Biomedical Engineering Reference
In-Depth Information
two-dimensional velocity fields in macroscale. The flow is first traced with particles. The measurement
is carried out by recording two digital images of particles at two different time instances. The digital
images are then divided into smaller interrogation windows where particle displacements are
evaluated. The displacement is evaluated using a two-dimensional cross-correlation of the two
corresponding interrogation windows. Because the cross-correlation can only resolve one pixel,
different curve fitting algorithms can be used. The same technique can be applied to microscale
velocity fields using a fluorescent microscope system-coupled laser illumination.
If the intensity matrices of the two corresponding interrogation windows are
I
1
(
i
,
j
) and
I
2
(
i
,
j
), the
cross-correlation function
R
(
m
,
n
) is determined as
X
N
X
M
R
ð
m
;
n
Þ¼
I
1
ð
i
;
j
Þ
$
I
2
ð
i
þ
m
;
j
þ
n
Þ
(8.15)
j
¼
1
i
¼
1
where
M
*
N
* is the size of the interrogation window measured in pixel. The position of a pixel in the
matrix is denoted by the coordinates
m
and
n
. The displacement vector of the particles in the inter-
rogation window is the vector between the origin of the coordinate system and the peak of the cross-
correlation function
R
(
m
,
n
). The quotient between the displacement vectors and the known time delay
between the acquisitions of the two images represents the velocity vector of the interrogation window.
Repeating this algorithm across the entire particle image results in the whole velocity field of the flow.
Utilizing fast Fourier transform (FFT) and inverse fast Fourier transform (FFT
-1
), Eqn
(8.15)
can
be formulated as follows:
FFT
1
n
FFT
o
R
ð
m
;
n
Þ¼
½
I
1
ð
i
;
j
Þ
FFT
½
I
2
ð
i
;
j
Þ
(8.16)
where FFT
Þ:
With the matrix
R
(
m
,
n
), the correlation peak can be identified and refined with subpixel accuracy.
The peak and its two neighboring points in the matrix
R
(
m
,
n
) are needed for the different subpixel
algorithms. The peak is detected by determining the maximum value
R
(
x
,
y
) in the correlation matrix
R
(
m
,
n
). Subsequently, the position (
x
,
y
) of the peak is stored. The four neighboring points
R
(
x
-1,
y
),
R
(
x
½
I
2
ð
i
;
j
Þ
is the conjugate of the complex array FFT
½
I
2
ð
i
;
j
1), as well as their positions, are also stored. Three points in each
direction are needed for the estimation of the peak position (
x
0
,
y
0
). The algorithms for estimating the
position (
x
0
,
y
0
) are called three-point estimators. The three basic estimators are the middle-point
estimator, the parabolic estimator, and the Gaussian estimator.
The middle-point estimator assumes the following fitting function:
þ
1,
y
),
R
(
x
,
y
- 1), and
R
(
x
,
y
þ
First order momentum
Second order momentum
:
f
ð
x
Þ¼
(8.17)
The corresponding refined position of the peak is
þ
ð
x
1
Þ
R
ð
x
1
;
y
Þþ
xR
ð
x
;
y
Þþð
x
þ
1
Þ
R
ð
x
þ
1
;
y
Þ
x
0
¼
x
;
R
ð
x
1
;
y
Þþ
R
ð
x
;
y
Þþ
R
ð
x
þ
1
;
y
Þ
(8.18)
þ
ð
y
1
Þ
R
ð
x
;
y
1
Þþ
yR
ð
x
;
y
Þþð
y
þ
1
Þ
R
ð
x
;
y
þ
1
Þ
y
0
¼
y
:
Rðx; y
1
ÞþRðx; yÞþRðx; y þ
1
Þ
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