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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)+2ϑ∗ <(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∗hzii∗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∗hzii∗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) expjπ

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 expjπ

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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