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Single Shot Digital Holography Using Iterative Reconstructionwith Alternating Updates of Amplitude and PhaseDennis J. Lee1,2,∗, Charles A. Bouman2, and Andrew M. Weiner2

1Sandia National Laboratories, 1515 Eubank Blvd. SE, Albuquerque, NM 87123, USA2School of Electrical and Computer Engineering, Purdue University, 465 Northwestern Avenue, West Lafayette, IN 47907, USA∗Corresponding author: [email protected]

AbstractWe present an image recovery approach to improve ampli-

tude and phase reconstruction from single shot digital holograms,using iterative reconstruction with alternating updates. This ap-proach allows the flexibility to apply different priors to amplitudeand phase, improves phase reconstruction in image areas withlow amplitudes, and does not require phase unwrapping for regu-larization. Phantom simulations and experimental measurementsof a grating sample both demonstrate that the proposed methodhelps to reduce noise and resolve finer features. The improvedimage reconstruction from this technique will benefit the manyapplications of digital holography.

IntroductionDigital holography has many versatile applications, includ-

ing microscopy [1], phase contrast [2], 3D displays [3], tomog-raphy [4], and terahertz imaging [5]. This technique enables themeasurement of both amplitude and phase, useful for quantify-ing path lengths, measuring index contrast, or viewing biologicalsamples [6, 7]. Various experimental methods can extract ampli-tude and phase from a measured interference pattern: for example,phase-shifting interferometry involves incrementally stepping thephase of the reference beam, from which the object phase can bededuced [8]. However, this method requires multiple images tobe recorded on a vibration-free optical table, typically with ex-pensive devices such as high frame rate cameras.

Off-axis interferometry, based on the interference of an ob-ject and reference beam slightly offset in angle, can compute am-plitude and phase with a single measurement. Conventional pro-cessing spatially filters the Fourier transform of the hologram [9].While simple, Fourier filtering suffers from some drawbacks: Thefilter window size is subjective, and good reconstruction requiresthe zero order and cross terms to be well-separated. A large sepa-ration between object and reference beams results in a high carrierfrequency hologram, in which the zero-order and sidebands arewell-separated in the spectrum. Although a high carrier frequencyis ideal for Fourier filtering, other factors may prevent perfect re-construction: The experimental configuration may not allow alarge angular separation of the object and reference beams, thesampling requirements of the camera pixels may limit the angularseparation, or the object may contain high spatial frequencies.

To overcome these drawbacks, other approaches include sub-tracting the zero order term from the hologram [10], iterativelysolving for the field in the spatial domain [11] and the frequencydomain [12], and applying a nonlinear filter [13]. Optimization-based approaches include formulating holography as a nonlinear

least squares problem [14], as constrained optimization [15], aspenalized likelihood with simulated data [16], or as a nonlinearinverse problem [17].

In this work, we propose a new image recovery approachthat calculates an object field from a single hologram, using it-erative reconstruction with alternating updates of amplitude andphase. To our knowledge, this work is the first application of thealternating update strategy to single shot digital holography. Thisapproach offers multiple advantages:

1. It allows prior knowledge, such as object smoothness, to beapplied separately to amplitude and phase. For example,although nearly transparent biological samples like cells aresmooth in amplitude, they exhibit sharp edges in phase.

2. Regularizing phase separately, rather than regularizing thefield Aeiφ , mitigates the effects of poor signal areas causedby low amplitudes, aiding phase reconstruction.

3. Our algorithm regularizes phase without requiring phase un-wrapping, as discussed in the Theory section.

This work is organized as follows. In the Theory section,we present continuous and discrete mathematical models of holo-graphic measurements, followed by an iterative reconstruction al-gorithm that recovers amplitude and phase via alternating updates.In the Experiment section, we discuss results from a phantom sim-ulation and from experimentally measured holograms of a gratingsample. Finally we provide some concluding remarks.

