We propose an experimental holographic imaging scheme combining compressive sensing (CS) theory with digital holography in phaseshifting conditions. We use the MachZehnder interferometer for hologram formation, and apply the compressive sensing (CS) approach to the holography acquisition process. Through projecting the hologram pattern into a digital micromirror device (DMD), finally we will acquire the compressive sensing measurements using a photodiode. After receiving the data of two holograms via conventional communication channel, we reconstruct the original object using certain signal recovery algorithms of CS theory and hologram reconstruction techniques, which demonstrated the feasibility of the proposed method.
I. INTRODUCTION
Over the past several years, the huge storage and bandwidth requirements for storing or transmitting holographic data has been a main limiting factor for application of digital hologram technology
[1

4]
. Fortunately, the newly developed theory known as compressive sensing (CS)
[5

7]
provides a new direction for imaging system designs. One of the most influential hardware realizations of imaging systems is the CSbased singlepixel camera
[8
,
9]
. Based on compressive sensing, many new applications have already sprung up in the realm of holography. The group of The Duke Imaging and Spectroscopy Program (DISP) first formulated hologram data compression as a compressive sensing problem
[10]
. Then we may mention online compressive holography
[11

13]
, offaxis compressive Fresnel holography
[14]
, compressive incoherent holography
[15

16]
, compressive single exposure on line holography
[17]
. In the optical domain, few applications of this singlepixel imaging setup in digital holography extending to the coherent regime have been reported although many successful applications of compressive sensing for holography have been demonstrated
[11

17]
.
To fully exploit the advantages of CS theory and holography, we develop a new holographic imaging scheme combining the CSbased singlepixelcamera with holography in the optical domain. Here, we use phaseshifting holography to generate holograms of an object, and then address the acquisition step by measuring the inner products between the hologram and the pseudorandom binary patterns generated by a digital micromirror device (DMD). With this method, we can directly acquire the compressed hologram using a photodiode, and never need the CMOS or CCD to record the hologram during the acquisition step. Then the acquired data is processed to produce a reconstruction of the object using reconstruction algorithms of CS. Our team has demonstrated the feasibility of the optical image recovery of fourstep phaseshifting holography based on compressive sensing
[18

19]
. A proofofconcept experiment evaluating the phase distribution of an ophthalmic lens with three step compressive phaseshifting holography is provided
[20]
. But it is necessary to record a lot of data with imaging architecture and spend a lot of time conducting an experiment. To overcome this drawback, on the basis of the method of fourstep phase shifting holography based on a CS, we propose the establishment of twostep phase shifting holography based on CS, in which half of the data is recorded to reconstruct the original object, speed of the reconstruction time is faster than the four step method. Moreover, when making use of our imaging system to simulate digital holographic imaging, we get holograms generated by the phaseshift digital hologram principle, in this process we use Fresnel to get similar holograms. In the process of recording data of holograms, we can directly acquire the compressed hologram using a photodiode. However, we reconstruct the original object using the compressed hologram rather than directly using holograms generated by the phaseshift digital hologram principle. The data need experience a lossy compression process. Because half of data is reduced in twostep phase shifting holographic method compared with fourstep phase shifting holographic method, in twostep phase shifting holographic method error caused by a lossy compression process will be less, realtime performance is also better than fourstep phase shifting holographic method. We reconstruct the original object using corresponding hologram algorithm instead of fourstep phase shifting algorithm, simulation results demonstate that reconstruction quality of image is better than fourstep phase shifting holographic method.
II. COMPRESSIVE SENSING
Compressive sensing combines sampling and compression into a single nonadaptive linear measurement process. It is mainly made up of three sections: sparse representation, random projection and reconstruction. This subsection presents a very brief CS overview necessary for the following sections.
Suppose that a basis matrix
φ
with size
N
×
N
can provide a Ksparse representation for a real value signal (e.g., image signal) X with length N. We can write
A with length N can be well approximated using only
K
□
N
nonzero entries. Let us define the acquisition model by
X is the input object, Y with size
M
×1 is our measurement, and Φ is an
M
×
N
measurement matrix, which is incoherent with
φ
. The goal is to reconstruct Y from few measurements. One can recover X from Y only from
M
=
O
(
K
log
N
), where
K
is the number of nonzero elements in
A
(
A
=
φ^{T} X
) and N is the total number of object pixels. As demonstrated in
[5

