Digital holographic phase imaging based on phase iteratively enhanced compressive sensing

Abstract

Digital holography has been widely applied in quantitative phase imaging (QPI) for monolayer objects within a limited depth. For multilayer imaging, compressive sensing is employed to eliminate defocused images but with missing phase information. A phase iteratively enhanced compressive sensing (PIE-CS) algorithm is proposed to achieve phase imaging and eliminate defocused images simultaneously. Linear filtering is first applied to the off-axis hologram in Fourier domain, and an intermediate reconstructed complex image is obtained. A periodic phase mask is then superimposed on the intermediate reconstructed image to iteratively eliminate the defocused images and recover the object with phase information. The experimental recovery of amplitude and phase of a two-layer sample with as little as 7% random measurement is demonstrated. The average phase error of the PIE-CS algorithm is analyzed, and the results show the feasibility for QPI.

Full text

Digital holographic phase imaging based on
phase iteratively enhanced compressive sensing
ZHENPENG LUO,1JIANSHE MA,1PING SU,1,*AND LIANGCAI CAO2
1Graduate School at Shenzhen, Tsinghua University, Shenzhen 518055, China
2Department of Precision Instrument, Tsinghua University, Beijing 100084, China
*Corresponding author: su.ping@sz.tsinghua.edu.cn
Received 30 November 2018; revised 31 January 2019; accepted 8 February 2019; posted 8 February 2019 (Doc. ID 353207);
published 12 March 2019
Digital holography has been widely applied in quantitative
phase imaging (QPI) for monolayer objects within a limited
depth. For multilayer imaging, compressive sensing is em-
ployed to eliminate defocused images but with missing phase
information. A phase iteratively enhanced compressive sens-
ing (PIE-CS) algorithm is proposed to achieve phase imaging
and eliminate defocused images simultaneously. Linear filter-
ing is first applied to the off-axis hologram in Fourier do-
main, and an intermediate reconstructed complex image is
obtained. A periodic phase mask is then superimposed on the
intermediate reconstructed image to iteratively eliminate the
defocused images and recover the object with phase informa-
tion. The experimental recovery of amplitude and phase of a
two-layer sample with as little as 7% random measurement is
demonstrated. The average phase error of the PIE-CS algo-
rithm is analyzed, and the results show the feasibility for
QPI. © 2019 Optical Society of America
https://doi.org/10.1364/OL.44.001395
Digital holography (DH) for quantitative phase imaging (QPI)
[1,2], three-dimensional (3D) particle imaging [3], and bio-
medical imaging [4,5] has been widely investigated in recent
years because of its advantages in performing quantitative wave-
front analysis and full-field measurement rather than mechani-
cal scanning and massive procedures. However, the quantitative
wavefront analysis of DH is limited to single-depth objects due
to the noises of defocused images.
Compressive holography [6] using compressive sensing (CS)
sampling protocols can retrieve multidimensional information
from a single captured hologram. Series of studies such as
time-resolved recording of moving events [7], object localization
with subpixel accuracy [8,9], particle detection [10], and 3D
tomography [11–14] have been reported. The 3D tomography
is one of the most interesting applications of compressive holog-
raphy. Nevertheless, all the reported work on 3D tomography
[11–13] adopts in-axis holography, and reconstruction of the ob-
ject field suffers from disturbing autocorrelation and conjugation
terms. These interference terms are removed by imposing a spar-
sity constraint, resulting in vanishing of the phase information.
In contrast, off-axis compressive holography can remove the
interference terms by linear filtering in the Fourier domain.
Therefore, applying sparsity constraint in off-axis compressive
holography may technically allow reconstruction of the complex
wavefront, and may also eliminate defocused images. Choi et al.
applied CS in off-axis holography for diffuse objects [14], and
Marim et al. also applied CS in off-axis holography under
frequency-shifting condition [15] and low-light illumination
condition [16] with spatial sparsity minimization constraint to
the reconstruction algorithm. Although the off-axis structure was
employed, both the reported algorithms aimed at reconstruction
of the intensities, resulting in amplitude-only results.
