Three-dimensional phase retrieval in propagation-based phase-contrast imaging

Abstract

We present a solution to the phase problem in near-field x-ray (propagation) imaging. The three-dimensional complex-valued index of refraction is reconstructed from a set of projections recorded in the near-field (Fresnel) setting at a single detector distance. The solution is found by an iterative algorithm based only on the measured data and the three-dimensional tomographic (Helgason-Ludwig) consistency constraint without the need for further a priori knowledge or other restrictive assumptions.

Full text

PHYSICAL REVIEW A 89, 033847 (2014)
Three-dimensional phase retrieval in propagation-based phase-contrast imaging
A. Ruhlandt, M. Krenkel, M. Bartels, and T. Salditt*
Institut f¨
ur R¨
ontgenphysik, Georg-August-Universit¨
at G¨
ottingen, Friedrich-Hund-Platz 1, G¨
ottingen, Germany
(Received 12 August 2013; published 25 March 2014)
We present a solution to the phase problem in near-field x-ray (propagation) imaging. The three-dimensional
complex-valued index of refraction is reconstructed from a set of projections recorded in the near-field (Fresnel)
setting at a single detector distance. The solution is found by an iterative algorithm based only on the measured
data and the three-dimensional tomographic (Helgason-Ludwig) consistency constraint without the need for
further aprioriknowledge or other restrictive assumptions.
DOI: 10.1103/PhysRevA.89.033847 PACS number(s): 42.30.Rx,87.57.Q−,87.59.−e
I. INTRODUCTION
X-ray phase contrast is an emerging imaging technique
which has seen considerable progress over the last 15 years
[1–4]. Owing to the penetration power of hard x rays, the
method is uniquely suitable for the investigation of bulk struc-
tures of unsliced, unstained, and (optically) nontransparent
specimen. The long-standing limitation of x-ray imaging, that
low-density and light-element tissues or samples were just
“too tranparent” and nonabsorbing, has been overcome in
revolutionary ways by the advent of phase-contrast techniques.
Phase contrast can be achieved in different ways, notably
by grating interferometry [5] or free propagation of partially
coherent beams behind the object [6–8], encoding phase infor-
mation into measurable intensities. While grating-based phase
contrast is probably the method of choice for macroscopic
medical imaging, propagation-based (near-field) imaging is
better suited for high-resolution imaging, covering length
scales from the centimeter range down to below 50 nm.
Aside from quantitative contrast and high resolution, the
main advantage of x rays over other probes (visible light
nanoscopy [9], electron microscopy) derives from its high
penetration power. Therefore the most relevant applications
will be in three dimensional (3D) structure analysis, enabled by
computed tomography (CT) and the fact that image formation
is sufficiently well described by line integrals through the tissue
(projection approximation) [10,11].
The main challenge of propagation (or holographic) imag-
ing is the phase-retrieval step based on intensity measurements
in the detection plane. Several phase-retrieval methods have
been developed, based for example on linearization of the
contrast formation leading to simplified versions of the
transport of intensity equation [12–14], or to an analytic form
of the free-space contrast transfer functions (CTFs) [7,15–18].
Nevertheless, all present phase-retrieval techniques, whether
based on Fourier filtering or iterative algorithms, require one
or several additional assumptions like negligible absorption,
slowly varying phase shifts, known relationships between
phase shift and absorption, or known compact support of the
object [19–21] to compensate the artifacts introduced by zeros
in the oscillatory CTFs. So far, the only way to overcome
these restrictions has been to use data sets with more than a
single object to detector distance [22–24]. What has so far been
missing is the capability to reconstruct objects with arbitrary
*Corresponding author: tsaldit@gwdg.de
and unknown composition and structure from the measured
intensity distribution in a single detection plane without restric-
tive additional assumptions. However, in the relevant case of
tomography, single distance projections for different angles of
the object are necessarily available. It is well known that these
projections are not independent of each other [25,26] because
the finite size of the sample imposes systematic correlations
in particular for the low spatial frequencies of neighbor-
ing projections. This so-called Helgason-Ludwig consistency
condition has been the starting point for reconstruction of
incomplete data (missing wedge) in classical absorption-based
CT [27]. The idea of the present work is to use this consistency
as an inherent constraint to solve the phase problem in the holo-
graphic imaging regime, eliminating the need of additional a
priori knowledge. The idea is implemented by a nested phase-
retrieval and tomographic-reconstruction scheme, which stabi-
lizes the reconstruction of low spatial frequencies which have
previously hampered reconstruction of single-image data, in
particular for specimens where no additional constraints could
be applied. More generally, consistency constraints relating
several data sets of an object (projections, single exposures
with overlap, images recorded at different photon energy) often
warrant successful reconstruction. The recent development
of ptychographic reconstruction techniques also forms an
excellent example for this strategy [28–30].
