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Off-axis compressed holographic microscopy in low light conditions Marcio Marim1,2, Elsa Angelini2, Jean-Christophe Olivo-Marin1 and Michael Atlan3 1Institut Pasteur, Unité d’Analyse d’Images Quantitative, CNRS URA 2582 25-28 rue du Docteur Roux. 75015 Paris. France 2Institut Télécom, Télécom 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 1 This article reports a demonstration of off-axis compressed holography in low light level imag- 1 ing conditions. An acquisition protocol relying on a single exposure of a randomly undersampled 0 diffractionmapoftheopticalfield,recordedinhighheterodynegainregime,isproposed. Theimage 2 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 1 and phase. In biological imaging, images are typically v compressible or sparse in some domain due to the ho- 5 mogeneity, compactness and regularity of the structures 3 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 2 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 3 perform random measurements of an optical field in a diffraction plane, and an iterative image reconstruction 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 minimization overN pixels leads to the hologrammagnitude mapgˆreportedinfig.2(a),while the same regularization constraint applied on R leads to the magnitude hologram 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. 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