TheoryContinuous Model

In digital holography, an object field, o(x) = A(x)eiφ(x), anda reference field, r(x) = A(x)eiφ(x), combine to form an interfer-ence pattern measured on a camera,

Iideal(x) = |o(x)+ r(x)|2

=∣∣∣A(x)eiφ(x)+ A(x)eiφ(x)

∣∣∣2= A2(x)+ A2(x)+2A(x)A(x)cos

[φ(x)− φ(x)

],

(1)

where x ∈ R2 is the spatial coordinate in the camera plane, andφ(x) typically represents linear phase. The measured hologramis generally noisy, which we can model as Gaussian noise thatcorrupts the interference pattern Iideal(x). Let I(x) denote themeasured hologram. It is also possible to model the measuredhologram using a Poisson distribution [18, 19]. We formulate theproblem as the minimization of a cost function with the general

©2016 Society for Imaging Science and TechnologyDOI: 10.2352/ISSN.2470-1173.2016.19.COIMG-158

IS&T International Symposium on Electronic Imaging 2016Computational Imaging XIV COIMG-158.1

formc(A,φ) = L(A,φ)+βAR(A)+βφ R(φ),

where L(A,φ) is the negative log-likelihood function correspond-ing to our model of the noisy hologram as a Gaussian distribu-tion, βA and βφ are scalar regularization parameters, and R(A) andR(φ) are roughness penalty functions for amplitude and phase, re-spectively. In the continuous formulation, the likelihood functionhas the form

L(A,φ) =12

∫R2

{A2(x)+ A2(x)

+2A(x)A(x)cos[φ(x)− φ(x)

]− I(x)

}2 dx.(2)

To reduce noise and impose prior knowledge such as edge-preserving smoothness on the object, we also minimize the totalvariation of the amplitude and phase with the regularizer terms

R(A) =∫R2

√|∇A(x)|2 + ε dx (3)

and

R(φ) =∫R2

√|∇eiφ(x)|2 + ε dx, (4)

where ε is a small constant to ensure differentiability. Other priormodels can also be used [20, 21]. Now the cost function becomes

c(A,φ) = L(A,φ)+βAR(A)+βφ R(φ)

=12

∫R2

{A2(x)+ A2(x)


]− I(x)

}2 dx

+βA

∫R2

√|∇A(x)|2 + ε dx

+βφ

∫R2

√|∇eiφ(x)|2 + ε dx.

(5)

Our goal is to solve for amplitude and phase by minimizing thiscost function:(

A, φ)= argmin

A,φc(A,φ). (6)

Discrete ModelWe discretize the amplitude A ∈ RN , phase φ ∈ RN , and in-

tensity I ∈ RN using vector notation:

A =[

A1 . . . Ai . . . AN]T

, (7)

φ =[

φ1 . . . φi . . . φN]T

, (8)

and

I =[

I1 . . . Ii . . . IN]T

, (9)

where N is the total number of pixels. We use similar vector nota-tions for the reference amplitude and phase A and φ . Accordingto the pixel pitch ∆x of the camera, we specify the sampled likeli-hood function as

L(A,φ) =12 ∑

m∈Z2

{A2(x)+ A2(x)


]− I(x)

}2∣∣∣x=m∆x

=12 ∑

i

[A2

i + A2i +2AiAicos

(φi− φi

)− Ii

]2.

(10)

Reconstruction Simulation ExperimentalTechniques Results Results

Fourier Filtering Figure 2(b) Figure 4(b)Iterative Reconstruction Figure 2(c) Figure 4(c)

Table 1. Comparison of reconstruction techniques.

We regularize the unknowns by minimizing the total variation,with

R(A) = ∑i

√(CA)2

i + ε (11)

and

R(φ) = ∑i

√∣∣[Ceiφ]

i

∣∣2 + ε. (12)

Here we use the shorthand vector notation

eiφ =[

eiφ1 . . . eiφi . . . eiφN]T

. (13)

C is a convolution matrix implementing the discretized firstderivatives of the nearest neighbors. For example, for

C =

[ChCv

], (14)

C ∈ R2N×N is a concatenation of Ch and Cv, which implementsfirst derivatives in the horizontal and vertical directions. Since weare examining derivatives of eiφ , no phase unwrapping is requiredfor regularization. The cost function becomes

c(A,φ) = L(A,φ)+βAR(A)+βφ R(φ)

=12 ∑

i

[A2

i + A2i +2AiAicos

(φi− φi

)− Ii

]2

+βA ∑i

√(CA)2

i + ε +βφ ∑i

√∣∣[Ceiφ]

i

∣∣2 + ε.

