Biomedical Engineering Reference
In-Depth Information
corresponds to the right-hand side of
Eq. (12.9)
. In mapping the experimental data, this
is equivalent to dividing the scattered field by the incident field
U
bg
(
x
,
y
;
θ
), which shifts
U
ð
S
Þ
in Fourier space. In other words, the scattered field used in the Fourier
diffraction theorem in the experiment is as follows:
ð
k
x
;
k
y
; θÞ
In the case of the first Born approximation:
U
ð
S
Þ
ðK
x
1
k
x
0
; K
y
1
k
y
0
; θÞ
5
ðUðx; y; θÞ
2
U
bg
ðx; y; θÞÞ=U
bg
ðx; y; θÞ
(12.10)
In case of the first Rytov approximation:
U
ð
S
Þ
ð
K
x
1
k
x
0
;
K
y
1
k
y
0
; θÞ
5
ln
ð
U
ð
x
;
y
; θÞ=
U
bg
ð
x
;
y
; θÞÞ
(12.11)
Rytov
Figure 12.5C and D
shows the results of this mapping on the (
K
x
,
K
y
,
K
z
5
0) and
(
K
x
,
K
y
5
0,
K
z
) planes, respectively. The data along the blue line in
Figure 12.5B
are
mapped onto the blue half circle on the (
K
x
,
K
z
) space of
Figure 12.5D
. Different angular
images are mapped onto different spaces such that they eventually cover a significant
portion of the (
K
x
,
K
y
,
K
z
) space of the object function
F
!
ð
Þ:
Looking at the frequency
spectrum of
Figure 12.5C and D
, ring patterns are clearly visible after mapping various
angular images, which is expected for the spherical shape of the sample. By taking the
inverse Fourier transform of the entire 3D frequency spectrum, the 3D distribution of
refractive index and absorption coefficient of the object can be obtained.
Even with the use of high NA object and condense lenses, an image within illumination
angles of up to
6
70
can be measured. As a result, the entire region of frequency space
cannot be filled as shown in
Figures 12.5C and D
. In other words, the inverse problem is
underdetermined. In the first round of reconstruction, zero values can be put in the missing
angle space. The reconstructed object function then exhibits negative bias around the sample
and the refractive index is smaller than the actual value (
Figure 12.6A
). To minimize the
artifact introduced by the missing angles, an iterative constraint algorithm
[18,19]
can be
applied based on the prior knowledge that the object function is non-negative for the live
cells. The index throughout the field of view, either inside or outside of the cell, is at least the
same or higher than the medium. As a first step, zero values are filled with the missing space
and then the inverse Fourier transform is taken to reconstruct a 3D map (
Figure 12.6D
).
In the reconstructed image, there are pixels whose index values are smaller than the index of
the medium (
Figure 12.6A
). These are forced to be the same as the index of the medium and
then the modified map is taken Fourier transformed. The index values in the Fourier space in
which zero values are assumed are no longer zero, and an approximate solution for the
missing angles is acquired (
Figure 12.6E
). But, at the same time, the data in the space which
contains measured data are now modified. Since the experimentally measured data are
accurate, we replace the modified data with the measured data. This procedure can be iterated
until the reconstructed object function converges (
Figure 12.6C and F
). Then, the negative
bias is removed and the reconstructed image becomes more accurate. For the case of the
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