Demonstration of single-shot digital holography using a Bayesian framework

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1 Reseach Aticle Vol. 35, No. / Januay 208 / Jounal of the Optical Society of Ameica A 03 Demonstation of single-shot digital hologaphy using a Bayesian fameok CASEY J. PELLIZZARI,, *MATTHIAS T. BANET, 2 MARK F. SPENCER, 3 AND CHARLES A. BOUMAN School of Electical and Compute Engineeing, Pudue Univesity, West Lafayette, Indiana 47907, USA 2 MZA Associates Copoation, Albuqueque, Ne Mexico 8706, USA 3 Ai Foce Reseach Laboatoy, Diected Enegy Diectoate, Kitland AFB, Ne Mexico 877, USA *Coesponding autho: casey.pellizzai@gmail.com Received Septembe 207; evised 6 Novembe 207; accepted 7 Novembe 207; posted 22 Novembe 207 (Doc. ID ); published 4 Decembe 207 In this pape, e pesent expeimental esults fo image econstuction, ith isoplanatic phase-eo coection, fom single-shot digital hologaphy data. We demonstate the utility of using a model-based iteative econstuction (MBIR) algoithm to jointly compute the maximum a posteioi estimates of the phase eos and the eal-valued object eflectance function. Specifically, e sho that the MBIR algoithm is obust to noise and phase eos ove a ange of conditions. 207 Optical Society of Ameica OCIS codes: (00.390) Invese poblems; ( ) Image econstuction-estoation; ( ) Wave-font sensing. INTRODUCTION Digital hologaphy (DH) can be used to sense both the amplitude and phase infomation etuning fom an actively illuminated object []. In pactice, DH systems ae sensitive to phase eos caused by index-of-efaction petubations in the atmosphee o optical systems. These eos can often be estimated diectly fom the DH data. Fo avefont sensing applications, the estimate of the phase eos is the desied senso output [2]. Altenatively, fo imaging applications, the phase eos must be estimated and coected to fom focused images. Conventional techniques used to estimate phase eos fom DH data involve maximizing an image shapness metic [3 5]. These image-shapening (IS) techniques econstuct the complex-valued eflection coefficient, g, given by the complexvalued atio of the eflected field to the incident field. Fo sufaces that ae ough elative to the illumination avelength, this leads to images ith high spatial-fequency vaiations knon as speckle. IS algoithms ae sensitive to speckle vaiations and equie incoheent aveaging of multiple data ealizations to estimate the phase eos ith lo eo [3]. Recently, e developed a model-based iteative econstuction (MBIR) algoithm fo jointly computing the maximum aposteioi (MAP) estimates of the phase eos, ϕ, and the eal-valued eflectance,, fom single-shot DH data [6]. The eflectance is given by the eal-valued atio of the eflected poe to the incident poe. Futhemoe, the eflectance can be expessed as E jgj 2, hee E indicates the expected value. The eflectance, in geneal, is smoothe and has highe spatial coelation as compaed to g. We ae accustomed to seeing in conventional images, and it is of geate inteest fo many imaging applications. Additionally, by econstucting, e can leveage its highe spatial coelation to bette constain the estimation pocess and poduce moe accuate estimates of the phase eos ith less data and less signal. In [6], the MBIR algoithm as shon to be obust to high noise and stong phase eos hen tested on synthetic data. In this pape, e analyze the pefomance of the MBIR algoithm using nonsynthetic, expeimental data. Fist, e descibe the laboatoy setup used to geneate DH data ove a ange of signal-to-noise atios (SNRs) and atmospheic tubulence stengths. We then compae econstuctions fom the MBIR algoithm to those fom an IS algoithm. 2. EXPERIMENTAL SETUP Figues and 2 descibe ou expeimental setup. Fo ou maste oscillato (MO) lase, e used an Oxxius lase souce ith 300 mw of continuous-ave poe and appoximately 50 m of coheence length at 532 nm. We used a half-ave plate and polaized beam-splitting cube to ceate to optical legs. In the fist optical leg, e coupled the beam fom the MO lase into a single-mode, polaization-maintaining (PM) fibe though a vaiable neutal-density (ND) filte and a half-ave plate. We then used a collimating lens and 20 optical beam expande to flood-illuminate the object. In the second optical leg, e coupled the beam fom the MO lase into a sepaate single-mode, PM fibe though a half-ave plate to ceate an off-axis local oscillato (LO). This off-axis LO ceated a quasiunifom efeence beam ith the appopiate tilt fo digital hologaphic detection in the off-axis image plane ecoding /8/ Jounal 208 Optical Society of Ameica