While the compressive acquisition of tomographic images
with high fidelity was successfully demonstrated by compressive
holography, the elimination of defocused images still remains
challenging. In optical scanning holography, several methods
including the inverse imaging algorithm [17], Wiener filter
[18], and Wigner distribution function [19] demonstrated sup-
pression of defocused images in sectional reconstruction. But
these methods came at the expense of estimator bias and in-
creased scanning time. Off-axis low-coherence holography [20]
has realized multidepth imaging without defocused images. In
this work, a partially coherent light source was employed, and
the light path was implemented to make sure that each object
layer could form a hologram separately. The information of
each layer was separated in the Fourier domain. Therefore, both
the depth of field and the number of layers were limited. The
device complexity and resolvable layer number became a trade-
off because of the slanted mirrors used to generate a different
optical path difference for each layer. It has been proven that
the iterative CS algorithm can filter out the conjugated image
due to its diffusivity of energy distribution [21]. Thus, if the real
or imaginary distribution of the defocused images has distinct
edges, the CS regularization function can only partially reduce
the noise. Increasing the diffusivity of the defocused images
may be beneficial to iteratively eliminate defocused images.
In this Letter, we propose a sparsity minimization algorithm
and linear filtering in off-axis compressive holography, for both
recovering complex samples and eliminating defocused images,
from random under-sampling of the hologram. In regard to the
successful elimination of defocused images, we superimpose
a periodic phase mask onto the intermediate reconstructed im-
age for increasing the diffusivity of the out-of-focus images.
Letter Vol. 44, No. 6 / 15 March 2019 / Optics Letters 1395
0146-9592/19/061395-04 Journal © 2019 Optical Society of America

The whole algorithm is named as phase iteratively enhanced CS
(PIE-CS).
The experimental system is shown in Fig. 1. The laser beam
(632.8 nm) is expanded and collimated, and then split by a beam
splitter BS1 into the sample and reference arm. The reference
beam travels through a tilted beam splitter BS2 and is incident
on the CCD at a small tilt angle. The sample beam travels
through the two-layer sample S, which is illustrated in Fig. 1(b).
The sample S is made up of two numbers: 9 and 2 in the USAF-
1951 negative resolution pattern (thickness: 2.3 mm). The spa-
tial resolution of the figures, Δo,isabout86μm. The size of the
numbers 9 and 2 is about 0.5mm×0.8mm, and the distance
between the two layers is 44.5 mm. The two numbers are trans-
parent while the background is black, and the two layers are
shifted to avoid occlusion.
The hologram is formed due to the interference between a
slanted reference plane wave and the two-layer object
Ox0,y0,z0, which is acquired by a CCD camera (1600 ×
1600, pixel pitch 3.45 μm, 8 bit, MV-EM510M/C,
Microvision Inc.). It can be mathematically represented by
IjERj2E2R2ERER,(1)
where Ris the reference wave, and the diffracted objective field,
E, can be obtained by
Ex,yZei2πz0
λF−1
2DfF2D fOx0,y0;z0gHfX0,fY0;z0gdz0,
(2)
where expi2πz0
λis the phase delay at the detector plane,
z00.Hexpi2πz0
λffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1−λfX02−λfY02
pis the trans-
fer function [22]. In Eq. (1), the terms E2,R2, and ERcan be
removed by setting an appropriate band-pass filter in the
Fourier domain of the off-axis hologram [23]. The filtered in-
terference pattern is
Ifilter ERnfilter,(3)
where nfilter is the filtering error. By filtering the hologram, most
of the noise can be directly removed. Also, linear filtering allows
reconstruction of the complex wavefront containing not only the
amplitude information, but also the phase information. This op-
eration can enhance the iterative reconstruction accuracy and
compressive capability of the CS algorithm. The sample interval
of CCD is ΔxΔyΔ3.45 μm. The sampling pitch, Δz,
in the zaxis equals the distance of the two layers, i.e., 44.5 mm.
