Numerical suppression of the twin-image in in-line holography of a volume of micro-objects

Abstract

We address the twin-image problem that arises in holography due to the lack of phase information in intensity measurements. This problem is of great importance in in-line holography where spatial elimination of the twin image cannot be carried out as in off-axis holography. A unifying description of existing digital suppression methods is given in the light of deconvolution techniques. Holograms of objects spread in 3D cannot be processed through available approaches. We suggest an iterative algorithm and demonstrate its efficacy on both simulated and real data. This method is suitable to enhance the reconstructed images from a digital hologram of small objects.

Full text

Numerical suppression of the twin-image in in-line
holography of a volume of micro-objects
L Denis1,2, C Fournier2, T Fournel2, C Ducottet2
1Ecole Sup´erieure de Chimie Physique Electronique de Lyon, 43 bd du 11 Novembre
1918, F-69616 Villeurbanne, France
2Laboratoire Hubert Curien (ex-LTSI) ; CNRS, UMR5516 ; Universit´e Jean Monnet;
18 rue Pr Benoˆıt Lauras, F-42000 Saint-Etienne, France
E-mail: loic.denis@cpe.fr
PACS numbers: 07.05.P, 42.40
Keywords: digital holography; in-line holography; deconvolution; twin-image.
Submitted to: Meas. Sci. Technol.
Abstract. We address the twin-image problem that arises in holography due to
the lack of phase information in intensity measurements. This problem is of great
importance in in-line holography where spatial elimination of the twin-image cannot
be carried out as in off-axis holography.
A unifying description of existing digital suppression methods is given in the light
of deconvolution techniques. Holograms of objects spread in 3D cannot be processed
through available approaches. We suggest an iterative algorithm and demonstrate its
efficacy on both simulated and real data. This method is suitable to enhance the
reconstructed images from a digital hologram of small objects.

Numerical suppression of the twin-image 2
1. Introduction
Twin (i.e. virtual) images were considered as a limitation of holography since its very
beginnings[1]. They are defocused images of the holographed objects that superimpose
on the real images of the objects in the optical or numerical reconstructions. Their
origin is tied to the principle of in-line holography and is recalled at the end of this
introductory section.
The twin-image problem has received considerable attention since Gabor’s
pioneering work on holography. It is still a concern today as the in-line holographic setup
is widely used in optical holography, soft X-rays holography and electronic microscopy.
Basically, two classes of approaches have been proposed to tackle this problem: a first
class which involves a change in the hologram-recording setup, and a second class that
attempt to numerically suppress the twin-image. The best known method in the first
category is off-axis holography, first suggested by Leith and Upatnieks[1]. An off-
axis hologram leads to spatially separated real and twin images at the reconstruction
step. The real image can then be selected by spatially filtering the reconstructed
planes[2]. The digital off-axis setup however proves less relevant for the study of a
collection of small objects due to sampling constraints[3] and practical considerations
(robustness to vibrations). In-line digital holography is therefore preferred in this
context and the twin-image problem has still to be addressed. By combining different
holograms of the same objects recorded at distances with λ/4 shifts, the phase
shifting technique[4] performs phase reconstruction and therefore achieves twin-image
suppression. Optical scanning holography[5] also reconstructs the phase thanks to an
additional modulation/demodulation step. These two techniques are however limited
to non-moving objects since the shifts or scans must be performed while the objects
remain fixed. They are therefore irrelevant for the study of objects in motion. In this
paper, we address the techniques adapted to the study of micro-objects in a volume.
We consider only the in-line Gabor setup in the following analysis.
Numerical techniques (i.e. methods of the second aforementioned category) attempt
to improve the system response Rof the holographic imaging technique (hologram
recording + reconstruction) by changing the reconstruction step. Many techniques have
been described in the literature as following a different approach. We provide in this
paper, a unifying description of some of them in the light of deconvolution techniques in
section 2. We show that the case of objects spread in a volume requires the development
of a specific algorithm. Section 3 describes the suggested algorithm as an extension of
existing iterative techniques. We show results on simulated and real data and discuss
the applications and limits of the algorithm in section 4. Before describing twin-image
suppression methods, we recall the origin of the twin-image phenomenon and give a
quantitative description of twin-images.
Twin-image origin and characteristics: Hologram formation corresponds to the
recording of the intensity pattern of the free propagated waves diffracted by the objects.