7]
, we can recover
x
from
y
via
1
_{l}
norm optimization problem:
III. COMPRESSIVE HOLOGRAPHIC IMAGING SYSTEM BASED ON COMPRESSIVE SENSING
Our holographic imaging setup based on CS theory is shown in
Fig. 1
. Where a linearly polarized laser beam is expanded, collimated, and then divided into an object beam and a reference beam. The phase of the reference wave is controlled by an electrooptic phase modulator. Then the two waves overlap to form interferograms. A traditional hologram generating device is different in that a singlepixel camera based on CS theory is used to collect signal and compress data in our system. We adopt a Digital Micromirror Device(DMD) for producing the measurement matrix Φ and computing random linear measurements of the interferograms
I_{H}
and the measurement matrix Φ. We can directly acquire the compressed hologram to obtain the compressed data Y using a photodiode.
Holographic imaging system.
In the case of the twostep algorithm
[23]
based on the traditional twostep quadrature phaseshifting holography
[21

22]
, when we set the phases of the reference wave in the first and second exposure to 0 and
respectively by regulating the electrooptic phase modulator, then two quadraturephase holograms
I
_{H1}
and
I
_{H2}
are recorded sequentially on the DMD and expressed as
I_{H}
(
x_{H} , y_{H}
) is a record of the interference between a reference field
u_{r}
and an object scattered field
u_{H} . u_{rk}
=
A_{r}
exp(
iφ_{rk}
) , k=1, 2,
A_{r}
and
φ_{rk}
are the amplitude and phase of the reference wave respectively.
I_{0}
is the zeroorder light given by
The DMD consists of many micromirrors, each mirror corresponds to a particular pixel in X and
ϕ_{m}
and can be independently oriented either towards lens (corresponding to one at that pixel in
ϕ_{m}
) or away from the lens (corresponding to a zero at that pixel in
ϕ_{m}
). Here
ϕ_{m}
can be considered as rows to form the random measurement matrix Φ(Φ = [
ϕ
_{1}
ϕ
_{2}
...
ϕ
_{M}
]
^{T}
∈
R
^{M×N}
). When the hologram
I_{H}
= (
x_{H}
,
y_{H}
) produced in our setup formed on the DMD, the biconvex lens is used to collect the reflected light, and then it is focused onto a photodiode that integrates the product
ϕ_{m}
X
to compute the measurement
y_{m}
=
ϕ_{m}
X
as its output voltage.
m
∈{1,2,3,...,
M
} .
To compute CS randomized measurements
Y
=ΦX=Φ
φ
A =[
y
_{1}
y
_{2}
...
y_{M}
]
^{T}
, we set the mirror orientations
φ_{m}
randomly using a pseudorandom number generator, measure
y_{m}
, and then repeat the process M times to obtain the measurement Y(Φ is the measurement matrix generated by the DMD).
Y is the measurement data,
Y
∈
R
^{M ×2}
,
1_{Hk}
∈
R
^{N ×1}
, Φ∈
R
^{M ×N}
. The optical signal was transformed to a digital signal by a photodiode and transmited through conventional channels to the computer. Then we can reconstruct the interference wave intensity by solving a CS reconstruction problem in the computer. According to CS, we can get the optimal estimation by solving the following convex optimization problem:
Since the hologram is piecewise constant with large gradient at the edge, we adopt a Total Variation(TV) algorithm to measure the
l
_{1}
norm as:
Here we adopt the
l
_{1 _}
magic
to solve this optimization problem. Then we need to compute
u_{H}
(
x_{H}
,
y_{H}
) ( the complex amplitude of the object optical field) by putting the optimal estimation
,
together. Finally we can get the original object image by the inverse Fresnel transformation. Now we will compute
u_{H}
(
x_{H}
,
y_{H}
) as follows. We show such an algorithm, which determines the reference wave intensity
directly from the hologram without the actual need to record the reference wave intensity at all. From holographic imaging principle,
should be within 0 and the maximum of
I
_{H1}
. Now we construct a 2D correlation coefficient, called CC
[23]
.
In Eq. (11), abs (.) denotes the amplitude of (.);
E_{r}
is the image decrypted by
I
_{H1}