The resolution of the CCD is NxNyN1600.fX0
x0∕NΔ2,fY0y0∕NΔ2. With these parameters, Eq. (3)
can be discretized as follows:
Ifiltern1Δ,n2Δ
nfilter Rn1Δ,n2ΔX
l
ei2π
λlΔzF−1
2D
×fF2DfOm1Δ,m2Δ,lΔzgHfX0,fY0,lΔzg:(4)
Complying with the standard CS theory, Eq. (4) can be rewritten
as a forward model as
gMRG−1
2DQG2DfΠf,(5)
where fwith the size Nx×Ny×Nz×1is the vector form of
O, which is the 3D sample; Πdenotes the measurement matrix;
G2D and G−1
2D denote the two-dimensional (2D) discrete
Fourier transform and inverse transform operator, respectively;
QbldiagP1P2PNzwith Plm1m2expi2π
λlΔz×
HfX0,fY0,lΔz,and“bldiag”means the block diagonal
matrix; Plm1,m2represents the element of the matrix Plat the
m1-th row and the m2-th column; RbldiagRRR
with Rm1,m2Rm1Δ,m2Δ;Rm1Δ,m2Δ
expfi2πΔ
λm1cos αm2cos βg is the reference wave, where
α90.66°andβ89.20° are the inclination angles of the
reference beam to the xaxis and the yaxis, respectively; Mwith
the size NM×Nx×Ny×Nzdenotes the down-sampling
matrix. In order to reconstruct the 3D volume, we employ the
two-step iterative shrinkage-thresholding algorithm (TwIST)
[24] to solve the following optimization problem:
ˆ
fargmin(1
2kg−Πfk2
2τX
Nz
l1X
Nx
x1X
Ny
y1
j∇frealfgx,y,lj
j∇fimagfgx,y,lj),(6)
where imagfand realfdenote the imaginary and real parts
of f,respectively,τ>0is a regularization parameter, and
PNz
l1PNx
x1PNy
y1j∇f·gx,y,ljis the total-variation (TV) [25]
regularization function.
Figures 2(a1) and 2(b1) illustrate the 3D datacube estimated
from the filtered hologram without down-sampling by conven-
tional back-propagation (BP) reconstruction and normal CS
reconstruction. In both of the reconstruction results, the num-
bers 2 and 9 are reconstructed at the appropriate distance. The
theoretical axial resolution of our experimental setup is Δzo ≈
4Δ2
o∕λ46.75 mm [11]. Because the axial distance of the
two layers (44.50 mm) is less than the axial resolution
(46.75 mm), defocused images appear in the reconstructed re-
sults, as can be seen in Figs. 2(a1) and 2(b1), of both the BP
and normal CS reconstructions. Note that the BP results are the
initial images for CS iteration, which are defined as the inter-
mediate reconstruction results. Normal CS iteration still cannot
eliminate defocused images. From the perspective of computa-
tional iteration, the intermediate reconstructed in-focus images
have clear boundaries, as shown in Fig. 3(a); thus, the gradient
values of in-focus images quickly converge to minimum under
the TwIST algorithm with the TV regularization function, and
the original information of the in-focus images is retained.
(b)
(a)
Laser
MO
P
L
BS1
BS2
M2
M1
Camera
S
Fig. 1. (a) Experimental setup of the off-axis holography. MO, mi-
croscopic objective; P, pinhole; L, collimation lens; BS1 and BS2,
beam splitters; M1 and M2, mirrors; S, two-layer sample. (b) The
two-layer sample has two numbers: 9 and 2.
1396 Vol. 44, No. 6 / 15 March 2019 / Optics Letters Letter

Although the defocused images have a certain degree of
dispersion, their boundaries still have visible gradient character-
istics under the TV regularization function. So, the gradient val-
ues of the defocused images will slowly converge to minimum,
which prevents the CS iteration from successfully eliminating the
disturbing images. Therefore, we propose to preprocess the
intermediate reconstructed image to enhance the diffusivity
of the defocused images, by multiplying a phase mask Φwith
the intermediate reconstructed image f.Φm1Δ,m2Δ
Ruexpi2πΔ
λum1cos αm2cos β, which is a discrete
phase matrix and has a periodical linear function with an adjust-
able frequency parameter u. Generally, the frequency νΦxand νΦy
of Φshould satisfy the following equation:
1
2Δo
<νΦx,Φy<1
2Δx,y
:(7)
We name this CS iteration algorithm with attached phase mask as
the PIE-CS algorithm. The phase mask converts the continuous
large gradient values of the defocused images into discrete smaller
gradient values, i.e., weakens the gradient characteristic of the
defocused images. Meanwhile, the gradient characteristics of the
in-focus images are kept well, as shown in Fig. 3(b).Thisis
the reason that the defocused images can be effectively removed
by the PIE-CS iterations. Accordingly, Eq. (5)turnsinto
˜gMRG−1
2DQG2DΘ−1Θf
˜
Π
˜
f,(8)
where
˜
fΘfwith ΘbldiagΦΦΦas the new object
and
˜
ΠΠΘ−1as the new measurement matrix. After the
PIE-CS iterations, fcan be directly retrieved by simply sub-
tracting the attached Φfrom the reconstructed images
˜
f.
Figure 2(c1) illustrates the 3D datacube estimated from the fil-
tered hologram without down-sampling by PIE-CS. The curves
in Fig. 2(c1) near the blue cross line show that the interference of
the defocused images has been effectively eliminated.
