A new estimation method for two-step-only quadrature phase-shiftingdigital holography

Shuqun Zhang a,n, Jianyang Zhou b

a Department of Computer Science, College of Staten Island, City University of New York, Staten Island, NY 10314, USAb Department of Electronic Engineering, Xiamen University, Xiamen, Fujian, China

a r t i c l e i n f o

Article history:Received 3 June 2014Received in revised form25 August 2014Accepted 29 August 2014Available online 22 September 2014

Keywords:Digital holographyPhase shiftingQuadrature phase shifting holography

a b s t r a c t

Two-step-only quadrature phase-shifting digital holography can reconstruct the original complex object usingonly two holograms without the additional recording of the reference wave intensity and the object waveintensity. Its success depends on the accurate estimation of the reference wave intensity from the two acquiredholograms. The previous estimation method is relatively computational expensive. In this paper, we present anovel simple and effective method for estimating the reference wave intensity with low computational load.Simulation results are presented to demonstrate the effectiveness and speedup of the proposed method.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

In digital holography, one of the most important tasks is toeliminate the undesired zero-order and twin image to obtain ahigh quality reconstruction. While there are several methods forzero-order and twin image elimination or suppression, phase-shifting holography (PSH) [1–3] has been the most widely usedtechnique for this task. The PSH requires the recording of multiplephase-shifting holograms and phase stability at different phaseshifts, where the phase shifts are usually taken as multiples of π/2.Multiple phase-shifting holograms can be recorded sequentially orin parallel [4,5]. Depending on the number of holograms acquiredand used in the reconstruction, there are four-step, three-step andtwo-step PSH, which require four, three and two holograms,respectively. Generally the reconstruction errors increase as thenumber of holograms increases when acquired sequentially owingto the stability requirement among different phase shifts. Two-step PSH [4–11] is thus preferred over the three-step and four-stepPSH because of fewer numbers of hologram recordings. Two-stepPSH is also referred as quadrature PSH (QPSH), and a zero phaseshift and a π/2 phase shift are usually used during recording.Although only two holograms are needed in the conventionalQPSH, it still requires the additional measurement of the intensityof the reference wave and the intensity of the object wave in orderto reconstruct the original complex object wave without the zero-order and twin image. We refer this method as the “standard”QPSH. Therefore, the number of exposures in the standard QPSH is

still four, the same as the four-step PSH but the setup of thestandard QPSH is easier than that of the four-step PSH because therecording of the intensity patterns is not sensitive to phasevariation between the reference wave and the object wave. Menget al. [10] showed that actually the intensity of the object light isnot needed in the reconstruction as long as the reference waveintensity is large enough, but we still need to acquire the intensityof the reference wave. Recently, Liu and Poon [11] proposed animproved QPSH called two-step-only QPSH, which uses only twoquadrature-phase holograms and doesn’t need to record either thereference wave intensity or the object wave intensity. Instead thereference wave intensity is obtained through a numerical estima-tion method rather than an actual measurement. This simplifiesthe QPSH setup. In Ref. [12], Liu et al. slightly improved theiroriginal trial-and-error reference wave intensity estimationmethod. However, their proposed estimation method is relativelycomplex and computational expensive. In this paper, we propose amuch simpler and faster method for estimating the intensity of thereference wave for the two-step-only QPSH. This is done byreducing the searching range for the reference wave intensityand simplifying the operations during the search.

2. Two-step-only quadrature phase-shifting digital holography

Fig. 1 illustrates a typical phase-shifting holographic setup. Linearlypolarized light from a coherent laser is expanded and collimated, andthen split by a polarizing beam splitter into reference and object beamstraveling in different directions. The object beam illuminates the object,and the reference arm creates different phase shifts through varyingthe wave plates’ orientations. The two beams are combined by a

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Optics Communications

http://dx.doi.org/10.1016/j.optcom.2014.08.0680030-4018/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author. Tel.: þ1 7189823178; fax: þ1 7189822856.E-mail address: shuqun.zhang@csi.cuny.edu (S. Zhang).