and the correct keys; E denotes the decrypted image with two holograms and the correct keys, which is calculated by the assumed values
(the value is gotten by searching from 0 to max (
I
_{H1}
) without actual measurements) of the reference light
Eq. (15) and Eq. (16). We then draw the curve of CC versus
A_{rc}
and set the criterion such that the minimum point of the curve locates the actual value of
A_{r}
. The algorithm can be expressed as follows:
A_{r}
in Eq. (4), Eq. (5) and Eq. (6) was replaced by
A_{rc}
in this algorithm, then we construct a complex hologram
I_{C}
as shown above. In the formula,
u_{H}
= Re(
uH
) +
i
Im(
u_{H}
) . By taking square of the absolute value of both sides of Eq. (12), we obtain the solution to a quadratic equation in
I
_{0}
:
Transposing and rearranging Eq. (4) and Eq. (5), we can write
The quantity, F(
F
=
A_{rc}
+ Re(
u_{H}
) + Im(
u_{H}
) ), will be positive everywhere if the intensity of reference light is large than that of the object wave intensity, which is generally true in practice
[23]
, so the minus sign should be chosen for Eq. (13). Then we can calculate
we can calculate the complex amplitude from Eq. (12)
Putting the optimal value
,
calculated from the above with CS theory into Eq. (16)
Finally we can get the original object image by the inverse Fresnel transformation.
IV. COMPUTER SIMULATIONS
In the following, we made a series of computer simulations to verify the effectiveness of our method. The object we used was grayscale image “K” as shown in
Fig. 2(a)
with 64× 64 pixels (hence, N=4096). The measurement matrix Φ generated by the DMD is random sequences of 0/1. The hologram
u_{H}
(
x_{H} , y_{H}
) was digitized shown in
Fig. 2(b)
. When we took 2048 measurements and 3276 measurements, we respectively got the reconstruction of the original image “K” shown in
Fig. 2(c)
and
Fig. 2(d)
using the method of fourstep phaseshifting holography based on CS. When we took 3686 measurements, we got the reconstruction of the original image “K” shown in
Fig. 2(e)
using the method of fourstep phaseshifting holography based on CS. When we took 2457 measurements and 3276 measurements, we respectively got the reconstruction of the original image “K” shown in
Fig. 2(f)
and
Fig. 2(g)
using the method of twostep phaseshifting holography based on CS. When we took 3686 measurements, we got the reconstruction of the original image “K” shown in
Fig. 2(h)
using the method of twostep phaseshifting holography based on CS. According to the simulation results, it is clear that the new holographic imaging method we propose is feasible. We can see that there is ambiguous noise in
Fig. 2(h)
, because we reconstruct the original object using the compressed hologram rather than directly using two holograms generated by the phaseshift digital hologram principle. The data experiences a lossy compression process.
The simulation results using our architecture. (a) Original image of letter K. (b) Digital hologram of letter K. (c) Reconstruction from 2048 (50%) measurements using the method of fourstep phase shifting holography based on compressive sensing (CS). (d) Reconstruction from 3276 (80%)) measurements using the method of fourstep phase shifting holography based on CS. (e) Reconstruction from 3686 (90%) measurements using the method of fourstep phase shifting holography based on CS. (f) Reconstruction from 2457 (60%) measurements using current method of twostep phase shifting holography based on CS. (g) Reconstruction from 3276 (80%) measurements using current method of twostep phase shifting holography based on CS. (h) Reconstruction from 3686 (90%) measurements using current method of twostep phase shifting holography based on CS.
Fig. 2.
