) Gray value distribution in a PIV recording pair Interrogation window shift

22. f 1 (i,j) f 2 (i,j) Recording 1 Recording 2 S‘ S ws2 = S/2 S ws1 = -S/2 Central Difference Interrogation (CDI) Evaluation samples of M  N pixels at point (x m,y m )

23. f 1 (i,j) f 2 (i,j) Recording 1 Recording 2 S‘ S ws2 = S/2 S ws1 = -S/2 Central Difference Interrogation (CDI) Evaluation Function

24. f 1 (i,j) f 2 (i,j) Recording 1 Recording 2 S ws2 = S/2 S ws1 = -S/2 Central Difference Interrogation (CDI) Particle image displacement (m*,n*)

25. True velocity to be determined (S o ) (x m,y m ) SoSo Central Difference Interrogation (CDI)

26. Window shift in the first recording (  -S o /2) (x m,y m ) SoSo Central Difference Interrogation (CDI)  -0.5S o

27. (x m,y m ) SoSo Central Difference Interrogation (CDI)  -0.5S o f 1 (i,j) Interrogation window in the first recording f 1 (i,j)

28. (x m,y m ) SoSo Central Difference Interrogation (CDI) f 1 (i,j) f 2 (i,j) Matched image pattern in the second recording f 2 (i,j)

29. (x m,y m ) SoSo Central Difference Interrogation (CDI) f 1 (i,j) f 2 (i,j) S CDI interrogation result (S)

30. (x m,y m ) SoSo Central Difference Interrogation (CDI) f 1 (i,j) f 2 (i,j) S SoSo S  CDI interrogation bias error (  )

31. (x m,y m ) SoSo Central Difference Interrogation (CDI) f 1 (i,j) f 2 (i,j) S SoSo S  Adjusted position (x’ m,y’ m ) CDI interrogation position deviation (x m -x’ m,y m -y’ m )

32. – smaller position deviation – smaller curvature flow bias – same computation speed (x m,y m ) SoSo Central Difference Interrogation (CDI) f 1 (i,j) f 2 (i,j) S SoSo S  Adjusted position (x’ m,y’ m ) Compare to FDI

33. Compare FDI and CDI in a four-roll mill test Top view Velocity field Experimental setup and flow velocity distribution

34. Compare FDI and CDI in a four-roll mill test PIV recording frames Animated PIV recordingsOverlapped PIV recordings

35. Compare FDI and CDI in a four-roll mill test FDI and CDI evaluation errors

36. Westerweel J, Dabiri D, Gharib M (1997) The effect of a discrete window offset on the accuracy of cross-correlation analysis of digital PIV recordings. Exp Fluids 23:20–28 Wereley ST Meinhart CD (2001) Second-order accurate particle image velocimetry. Exp Fluids 31:258–268 Gui L and Wereley ST (2002) A correlation-based continues window shift technique for reducing the peak locking effect in digital PIV image evaluation. Exp. Fluids 32: 506-517 Wereley ST and Gui L (2003) A correlation-based central difference image correction (CDIC) method and application in a four-roll-mill flow PIV measurement. Exp. Fluids 34, 42-51 References