Optics Communications 335 (2015) 183–188

second beam splitter, and the resulting interference pattern is capturedby a CCD camera and finally stored in an image processing system. Inthe standard two-step QPSH, we record two quadrature-phase on-axisholograms sequentially. One hologram I1(x, y) is the interferencepattern between the object wave and the reference wave, and theother hologram I2(x, y) is the interference pattern between the objectwave and the π/2 phase-shifted reference wave. Let R and O be thereference plane wave and complex object wave at the CCD plane,respectively. The two holograms can be written as

I1 ¼ RþOj j2 ¼ Rj j2þ Oj j2þR OþOn� �

; ð1Þ

I2 ¼ ejπ=2RþO�� ��2 ¼ Rj j2þ Oj j2� jR O�On

� �; ð2Þ

where the asterisk denotes the complex conjugation. The zero-orderterm, designated as I0¼ |R|2þ |O|2, can be removed from the twoholograms as follows

I1� I0 ¼ R OþOn� �

; ð3Þ

I2� I0 ¼ � jR O�On� �

; ð4ÞThen the complex object wave at the CCD plane can be recon-

structed by adding Eqs. (3) and (4) multiplied by j as

O¼ I1� I0ð Þþ j I2� I0ð Þ2R

: ð5Þ

In this reconstruction using Eq. (5), we need to know the zeroorder term I0 and the reference wave R in order to obtain a zero-order- and twin-image-free reconstruction. This requires measuringthe intensity of the reference wave and that of the object wave toproduce I0, and so there are four recordings in total for the standardQPSH: two phase-shifted holograms, the reference wave intensity andthe object wave intensity. In order to reduce the number of recordingsfrom four to two, Liu et al. [11,12] proposed a two-step-only QPSHwhich doesn’t need the actual measurement of the reference waveintensity and the object wave intensity. Instead they first numericallyestimate the reference wave intensity and then use it to calculate thezero-order term I0 since I0 can be derived as a function of R2, I1, and I2as represented by the following:

2I20� 2I1þ2I2þ4R2� �

I0þ I21þ I22þ4R4� �

¼ 0: ð6Þ

Laser

CCDr

Polarizing Beam splitter

Beam splitter

Mirror

Mirror /4/2/2

Lens

Object

Spatial Filter

λ λλ

Fig. 1. Phase-shifting digital holography setup.

Fig. 2. Object and holograms: (a) the object used in the experiment, (b) the inline hologram obtained with zero phase shift, and (c) the hologram obtained with π/2phase shift.

S. Zhang, J. Zhou / Optics Communications 335 (2015) 183–188184

The object wave intensity is not needed in calculating the zero-order I0. There are two solutions to the above quadratic equation

I0 ¼I1þ I2þ2R2

27

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI1þ I2þ2R2� �2

�2 I21þ I22þ4R4� �r

2: ð7Þ

In this equation, the minus sign is chosen when the intensity ofthe reference light is larger than the object wave intensity, whichis generally true in practice. The idea of the two-step-only QPSH isbased on the following. If the reference wave intensity R2 can beestimated numerically from the two recorded holograms I1, and I2,we can use it to obtain I0 using Eq. (7) and then apply in Eq. (5) toretrieve the desired object wave O. Therefore, the key to thesuccess of the two-step-only QPSH is to estimate the referencewave intensity R2. Liu et al.’s algorithm [12] searches all possiblevalues of R2 from 0 to the maximum of I1. In the searching process,it goes through the following steps:

(1) Calculate the target hologram HT using the following equation:

HT ¼1� jð Þ2

I1� I2ð Þ: ð8Þ

(2) For the test reference intensity R2testfrom 0 to max(I1) by some

small step ΔR2, do the following:i. Calculate I0 using Eq. (7) with the test reference R2

test andthen use the resulting I0 to compute the complex zero-order-and twin-image-free hologram using the following equation

HR ¼I1� I0ð Þþ j I2� I0ð Þ

2: ð9Þ

ii. Calculate a parameter called normalized correlation peak(NCP) which is defined as

NCP¼ real HT � HR½ �max� HR � HR½ �max

; ð10Þ

where real{.} denotes the real part of {.}, and the symbol“�” denotes a cross-correlation operation.

iii. Calculate the relative intensity of the reference light definedas R2

test/max(I1).(3) Plot the curve of the relative intensity of the reference R2

test/max(I1) versus NCP, and locate the peak to find the actual value of R2.

The above described algorithm for estimating the reference waveintensity is relatively complex since in every step search of R2, itinvolves two complex two-dimensional image cross-correlationsand two maximum operations for searching the correlation peaks.In the following, we propose a simpler and faster method.