2 04 Vol. 35, No. / Januay 208 / Jounal of the Optical Society of Ameica A Reseach Aticle Fig.. Simplified diagam of ou expeimental setup. The output fom a 532 nm lase as split into to paths. One path as attenuated by a vaiable ND filte and sent though a beam expande to illuminate the object. The othe path as intefeed ith the scatteed signal using an off-axis IPRG. A phase sceen as placed in font of the imaging lens to simulate isoplanatic atmospheic tubulence. Fig. 3. Example of eak-tubulence (top o) and stongtubulence (bottom o) data. The left column shos the a digital hologams, h. The second column shos the spectum magnitude of h along ith the egions of inteest. The geen dashed line shos the subspace used fo the complex pupil image, y, having a magnitude shon in the thid column. The hite solid line shos the bounday of the binay apetue tansmission function, a, used fo pocessing, and the ed dashed line shos a egion of the data, y n, hich contains pimaily measuement noise used fo computing the SNR. Note that the cente 00 pixels 00 pixels of the spectum have been masked fo plotting puposes only. Fig. 2. Images of the tansmitte and eceive optics (left) and the object (ight) used in ou expeiment. The ed-dashed box on the object shos the appoximate field of vie (FOV) fo the imaging system. geomety (IPRG) [2]. Fo detection, e used a single lens to image the scatteed light fom the object onto the focal-plane aay (FPA) of a Guppy PRO f-25 FieWie camea. We saved the ecoded digital hologams in a , 6-bit TIFF file. We positioned an Ai Foce esolution chat, backed by cadstock, 9 ft aay fom the imaging lens and ensued that it as unifomly illuminated. In addition, e positioned the beam expande to minimize specula eflections fom the object. Afte setting up the imaging system ith an image distance of 8 00, e simulated both eak and stong isoplanatic tubulence by placing diffeent phase sceens diectly in font of the imaging lens. A clea piece of plastic as used to simulate the eak tubulence, and a Lexitek phase sceen as used fo the stong tubulence. We ecoded digital hologams at fou diffeent signal levels, hich e vaied ith the adjustable ND filte. The digital hologam, h, must be demodulated and filteed to obtain the data, y, used fo image econstuction. Figue 3 shos the steps used to obtain y fo both the eak-tubulence (top o) and stong-tubulence (bottom o) cases. In looking at the spectum of the stong-tubulence data, e see an iegulaity toads the top of the pupil-plane image, hich may have been caused by a eflection fom the Lexitek phase sceen. To avoid this iegulaity, e used a smalle subset of the pupil-plane image samples, as indicated by the geen dashed squae. The esulting, smalle pupil-plane image data, y, as pixels compaed to pixels fo the eak-tubulence data. The solid hite lines in Fig. 3 indicate the bounday of the binay apetue tansmission function, a, used fo image pocessing (i.e., a inside the hite line and zeo elsehee). We used a cicula apetue fo the eak-tubulence data that matched the pupil function of ou imaging system. Fo the stong-tubulence data, e used a squae function that filled the entie data indo. The ed dashed line in Fig. 3 shos a egion of the data, y n, hich contains pimaily measuement noise. To quantify the SNR, e define the samples in the solid hite egion, hich contains the signal, as y s and compute SNR accoding to SNR s2 y s s 2 y n s 2 ; () y n hee s 2 computes the sample vaiance of the agument. 3. ESTIMATION FRAMEWORK The output data, indicated by a geen dashed line in Fig. 3,isa complex-valued pupil-plane image and can be epesented by a vecto, y C N.In[6] e shoed that y can be fomulated using an additive noise model given by y Af : (2) Hee, f C N is the field in the object plane given by f Γg, hee g C N is the object s complex-valued eflection coefficient and Γ C N N is a diagonal matix that applies the object-plane, quadatic-phase facto fom the Fesnel popagation integal [7]. Given a eflectance function,, the eflection coefficient, g, can be modeled using a complex nomal distibution given by p gj CN 0; D, hee D denotes an opeato that poduces a diagonal matix fom its vecto agument. The vecto C N in Eq. (2) is the measuement noise that has a complex nomal distibution, p CN 0; σ 2 I,

