3D WAVE FIELD PHASE RETRIEVAL FROM MULTI-PLANE
Artem Migukin, Vladimir Katkovnik, Jaakko Astola
Department of Signal Processing, Tampere University of Technology (TUT)
Korkeakoulunkatu 10, FI-33720, Tampere, Finland
e-mail: artem.migukin@tut.?, vladimir.katkovnik@tut.?, jaakko.astola@tut.?
wave ?eld from a number of intensity observations, ob-
The proposed algorithm can be treated as a multiple plane
iterative Gerchberg-Saxton algorithm . It is obtained
from the best linear estimate of the complex-valued object
distribution derived for the complex-valued observations.
This estimator is modi?ed for the intensity measurements
in the sensor planes. The algorithm is studied by numeri-
cal experiments performed for amplitude and phase object
distributions. It is shown that the proposed method allows
reconstructing the whole 3D wave ?elds for different setup
parameters. This technique can be applied for 3D imag-
ing. The comparison versus the successive iterative method
 shows an accuracy advantage of the proposed algorithm
provided that the type of modulation in the object plane is
Index Terms — Imaging, information retrieval, holog-
raphy, iterative methods, inverse problems, phase estima-
The reconstruction of the whole wave ?eld (both ampli-
tude and phase) is an important problem utilized in differ-
ent technical and scienti?c applications, in particular for
3D imaging or nondestructive testing. The phase can not
be measured directly, thus we recover the phase from a
number of intensity measurements. There are two groups
of the wave ?eld reconstruction methods: interferometric
one with a reference beam and methods without a reference
beam (phase retrieval). The phase retrieval techniques are
much more reliable and technically simpler than the inter-
ferometric ones, in particular, because of the simplicity of
the optical setup. Furthermore, the phase retrieval approach
is more robust with respect to various disturbances (e.g. vi-
This research is supported by the Academy of Finland, project No.
213462 (Finnish Centre of Excellence program 2006 - 2011), and the
post graduate work of Artem Migukin is funded by the Tampere Graduate
School in Information Science and Engineering (TISE).
retrieval from the magnitude measurements is not a trivial
problem. In this paper we study a novel technique based on
the parallel usage of the observations from all sensor planes
simultaneously for the reconstruction of the 2D wave ?eld
in the object plane and 3D wave ?eld distributions in the
observation planes. The proposed technique is very dif-
ferent from the established successive methods where the
propagation successively from one plane to the next follow-
The planar laser beam scattered by an object propagates
through the space. The intensity of the resulting wave ?eld
distribution is registered by digital sensors in the sensor
planes parallel to the object plane (see Fig.1).
The Gerchberg-Saxton-Fienup iterative algorithm is the
most popular phase recovery method, initially proposed for
a single measurement plane. It is based on the essential us-
age of a prior knowledge on the object size and object type
(phase or amplitude modulation of the wave ?eld) , .
The further development and generalization of this tech-
nique results in various modi?cations for different appli-
cations (e.g. , ). For instance, in  a multi-plane
modi?cation of the algorithm is developed in order to ob-
tain a desired wave ?eld distributions in different planes.
In this work we consider the wave ?eld distribution in
the object plane as the only unknown of the problem which
one-to-one de?nes the wave ?elds for sensor planes. In this
approach prior information on the object such as the size
and modulation type is used in order to improve the ac-
curacy of the wave ?eld reconstruction. The main contri-
bution of this paper concerns the development of the 3D
wave ?eld phase retrieval algorithm and numerical compar-
ative analysis of the proposed parallel phase retrieval algo-
rithm versus the successive method presented in . The
in?uence of the prior information in the object plane on the
wave ?eld reconstruction accuracy is analyzed. In the re-
construction algorithms the wave ?eld propagation is per-
formed using the conventional angular spectrum decompo-
Figure 1. Multiple plane wave ?eld reconstruction scenario: u0[k] and
uzl[k] are discrete complex amplitudes in the object and measurement
planes respectively, l = 1;::::;L.
2. WAVE FIELD PROPAGATION MODEL
Let u0(x) and uzl(x), x 2 R2; l = 1;:::;L, denote the
complex-valued wave ?eld distributions in the object and
sensor planes, respectively. zl = z1+ (l ? 1) ? ?z in-
dicates a distance between the parallel object and l ? th
sensor planes, ?zis a distance between two sensor planes,
z1is a distance from the object to the ?rst measurement
plane and L is a number of the observation planes (sensor
positions). We assume that the wave ?eld distributions in
the object and sensor planes are pixel-wise invariant. Be-
cause of this pixelation we obtain the sampled version of
the continuous wave ?eld distributions: u0(x) ! u0[k],
uzl(x) ! uzl[k], where k = (kx;ky) 2 Z2is a two di-
mensional vector with integer components. In Fig.1 this
multi-plane phase retrieval model is presented.
