Off-axis compressed holographic microscopy in low-light conditions

0.31 MB·English

by  Marcio M. Marim

Off-axis compressed holographic microscopy in low light conditions
Marcio Marim1,2, Elsa Angelini2, Jean-Christophe Olivo-Marin1 and Michael Atlan3
1Institut Pasteur, Unit´e d’Analyse d’Images Quantitative, CNRS URA 2582
25-28 rue du Docteur Roux. 75015 Paris. France
2Institut T´el´ecom, T´el´ecom ParisTech,
CNRS LTCI. 46 rue Barrault. 75013 Paris. France
3 CNRS UMR 7587, INSERM U 979, UPMC,
UP7, Fondation Pierre-Gilles de Gennes.
Institut Langevin. ESPCI ParisTech - 10 rue Vauquelin. 75005 Paris. France
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This article reports a demonstration of off-axis compressed holography in low light level imag-
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ing conditions. An acquisition protocol relying on a single exposure of a randomly undersampled
0
diffractionmapoftheopticalfield,recordedinhighheterodynegainregime,isproposed. Theimage
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acquisition scheme is based on compressed sensing, a theory establishing that near-exact recovery
n of an unknown sparse signal is possible from a small numberof non-structured measurements. Im-
a
age reconstruction is further enhanced by introducing an off-axis spatial support constraint to the
J
imageestimationalgorithm. Wereportaccurateexperimentalrecoveringofholographicimagesofa

0 resolution target in low light conditionswith a frame exposureof5 µs, scaling down measurements
1 to9% of random pixels within thearray detector. OCIS : 070.0070, 180.3170

]
s
c Off-axis holography is well-suited to dim light imag-
i ing. Shot-noise sensitivity in high optical gain regime
t
p can be achieved with few simple setup conditions [1].
o
Holographic measurements are made in dual domains,
.
s where each pixel exhibits spatially dispersed (i.e. multi-
c
plexed) information from the object. The measurement
i
s domain and the image domain are “incoherent”, which
y
is a requirement for using compressed sensing (CS)
h
sampling protocols [2]. In particular, CS approaches
p
using frequency-based measurements can be applied to
[
holography sampling the diffraction field in amplitude