Numerical suppression of the twin-image 3
Hologram reconstruction however does not exactly solves the inverse propagation
problem but rather propagates further the intensity pattern. The complete system
response Rof a point object in the focus plane therefore is not punctual but is the sum
of a (real) punctual response and a (twin) defocused pattern[6]∗:
R(x, y) = δ(x, y)+2<h2z(x, y),(1)
with δ(x, y) denoting Dirac’s distribution, <the real part, and zthe distance separating
the point object and the hologram. The symbol hzis Fresnel function of parameter z
and is the convolution kernel that models, under Fresnel’s approximation, the diffraction
at distance z. It is defined by:
hz(x, y) = 1
jλz exp j π x2+y2
λz .(2)
The symbols xand yare the spatial coordinates in transverse planes (i.e. orthogonal to
the optical axis) and zis the depth coordinate (along the optical axis). The symbol λ
is the radiation wavelength and j=√−1. By convention, we underline complex-valued
variables in this paper.
It is worth noting that a point object is recorded, in the hologram plane, as an
intensity pattern which is classically approximated by the difference: 1 −2<(hz) [6].
This difference can be expanded into three terms: 1 −2<(hz) = 1 −hz−h−zdue to
the conjugation property conj(hz) = h−z. The hologram intensity can then be seen as
the diffraction amplitude created by two symmetrical objects (resp. real and virtual)
located on both sides of the hologram, hence the twin-image can be interpreted as the
diffraction pattern of a virtual object.
In a focus plane, the amplitude of the twin-image depends on that of the real image,
on the size of the holographed object and on the recording distance. Let ϑbe the
object’s aperture. In its focus plane, the reconstructed amplitude is given under a linear
system approximation by the convolution with the system response:
[ϑ∗R] (x, y) = ϑ(x, y)+2ϑ∗ <(h2z)(x, y).(3)
We derive in Appendix A an approximation of the twin-image for small and symmetrical
objects (i.e. whose largest dimension dverifies the condition d√2λz):
[ϑ∗R] (x, y)≈ϑ(x, y)+2F[ϑi]( x
2λz ,y
2λz )<h2z(x, y).(4)
This approximation is useful to quantify the amplitude of the twin-image with respect
to the real image. The twin-image approximate expression given in equation 4 involves
the real part of Fresnel’s function with parameter 2zand the Fourier transform of the
object’s aperture scaled by a factor 1/(2λz). The recording distance plays the role of a
scaling factor: the twin-image is dilated when the recording distance increases. Then,
∗Note that we do not consider here the effects due to sampling (pixel integration) and finite sensor
size. In practice these affect the system response (i.e. perform a low-pass filtering).

Numerical suppression of the twin-image 4
the larger the recording distance, the larger the spatial separation between the real
image of the object and its superimposed twin-image.
The signal to noise ratio between the real image of an object and its corresponding
twin image depends both on the recording distance and the object’s size. For a
given recording distance z, a dilation of the object by factor aleads — according to
the 2D Fourier transform scaling property — to a multiplication of the twin-image’s
amplitude by a2. The wider the object, the higher the corresponding twin-image
amplitude. Conversely, for a given object, the twin-image amplitude is proportional
to the inverse of the recording distance zthrough Fresnel’s function. Hence, the larger
the recording distance, the smaller the amplitude of the twin-image.
2. Numerical suppression of the twin-image in the light of deconvolution
techniques
The many numerical techniques introduced in literature for twin-image suppression
can be classified into distinct families of approaches. We will first briefly sum up
these approaches and then show that there are strong connections between physically-
based, filtering and iterative approaches. Image deconvolution provides an interesting
framework to give a unified description of these methods. On the basis of this
description, we derive in section 3 an algorithm adapted to holograms of micro-objects
spread in a volume.
2.1. Classes of existing approaches
Physically-based techniques: These techniques date back to the origins of holography.
Bragg and Rogers[7] suggested the use of a second (auxiliary) hologram recorded at
twice the distance of the main hologram to subtract the twin image. A similar approach
can be conducted by subtracting from the reconstructed plane an estimate of the twin-
image obtained by reconstructing an out-of-focus plane located at three times the in-
focus distance. The twin-image is then pushed back at twice its original distance and
its amplitude reduced accordingly. This approach has been followed by authors using
one[6, 8] or two[9, 10, 11] holograms.
Inverse filtering approaches: Some authors[12, 13] try to invert the system response R
by designing an inverse filter in the Fourier domain. Different strategies are proposed
to circumvent the filter singularities problem.
Iterative approaches: Iterative techniques have been designed[14, 15, 16] to improve
the reconstructed amplitude, using a finite support constrain.
Phase retrieval techniques: With two or more holograms, more general phase
reconstruction algorithms may also be used. Gerchberg-Saxton algorithm[17, 18] has