Using the ASD modeling the link between u0[k] =
ju0[k]j?exp(j??0[k]) and uzl[k] = juzl[k]j?exp(j??zl[k])
is given in the frequency domain as
Uzl[f] = ASDzl;zo[f] ? Uo[f].
Here f = (fx;fy) 2 Z2is the spatial frequency, Uzl[f]
and Uo[f] are calculated as the 2D Fourier transform of
uo[k] and uzl[k] using FFT and the ASD discrete transfer
function is given analytically as :
ASDzl;zo[f] = exp(j2?zl=??
l? ?2jjfjj2); (2)
wherejjfjj2istheEuclideannorm, jjfjj2< (N2
? is the wavelength, ? is the pixel size (we assume that the
pixels are square ???) and Nl?Nlis the size (in pixels)
of the wave ?eld distribution in the l?th observation plane.
3. PHASE RETRIEVAL ALGORITHM
Let us assume for a while that in the sensor planes the
complex-valued observations are available. Then the best
least square estimate of Uo[f] from the observed Uzl[f] is
calculated from the following optimization problem
^Uo[f] = arg min
jjUzl[f] ? ASDzl;zo[f] ? Uo[f]jj2
It can be shown that the optimal estimate of^Uo[f] is of
zl;zo[f] ? Uzl[f];
where '?' stands for a complex-conjugate variable.
when only the intensities of the complex-valued distribu-
tions are measured.
Assume that the observations are de?ned as
where "l[k] ? N(0;?2) is the Gaussian noise.
The proposed phase retrieval algorithm is recursive and
de?ned for ASD by the equations:
? FFT fv(t?1)
0;l[k] = FFT?1f^Q(t)
0;l[f]g; ^ u(t)
zl[k] = FFT?1fASDzl;zo[f] ? FFT f^ u(t)
zl[k] = angle(^ u(t)
l[k] = ozl[k] ? exp(j ?^?
The expressions (6) de?ne the iterative multiple plane
parallel algorithm, which can be treated as a generalization
of the Gerchberg-Saxton algorithm. The ?rst two equations
of this algorithm de?ne the object wave ?eld estimate, ob-
tained from estimates in the sensor planes (backward prop-
agation). The complex-valued ^ u(t)
cording to a prior information about the object distribution.
The estimates in the sensor planes are exploited in paral-
lel for the calculation of the object estimate. The estimate
in the object plane is used for prediction (forward propaga-
tion) in the sensor planes (third equation). The intensities
of these predictions are corrected by the given intensity ob-
servations (the ?fth equation in (6)).
In the single-beam multiple-intensity phase reconstruc-
tion (SBMIR) algorithm  the phase reconstruction is
produced by the wave ?eld propagation modeling from one
sensor plane to the next following one with a circle loop go-
ing from the last sensor plane to the ?rst one. The study of
this algorithm demonstrates the ef?ciency of this technique
in simulations and for real experimental data . The pro-
posed technique (6) is essentially different from SBMIR
by its structure because the observations from all planes are
zl[k]); t = 1;2:::
o [k] can be corrected ac-
Figure 2. The object wave ?eld reconstruction, ? = 0:01; L = 10, z1=
1:5?zf: (a) j^ u0j, AM, RMSE = 0:011, (b)^?0, PM, RMSE = 0:11,
(c) j^ u0j, AM, RMSE = 0:044, (d)^?0, AM, RMSE = 0:198, (e)
j^ u0j, PM, RMSE = 0:147, (f)^?0, PM, RMSE = 0:175.
processed in parallel while in SBMIR a plane-to-plane
phase reconstruction is used. The use of the object dis-
tribution as the only estimated variable enables the paral-
lel algorithm to involve a prior information on the type of
the object distribution (the amplitude or phase distribution
of uo[k]) as well as the object size. This additional infor-
mation has a signi?cant in?uence on the reconstruction ac-
curacy. Simulation con?rms the advantage of this parallel
processing, when the prior information (that the object is of
the amplitude or phase type) is used in the algorithm. We
assume that the size of the reconstructed object is known.
4. NUMERICAL EXPERIMENTS
ages are square N ? N, N = 256 with the square pixels
? ? ? of the same size in the object and sensor planes,
? = 6:7?m, the wavelength ? = 632:8 nm. The "in-
focus" distance is calculated as zf = N?2=? = 18:16
mm. It is assumed that the additive noise in (5) is zero-
mean Gaussian with ? = 0:01 and ? = 0:05. The number
Figure 3. The RMSE reconstruction accuracy of the object magnitude
versus the number of planes L, AM, ? = 0:01, z1 = 1:5 ? zf: the
proposed parallel algorithm versus SBMIR.
of measurement planes varies: L = [1;20]. The results
are shown for 100 iterations of the algorithm. The distance
between the measurement planes is ?xed: ?z = 0:5mm.
