Biomedical Engineering Reference
In-Depth Information
FIgurE 9.7
Elimination of zero-order terms and twin image by spatial frequency filtering, in an off-axis con-
figuration. (a) Masked Fourier spectrum, (b) convolution, and (c) single Fourier transform.
method, but at the expense of introducing some noise in the reconstructed images. To this day, other
Fourier-based methods have been proposed to eliminate the zero-order and the twin image terms, even
when their respective spectrum overlaps. These methods include nonlinear filtering (Pavillon et al
.
, 2009)
and reconstruction using Gabor wavelet transform algorithms (Weng et al
.
, 2008), to cite only two.
Once the zero-order and twin image terms have been filtered out, the retrieved imaging term still
has to be demodulated. Here, demodulation is the process of extracting the phase of the initial object
wavefront by removing the linear phase gradient introduced in
r
by the off-axis configuration. One must
recall that the imaging term is made of the product of
o
with
r
*
(or
o
*
with
r
) and that its phase is there-
fore the sum of the phases of
o
and
r
*
(or
o
*
with
r
). For an optical configuration in which the reference
wave has a wavevector
k
0
subtending an off-axis angle θ with the optical axis, a large tilt corresponding
to
xy
projection of
k
0
will be introduced in the retrieved phase. It is common practice to demodulate this
phase contribution, as detailed in Section 9.4.3.
Results of off-axis spatial frequency filtering and subsequent demodulation are presented in Figure 9.7.
9.4.2.2 Phase-Shifting Holography
Phase-shifting holography was proposed by Yamaguchi and Zhang (1997) and relies on combinations of
multiple holograms to eliminate the zero-order and twin image terms.
In its original form, it required that four different holograms are recorded, each with exactly a π/2
phase shift compared to the previous. Assuming that
I
0
,
I
π/2
,
I
π
, and
I
π/2
are four such holograms, then
the complex object field ψ
0
can be retrieved by the following combination:
(9.12)
ψ
=
(
I
−
I
)
+
i I
(
−
I
)
=
Ae
i
ϕ
,
0
3
π
/
2
π
/
2
0
π
where the amplitude
A
and phase φ of ψ
0
are, respectively:
I
−
−
I
(9.13a)
ϕ
=
tan
−
1
3
π
/
2
π
π
/
2
I
I
0
(9.13b)
A
=
(
I
−
I
)
2
+
(
I
−
I
) .
2
3
π
/
2
π
/
2
0
π
The phase shifts are generally induced by translating an optical component, for example, a mirror,
with a piezoelectric transducer. Other schemes have been proposed and include rotating a wave plate,
using a diffraction grating or an acousto-optic modulator. No matter how they are generated, the phase