In previous combinations of tomographic reconstruction
and near-field phase retrieval [31–33], the process is separable
into a first step of phase retrieval for each projection indepen-
dently, followed by tomographic reconstruction, performed
by filtered back projection (FBP), in a second step. The
tomographic reconstruction is thus carried out on a set
of projections for which the preceding phase retrieval is
considered as final. In far-field coherent diffractive imaging,
3D phase retrieval by use of the 3D fast Fourier transform
is easily implemented to the advantage of better convergence
properties, but still necessitates a support constraint [34,35]. In
contrast, in this work the representation of the object is strictly
three dimensional and a coupled algorithm based on iterative
phase retrieval and an algebraic tomographic reconstruction
technique (ART) [36] is proposed, automatically exploiting
the consistency condition. Based on simulated as well as on
experimental data, we show that the combined (or coupled,
nested) phase retrieval and tomography approach yields
significantly enhanced reconstructions with considerably less
input of aprioriknowledge on the sample, even in the presence
of noise. To demonstrate this, we present a specific algorithmic
implementation, as given below.
1050-2947/2014/89(3)/033847(8) 033847-1 ©2014 American Physical Society

A. RUHLANDT, M. KRENKEL, M. BARTELS, AND T. SALDITT PHYSICAL REVIEW A 89, 033847 (2014)
II. PROPAGATION-BASED PHASE-CONTRAST
TOMOGRAPHY
The principle of phase-contrast tomography based on free-
space propagation is sketched in Fig. 1(a). The object is
rotated around the yaxis, perpendicular to the propagation
direction zof a monochromatic plane wave I(z)=I0exp(ikz)
with wave number k. The interaction between object and
wave is described by the complex index of refraction, which
for x rays is commonly written as n=1−δ+iβ, where
δcharacterizes the phase shift and βthe absorption of the
material. Note that for hard x rays around 10 keV and soft
tissue, δis on the order of 10−6and βis up to three orders of
magnitude smaller. The exit wave of an object rotated by an
angle αin the xy plane directly behind (z=0) the object can
be expressed in the projection approximation [2]as
(x,y)α=I0exp ik 
n(x,y,z)αdz
∝exp[−iP(˜
δα)] exp[−P(˜
βα)],(1)
with the projection operator P(˜
δα)(x,y)=˜
δα(x,y,z)dz
for ˜
δα:=δαkand P(˜
βα), respectively, which is in principle
the two-dimensional Radon transform of ˜
δand ˜
β.Forthe
simulation below, the distance dzbetween sample and detector
is chosen such that the Fresnel number F:=a2/(λdz)≈1,
corresponding to holographic contrast formation, where ais a
characteristic size of the object and λ=2π/k the wavelength.
However, the main conclusions do not depend on the particular
choice of F. Tomography can be well described in Fourier
space via the projection-slice theorem [37], which identifies
the Fourier transform of a projection with a radial slice
through reciprocal space under the angle αcorresponding to
the projection. All slices share common values at the origin.
For objects or reconstruction volumes of finite size L(band
limited), convolution with the object shape function broadens
each Fourier slice to a thickness d∝1/L, as sketched in
Fig. 1(b), and depending on the edges may also introduce long-
range tails, for example in the form of a sinc function for the
simple case of a quadratic object. A mathematical formulation
of the coupling between the projections in real space is given
by the Helgason-Ludwig consistency condition [25–27]. This
coupling of different projections is ideally suited to improve
phase retrieval in propagation imaging, in particular for low
spatial frequencies which are poorly transferred and otherwise
only loosely constrained. In order to use this advantage, it is
necessary to couple phase retrieval with tomographic recon-
struction, i.e., to combine the two previously sequential steps.