Numerical suppression of the twin-image 5
been applied to digital holography in the work of Liu and Scott[19], and more recently
in the work of Zhang et al. [20]. It iteratively reconstructs the phase in each hologram
plane by performing back and forth propagation and enforcing at each step a constraint
based on the measured hologram intensities. An improved version[21] of Gerchberg-
Saxton algorithm has been suggested to account for the numerical errors (hz∗h−z6=δ
due to windowing effects).
Maximum a posteriori approach: Sotthivirat and Fessler suggested[22] a reconstruction
of the image of a plane object using a maximum a posteriori approach (i.e. regularized
image restoration). In the case of the study of small objects spread in a plane, the
definition of an appropriate prior model is however difficult.
Other approaches: A quite different approach has been followed in [23]: the hologram
is decomposed into blocks and reconstructed as a combination of off-axis sub-holograms.
Tiller et al. suggest using two close holograms to deduce the phase on the basis of the
connection between intensity deviation between two close planes and phase value.
2.2. The deconvolution point of view
Twin-image suppression has already been identified as a deconvolution problem by other
authors (see [6]). We show here that the deconvolution provides a framework that reveals
previously unnoticed connections between various approaches.
Let us consider a collection of Nobjects defined by their apertures (ϑi)i=1..N . In
a plane located at distance zfrom the hologram, the complex amplitude due to the
objects is az=PN
i=1 ϑi∗δxi,yi∗hzi−z, where (xi, yi) is the transversal location of object
iand ziis the distance between object iand the hologram plane. In the case of a single
real-amplitude object, az=ϑas is the case for objects spread in a unique plane located
at the recording distance z. Note that phase objects such as micro-lenses or biological
structures may also be considered, the aperture ϑthen is complex-valued.
The intensity recorded in the hologram plane (after suppression of the continuous
background constant) is given by [6]:
IH=−2<az∗hz=−az∗hz−conj(az)∗h−z.(5)
The numerically reconstructed amplitude in the plane zis obtained[24] by convolving
the hologram with Fresnel’s kernel of parameter −z:
AR=IH∗h−z=−az−conj(az)∗h−2z(6)
The common principle of all methods for numerical suppression of the twin-image
is expressed either: (i) in the hologram plane (Eq. 5), or (ii) in the reconstructed plane
(Eq. 6).

Numerical suppression of the twin-image 6
2.2.1. Objects within a plane In this section, we consider that all objects are located
at the same distance z(the recording distance) from the hologram and that their
transmittance 1 −ϑiis real-valued. The amplitude aztherefore is real and equations 5
and 6 become:
IH=−2az∗ < h−z,and (7)
AR=−az∗δ+h−2z.(8)
The twin-image suppression problem then corresponds to a deconvolution problem in
both cases: (i) recover azfrom IH; and (ii) recover azfrom AR. We describe in the
following the main twin-image suppression techniques and underline the connections
between each of them using the common framework of deconvolution techniques.
Fourier-quotient method: The basic technique to perform deconvolution is to design a
filter in the Fourier domain that inverses the convolution kernel. The transfer function
of the inverse filters corresponding to approaches (i) and (ii) are:
F[F(i)
I](u, v) = 1
cos [λzπ(u2+v2)],and (9)
F[F(ii)
I](u, v) = 1
1 + exp −2jπλz(u2+v2).(10)
These filters have a singularity for each zero of their denominator. Different solutions
have been suggested:
•the transfer functions are set to one whenever a singularity is reached[12],
•a positive constant is added to the denominator[13],
•the filter is expanded into a series and only the first terms of the series are kept[6, 8].
We give here the series expansion used in [8] as it will provide a direct connection with
iterative approaches. It is based on the classical expansion 1/(1 + x) = 1 −x+· ·· +
(−1)nxn:
F[F(ii)
I](u, v) = 1 −exp −2jπλz(u2+v2)+ exp −4jπλz(u2+v2)− · ··
+(−1)nexp −2njπλz(u2+v2),(11)
and can be physically interpreted as pushing back the twin-image at distance (2n+ 2)z:
−AR∗F(ii)
I=−AR∗δ−h−2z+h−4z− · ·· + (−1)nh−2nz 
=az+ (−1)naz∗h−(2n+2)z.
(12)
Iterative approaches: The inverse filter F(ii)
Iobtained by series expansion can be put
into the following recursive form:
(a0=AR
an+1 =AR−an∗h−2z
(13)