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and phase. In biological imaging, images are typically
v
compressible or sparse in some domain due to the ho-
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mogeneity, compactness and regularity of the structures
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7 of interest. Such property can be easily formulated as FIG. 1: Experimental image acquisition setup.
1 mathematical constraints on specific image features. CS
. can be viewed as a data acquisition theory for sampling
1
0 and reconstructing signals with very few measurements acquisition setup consists of a frame exposed with the
1 [2–4]. Instead of sampling the entire data domain and reference beam alone and subtracted to a frame exposed
1 then compress it to take advantage of redundancies, with light in the object channel, beating against the
v: CS enables compressed data acquisition from randomly reference, to yield the holographic signal. This setup
i distributed measurements. Image reconstruction relies prevents any object motion artifact that would poten-
X on an optimization scheme enforcing some specific tially occur with phase-shifting methods [9]. The CS
r sparsity constraints on the image. CS was used recently imagereconstructionalgorithmreliesonatotalvariation
a
to improve image reconstruction in holography by minimization constraint restricted to the actual support
increasing the number of voxels one can infer from of the output image, to enhance image quality.
a single hologram and canceling artifacts [5–7]. CS
was also used for image retrieval from undersampled
Weconsidertheholographicdetectionofanobjectfield
measurements in millimeter-wave holography [8] and
E of small amplitude with a reference (or local oscilla-
off-axis frequency-shifting holography [9].
tor)fieldE ofmuchlargeramplitude,toseeklow-light
LO
detection conditions, using the Mach-Zehnder interfer-
In this work, we describe an original acquisition ometer sketched in fig. 1. The main optical radiation
protocol to achieve off-axis compressed holography in comesfromasinglemodecontinuouslaseratwavelength
low-light conditions, from undersampled measurements. λ = 532 nm. Lenses with short focal lengths are used
The main result presented in this article is an experi- in both channels to create point sources. In the object
mental demonstration of accurate image reconstruction channel,anegativeU.S.AirForce(USAF)resolutiontar-
from very few low-light holographic measurements. The getisilluminatedintransmission. Theamountofoptical
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powerintheobjectchannelistunedwithasetofneutral Fresnel transform [10]
densities. The interference pattern of E beating against
wEiLtOhincotrhreescpeonntrdasltroegiIon=of|Ea S+onEyLOIC|2X. 2I8t5AisLmCeCasDuraerd- fk,l = λ∆izeiπλ∆z(Nkx22d2x+Nyl2d2y)
rcaoyundte,teNcto=rN(gxai×nGNCyCeDle=me3n.8tsp,hwohteor-eeleNcxtro=nsNpyer=d1ig0i2ta4l, ×XNx XNy Fp,qeiλ∆πz(p2d2x+q2d2y)e−2iπ(Nkpx+Nlqy) (2)
pixel size d = d = 6.7µm, quantum efficiency ∼ 0.6).
x y p=1q=1
The frame rate is set to 12 Hz and the exposure time to
5 µs. A mechanical chopper is used to switch the ob- where i2 = −1, and (k, l), (p, q) denote pixel indexes.
ject illumination on-and-off from frame to frame. The The quadratic phase factor depends on a distance
recorded interference pattern takes the form parameter ∆z. Standard holographic reconstruction, as
reported in figs. 3(d-f), consists in forming the intensity
image of the object g = |f|2 from the measurements of
F over the whole detection array, with eq. 2.
I =|E |2+|E|2+EE∗ +E∗E (1)
LO LO LO
Wewanttorecoverg fromasmallnumberofmeasure-
ments F| = Φf in the detector plane, where F| ⊂ F.
Γ Γ
Φisa(M×N)sensingmatrixencodingtheFresneltrans-
form(eq. 2) andthe sampling of a subset Γ of M pixels,
where ∗ denotes the complex conjugate. In our setup
randomly distributed among the N pixels of the detec-
we have |E|2 ≪ |E |2. Let’s define n and n , the
LO LO tion array. We want M to be as small as possible, to
number of photo-electrons released at each pixel, from
benefit from the best compression ratio M/N with re-
light in the object and LO channel respectively, imping-
specttonon-CSholography,andenhancethethroughput
ing on the detector. The reference beam intensity is ad-
savingsparameter1−M/N. Forasuccessfulreconstruc-
justed so that the LO shines the detector to half satu-
tion, the sensing matrix Φ must be incoherent with the
ration of the pixels’ dynamic range, on average. This
sparsifying basis Ψ enforced on the reconstructed image
amounts to hn i/G ∼ 2000 digital counts. The
LO CCD [3]. This is the case for measurements from the Fresnel
brackets h·i denote the average over N pixels. Hence
transform and a sparsity constraint minimizing the to-
hn i = 7.6×104 e (photo-electrons) per pixel. In the
LO talvariation(TV)measureofthereconstructedimageg.
object channel, three optical densities D = 0, D = 0.5,
The TV is measured on the gradient map of the image
andD =1,aresetsequentiallytoreachverylowhnival-
as: k∇g k = |∇g |. Since the target is piecewise
ues. The average number of digital counts in 50 consec- ℓ1 Pk,l k,l
constant with sharp edges (such as most microscopy im-
utive frames recorded in these conditions, while the LO
ages), its spatial gradient is sparse k∇g k < M ≪ N.
beam is blocked, are reported in figs. 3(a)-(c). The de- ℓ0
The existence of a sparse representation means that g
tection benefits from a holographic (or heterodyne) gain
GH = h|EEL∗O|i/h|E|2i = (hnLOi/hni)1/2, which ranges rheacsoantstmruocsttiokn∇wgekℓm0udsetgrpeeersfoorfmfreaetdloemas.tFMora>sukc∇cegsskful
from GH =177 (D =0) to GH =563 (D =1). The spa- ℓ0
tialsupportofthesignaltermEE∗ isacompactregion measurements,butmuchlessthanN. Givenpartialmea-
LO surements F| , we seek an estimate gˆ with maximum
R of P = 400×400 pixels. In such high gain regimes, Γ
the object field self-beating contribution |E|2, spreading sparsity (i.e. with minimal norm k∇g kℓ1) whose Fres-
