Correlation Interrogation

Report
Imaging Techniques for Flow and Motion Measurement
Lecture 7
Correlation Interrogation
& FFT Acceleration
Lichuan Gui
University of Mississippi
2011
1
Correlation Interrogation

Correlation interrogation basics
Interrogation grid & interrogation window
PIV recording
Mg
Interrogation grid (MgNg)
M
Ng
N
Interrogation window (MN)
2
Correlation Interrogation

Correlation interrogation basics
Evaluation sample
PIV recording
Evaluation sample
3
Correlation Interrogation

Correlation interrogation basics
Coordinate systems
y
PIV recording (nxny pixels)
Evaluation sample (MN pixels)
G(x,y)
j
g(i,j)
o
o
i
x=1,2,•••,nx
i=1,2,•••,M
y=1,2,•••,ny
j=1,2,•••,N
x
4
Correlation Interrogation

Correlation interrogation basics
Interrogation window overlap
ox  1 
Mg
-
oy  1 
Ng
Grid distance smaller than window side length
Enable reuse of particle images
Over sampling requires too much evaluation time
ox=oy=50%
M
N
ox=oy= 75%
5
Correlation Interrogation

Evaluation function
Auto-correlation function
g(i,j)
One double exposed evaluation sample, i.e. g(i,j)=g2(i,j)=g1(i,j)
Displacement determined by positions of the secondary maxima
Two possible velocity directions
M
N
1
 m, n     g i, j   g i  m, j  n 
0.8
(m,n)
-
i 1 j 1
0.6
0.4
0.2
0
-30
-30
-20
-20
-10
-10
n
0
0
10
10
20
30
(m*=10, n*=-5)
m
20
30
(m*=-10, n*=5)
6
Correlation Interrogation
Evaluation function
Cross-correlation function
-
Two single exposed evaluation samples, i.e. g2(i,j) & g1(i,j)
Displacement determined by position of the maximum
Velocity direction clear
M
g1(i,j)
N
 m, n     g1 i, j   g 2 i  m, j  n 
i 1 j 1
1
0.8
(m,n)

0.6
0.4
0.2
0
-30
-30
-20
g2(i,j)
-20
-10
-10
n
0
0
10
10
20
m
20
30
30
(m*=3, n*=5)
7
Correlation Interrogation

Fast computation of evaluation function
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
M
N
 m, n     g1 i, j   g 2 i  m, j  n 
i 1 j 1
g1 i, j 
FFT
g 2 i, j 
FFT
gˆ1 u, v
gˆ 2 u, v
Complex conjugate
gˆ 2* u, v
Changing the sign of the image part
m, n
FFT-1
ˆ u, v  gˆ1 u, vgˆ 2* u, v

8
Correlation Interrogation

Fast computation of evaluation function
Acceleration with radix-2 based FFT algorithm
- Periodical padding when using FFT
M
N
m, n   g1 i, j   g 2 i  m, j  n 
i 1 j 1
  A   B  C   D
A:
Effective correlation region
B,C,D: Periodically Padded regions
9
Correlation Interrogation
Performance of FFT-based correlation
- 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
Reliability:
 
N valid
N total
RMS error:

1
N valid
N valid
k 1
2
 Sk  Sk 
S : measured value
S’: real value
Some test results with synthetic images


1
0.5
0.8
0.4
f =64X64
f =16X16
f =32X32
0.3
0.6
f =32X32
0.4
16X16
0.2
0.1
S
0
0
4
f =64X64
0.2
8 12 16 20 24 28 32
S


0
0
4
8 12 16 20 24 28 32
f: interrogation window
10
Correlation Interrogation

Arbitrarily sized interrogation window
M *  2
 *

N  2
2 1  M  2
for  1
2  N  2
g * i, j   g i, j 
g * i, j  
1 i M , 0  j N
M
N
1
 g  p, q 
MN p 1 q 1

M  i  M * , 1  j  N *

or

*
1  i  M , N  j  N *


g * i  kM * , j  lN *  g * i, j  for k , l  0, 1, 2, 3  
M * N*
 m, n   g1* i, j g 2* i  m, j  n
*
i 1 j 1


1
0.5
Test results with synthetic
0.8
0.4
0.6
0.3
images (f=32x32)
1  original
2  padded to f=64x64
0.4
1
2
0.2
2
0.2
1
S
0
0
5
10
15
20
25
0.1
S
0
0
5
10
15
20
25
11
Correlation Interrogation

Fast computation of evaluation function
Acceleration with radix-2 based FFT algorithm
- Computation time test
12
Correlation Interrogation

Effect of linear transformation of evaluation samples
g1 i, j , g 2 i, j 
g1 i, j , g 2 i, j 
g1 i, j   A1 g1 i, j   B1
g 2 i, j   A2 g 2 i, j   B2
M
N
M
m, n   g1 i, j  g 2 i  m, j  n
i 1 j 1
M
N
m, n    g1 i, j   g 2 i  m, j  n 
i 1 j 1
N
M
N
m, n  A1 A2m, n  A1B2  g1 i, j   A2 B1  g 2 i  m, j  n  B1B2
i 1 j 1
i 1 j 1
m, n  Cm, n  D
13
Homework
– 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
14

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