Numerical suppression of the twin-image 7
This corresponds exactly to the iterative algorithm of van Cittert[25, 26] applied to
deconvolve equation 8.
In the case of small objects, an a priori spatial constraint can be added if the
support of the objects is known. Let us note Mas the binary function indicating
whether point (x, y) is in an object (M(x, y) = 1) or not (M(x, y) = 0). This spatial
constraint can be enforced at each iteration (i.e. by projection on the sub-space of
solutions that satisfy the constraint):
(a0=M·AR
an+1 =M·hAR−an∗h−2zi(14)
This algorithm corresponds to the one introduced by Lannes[14].
Koren et al. suggested[15, 16] another iterative algorithm also using a spatial
constraint:





a0(twin) = (1 −M)·AR+M·¯
AR,with ¯
ARthe mean value of AR
an+1(twin) = (1 −M)·a0(twin) +M·(an∗h−2z)
an=γ(1 −M)·(an(twin) ∗h2z) + M·(an(twin) ∗h2z),with γ∈[0,1].
(15)
This algorithm goes back and forth between two planes corresponding respectively to
the twin image a(twin) and the real one a. The underlying idea of this algorithm is that
the real and virtual images carry the same information and can be interchanged through
a convolution with h2zand h−2z. Getting a real image without a spurious twin-image is
indeed equivalent to getting a twin-image with no associated real image. The twin-image
is well known outside the support of the real images. Inside this support, however, the
twin and real images are superimposed. The algorithm therefore reconstructs iteratively
the twin-image: at each step, the twin-image is updated inside the support Mand the
real image is reduced outside M. This algorithm requires a good knowledge of the
support M.
2.2.2. Objects in a volume When the studied objects are spread in a volume, the twin-
image suppression problem expressed in equations 5 and 6 is no longer a deconvolution
problem. However, if two holograms of the same objects — recorded at different
distances or with a different wavelength — are available, they can be combined to
obtain the classical deconvolution formulation. Let I1and I2be two holograms of the
same objects respectively recorded at distances zand z+d. Then, according to equation
5, they are related to the complex amplitude of the objects through:
−I1+ (I2∗hd) = az∗hz∗(δ−h2d).(16)
Several authors[12, 9, 10, 11] applied a deconvolution to equation 16 using similar
approaches as those described as the Fourier-quotient methods in the previous
paragraph.
Multi-hologram approaches (either deconvolution-based or performing phase-
retrieval) are, however, limited in practice (except some specific setup like multi-
wavelength holography[27] or semi-transparent holograms in X-ray holography[28]) to

