Biomedical Engineering Reference
In-Depth Information
following section, it is explained how the sample-induced phase change
Δϕ
s
(
x
,
y
,
z
IP
) in the
image plane
z
5
z
IP
is retrieved from the intensity distribution in
Eq. (6.1)
.
6.3.2 Spatial Phase Shifting-Based Reconstruction of Digital Holograms
For the numerical reconstruction of temporal phase shifted and digital off-axis holograms,
various methods have been developed (for an overview, see Refs.
[12,26,37
41]
). Here, the
numerical reconstruction is performed in two steps. First, the complex object wave
O
is
reconstructed in the hologram plane. This is performed by spatial phase shifting. Spatial
phase shifting provides the retrieval of the object wave without the disturbing terms “twin
image” and “zero-order intensity.” In temporal phase shifting holography, several
holograms for known (or unknown but equidistant) phase shifts of the reference wave (or
alternatively of the object wave) are recorded to retrieve the object wave (for illustration,
see Refs.
[42
44]
). In contrast, in spatial phase shifting holography, neighboring pixels of
single off-axis holograms are evaluated. Therefore, the same numerical algorithms as for
temporal phase shifting
[45]
can be used
[46]
. However, the interferogram equation
(
Eq. (6.1)
) can also be solved pixel-wise within a squared area of pixels (in practice usually
5
3
5 pixels) around a given hologram pixel using the least squares principle. By definition
of appropriate substitutions, the resulting nonlinear problem is transferred to a form that can
be solved by linear algorithms
[9,47]
. The resulting robust reconstruction method for
O
was
found particularly suitable for the application in DHM and has been reported to be applied
successfully for the analysis of living cells in Refs.
[9,29]
.
In case that objects have been recorded out of focus, numerical refocusing is required.
Thus, in an optionally subsequent step,
O
is numerically propagated to the image plane.
This is typically performed by the numerical implementation of the Fresnel
Huygens
principle
[38,48]
. Here, the numerical wave propagation is illustrated by an approach of the
convolution method in which the convolution theorem is applied after the Fresnel
approximation
[29,49]
. The propagation of
O
(
x
,
y
,
z
H
) to the image plane
z
IP
that is located
at
z
IP
5
z
H
1
Δz
in the distance
Δ
z to the hologram plane (see
Figure 6.1
) is performed by
using the following equation:
zÞ
5
F
2
1
2
2
Oðx
;
y
;
z
IP
5
z
H
1
Δ
f
F
fOðx
;
y
;
z
H
Þg
exp
ð
i
πλΔ
zðν
1
μ
ÞÞg
(6.2)
λ
ν
μ
In
Eq. (6.2)
,
are the coordinates
infrequency domain, and F denotes a Fourier transformation. The advantage of this
approach is that the size of the propagated wave field is preserved during the refocusing
process. This is a particular advantage for numerical autofocusing as described in
Section 6.3.4
because it simplifies the comparison of the image definition in different focal
planes. Furthermore, in contrast to the propagation by digital Fresnel transformation,
Eq. (6.2)
allows the refocusing of only slightly defocused images of the sample near the
is the wave length of the applied laser light,
and
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