(15)

Our goal is to solve for A and φ by minimizing this cost function:(A, φ

)= argmin

A,φ∈RNc(A,φ). (16)

We estimate A and φ by alternatively updating them in each iter-ation as

A(n+1) = argminA∈RN

c(A,φ (n)) (17)

and

φ(n+1) = argmin

φ∈RNc(A(n+1),φ). (18)

Iterative Reconstruction AlgorithmTo solve the reconstruction problem in Eq. (16), we em-

ploy the alternating update strategy in Eqs. (17) - (18), detailed inAlgorithm 1. First fixing φ , we update A through a gradient de-scent approach [22], until the cost function reduces to a specified



http://dx.doi.org/10.2352/ISSN.2470-1173.2016.19.COIMG-158


Algorithm 1 Iterative Reconstruction with Alternating Updatesof Amplitude and Phase

Input: Measured hologramOutputs: Amplitude A and phase φ

1: n← 02: Initialize A(0) and φ (0) by Fourier filtering.3: while c(A(n),φ (n))> ε1 do

4: i← 05: A(n, 0)← A(n)

6: while c(A(n, i),φ (n))> ε2 do7: d←−∇Ac(A,φ (n))

∣∣∣A=A(n, i)

= −∇AL(A,φ (n))∣∣∣A=A(n, i)

−βA∇AR(A)∣∣∣A=A(n, i)

8: α∗← argminα∈R c(

A(n, i)+αd,φ (n))

9: A(n, i+1)← A(n, i)+α∗d10: i← i+111: end while

12: A(n+1)← A(n, i)

13: i← 014: φ (n, 0)← φ (n)

15: while c(A(n+1),φ (n, i))> ε3 do16: d←−∇φ c(A(n+1),φ)

∣∣∣φ=φ (n, i)

= −∇φ L(A(n+1),φ)∣∣∣φ=φ (n, i)

−βφ ∇φ R(φ)∣∣∣∣φ=φ (n, i)

17: α∗← argminα∈R c(

A(n+1),φ (n, i)+αd)

18: φ (n, i+1)← φ (n, i)+α∗d19: i← i+120: end while

21: φ (n+1)← φ (n, i)

22: n← n+123: end while

24: A← A(n)

25: φ ← φ (n)

26: return A, φ

Experimental Data Corresponding FigureHologram with a Figure 3(a)

low carrier frequencyHologram with a Figure 3(d)

high carrier frequency

Table 2. Experiment data and corresponding figures.

threshold ε2; updates for φ proceed similarly, using the new valueof A. The alternating updates repeat, enclosed in an outer loop,terminating when cost reduces to a threshold ε1.

The update steps require various derivatives to be computed.For reference, we list the gradients of the likelihood functions,

[∇AL(A,φ)]i =[A2

i + A2i +2AiAi cos

(φi− φi

)− Ii

]·[2Ai +2Ai cos

(φi− φi

)] (19)

and [∇φ L(A,φ)

]i =−

[A2

i + A2i +2AiAi cos

(φi− φi

)− Ii

]·[2AiAi sin

(φi− φi

)],

(20)

and the Laplacians of the likelihood functions, diagonal matriceswith entries[

∇2AL(A,φ)

]ii=[2Ai +2Aicos(φi− φi)

]2+2[A2

i + A2i +2AiAicos(φi− φi)− Ii

] (21)

and[∇

2φ L(A,φ)

]ii=[2AiAisin(φ − φ)

]2+{

Ii−[A2

i + A2i +2AiAicos

(φi− φi

)]}·2AiAicos(φi− φi).

(22)

ExperimentSimulation Results

For simulation, we create a test object consisting of uniformamplitude A(x), equal to 1 everywhere. We set the phase φ(x)to have the form of the Shepp-Logan phantom, shown in Fig.1(a). The resulting object field, o(x) = eiφ(x), may simulate anearly transparent biological sample like a cell. In an off-axisconfiguration, the reference beam is r(x) = ei2πf·x. We set thespatial frequency f = (0.16,0), with units of (pixel)−1. Figure1(b) shows the interference pattern I(x) produced by o(x) andr(x). The Fourier filtering technique applies a filter to the Fouriertransform of I(x) to reconstruct the object. Figure 1(c) highlightsthe filter, indicated by the black circle, superimposed on the powerspectrum of I(x).

Figure 2 shows the phase reconstruction results, where wecompare Fourier filtering and our iterative reconstruction ap-proach as described in Table 1. The phantom in Fig. 2(a), theground truth, has a high phase value of 2 rad at the outer edge,which contributes to the high frequency content in the power spec-trum in Fig. 1(c). Since the filter cuts off high frequencies, ringingoccurs in the retrieved phase in Fig. 2(b). Our iterative recon-struction algorithm overcomes the limitations of filtering whilereducing noise, as demonstrated in the reconstruction in Fig. 2(c).As a more quantitative comparison, we plot line-outs across thedashed white line in Fig. 2(a) and across the same points in Figs.2(b) and 2(c). The line-out in Fig. 2(d) displays the ringing inthe phase computed by Fourier filtering, contrasting with the pro-posed method, which exhibits greater fidelity to the true phase anda better ability to resolve finer features.