Numerical suppression of the twin-image 8
(a) (b)
Figure 1. Twin-image appearing at the reconstruction step: (a) “twin-image free”
reconstruction using the phase; (b) classical reconstruction from the hologram, showing
the twin-image
the study of still objects as they require a recording of two holograms at different
distances. We therefore suggest a new algorithm in section 3 that extends iterative
approaches with a support constraint to the case of a single hologram of 3D objects.
3. Algorithm for twin-image suppression in the case of 3D objects
The use of a single hologram to reconstruct a twin-image free volume of objects prevents
us formulating the problem directly as a deconvolution problem (see section 2.2.2). An
iterative approach with finite-support constraint is however possible. It can then be
interpreted in the context of wavelet denoising. Classical reconstruction gives images
corrupted by the twin-images. Twin-image noise can be spatially localized by using
a Fresnel transform with opposite parameter. The Fresnel transform is a redundant
scaling transform[29] that separates, at some scales, signal (i.e. real images) from
noise (i.e. twin-images). We can therefore apply the technique of wavelet coefficient
thresholding[30].
In the following paragraphs we the algorithm that we suggest for twin-image
suppression. Section 4 then illustrates some results on both simulated and real
holograms.
3.1. Principle
The aim of the algorithm is to spatially separate the real and virtual images by masking.
Figure 1 illustrates the twin-image phenomenon. Small diffracting objects were used to
simulate the complex amplitude in a recording plane. The intensity of the field has been
stored and two reconstructions have been performed. Image (a) has been computed from
the simulated complex amplitude. It is therefore free of the twin-image phenomenon.
Only two of the three objects are in focus. The third one is slightly out-of-focus. Image
(b) is the result of the simulated hologram reconstruction. The lack of phase information
results in the presence of the twin-image. Low frequency fringes appear in image (b) that
could not be seen in image (a): this is the so-called “twin-image”. In-focus images can
easily be spatially separated from their twin-image. Out-of focus objects, on the other

Numerical suppression of the twin-image 9
hand, cannot be separated. Twin-image suppression, in the case of objects in a volume,
must be performed in 3D. In the following, we consider a set of planes (z1, . . . , zp) that
coarsely sample the volume. The hard-thresholding wavelet coefficient algorithm can
then be written:





aH
0=IH
an(twin)(zi) = (1 −M(zi)) ·anH∗hz
an+1H=1
pPp
i=1 an(twin)(zi)∗h−zi,
(17)
with Mbeing a 3D binary mask that indicates the support of the objects, and with
arg(a) being the phase of complex number a. We describe its construction in the next
paragraph. The algorithm iteratively reconstructs the complex amplitude aHin the
hologram plane IH. The noise is suppressed in each plane a(twin)(zi) of the twin half-
space.
The total redundancy of Fresnel transform is a drawback of the algorithm efficacy. If
we consider the sampling of the volume into pplanes and that the twin-image of a given
object is suppressed only in one of the pplanes by mask M, then the amplitude of this
twin-image is reduced by factor (p−1)/p after summing back all filtered planes. Direct
use of the wavelet denoising procedure, therefore, is not usable. Instead of denoising
the transformed planes, it is more efficient to suppress the phase corresponding to the
twin-object in the hologram plane. This leads to the modified version of the algorithm
illustrated on figure 2, for i= 1 to p:





















a0H=IH



an(0)
H=anH
an(i)
H=an(i−1)
H−hM(zi)·an(i−1)
H∗hzii∗h−zi
an+1H=an(p)
H.
(18)
This time, the complex amplitude in the hologram plane aHis modified ptimes per
iteration. The complex amplitude of the twin objects that are in focus in plane zi:
hM(zi)·an(i−1)
H∗hzii∗h−ziis subtracted from the hologram amplitude aH. The
subtraction takes place before processing the next plane zi+1 which avoids multiple
suppression of a virtual object.
3.2. Iterative cleaning
The mask Mis obtained by rough segmentation of the real-objects half-space. In the
case of amplitude-only objects, thresholding is sufficient to perform an approximate
segmentation. Phase objects on the other hand might be segmented based on a focusing
criteria such as that suggested by Dubois et al. [31] and on the phase in the reconstructed
planes. Note however that we have not checked in this study the efficacy of the method
on holograms in this latter case.

Numerical suppression of the twin-image 10
Figure 2. Twin-image cleaning scheme
The mask can be either computed once, before algorithm 18, or updated during the
algorithm execution. In case of under-segmentation (i.e. non-detection of some objects)
the twin-image suppression will be less efficient since the twin-images corresponding to
the non-detected objects will remain.
To soften the constraints on the quality of the segmentation Mand to reduce the
zsampling steps, we suggest modifying the binary mask obtained by segmentation by:
(i) dilating it with a disk a few pixels wide, (ii) low-pass filtering the result (11 ×11
Gaussian kernel with 2 pixels standard deviation). It is then possible to suppress a twin-
image even if the considered plane is not exactly in-focus. The low-pass filter is aimed at
reducing sharp boundaries that could create ripples after Fresnel transform. The process
of cleaning a twin-object from a slightly out-of-focus plane is illustrated in figure 3(a).
A hologram of a disk-shaped object of 90µm diameter, recorded at 120mm from the
object, has been simulated. The cleaning plane is located at 2mm from the focus-plane
in order to prove that the zsampling may be loose. The virtual object is efficiently
suppressed in few iterations. The evolution of the relative energy of the virtual object
is depicted in figure 3(b). The rapid decrease in the first iterations proves the efficiency
of the cleaning algorithm. We found in practice that the twin-image suppression was
satisfying after less than 10 iterations in the examples shown in section 4.