3. The proposed algorithm

Our proposed method improves the estimation of thereference wave intensity in two places: reducing the searchrange for R2 from [0, max(I1)] to a much smaller range, andsimplifying the operations in each step search for R2. From

1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 105

Reference wave intensity R2

Sum

mat

ion

of th

e ca

lcul

ated

obj

ect w

ave

inte

nsity

Fig. 3. The curve of summation of the calculated object wave intensity versus thereference wave intensity.

Fig. 4. Reconstruction results: (a) the reconstructed object wave at the CCD plane,and (b) the reconstructed object.

S. Zhang, J. Zhou / Optics Communications 335 (2015) 183–188 185

common sense, R2 should be within 0 and the maximum of I1.However, we can derive a smaller bound for R2 from Eq. (7)since a valid R2 should make I0 real numbers everywhere in theimage. Accordingly, the expression inside the square root inEq. (7) must be greater than or equal to zero in order to havereal number solutions for I0, i.e.

I1þ I2þ2R2� �2

�2 I21þ I22þ4R4� �

Z0: ð11Þ

Eq. (11) can be expanded as a quadratic inequality in R2 as follows

�4R4þ4 I1þ I2ð ÞR2� I1� I2ð Þ2Z0: ð12ÞThe solution to this inequality gives the range of R2 as

I1þ I2ð Þ=2�ffiffiffiffiffiffiffiffiI1I2

prR2r I1þ I2ð Þ=2þ

ffiffiffiffiffiffiffiffiI1I2

p; ð13Þ

or

L¼ffiffiffiffiI12

r�

ffiffiffiffiI22

r !2

rR2rffiffiffiffiI12

rþ

ffiffiffiffiI22

r !2

¼H; ð14Þ

where L and H are used to denote the low and high limits of therange. Since the range must be true for all pixels of I1 and I2, therange of R2 must be between max(L) and min(H). Considering thatR2 should be also less than the maximum of I1, the high limit of therange is modified to be min[min(H), max(I1)]. The above-derivedrange for R2, maxðLÞ; min fminðHÞ; maxðI1Þg½ �will be much smallerthan the original range [0, max(I1)] in most cases.

Our second improvement is on simplifying the operations inevery step search of the reference wave intensity. The criterion forour search is as simple as follows. The best estimated R2 should makethe reconstructed object wave intensity the smallest. The reason isthat, if the estimated reference wave intensity is the same as the oneactually used in the recording, then the reconstructed object wavewill be clean without the zero-order and twin image. Otherwise, ifthe estimated reference wave intensity is different from the oneactually used in the recording, then the reconstructed object wavewill contain the original object wave plus some uneliminated zero-order and twin image. So the intensity of the reconstructed objectwave is the smallest with the correct reference light intensity used

Fig. 5. Results of the second simulation: (a) the hologram obtained with zero phase shift, (b) the hologram obtained with π/2 phase shift, (c) the reconstructed object wave atthe CCD plane, and (d) the reconstructed object.

S. Zhang, J. Zhou / Optics Communications 335 (2015) 183–188186

for reconstruction (because of no zero-order and twin image), and itwill become larger as the reference intensity used for reconstructiondeviates from the correct value (because of the added zero-order andtwin image). While we can calculate the intensity of reconstructedobject wave using Eq. (5), we can also easily compute it using I0 since

jO2j ¼ jI0�R2j:Therefore, in our searching for R2, we only need tocompute the summation of the reconstructed object wave intensity,

i.e. ∑Mi ¼ 1∑

Nj ¼ 1jI0�R2

testj; where M�N is the hologram size. If

RtestaR;the summation ∑Mi ¼ 1∑

Nj ¼ 1jI0�R2

test j contains the original

object wave intensity and some other terms. Only when Rtest ¼ R; the

summation ∑Mi ¼ 1∑

Nj ¼ 1jI0�R2

test j contains the original object wave

intensity only, which is the smallest. The proposed method thusundergoes only the following two steps:

(1) For the test reference intensity R2test from max(L) to min

fminðHÞ;maxðI1Þg by some small step ΔR2, calculate I0 usingEq. (7) and then compute ∑M

i ¼ 1∑Nj ¼ 1jI0�R2

test j using theresulting I0.

(2) Plot the curve of the tested reference intensity R2test versus

∑Mi ¼ 1∑

Nj ¼ 1jI0�R2

test j; and locate the minimum to find the actualvalue of R2.