III. RECONSTRUCTION ALGORITHM
As a reference, we use a sequential procedure that first
reconstructs the phase of each detector image and afterwards
combines all results into a 3D object. The phase retrieval
is based on an iterative algorithm which cycles between
detection plane and object plane, as sketched in Fig. 1(c).The
procedure is similar to but more general than the well-known
Gerchberg-Saxton (GS) algorithm [19,38] and is therefore
denoted as a modified GS (MGS) scheme. Starting with
the real-valued modulus of the holographic images |0|in
the detection plane, all images are propagated back to the
object plane using the inverse Fresnel propagator D−zd.This
kind of holographic reconstruction leads to severely perturbed
reconstructions due to the fact that the phase information
is unknown. Since tomography requires the validity of the
projection approximation, energy conservation and positivity
of the electron density result in ||1 and arg()0,
which is equivalent to δ0 and β0. After applying
these unrestrictive constraints, the ith iterate iis propagated
forwards to the detection plane, yielding 
i. Finally, the
amplitude constraint i+1=
i|0|/|
i|changes the image
amplitude to the measured values |0|while keeping the
phase information. This procedure is then repeated several
times and is called the outer loop in the following. Instead of
imposing the restrictive condition β=0 valid only for a pure
phase object or β/δ =const (single-material assumption), this
approach imposes only conditions which are applicable to all
objects. Note that the constraint β0 has been shown before
to improve the quality of holographic reconstructions [39].
While convergence and fixed-point properties of similar
iterative phase-retrieval algorithms have been well studied and
have seen continuous adaptations and generalizations with
FIG. 1. (Color online) (a) Schematic of holographic phase-contrast imaging. A plane wave illuminates the object ,leadingtoanexitwave
in the xy plane as given by the projected optical indices, followed by free-space propagation, resulting in holographic phase-contrast formation as
recorded in the detection plane at distance zd. (b) The finite cross section Lof the reconstruction volume leads to an effective width d∝1/L of
the Fourier slices, characterizing correlated areas in reciprocal space. This width results in a coupling between different projections in particular
at low spatial frequencies, providing a powerful constraint for phase retrieval. (c) Sketch of the iterative reconstruction algorithm devised to
retrieve both the phase information and the three-dimensional distribution of the optical indices of the object. The Gerchberg-Saxton-like step
explained in the text is represented by the dashed line.
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THREE-DIMENSIONAL PHASE RETRIEVAL IN . . . PHYSICAL REVIEW A 89, 033847 (2014)
respect to the employed constraints, one decisive difference
with respect to earlier work has to be stressed. Rather than
being performed independently for each projection, the update

α→αcan be performed by an inner cycle of tomographic
reconstruction followed by reprojection from the 3D volume
to the exit plane corresponding to α. This effectively imposes
the consistency between different projections.
Combining the previously sequential steps of phase re-
trieval and tomography, in each outer loop the complex-valued

αare separated into a phase-shift part 
α,˜
δ=−|arg(
α)|
and an absorption part 
α, ˜
β=−ln[min(
α,1)], applying the
respective constraints ensuring δ0 and β0. The three-
dimensional volumes ˜
δ(x,y,z) and ˜
β(x,y,z) are reconstructed
independently using a modified ART. Afterwards, the volumes
are projected and combined into new α. We denote this
nested iterative 3D reconstruction scheme which intrinsically
obeys the consistency constraint as “iterative reprojection
phase retrieval” (IRP). Stated otherwise, instead of a set of
2D projections, the reconstruction volume is strictly three
dimensional throughout the entire reconstruction process. The
exact ratios of outer phase retrieval and inner 3D reconstruction
steps, however, can be chosen, i.e., optimized during the
reconstruction, as discussed below.