Numerical suppression of the twin-image 11
(a)
relative energy
1 2 3 4 5
10−3
10−2
10−1
100
iteration step
(b)
Figure 3. Illustration of the iterative suppression of a virtual object: (a) evolution
of the twin-object in its in-focus plane (first row) and in the cleaning plane (second
row) (please note that the contrast of the figure has been increased for visualization
purposes); (b) relative energy of the twin-object at each iteration.
4. Results
We give some illustrations of the algorithm described in section 3 in three different cases:
a simulated hologram of a large depth volume with some small objects; a simulated
hologram of a uniform collection of particles; and an experimental hologram of water
droplets.
Synthetic hologram of a simple three dimensional distribution of small objects The
objects studied here are spread over three parallel planes each located at a 20 mm
distance. The simulated hologram is recorded at 100mm from the closest plane for a
1024 ×1024 sensor with 6.7×6.7µm2pixels (8 bit quantization depth). Figure 4(a)
gives a diagram representing the object. The corresponding computed hologram is

Numerical suppression of the twin-image 12
displayed in figure 4(b). Figures 4(c) to (e) show the three reconstructed planes using the
classical reconstruction algorithm. The reconstructed planes after twin-image cleaning
are given in figures 4(f) to (h). The twin-image reduction is very efficient on this simple
experiment.
Simulated hologram of opaque particles One hundred opaque spheres (diameter 90µm)
were spread in a 6 ×6×6×mm3volume. A hologram was numerically simulated
at a distance z= 120mm for a 1024 ×1024 sensor with 6.7×6.7µm2pixels (8 bit
quantization depth). Figure 5(a) displays the hologram. Classically reconstructed and
twin-image cleaned planes are displayed in figures 5(b) and 5(c) respectively. Five
“cleaning” planes were used in the algorithm (p= 5 in algorithm 18). Figure 5(d)
gives the signal to noise ratio in each 5 plane before (lowest curve) and after (highest
curve) twin-image reduction. The signal to noise ratio have been computed by using
the complex amplitude in the hologram plane as it is available in numerical simulations.
Let us write I(th)
Rzthe theoretical intensity (i.e. the twin-image free computed with the
complex amplitude) and I(comp)
Rzthe intensity obtained with a reconstruction algorithm
(i.e. classical reconstruction, or twin-image suppression using algorithm 18 prior to
reconstruction). The signal to noise ratio shown on figure 5(d) is:
SN R = 10 log10 


1
NX
xX
y

I(th)
Rz(x, y)
I(th)
Rz(x, y)−I(comp)
Rz(x, y)

2


,(19)
with Nthe number of pixels. In this case, the twin-image reduction algorithm has
increased the SNR by 10dB.
Experimental hologram of a water spray A cloud of water droplets was produced by
a water spray. A hologram was recorded on a PCO Sensicam camera (1280 ×1024
6.7×6.7µm2pixels with 12 bit quantization depth) using a YAG pulsed laser (lambda =
532nm, pulse duration= 7ns). The droplets were located within a bounding volume of
60mm depth. Their size is roughly 90µm. Figure 6(a) shows the experimental hologram.
The classical reconstruction is given in figure 6(b). The reconstruction after twin-image
reduction is displayed in figure 6(c). The twin-image has been strongly attenuated. The
algorithm seems robust against noise due to hologram acquisition (background noise
due to spurious interference patterns).