It is obvious that the above-proposed method is much simplerthat Liu et al.’s method described in Section 2 owing to smaller searchrange and very simple searching operation. Once the reference waveintensity is obtained, we can reconstruct the zero-order- and twin-image- free object wave at the CCD plane using Eq. (5), and furtherretrieve the original object information using a Fresnel transform.

4. Simulation results

To verify the proposed method, computer simulations of theexperimental process were performed using Matlab. The CCDimage sensor has 512�512 pixel with pitch size of 10 μm�10 μm.The light source is a laser with a wavelength of 0.6328 μm, and thedistance from the object plane to the CCD array plane is 20 cm. Theamplitudes of the objects in the simulations are normalized tothe range of [0, 1] and the intensity of the reference wave is set tofour. An object to be recorded is given as shown in Fig. 2(a). Fig. 2(b) and (c) are the two inline holograms recorded by the CCDcamera with zero and π/2 phase shifts, respectively. Based on theproposed method, we first computed the search range for thereference wave intensity R2 from the two holograms as [1.69, 4.55],which is much smaller than the original range [0, 9.41] computedusing the method in Refs. [11,12]. The curve of the summation ofthe reconstructed object wave intensity for different referencevalues is obtained as shown in Fig. 3. The valley value of the curvecorrectly locates the setup value of R2 used in the simulation.Using the estimated R2, we reconstruct the object wave at the CCDplane and the original object as shown in Fig. 4(a) and (b). It isseen that the original object is perfectly reconstructed.

Another simulation with a different object was performed andthe results were obtained as shown in Fig. 5. We record the twoholograms with zero and π/2 phase shifts as shown in Fig. 5(a) and(b). The new search range for R2 is calculated to be [0.87, 4.86],which is reduced from the original range [0, 5.78]. The referencewave intensity is then obtained from the basin value of the curveof the reference wave intensity versus the summation of thecalculated object wave intensity as shown in Fig. 6. The objectwave at the CCD plane and the original object are finally recon-structed with the estimated reference wave intensity, as given inFig. 5(c) and (d), respectively. Again, the reference wave intensityis estimated accurately and the object is well reconstructed.

Next we compare the computation time of the proposed methodwith Liu et al.’s method presented in Ref. [12]. Both algorithms areimplemented using Matlab, where fast Fourier transform has beenused to speed up the 2D image cross-correlation operation. Thecomputer used is a Lenovo X61 laptop with 1.6 GHz Intel Core2 Duo CPU and 4 GB RAM. It is obvious that both the computationtime and the accuracy of the estimation depend on the search stepsize ΔR2 in the trial. The smaller step size, the more accurate theestimation of reference wave intensity and the final reconstruction.For step size ΔR2¼0.001, we measured the computation time for theproposed method and Liu et al.’s method as 52.3 s and 596.9 s,respectively. That is, the proposed method is more than 10 timesfaster than Liu et al.’s method. For larger step size ΔR2¼0.01 (i.e.,lower accuracy), the proposed method takes 5.1 s while Liu et al.’smethod needs 47.9 s, which is also near 10 times faster for theproposed method. It is not difficult to see that the smaller step ofthe trial (or the higher estimation accuracy), the higher speedup canbe obtained using the proposed method.

5. Conclusions

Compared to the standard two-step QPSH requiring four record-ings, the two-step-only QPSH finds advantage in requiring only tworecordings without additionally measuring the reference waveintensity and the object wave intensity. In the two-step-only QPSH,the reference wave intensity is obtained by numerical estimationinstead of actual measurement. We have reported a new referencewave estimation method for the two-step-only QPSH, which ismuch simpler and faster than the previous estimation method. Thisis done by using a small search range and simple search operation.The effectiveness and speedup of the proposed method have beenverified by simulation results.

Acknowledgments

This work was supported by a grant from National KeyTechnology R&D Program of MOST of China (2012BAI07B06) anda grant from The City University of New York PSC-CUNY ResearchAward Program.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

3.5

4x 105

Reference wave intensity R2

Sum

mat

ion

of th

e ca

lcul

ated

obj

ect w

ave

inte

nsity

Fig. 6. The curve of summation of the calculated object wave intensity versus thereference wave intensity in the second simulation.

S. Zhang, J. Zhou / Optics Communications 335 (2015) 183–188 187

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