IV. SIMULATIONS AND RESULTS
To implement the tomographic reconstruction, we have
devised an algorithm which can be considered as a modified
ART. Compared to the FBP it has the advantage of introducing
fewer high-frequency artifacts, which would inhibit the overall
reconstruction [40]; see Appendix BThe algorithm employs
an envelope M0=min[P−1(
α)] ∀αas initial guess for the
3D volume, where each voxel is set to the maximum value
which it can possibly have, considering all Nprojections (and
analogously for ˜
β). The back-projection operator is defined as
P−1
α:M(x,y,z)α=(x,y)α. The next approximation Mi+1
to the 3D volume for either ˜
βor ˜
δis then calculated as follows:
Mi+1=1
N
α
MiP−1
α
P(Mi)α.(2)
By this update step it is ensured that for each angle the
projection of the volume fulfills the current projection guess

α. By averaging over all Nprojections, it is guaranteed
that this algorithm converges to the least-squares solution for
inconsistent projections [40]. This inner loop is repeated κ
times and afterwards the outer loop is continued, producing a
new guess in the detection plane.
The quality of the reconstruction is monitored by an error
metric according to
σ2:=1
Nn
xny
α,x,y
[(x,y)α−0(x,y)α]2,(3)
comparing the modified to the recorded detector images
0. Here, nxnydenotes the number of detector pixels. Thus,
σrepresents the average error per pixel in the detection plane,
considering all Nprojections. We have made the observation,
that the quality of the tomographic reconstruction needs to
be adapted to the phase-retrieval quality as a small number
of iterations in the inner loop can limit the quality of the
outer loop. One way to assure this is to double the number κi
of tomographic iterations when σ :=σi−1−σi<ε, starting
with κ0=1 (the first inner-loop iteration). A stopping criterion
for the outer loop can thus be a maximum number of inner
iterations κmax, as used in this work.
To test the validity and performance of the algorithm,
we have used a phantom composed of 1303cubic voxels,
representing a superposition of several ellipsoids that differ
in scale and value. Holographic intensity distributions were
simulated for N=90 equidistant angles in the range 0◦α<
180◦. One exemplary projection image and one holographic
detector image are shown in Fig. 1(a). Figure 2illustrates
the advantages of IRP compared to the MGS scheme and
thereby the quality of the consistency constraint. Figure 2(a)
shows a slice through the phase volume and a histogram of the
phantom depicting the frequency of the different phase-shift
values. For a pure phase object (β=0), both the sequential
reconstruction approach (MGS) shown in Fig. 2(b) and the
coupled approach (IRP) shown in Fig. 2(c) yield quite good
3D phase reconstructions. However, the latter shows fewer
image artifacts. As quantified by the histograms, the IRP
reconstruction allows for a better separation of the different
phase-shift values. In particular, the lowest value of ˜
δcan
be distinguished from the noisy underground only in the
histogram of the IRP reconstruction. For more realistic cases,
βis always at least slightly larger than zero. Simulations
show that already small values β/δ ≈0.003, in the range
expected for soft biological tissues, lead to a decrease of the
reconstruction quality in the separate reconstruction scheme
(MGS) and to systematic errors by wrongfully forcing ||=1.
If the ratio β/δ is known and the value is used as an
additional constraint, applicable to samples of varying density
but identical stochiometry (single material), the conventional
approach is also able to reconstruct the volume sufficiently well
[24]. However, in most cases, β/δ ratios are not known or the
object consists of more than one material, leading to spatially
varying β/δ ratios. For a simulated object with different
β/δ ratios for each ellipsoid (ranging from 0.01 to 0.075),
the conventional approach (MGS) shows significant artifacts,
especially for lower spatial frequencies; see, for example, the
large hole in the center of the reconstructed phantom as shown
in Fig. 2(d). IRP reconstruction, on the other hand, shows only
slight deviations, as shown in Fig. 2(e), and is still capable
of separating the materials based on the better fidelity in
voxel values (histogram). Even completely uncorrelated β/δ
distributions can be reconstructed, demonstrating the power of
the 3D consistency constraint (see the Appendix Cfor further
details). Finally, IRP converges much faster with respect to
the number of MGS loops required (400 vs 10 000), but it
should be mentioned that the tomography step in each iteration
is computationally very expensive, at least if ART is used.