Furthermore, we analyze the phase error of CS and PIE-CS
in off-axis DH. The average phase error of the iteration algo-
rithms is defined as δA jϕA−ϕBPj, where ϕAand ϕBP are the
phases of the reconstructed object by the iteration algorithms
and by BP in 100% sampling rate, respectively. The ¯· is the
average operation, and the evaluation area is the two numbers’
Sampling
Rate
100%
amplitude
7%
amplitude
100%
phase
7%
phase
(b1) (c1)
(a1)
(a2) (b2) (c2)
1
0
0
-0.8
-0.5
1.5 0
-1.2 -0.2
1
-1
0.2
-1
0.5
1.5
-0.5
-1.2
0.2
1.5
-1.5
1.5
-1.5
1.5
-1.5
Fig. 2. Amplitude and phase recovery of the two-layer sample by BP: (a1)(a2), CS: (b1)(b2), and PIE-CS: (c1)(c2), under sampling rate of 100%
(a1)(b1)(c1) and 7% (a2)(b2)(c2). In PIE-CS, u4when sampling rate is 100%, and u1when sampling rate is 7%. The red scale bar in (b1)
represents 345 μm on the sample. The yellow and red rectangles represent defocused images and in-focus images, respectively. The blue curves in
each amplitude recovery image exhibit the amplitude distribution along the blue cross lines.
Real
part
Imaginary
part
(a) (b)
10
0
-5
10
0
15
-15
15
0
12
-10
10
-2
10
-5
10
-8
0
-20
5
-15
15
-10
10
-20
20
-10
15
-10
15
-20
20
Fig. 3. Real and imaginary parts of the intermediate reconstructed
two-layer sample without (a) and with (b) a phase mask (u1) at two
depths. Background: intermediate reconstructed real or imaginary im-
age; insets: in-focus and defocused images.
Letter Vol. 44, No. 6 / 15 March 2019 / Optics Letters 1397

area rather than the whole image. δAof PIE-CS is affected by
the frequency parameter uof the phase mask. According to
Eq. (7), the available range of ucan be obtained. δAof PIE-
CS for the numbers 9 and 2 are calculated using different u
values and under different sampling rates. The results are com-
pared with δCS, as shown in Fig. 4. As concluded in Fig. 4,
although the choice of higher uvalue can achieve better results
in high sampling rate cases, the reconstructed results are much
worse in low sampling rate cases. Therefore, u1is the op-
timal choice for PIE-CS. The results show that our PIE-CS
algorithm works for QPI.
In addition, PIE-CS demonstrates strong recovery capacity
from randomly and rarely under-sampled acquisition. We show
7% random down-sampling results in Figs. 2(a2),2(b2), and
2(c2). It can be seen that the two in-focus numbers on the two
layers can be reconstructed without the defocused images, by
PIE-CS but not with traditional CS, while BP reconstruction
totally fails to estimate the 3D datacube. We quantitatively an-
alyze the effect of the sampling rate on the phase error of the
PIE-CS algorithm. As shown in Fig. 4(b) for 30% sampling and
Fig. 4(c) for 7% sampling, δAof PIE-CS has minimum values
when u1, while δAof CS is worse. This is because in this
work, the frequency of the reference light is adjusted optimally
for spectral separation and computational resolution. Conversely,
the δAof PIE-CS is larger than δAof CS in 7% sampling, as
shown in Fig. 4(c) when u>1, because the PIE-CS algorithm
cannot recover high-frequency phase information of
˜
fin
down-sampling cases.
In summary, we proposed a PIE-CS algorithm and exper-
imentally demonstrated off-axis compressive holography, which
enables the simultaneous reconstruction of complex samples
and elimination of defocused images in a sparsely compressive
acquisition. Phase reconstruction depends on linear filtering of
the hologram in Fourier domain. Since the gradient character-
istic of defocused images affects the elimination of defocused
images by CS iteration, a periodic linear phase mask is multi-
plied with the intermediate reconstructed image, and the de-
focused images are then successfully eliminated by using the
CS algorithm. The CS algorithm with phase mask attached
is named the PIE-CS algorithm. A two-layer object without
overlapping is successfully reconstructed with amplitude and
phase at each depth for as little as 7% down-sampling. We
quantitatively analyzed the effect of the frequency of the peri-
odic phase mask and the sampling rate on the average phase
error of the PIE-CS algorithm. The average phase error of
the PIE-CS algorithm is much less than that of the traditional
CS algorithm under high sampling rate conditions. The mini-
mum average phase error in the experiment is about 3.42 nm,
i.e., 0.0054λ, which verifies that our PIE-CS algorithm works
for QPI. The proposed off-axis holographic setup and PIE-CS
algorithm have potential applications in biomedical phase im-
aging of multiple sections, tracking fast moving objects in 3D
space, and special optical metrology.
