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3D WAVE FIELD PHASE RETRIEVAL FROM MULTI-PLANE

OBSERVATIONS

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.?

ABSTRACT

Wereconstructaspatiallydistributedthree-dimensional(3D)

wave ?eld from a number of intensity observations, ob-

tainedindifferentsensorplanes, paralleltotheobjectplane.

The proposed algorithm can be treated as a multiple plane

iterative Gerchberg-Saxton algorithm [1]. 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

[2] shows an accuracy advantage of the proposed algorithm

provided that the type of modulation in the object plane is

known.

Index Terms — Imaging, information retrieval, holog-

raphy, iterative methods, inverse problems, phase estima-

tion

1. INTRODUCTION

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-

brations).

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).

However, mathematicallyandcomputationallythephase

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

missingphasedataarereconstructedbymodelingwave?eld

propagation successively from one plane to the next follow-

ing one.

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) [1], [3].

The further development and generalization of this tech-

nique results in various modi?cations for different appli-

cations (e.g. [4], [5]). For instance, in [6] 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 [2]. 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-

sition (ASD).

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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 [8]:

(1)

ASDzl;zo[f] = exp(j2?zl=??

s

1 ?

?2

l? ?2jjfjj2); (2)

N2

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.

l??2)=(?2);

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

Uo[f]

L

X

l=1

jjUzl[f] ? ASDzl;zo[f] ? Uo[f]jj2

2.

(3)

It can be shown that the optimal estimate of^Uo[f] is of

the form

^Uo[f] =1

L

L

X

l=1

ASD?

zl;zo[f] ? Uzl[f];

(4)

where '?' stands for a complex-conjugate variable.

Letusapplythesolution(4)fortheconsideredscenario,

when only the intensities of the complex-valued distribu-

tions are measured.

Assume that the observations are de?ned as

p

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:

ozl[k] =

juzl[k]j2+ "l[k];

(5)

^Q(t)

0;l[f] =ASD?

zl;zo[f]

L

? FFT fv(t?1)

l

[k]g;

L

X

(6)

^ q(t)

0;l[k] = FFT?1f^Q(t)

0;l[f]g; ^ u(t)

0[k] =

l=1

^ q(t)

0;l[k];

^ u(t)

zl[k] = FFT?1fASDzl;zo[f] ? FFT f^ u(t)

^?

zl[k]);

0[k]gg;

(t)

zl[k] = angle(^ u(t)

v(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 [2] 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 [7]. The pro-

posed technique (6) is essentially different from SBMIR

by its structure because the observations from all planes are

(t)

zl[k]); t = 1;2:::

o [k] can be corrected ac-

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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

Thenumericalexperimentsareperformedforamplitudeand

phaseobjectdistributionswiththetest-imagelena. Theim-

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-

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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

thesensorplanes(fortheproposedmethod)versusthenum-

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-

creases).

Itisfoundthatalargernumberoftheobservationplanes

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-

jectwave?eldtheconcordanceofthephaseestimates^?zl[k]

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.

5. CONCLUSIONS

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-

utions. Theimprovementofthereconstructionaccuracyde-

pendingontheincreaseofthenumberofobservationplanes

is shown. The prior information on the object type allows

the proposed algorithm to yield better accuracy with respect

to the successive SBMIR algorithm.

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