The in?uence of the quantization of the observations on the
wave ?eld reconstruction accuracy is out of the scope in
this work, and we assume that a high precision data from a
sensor is given.
Imaging of the reconstructed amplitude j^ u0j and phase
^?0distributions in the object plane is shown in Fig.2. These
images correspond to the amplitude (AM) and phase mod-
ulation (PM) of the object distribution. We demonstrate
the in?uence of the knowledge of the type modulation pri-
ori on the quality of imaging. Here we show the recon-
struction for the amplitude (Fig.2 (a)) and phase (Fig.2 (b))
object distributions provided that it is known in advance the
corresponding object modulation. If the type of the distrib-
ution is unknown in advance, both the amplitude and phase
are estimated in the object plane. We show the amplitude
and phase estimates for the AM in Fig.2 (c) and (d), and
for PM in Fig.2 (e) and (f) respectively. Here and further
the wave ?eld reconstruction accuracy is given via the root
mean square error (RMSE).
In Fig.3 we compare the object wave ?eld reconstruc-
tion accuracy (for AM), obtained by the proposed algo-
rithm (6) and by the successive SBMIR. The original
successive iterative process of SBMIR has no direct con-
nection to the object plane, and it is not able to use the prior
information on its distribution (see "SBMIR;complex").
We have modi?ed this algorithm and included the object
plane in this successive recursive procedure. The corre-
sponding result is shown as "SBMIR;abs". The curves
in Fig.3 show that the proposed algorithm gives a better
accuracy for the amplitude object, when the type of the ob-
ject distribution is used for the estimation ("ASD;abs").
If we do not use the prior information on the object distri-
bution and estimate the object distribution as a complex-
Figure 4. The reconstruction accuracy of the complex-valued wave ?eld
(the amplitude juzj and phase ?z) in the sensor planes, obtained using the
proposed algorithm for different noise levels ?, AM: the mean values of
RMSE for L planes, z1= 1:5 ? zf.
valued one, the SBMIR algorithm gives a better accu-
racy than the proposed algorithm. In this case the result
for the parallel algorithm is marked as "ASD;complex".
The SBMIR algorithm converges very quickly and the in-
crease of the number of iterations does not yield to a sig-
ni?cant improvement in accuracy. The accuracy value for
the proposed method increases monotonically, but slower
than for SBMIR. For large number of iterations and L the
proposed algorithm demonstrates better accuracy.
In Fig.4 the quality of the wave ?eld reconstruction in
ber of the observation planes L is illustrated. The accuracy
of 3D wave ?eld for all measurement planes is considered
with calculation of RMSE for the phase and amplitude
separately. We present the mean value of RMSE over
all measurement planes, because these values for differ-
ent planes are quite close with the standard deviation from
the mean values not more than 6%. It is seen that if ? is
larger, the improvement of wave ?eld reconstruction is not
so essential for larger L (a slope of the RMSE curves de-
L results in monotonically better accuracy for both the ob-
ject and sensor planes. This improvement is valuable for
small L (say, L = 2;3) and not essential for larger L: For
instance (for AM), two additional planes from L = 3 to
L = 5 improves the accuracy in RMSE values by approx-
imately 40% for the object wave ?eld reconstruction and
approximately 20% for the wave ?eld reconstruction in the
sensor planes. L = 10 results in the further improvement:
more than 90% and 40% in comparison with L = 3 for the
object and sensor planes respectively.
In the reconstruction of the whole complex-valued ob-
is of very importance, because the phase component of the
?nal object estimate (the sum in the second equation of the
algorithm (6)) should have close values. The wrapping ef-
fect could lead to a quite strong damage for the parallel al-
gorithm, poor quality of reconstruction and imaging of the
?nal estimates. In Fig.2 (d) an example of this wrapping ef-
fect for AM can be seen. The phase wrapping effects in the
reconstructed phase distributions are seen as bright white
sports. Note that it also results in the worse reconstruction
of the object amplitude.
In this work we present a phase retrieval algorithm based
on simultaneous processing of the data, obtained in a num-
ber of parallel observation planes. Numerical experiments
demonstrate the applicability of the proposed algorithm for
the reconstruction of the complex-valued wave ?eld distrib-
is shown. The prior information on the object type allows
the proposed algorithm to yield better accuracy with respect
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