3 Reseach Aticle Vol. 35, No. / Januay 208 / Jounal of the Optical Society of Ameica A 05 hee σ 2 is the noise vaiance, and I is the identity matix. Finally, the matix A accounts fo the popagation and measuement geomety and can be decomposed as A D a D expfjϕg F: (3) In Eq. (3), a R N is the entance-pupil tansmission function and ϕ is the phase-eo function. Finally, e choose the econstuction paametes such that F C N N is a todimensional discete Fouie tansfom (DFT) matix scaled so that F H F I, hee the supescipt, H, indicates the Hemitian tanspose. Ou goal is to jointly compute the MAP estimates of and ϕ fom y, hich ae given by ˆ; ˆϕ agminf log p yj;ϕ p log p ϕ g; (4) ;ϕ Ω Fig. 4. EM algoithm fo joint MAP estimation of and ϕ. Hee, j j 2 indicates the element-ise magnitude squae of a vecto. hee Ω epesents the jointly feasible set. Diect optimization of Eq. (4) is not pactical, because it equies computing the deteminate and invese of a dense matix. Instead, e use the expectation maximization (EM) algoithm to eplace the cost function ith a suogate function given by Q ;ϕ; 0 ;ϕ 0 E log p yjf ;ϕ p f j p log p ϕ jy; 0 ;ϕ 0 ; (5) hee 0 and ϕ 0 ae the cuent values of and ϕ, espectively, and the expectation is taken ove the andom vecto, f, to fom the Q function [6]. Evaluation of the expectation in Eq. (5) constitutes the E-step of the EM algoithm. Figue 4 shos the altenating minimization appoach used fo implementing the M-step. We used iteative coodinate descent (ICD) to minimize Q ith espect to and ϕ [8]. Appendix A povides details on the exact fom of Q as ell as the pio models and paametes used fo this expeiment. We use the iteative initialization pocess fo ϕ descibed in [6]. The EM algoithm is un fo N K iteations. We then estat the pocess by using ou last estimate of ϕ as the ne initial estimate, but einitializing as 0 ja H yj 2. This pocess is epeated N L times. On the N L th time, e allo the algoithm to un until k k k k k k k 0 4, hee k is the iteation index and k k indicates the L nom of a vecto. Fo this ok, e set N K 0 and N L 50 fo the eak-tubulence econstuctions and N K 00 fo the stong-tubulence econstuctions. Fig. 5. (a) eak-tubulence econstuctions, and (b) stong-tubulence econstuctions. Fo both subplots, the left column shos the oiginal bluy image, the middle column shos the image-shapening esults, and the thid column shos the MBIR esults. It is impotant to note that these esults ae shon using a log-based decibel scale.

4 06 Vol. 35, No. / Januay 208 / Jounal of the Optical Society of Ameica A Reseach Aticle 4. RESULTS AND CONCLUSIONS We chose to compae the MBIR algoithm to the pointby-point IS appoach pesented in [3] using the M 2 shapness metic. The algoithm computes the phase-eo estimate accoding to ˆϕ agmaxf k jf H D expfjϕg H yj k g; (6) ϕ hee indicates the application of an exponent to each vecto element. Folloing the pocess descibed in [3], e used 20 iteations of conjugate gadient to optimize Eq. (6), and the algoithm as initialized using a 5th-ode Zenike polynomial estimate obtained using an iteative method to estimate only up to the 3d-ode tems, then up to the 4th, and so on, continuing up to 5th ode. Figue 5 shos econstuctions fo both the IS and MBIR algoithms along ith the oiginal bluy images fo fou diffeent SNRs and to tubulence stengths. To compess the lage dynamic ange that occus in coheent images, e pesent the images using a log-based decibel scale given by db 0 log 0,hee 0; is the nomalized eflectance function. Figue 6 shos the phase-eo estimates coesponding to the econstuctions in Fig. 5. The eak-tubulence esults sho that the IS algoithm is able to coect most of the phase eos; hoeve, the esultant images ae speckled, they have lo contast, and thee is esidual bluing. On the othe hand, the MBIR algoithm poduces images that have high contast, ith most of the object nea the peak image value and the backgound moe than 30 db belo that. Futhemoe, the MBIR algoithm poduces images that have much less speckle vaiation and esidual bluing compaed ith the IS algoithm. At the loest SNR, the MBIR algoithm makes it easie to distinguish the object fom