V. RECONSTRUCTION OF NOISY AND
EXPERIMENTAL DATA
Figure 3illustrates the influence of noise on the re-
construction. For the case shown, Gaussian noise with a
standard deviation of σ=0.01 was added to every simulated
detector intensity image for the case of a pure phase object
[β=0, Fig. 3(a)]. The IRP reconstruction shown in Fig. 3(c)
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A. RUHLANDT, M. KRENKEL, M. BARTELS, AND T. SALDITT PHYSICAL REVIEW A 89, 033847 (2014)
FIG. 2. (Color online) Reconstruction quality for the standard approach of separated phase and tomographic reconstruction compared to
the coupled reconstruction according to IRP, each for the case of a pure phase object and a mixed object. (a) Phase shift ˜
δof the reference
phantom. (b) Reconstructed phase shift of a pure phase object β=0 using the MGS algorithm (20 000 GS loops, followed by 128 ART
iterations). The pure phase constraint has been enforced in the reconstruction. (c) Same as in (b) but with IRP reconstruction (κmax =128 ART
iterations), and without using the fact that the phantom was a phase object. (d) Reconstructed phase shift of an object with fixed β/δ =0.02
coupling, using the modified GS algorithm (20 000 GS loops, followed by 128 ART iterations). As in (b) the coupling value was enforced. (e)
As in (d) but with IRP reconstruction (κmax =128 ART iterations), and without using any preknown coupling values. Slices of reconstructed
volumes in the yz plane (top row) are shown to illustrate the difference in data quality, along with the histograms of the ˜
δvalues for the entire
volume (bottom row). The scale bar applies to the entire row, the color bars to all images and the histograms. The frequency of occurrence in
the histograms is shown in linear scale from 0 to 5000 counts.
substantially outperforms the result obtained by the conven-
tional sequential scheme (MGS), depicted in (b). To test the
IRP algorithm on experimental data, we finally applied the
reconstruction to a phase-contrast data set of Deinococcus
radiodurans bacterial cells, recorded by tomographic x-ray
propagation microscopy [21], i.e., tomography in a cone beam
with geometric magnification and phase-contrast formation
by free-space propagation. The freeze-dried bacteria were
positioned 8 mm behind a quasipoint source formed by an
x-ray waveguide, resulting in an effective pixel size of 83 nm.
A typical holographic intensity image (experimental data) is
showninFig.3(d). Previously the data were treated in the
sequential mode, first using phase retrieval based on a mod-
ified hybrid-input-output algorithm (MHIO) with a support
constraint, followed by FBP of all phased projections. A major
challenge of previous phase retrieval was the automatic support
determination for all projection angles [21]. A slice through
the reconstruction volume is shown in Fig. 3(e) for the original
sequential reconstruction (MHIO) and in (f) for IRP, showing
an excellent quality with a more evenly distributed signal than
before without any additional support information.
VI. SUMMARY
In summary, we have shown that by coupling the phase
retrieval and tomographic reconstruction, the phase problem
FIG. 3. (Color online) Influence of Gaussian noise and reconstruction of experimental data. (a) xy slice of the undisturbed pure phase
object. (b) xy slice of the MGS reconstruction from noisy pure phase object images (σ=0.01, 20 000 loops, followed by 128 ART iterations).
(c) xy slice of the IRP reconstruction from the same detector images as in (a) (κmax =128 ART iterations). (d) Typical holographic diffraction
pattern (detector plane) of a bacterial cell [21]. (e) Slice though the modified HIO reconstruction as described in [21]. (f) Slice through the IRP
reconstruction from the experimental data, based on 83 projections.
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THREE-DIMENSIONAL PHASE RETRIEVAL IN . . . PHYSICAL REVIEW A 89, 033847 (2014)
in full field (Fresnel) x-ray propagation imaging can be solved
beyond the current restrictions. In particular, no assumptions
either on the phase-shifting and absorption properties of the
object or on the support are necessary. Instead, redundancy
of the data in view of the unknowns is derived from the
tomographic consistency condition, i.e., the correlations in
different projections which are inherent in finite-size volumes,
but had not been exploited for near-field phase retrieval
before. While the general principle of the nested or coupled
reconstruction holds for many different implementations of
the phase retrieval and tomography, the specific algorithmic
example IRP exhibits a particularly high performance; how-
ever, at the cost of significant computational effort. It can be
expected that many corrections to imperfect data as well as
regularization strategies can be accommodated in this scheme.