Numerical suppression of the twin-image 13
z
7mm
7mm
20mm
(a) (b)
(c) (d) (e)
(f) (g) (h)
Figure 4. Twin-image cleaning applied on a simple synthetic hologram

Numerical suppression of the twin-image 14
Figure 5. Twin-image cleaning applied on a synthetic hologram of 100 particles: (a)
hologram; (b) a reconstructed plane; (c) the same plane after twin-image suppression;
(d) comparison of the signal to noise ratio (SNR) for planes that sample the volume,
the lower curve corresponds to classical reconstruction, the upper curve to twin-image
suppression.
5. Conclusion
The twin image occurs because the traditionally used reconstruction does not invert
the recording step. Numerical techniques for twin-image suppression try to improve the
system response. We have shown that identifying the twin-image suppression problem
as a deconvolution problem provides a unifying framework. We have set bridges between
different methods previously described in the literature as being distinct approaches.
Existing algorithms cannot be applied to the case of small objects spread in a
volume. We therefore derived an iterative algorithm that is both an evolution of finite-

Numerical suppression of the twin-image 15
Figure 6. Twin-image cleaning applied on an experimental hologram of a water
spray: (a) experimental hologram; (b) a reconstructed plane; (c) the same plane after
twin-image suppression

Numerical suppression of the twin-image 16
support techniques for plane objects and is inspired from wavelet denoising. The first
results on simulated and real holograms are encouraging as the twin-images are efficiently
cleaned. More generally, further experimental evaluation in various conditions should
be completed to clarify the potential of the method.
It seems that the method could also be applied to holograms of phase-objects. An
experimental validation on such holograms should however be considered.
The twin-image suppression technique we described might be usefull in particle
holography for fluid dynamics. Its application for correlation-based measurements of
particle holograms would suppress the indetermination due to the twin-image[32]. The
algorithm could also be extended to twin and real objects suppression and possibly
provide an alternative to Monnom et al. technique for out-of-focus object influence
reduction[33]. It could then be used in applications such as interrogation cell correlation
for velocity measurements[34] or multi-plane velocimetry[35]. Note that the described
twin-image cleaning algorithm is limited to low or medium concentrated holograms as
it relies on the spatial separation between real images and the surrounding twin-images.
Furthermore, objects close to the hologram boundaries get blurred by the incorrect
twin-image cleaning due to border effects. This problem requires further work and can
be reduced by windowing[36].
Denoising and segmentation appear as tightly linked problems. For efficient twin-
image suppression, denoising must be performed jointly with volume segmentation. The
study of small objects in a volume (i.e. finite support objects surrounded by empty
regions) suggests the use of adapted image processing techniques. Enforcing a finite
support constraint as we do in this paper is a possible approach. In the case of well
defined objects, such as spherical particles, a parametric estimation approach in an
inverse problems framework proves to result in superior performance[37, 38].
Appendix A. Approximate expression of the twin-image of a small object
We derive here an approximate expression of the twin-image in the focus plane of the
real image. We consider small objects, a situation common to particle holography and
three dimensional imaging of small elements such as micro-organisms in water[39]. The
twin-image amplitude is given (eq. 3) by:
ϑ∗ <(h2z)(x, y) = <ϑ∗(h2z)(x, y)
=<h1
2jλz RRϑ(ξ, η) expjπ
2λz [(x−ξ)2+ (y−η)2]dξdηi,(A.1)
as we consider objects described by a real-valued aperture ϑ.
This expression can be approximated by:
ϑ∗ <(h2z)(x, y)≈ <1
2jλz expjπ
2λz [x2+y2]·
ZZ ϑi(ξ, η) exp−jπ
λz [xξ +yη]dξdη(A.2)

Numerical suppression of the twin-image 17
for small objects such that 1
2λz [ξ2+η2]max 1, i.e. whose largest dimension dverifies
the condition d√2λz.
The integration corresponds to the Fourier transform of ϑup to a spatial scaling
transform with a scale factor of 1/(2λz):
ϑ∗ <(h2z)(x, y)≈ <h2z(x, y)F[ϑi]( x
2λz ,y
2λz ).(A.3)
For symmetrical objects, the aperture ϑis even and the Fourier transform is real. The
twin-image is then given by:
ϑ∗ <(h2z)(x, y)≈ <h2z(x, y)F[ϑi]( x
2λz ,y
2λz ),(A.4)
and appears clearly as a modulated signal in two ways:
•the frequency is linearly modulated with rate r/(2λz), with rthe distance to the
object center,
•the amplitude is modulated according to the Fourier transform of the object
aperture.
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