Figure 1. Reconstruction by Fourier filtering on simulated data. (a) Simulated phase image (rad). (b) Simulated hologram using (a). (c) Power spectrum of (b).

Figure 2. Phase reconstruction of the simulated phantom. (a) Ground truth phase (rad). (b) Reconstructed phase using Fourier filtering. (c) Reconstructed

phase using iterative reconstruction, the proposed method. (d) Line-outs across the dashed white line in (a) with comparison to (b) and (c).





Figure 3. Reconstruction by Fourier filtering on experimental data. (a) Hologram with a low carrier frequency. (b) Power spectrum of (a). (c) Reconstructed

phase using Fourier filtering on the low carrier frequency hologram. (d) Hologram with a high carrier frequency. (e) Power spectrum of (d). (f) Reconstructed

phase using Fourier filtering on the high carrier frequency hologram. The scale bar lengths are 10 µm.

Figure 4. Phase reconstruction of a grating sample. (a) Ground truth phase (rad). (b) Reconstructed phase using Fourier filtering on the low carrier frequency

hologram. (c) Reconstructed phase using iterative reconstruction, the proposed method, on the low carrier frequency hologram. (d), (e) Line-outs across the

middle and rightmost gratings in (a)-(c). The scale bar lengths are 10 µm.





Experimental ResultsWe test our proposed algorithm on experimentally measured

data, which consists of holograms with low and high carrier fre-quencies as described in Table 2. Our sample consists of gratingspatterned on PMMA film (n = 1.49) with a height contrast of about1.5 µm. The camera measures an interference pattern, capturedin Fig. 3, with the reference beam at a slight angular offset fromthe object beam, under coherent illumination at 632.8 nm. Exper-imentally, we are able to vary the angular offset of the beams; asmall offset leads to the hologram with a low carrier frequency inFig. 3(a), while a larger offset produces the hologram with a highcarrier frequency in Fig. 3(d). A higher carrier frequency enableslarger sideband separation in the power spectra, evident in a com-parison of Figs. 3(b) and 3(e), where the frequency content of thegratings appear aligned along the grating vectors.

In the Fourier filtering technique, a filter isolates one of thesidebands, depicted as black circles in Figs. 3(b) and 3(e), forobject reconstruction. The filter may not be able to capture theentire frequency content of the object, as in Fig. 3(b), resulting inthe blurrier reconstruction in Fig. 3(c) compared to Fig. 3(f).

Figure 4 compares reconstruction results. For ground truth,we measure a hologram with a high carrier frequency, from whichwe reconstruct the high resolution phase image in Fig. 4(a), us-ing Fourier filtering. We measure a hologram with a low carrierfrequency, our test data, with the goal of reconstructing a highquality image. Fourier filtering fails to capture high frequencydetails, as shown in Fig. 3(b), resulting in the blurry reconstruc-tion in Fig. 4(b). In contrast, our proposed method recovers thehigh frequency details, displayed in Fig. 4(c), that are lost withconventional Fourier filtering. Line-outs across the middle andrightmost gratings, plotted in Figs. 4(d) and 4(e), quantitativelydemonstrate that our algorithm can reconstruct images in closeagreement with the ground truth.

ConclusionWe have developed an image recovery approach to improve

amplitude and phase reconstruction from single shot digital holo-grams, using iterative reconstruction with alternating updates.This approach allows the flexibility to apply different priors toamplitude and phase, improves phase reconstruction in image ar-eas with low amplitudes, and does not require phase unwrappingfor regularization. Phantom simulations and experimental mea-surements of a grating sample both demonstrate that the proposedmethod helps to reduce noise and resolve finer features. The im-proved image reconstruction from this technique will benefit themany applications of digital holography.

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Author BiographyDennis J. Lee received his Ph.D. in Electrical and Computer Engi-

neering from Purdue University in West Lafayette, IN. He is a researchscientist with Sandia National Laboratories in Albuquerque, NM. His re-search interests include computational imaging, signal and image pro-cessing, and optics.





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