Biomedical Engineering Reference
In-Depth Information
In (9.4)
m
=1
, ..., M
and
n
=1
, ..., N
are the indices of the CCD pixel
co-ordinates
x
m
,y
n
.
I
0
(
x
m
,y
n
)=
I
O
(
x
m
,y
n
)+
I
R
(
x
m
,y
n
) is the sum of the
int
ensities of object wav
e and reference wave. The parameter
γ
(
x
m
,y
n
)=
2
I
O
(
x
m
,y
n
)
I
R
(
x
m
,y
n
)
/I
0
(
x
m
,y
n
) denotes the modulation of the intensity
patterns. The term sinc(
Φ/
2) = sin (
Φ/
2)
/
(
Φ/
2) describes the intensity inte-
gration on a single CCD pixel in the angle range
Φ
.
φ
(
x
m
,y
n
) is the distrib-
ution of the object wave phase and
C
an additional constant phase off-set.
For the calculation of the object wave phase from each recorded intensity
pattern, in addition to Fourier transformation methods [15-17], a variable
three-step algorithm [12] can be applied by taking into account three intensity
values
I
k−
1
,
I
k
,
I
k
+1
(in horizontal or vertical orientation) of neighboring CCD
pixels:
φ
k
+
kβ
+
C
= arctan
1
−
cos
β
sin
β
I
k−
1
− I
k
+1
2
I
k
− I
k−
1
− I
k
+1
modulo
2
π.
(9.5)
The algorithm requires for a correct phase evaluation that the mean speckle
size is at least the size of three pixels of the digitized interferograms. For the
parameter
β
, either
β
x
or
β
y
may be chosen. The adjustment of the phase
gradients can be performed by analysis of the 2D frequency spectrum of the
carrier fringe pattern in the interferograms by 2D digital fast Fourier transform
(FFT) [13, 16].
The phase difference ∆
φ
k
modulo 2
π
between two phase states
φ
k
,
φ
k
of
the object wave front is calculated:
∆
φ
k
=(
φ
k
+
kβ
+
C
)
(
φ
k
+
kβ
+
C
)=
φ
k
− φ
k
.
−
(9.6)
Figure 9.2 illustrates the evaluation process of spatial phase-shifted inter-
ferograms for displacement detection. Figure 9.2a,b shows the spatial phase-
shifted interferograms
I
,
I
obtained from two different displacement states of
a tilted metal plate. In Fig. 9.2c,d the corresponding phase distributions
φ
,
φ
(mod 2
π
), calculated by (9.5) from Fig. 9.2a,b, are depicted. The correspond-
ing correlation fringe pattern calculated by (9.3) is depicted in Fig. 9.2e. The
resulting phase difference distribution ∆
φ
mod 2
π
obtained by subtraction of
Fig. 9.2c,d modulo 2
π
as well as the according filtered phase difference distri-
bution is shown in Fig. 9.2f,g. Figure 9.2h,i finally represents the unwrapped
phase difference after removal of the 2
π
ambiguity and the related pseudo
3D representation of the data. From the data in Fig. 9.2h the underlying dis-
placement of the plate is calculated by taking into consideration recording
and imaging geometry of the experimental setup by (9.2).