Funding. The 863 Program of China (2013AA014402);
National Natural Science Foundation of China (NSFC)
(61827825); Basic Research Program of Shenzhen
(JCYJ20170412171744267).
REFERENCES
1. Y. K. Park, C. Depeursinge, and G. Popescu, Nat. Photonics 12, 578
(2018).
2. E. Cuche, P. Marquet, and C. Depeursinge, Opt. Lett. 38, 6994 (1999).
3. J. Garcia-Sucerquia, W. Xu, S. K. Jericho, P. Klages, M. H. Jericho,
and H. J. Kreuzer, Appl. Opt. 45, 836 (2006).
4. G. Popescu, L. P. Deflores, J. C. Vaughan, K. Badizadegan, H. Iwai,
R. R. Dasari, and M. S. Feld, Opt. Lett. 29, 2503 (2004).
5. N. Chen, C. Zuo, E. Y. Lam, and B. Lee, Sensors 18, 3711 (2018).
6. D. Brady, K. Choi, D. Marks, R. Horisaki, and S. Lim, Opt. Express 17,
13040 (2009).
7. Z. Wang, L. Spinoulas, K. He, L. Tian, O. Cossairt, A. K. Katsaggelos,
and H. Chen, Opt. Express 25, 250 (2017).
8. Y. Liu, L. Tian, J. W. Lee, H. Y. H. Huang, M. S. Triantafyllou, and G.
Barbastathis, Opt. Lett. 37, 3357 (2012).
9. Y. Liu, L. Tian, C. H. Hsieh, and G. Barbastathis, Opt. Express 22,
9774 (2014).
10. W. S. Chen, L. Tian, S. Rehman, Z. Y. Zhang, H. P. Lee, and G.
Barbastathis, Opt. Express 23, 4715 (2015).
11. C. Fernandez Cull, D. A. Wikner, J. N. Mait, M. Mattheiss, and D. J.
Brady, Appl. Opt. 49, E67 (2010).
12. K. Choi, R. Horisaki, J. Hahn, S. Lim, D. L. Marks, T. J. Schulz, and
D. J. Brady, Appl. Opt. 49, H1 (2010).
13. J. Hahn, S. Lim, K. Choi, R. Horisaki, and D. J. Brady, Opt. Lett. 19,
7289 (2011).
14. X. Y. Wu, Y. J. Yu, W. J. Zhou, and A. Asundi, Opt. Express 22, 19860
(2014).
15. M. Marim, M. Atlan, E. Angelini, and J.-C. Olivo-Marin, Opt. Lett. 35,
871 (2010).
16. M. Marim, E. Angelini, J.-C. Olivo-Marin, and M. Atlan, Opt. Lett. 36,
79 (2011).
17. X. Zhang, E. Y. Lam, and T.-C. Poon, Opt. Express 16, 17215 (2008).
18. T. Kim, Appl. Opt. 45, 872 (2006).
19. H. Kim, S.-W. Min, B. Lee, and T.-C. Poon, Appl. Opt. 47, D164
(2008).
20. L. Wolbromsky, N. A. Turko, and T. Shaked, Opt. Lett. 43, 2046
(2018).
21. W. H. Zhang, L. C. Cao, D. J. Brady, H. Zhang, J. Cang, H. Zhang, and
G. F. Jin, Phys. Rev. Lett. 121, 093902 (2018).
22. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and
Company Publishers, 2005).
23. E. Cuche, P. Marquet, and C. Depeursinge, Appl. Opt. 39, 4070
(2000).
24. J. B. Dias and M. Figueiredo, IEEE Trans. Image Process 16, 2992
(2007).
25. A. Chambolle, J. Math. Imaging Vis. 20, 73 (2004).
uu
u
2
9
Sampling
rate 100%
Sampling
rate 30%
(a) (b)
Sampling
rate 7%
(c)
2
9
2
9
Fig. 4. Effect of the frequency parameter uon the average
phase error δPIE-CS when the sampling rate is (a) 100%, (b) 30%,
and (c) 7%. The horizontal dashed lines are the comparison of δCS
from CS.
1398 Vol. 44, No. 6 / 15 March 2019 / Optics Letters Letter