A first investigation with noisy data proved the robustness of
the IRP algorithm. Towards faster reconstruction, in particular
needed for images with larger pixel number, one should
consider a coupled phase retrieval and tomography scheme
which replaces ART by FBP. For experimental data, good
reconstructions have been achieved without additional support
information or assumptions about the sample. With the pre-
sented approach, x-ray phase-contrast propagation imaging is
now equipped with a means to treat images recorded at a single
detector distance, and for the most general case of objects.
An example program performing the IRP reconstruction is
included as Supplemental Material [41].
ACKNOWLEDGMENTS
We thank Markus Osterhoff for helpful discussions and in
particular for his advice on numerical issues, and acknowledge
support by the Collaborative Research Center 755 Nanoscale
Photonic Imaging of the German science foundation (DFG).
APPENDIX A: PHANTOM DEFINITION
The phantom used in this work consists of 130 ×130 ×130
voxels, representing a sum of 17 ellipsoids with different sizes
and values, each defined by the formula
˜
δ(x,y,z)=viif x−cx
a2+y−cy
b2+(z−cz)2<r
2,
0else.
(A1)
TABLE I. Parameters of all ellipsoids that make up the phantom.
The unit vx indicates the size of one voxel.
Ellipsoid cx(vx) cy(vx) cz(vx) r(vx) a(vx) b(vx) vi(vx)
1 0 0 0 34.18 0.55 0.75 0.005
2−8.92 −14.86 0 14.86 0.5 0.7 0.005
3−8.92 14.86 0 14.86 0.5 0.7 0.005
4 8.92 −14.86 0 14.86 0.5 0.7 0.005
5 8.92 14.86 0 14.86 0.5 0.7 0.005
6 8.92 14.86 0 7.43 0.5 0.7 0.005
7−2.97 −14.86 14.86 14.86 0.5 0.7 0.005
8−2.97 14.86 14.86 14.86 0.5 0.7 0.005
9 2.97 −14.86 14.86 14.86 0.5 0.7 0.005
10 2.97 14.86 14.86 14.86 0.5 0.7 0.005
11 −0.74 −14.86 −18.28 4.46 0.5 0.7 0.01
12 −0.59 14.86 −18.58 2.97 0.5 0.7 0.012
13 0.45 −29.72 −20.36 1.49 0.5 0.7 0.015
14 0.30 44.58 −18.43 5.94 0.5 0.7 0.02
15 −0.74 14.86 −18.28 4.46 0.5 0.7 0.01
16 14.86 7.43 −29.72 3.72 8 5.5 0.0025
17 0 0.00 −29.72 3.72 1 1 0.005
The parameters cx,cy,cz,a,b, and rare listed in Table I.Due
to the overlap of some ellipsoids, the phantom shows nine
different values from 0.005 to 0.027.
APPENDIX B: COMPARISON OF TOMOGRAPHIC
RECONSTRUCTIONS
The deviations of the reconstructed volumes Mrec from the
ideal phantom Mph are measured by the root mean square
deviation
σ:=



1
N
N

i=1
(Mph,i −Mrec,i )2,(B1)
where Ndenotes the number of voxels (here N=1303).
The phantom was projected to 90 equidistant angles fom
0to180
◦and reconstructed with the presented algebraic
reconstruction technique implementation and the common
filtered back projection. The results are compared in Fig. 4for
a slice perpendicular to the rotation axis. The ART shows less
high-frequency noise, leading to σART =8.19 ×10−5after
128 iterations. The FBP yields σFBP =2.88 ×10−4.Evenif
FIG. 4. (Color online) Comparison of tomographic reconstruction (no phase retrieval) by ART and FBP. (a) shows a slice through the
phantom perpendicular to the axis of rotation. The volume was projected onto 90 equidistant angles fom 0 to 80◦and reconstructed using (b)
ART (128 iterations) and (c) FBP.