overaregiontwiceaslargeasRalongeachspatialdimen- nel coefficients Fˆ|Γ match the observations F|Γ within
sion, can be neglected in comparison to the magnitude some error δ. For numerical reasons, the norm kk was
ℓ0
of EE∗ and E∗E in eq. 1. In off-axis configuration, approximated with the norm kk in the formulation of
LO LO ℓ1
the term of interest EE∗ is also shifted awayfrom |E|2 the reconstruction problem
LO
and E∗E , which improves the detection sensitivity at
the expenLsOe ofspatialbandwidth. Forthe currentsetup, gˆ=argmRink∇g kℓ1 s.t. kFˆ|Γ−F|Γ kℓ2 ≤δ (3)
theratioofavailablebandwidthbetweenoff-axisandon-
axis holography is equal to P/N ∼ 15%. To cancel- δ is a constraintrelaxationparameterintroduced to bet-
out the LO flat-field fluctuations within the exposure ter fit noisy measurements. Contrary to our initial im-
time, a frame acquired without the object I = |E |2 plementation [9], in which the TV measure k∇gˆk was
0 LO ℓ1
is recorded. The difference of two consecutive frames minimized overN reconstructedpixels in the spatialdo-
I −I ≃ EE∗ +E∗E yields a measure of the holo- main,itisnowonlyminimizedwithintheoff-axisspatial
0 LO LO
graphicsignalF =EE∗ . F whichisproportionaltothe support R, which bounds are illustrated in Fig. 2d by
LO
diffractedcomplex fieldE, willnow be referredto as the the white dashed square. This restriction on the spa-
opticalfield itself. Eachmeasurementpointonthe array tial support being constrained leads to a more accurate
detectorF ,wherep=1,...,N andq =1,...,N ,corre- estimate of gˆ, actually reducing the number of relevant
p,q x y
sponds toa pointinthe Fresnelplane ofthe object. The degreesoffreedomto estimate,andhence the number of
optical field F, measured in the detection plane yields samples M required. For comparison purposes, CS re-
the field distribution in the object plane f via a discrete constructions without and with support constraint from
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perform random measurements of an optical field in a
diffraction plane, and an iterative image reconstruc-
tion enforcing sparsity on a bounded image support.
Compressed off-axis holography is a powerful method
to retrieve information from degraded measurements
at high noise levels. We demonstrated single-shot
imaging in high heterodyne gain regime at 5 µs frame
exposure around one photo-electron per pixel in the
FIG. 2: Compressed holographic reconstruction of g without
supportconstraint(a). ReconstructionwithTVminimization
overtheregion R(b). Inbothcases,hni=2.4e(D=0)and
M/N =9%. Magnified views over 330×330 pixels (c,d).
the same original frame are reported in fig.2. TV mini-
mization overN pixels leads to the hologrammagnitude
mapgˆreportedinfig.2(a),while the same regularization
constraint applied on R leads to the magnitude holo-
gram reported in fig.2(b). Magnified views in figs. 2(c) FIG. 3: Amount of digital countsin theobject channel aver-
and 2(d) show a clear increase in image sharpness with aged over N pixels, for three different attenuations : D = 0
boundedspatialregularization. Noiserobustnessofcom- (a),D=0.5(b),D=1(c). TheLObeamisturnedoff. The
optical field E impinges onto the detector (i) and is blocked
pressed holography versus standard holography is illus-
(ii) sequentially by the optical chopper, from one frame to
trated in fig. 3, in conditions of low-lightillumination of
the next. The horizontal axis is the frame number, the ver-
the target. Standard Fresnel reconstructionfrom N pix-
tical axis is the average number of counts per pixels. Stan-
elsleadstotheimagesreportedinfigs. 3(d)-(f),recorded
dard holographic reconstructions at D = 0 (d), D = 0.5 (e),
at hni = (2.4,0.75,0.24) for figs.3 (d,e,f). CS image re-
D = 1 (f). CS reconstructions at D = 0 with M/N = 9%
constructions of gˆ with bounded TV regularizationfrom (g), at D = 0.5 with M/N = 13% (h), and at D = 1 with
the same data are reported in figs. 3 (g)-(i). Highly ac- M/N =19% (i).
curate image reconstruction is achieved, at compression
rates of 9% in fig. 3(g), 13% in fig. 3(h), and 19% in fig.
3(i), i.e. from much less measurements than needed for
object channel. In these conditions, throughput savings
Fresnel reconstruction.
from 81% to 91% can be reached.
In conclusion, we have presented a detection scheme
for coherent light imaging in low-light conditions suc- This work was funded by Institut Pasteur, DGA,
cessfully employing compressed sensing principles. It ANR, CNRS, and Fondation Pierre-Gilles de Gennes.
combines a single-shot off-axis holographic scheme, to
[1] M.GrossandM.Atlan.Digitalholographywithultimate complete frequency information. IEEE Trans. Inform.
sensitivity. OpticsLetters,32:909–911, 2007. Theory, 2004.
[2] E.Candes,J.Romberg,andT.Tao. Robustuncertainty [3] E. Candes and J. Romberg. Sparsity and incoherence in
principles: Exact signal reconstruction from highly in- CompressiveSampling.InverseProblems,23(3):969–985,
4
2006. [8] Christy Fernandez Cull, David A. Wikner, Joseph N.
[4] D. L. Donoho. Compressed Sensing. IEEE Trans. on Mait, Michael Mattheiss, and David J. Brady.
Information Theory,52(4):1289–1306, April 2006. Millimeter-wave compressive holography. Appl. Opt.,
[5] David Brady, Kerkil Choi, Daniel Marks, Ryoichi Ho- 49(19):E67–E82, 2010.
risaki,andSehoonLim. Compressiveholography. Optics [9] Marcio Marim, Michael Atlan, Elsa Angelini, and Jean-
Express,17:13040–13049, 2009. Christophe Olivo-Marin. Compressed sensing with off-
[6] Lo¨ıcDenis,DirkLorenz,EricThibaut,CorinneFournier, axis frequency-shifting holography. Optics Letters,
and Dennis Trede. Inline hologram reconstruction with 35(6):871–873, March 2010.
sparsityconstraints. Opt.Lett.,34(22):3475–3477, 2009. [10] U. Schnars and W. Juptner. Direct recording of holo-
[7] Kerkil Choi, Ryoichi Horisaki, Joonku Hahn, Sehoon grams by a ccd target and numerical reconstruction.
Lim, Daniel L. Marks, Timothy J. Schulz, and David J. Appl. Opt.,33:179–181, 1994.
Brady. Compressiveholographyofdiffuseobjects. Appl.
Opt.,49(34):H1–H10, 2010.