the backgound. The stong-tubulence esults sho that the IS algoithm is not able to estimate stong phase eos fom single-shot data. Convesely, the MBIR algoithm is able to poduce highly focused images ith educed speckle vaiations and high Fig. 6. IS and MBIR phase eo estimates, ˆϕ, fo (a) the eaktubulence cases and (b) the stong-tubulence cases, coesponding to Fig. 5. contast at all but the loest SNR. These expeimental esults closely esemble the simulated esults in [6]. In summay, e have expeimentally demonstated the utility of the MBIR algoithm fo image econstuction fom single-shot DH data. The MBIR algoithm as shon to outpefom an IS algoithm ove a ange of SNRs and tubulence stengths. Oveall, the MBIR algoithm as able to poduce highly focused images ith educed speckle vaiation and high contast fo all but the loest SNR cases. APPENDIX A: EM SURROGATE FUNCTION Folloing [6], the EM suogate function is given by Q ;ϕ; 0 ; ϕ 0 σ 2 2Refy H A ϕ μg log jd j XN i i C i;i jμ i j 2 X jδ b j p i;j fi;jg P pσ p Δ! q p T σ Δ q p T σ X jδ ϕ j 2 b i;j : (A) fi;jg P 2σ 2 ϕ Hee, Ref g indicates the eal pat of the agument, the supescipt, H, indicates the Hemitian tanspose, and the subscipt, ϕ, indicates the dependence of A on ϕ. The vaiables μ and C ae the mean and covaiance matix fo the complex nomal posteio distibution, p f jy; ; ϕ, and ae given by μ C σ 2 A H ϕ 0y (A2) and! C σ 2 A H ϕ 0A ϕ 0 D D σ 2 σ2 : (A3) Fo cicula apetues, e appoximate the covaiance matix as shon in Eq. (A3), since it simplifies computations, and e have found that it oks ell. Fo squae apetues, the appoximation in Eq. (A3) is exact. To educe the numbe of unknons, e allo the phaseeo function, ϕ, to be modeled on a gid that has loe esolution than the measued data, denoted as ϕ. To scale ϕ to the esolution of ϕ, e use a neaest-neighbo intepolation scheme given by ϕ Pϕ, hee P is an N N n 2 b intepolation matix ith elements in the set {0, }, and n b is the facto of subsampling used in both dimensions. In this ok, e used n b 4. The last to tems in Eq. (A) esult fom using Makov andom field pio models fo and ϕ. The vaiable b i;j is the eight beteen neighboing pixel pais ( i and j,oϕ i and ϕ j ), and P is the set of all pai-ise cliques falling ithin the same neighbohood. We used a 3 3 Gaussian kenel ith standad deviation 0. pixels fo b. Δ i j and Δ ϕ ϕ i ϕ j ae the diffeence beteen pixel-pai values, T is a unitless theshold value that contols the tansition of the potential function fom having the exponent, q, to having the exponent, p [9]. In this ok, T 0., q 2, and

5 Reseach Aticle Vol. 35, No. / Januay 208 / Jounal of the Optical Society of Ameica A 07 p.. The vaiable, σ, contols the vaiation in ˆ, and σ ϕ contols the vaiation in ˆϕ. We set σ ϕ 0.5 ad fo eak tubulence and σ ϕ.0 ad fo stong tubulence. The values of σ and σ ee set accoding to σ γ s and σ N y H y 0.5, hee 0 is the initialized value of, γ is a unitless paamete intoduced to tune the amount of egulaization in, and s 2 computes the sample vaiance of a vecto s elements. We set γ 2 fo N and γ.5 fo N REFERENCES. T.-C. Poon and J.-P. Liu, Intoduction to Moden Digital Hologaphy: With MATLAB (Cambidge Univesity, 204). 2. M. F. Spence, R. A. Rayno, M. T. Banet, and D. K. Make, Deeptubulence avefont sensing using digital-hologaphic detection in the off-axis image plane ecoding geomety, Opt. Eng. 56, 0323 (207). 3. S. T. Thuman and J. R. Fienup, Phase-eo coection in digital hologaphy, J. Opt. Soc. Am. A 25, (2008). 4. J. C. Maon, R. L. Kendick, N. Seldomidge, T. D. Go, and T. A. Höft, Atmospheic tubulence coection using digital hologaphic detection: expeimental esults, Opt. Expess 7, (2009). 5. A. E. Tippie and J. R. Fienup, Multiple-plane anisoplanatic phase coection in a laboatoy digital hologaphy expeiment, Opt. Lett. 35, (200). 6. C. Pellizzai, M. F. Spence, and C. A. Bouman, Phase-eo estimation and image econstuction fom digital-hologaphy data using a Bayesian fameok, J. Opt. Soc. Am. A 34, (207). 7. J. W. Goodman, Intoduction to Fouie Optics (Robets & Company, 2005). 8. C. A. Bouman, Model based image pocessing, 203, engineeing.pudue.edu/~bouman/publications/pdf/mbip-book.pdf. 9. J. B. Thibault, K. Saue, C. Bouman, and J. Hsieh, A thee-dimensional statistical appoach to impoved image quality fo multi-slice helical CT, Med. Phys. 34, (2007).