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

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

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

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

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*)

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

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

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 )

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

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

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 ‘

Forward Difference Interrogation (FDI) Typical curvature flow

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

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

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

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

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)

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

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

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 )

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

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 )

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

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*)

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

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

(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)

(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)

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

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

(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 )

– 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

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

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

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

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