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A. RUHLANDT, M. KRENKEL, M. BARTELS, AND T. SALDITT PHYSICAL REVIEW A 89, 033847 (2014)
FIG. 5. (Color online) Influence of the β/δ ratio (rows) on the reconstruction results for the MGS and IRP algorithms (columns).
the nonphysical negative values are set to zero, the standard
deviation is σFBP>0=1.63 ×10−4.
APPENDIX C: OBJECTS WITH MIXED PHASE
AND AMPLITUDE CONTRAST
This appendix presents more simulation results on objects
with mixed phase and amplitude contrast. Several established
phase-retrieval algorithms perform very well for pure phase
objects. However, even for hard x rays and soft tissues a
small contribution of the absorption is expected and can spoil
proper reconstruction, highlighting the importance of phase
retrieval for more general mixed objects. The influence of
asmallβ/δ ratio on the reconstruction results is illustrated
in Fig. 5. The results for the conventional sequential approach
(MGS) is compared to IRP for cases β/δ =0uptoβ/δ =0.2.
In all cases, the ratio was unknown to the algorithms and
hence not used as a constraint. In both algorithms, only the
“mild” constraints ||1 and arg()0 were used. The σ
values for the different results shown in Fig. 5are presented in
Table II. It can be seen that the consistency condition in IRP
leads to better results in all cases.
TABLE II. Standard deviation σ×103for different reconstruc-
tion methods (rows) and different β/δ ratios (columns). The ART
was stopped after 128 iterations, IRP was carried out until the inner
loop reached 128 iterations.
β/δ ratio 0.2 0.02 0.0035 0
MGS and FBP 1.18 0.59 0.40 0.37
MGS and ART 1.12 0.53 0.38 0.36
IRP 0.34 0.23 0.21 0.21
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THREE-DIMENSIONAL PHASE RETRIEVAL IN . . . PHYSICAL REVIEW A 89, 033847 (2014)
FIG. 6. (Color online) Influence of neglected weak absorption on the reconstruction. (a) Phantom xz plane, perpendicular to the axis of
rotation. (b) Reconstructed xy plane using 20 000 iterations of MGS with the constraint ||=1, followed by 128 iterations of ART. (c)
Phantom xy plane. (d) Reconstructed xy plane.
Next, we address the question of whether the reconstruction
quality can be improved by imposing the constraint of a
pure phase object (i.e., forcing the amplitude to 1), if the
object is weakly absorbing. In other words we check the
validity of neglecting absoprtion for the purpose of stabilizing
phase retrieval. Figure 6shows reconstructions for the case
of β/δ =0.02, resulting in a maximum absorbtion of 1.42 %
and a mean absorption of 0.33 %. 90 angles were simulated
and reconstructed using MGS (20 000 iterations) with the
additional condition ||=1, i.e., no absorption. The three-
dimensional object was reconstructed using ART with 128
iterations. The results show much fewer distortions in the
object shape and features [compare to Fig. 5(c) MGS] but
a significant error in the values. Please note that this effect
cannot be corrected for by a constant offset. The comparison
with the phantom yields σ=6.76 ×10−4, which is inferior
to all reconstructions shown in Table II. This shows that
neglecting absorption even for very weakly absorbing objects
yields inconsistent phase distributions.
Finally, we compare the reconstruction results for com-
pletely independent β/δ ratios. Figure 7shows the results for
˜
δand ˜
βfor a simulation with both the original phantom for
˜
δand an independent absorption phantom. For the latter, the
original phantom was rotated, enlarged, and set to lower values.
Note that this includes the unphysical case of absorption
without a phase shift. 180 equidistant angles from 0 to
180◦were simulated and reconstructed using MGS (14 000
iterations) and IRP (up to 128 inner ART loops). ˜
δcan be
reconstructed to a nearly quantitative level using IRP and even
the contour of the much weaker ˜
βpart can be retrieved. In
contrast, the MGS reconstruction leads to heavily distorted
volumes.
FIG. 7. (Color online) Reconstruction of completely independent β/δ values. The top row shows the ˜
βpart, the bottom row the ˜
δpart of
the same volume. (a),(b) Phantom xy plane. (c),(d) Same plane, MGS results. (e),(f) Same plane, IRP results.
033847-7

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