Biomedical Engineering Reference
In-Depth Information
From geometry, we obtain:
q
R
2
q
R
2
q
R
2
q
R
2
2
2
2
2
1
x
0
1
y
0
d
5
1
jr
2
r
0
j
1
jr
0
j
1
ðx
2
x
0
Þ
1
ðy
2
y
0
Þ
(7.7)
2
5
2
The difference can be positive or negative, depending on the angle of the curvature we are
compensating. Finally, the curvature corrected field:
q
R
2
q
R
2
2
2
1
x
0
1
y
0
Eðx; y
;
0
Þ
5
E
0
ðx; y
;
0
Þ
exp i
k
1
ðx
2
x
0
Þ
1
ðy
2
y
0
Þ
(7.8)
6
2
Equation (7.8)
agrees with the approximation from Ref.
[5]
, in the case
Rcr
and
r
0
-
0:
"
#
r
1
1
p
R
2
r
2
R
2
r
2
2
R
2
2
1
x
2
1
y
2
2
R
2
πR
λ
2
π
λ
k
1
r
2
2
R
5
kR
2
1
1
1
(7.9)
5
5
This is a known expression for Newton's rings, which means that if the object is a plane
mirror, the resulting interference pattern would be a set of concentric rings of radius of
p
mRλ
, where
m
5
0, 1, 2,
...
.
Figure 7.4
shows the image of the USAF resolution target covered with a layer of
aluminum to make it entirely reflective. The pattern on the resolution target is elevated
approximately 100 nm above the flat background.
Figure 7.4A
shows the reconstructed
image before the curvature correction and
Figure 7.4B
is the same image after the curvature
correction was applied. In general, if the parameters are chosen correctly, even a substantial
curvature mismatch can be compensated.
7.2.4 Additional Phase Background Removal
In practice, numerically compensating the wave front curvature works well for smaller
image frames. For larger images, even a small amount of uncompensated background or tilt
can be a problem, especially if, for example, many cells are imaged simultaneously with the
goal of measuring their total volume. In that case, even a small variation of the background
due to not fully compensated curvature will drastically affect the volume measurements
[6]
.
Figure 7.5A
shows several simulated cells on the flat substrate. The simulated “imaging”
here is done with two wavelengths and the final unwrapped images were obtained using the
linear regression method (see later). Even a small amount of uncompensated curvature in a
single wavelength image (
Figure 7.5B)
is present in the final unwrapped image (see
Figure 7.5C)
. To properly calculate the total volume of all cells, this curvature must be
completely removed. The ideal method for background subtraction will remove the
uncompensated curvature with the minimal user intervention, while retaining the cells'
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