The simulation results using our architecture. (a) Original image of letter K. (b) Digital hologram of letter K. (c) Reconstruction from 2048 (50%) measurements using the method of fourstep phase shifting holography based on compressive sensing (CS). (d) Reconstruction from 3276 (80%)) measurements using the method of fourstep phase shifting holography based on CS. (e) Reconstruction from 3686 (90%) measurements using the method of fourstep phase shifting holography based on CS. (f) Reconstruction from 2457 (60%) measurements using current method of twostep phase shifting holography based on CS. (g) Reconstruction from 3276 (80%) measurements using current method of twostep phase shifting holography based on CS. (h) Reconstruction from 3686 (90%) measurements using current method of twostep phase shifting holography based on CS.
V. CONCLUSION
In this paper, we reconstruct an original image by recording two holograms in the optical domain using certain signal recovery algorithms of CS theory and hologram reconstruction techniques, which greatly enhances the hologram photography and recovery efficiency. First, we can directly acquire the compressed hologram using a photodiode and never need the CMOS or CCD to record the hologram during the acquisition step, which reduces the complexity and size of holographic imaging devices. In addition, speed of the reconstruction time is faster than the four step method. Realtime performance in twostep phase shifting is also better than that in fourstep phase shifting. It will make coherent dynamic imaging possible. Our method is useful for substantially reducing the bandwidth or storage requirements of detector pixels and the scanning effort in the acquisition step. The most significant advantage about the holographic imaging system design is the enhancement of detector efficiency, because only significant components in the images’ transformed space, which are important for people’s visual perception, are collected. Our current work involves improving the quality of the image reconstruction according to the feature of hologram and CS theory, in the meantime, we are building an experimental platform for verification.
Acknowledgements
This work was supported by the Project of Department of Education of Guangdong Province, China (Research on method and application of compressive imaging with coherent light illumination). The authors wish to express their deepest thanks to the editor and reviewers for giving us constructive suggestions that have helped us both in the English accuracy and in the depth of discussion to improve the quality of this paper.
Naughton T. J
,
Javidi B
2004
“Compression of encrypted threedimensional objects using digital holography,”
Opt. Eng
43
2233 
2238
DOI : 10.1117/1.1783280
Darakis E
,
Soraghan J. J
2006
“Use of fresnelets for phaseshifting digital hologram compression,”
IEEE Trans. Image Process
15
3804 
3811
DOI : 10.1109/TIP.2006.884918
Shortt A. E
,
Naughton T. J
,
Javidi B
2007
“Histogram approaches for lossy compression of digital holograms of threedimensional objects,”
IEEE Trans. Image Process
16
1548 
1556
DOI : 10.1109/TIP.2007.894269
Seo Y. H
,
Choi H. J
,
Yoo J. S
,
Lee G. S
,
Kim C. H
,
Lee S. H
,
Kim D. W
2010
“Digitalhologram compression technique by eliminating spatial correlations based on MCTF,”
Opt. Commun.
283
4261 
4270
DOI : 10.1016/j.optcom.2010.06.052
Candes E. J
,
Romberg J
,
Tao T
2006
“Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,”
IEEE Trans. Inf. Theory
52
489 
509
DOI : 10.1109/TIT.2005.862083
Duarte M. F
,
Davenport M. A
,
Takhar D
,
Laska J. N
,
Sun T
,
Kelly K. F
,
Baraniuk R. G
2008
“Singlepixel imaging via compressive sampling,”
IEEE Signal Process. Mag.
25
83 
91
DOI : 10.1109/MSP.2007.914730
Takhar D
,
Laska J. N
,
Wakin M. B
,
Duarte M. E
,
Baron D
,
Sarvotham S
,
Kelly K. E
,
Baraniuk R. G
,
Bouman C. A
,
Miller E. L
,
Pollak I
2006
“A new compressive imaging camera architecture using opticaldomain compression,”
Brady D. J
,
Choi K
,
Marks D. L
,
Horisaki R
,
Lim S
2009
“Compressive holography,”
Opt. Express
17
13040 
13049
DOI : 10.1364/OE.17.013040
Marim M
,
Angelini E
,
OlivoMarin J. C
,
Atlan M
2011
“Offaxis compressed holographic microscopy in lowlight conditions,”
Opt. Lett.
36
79 
81
DOI : 10.1364/OL.36.000079
Horisaki R
,
Tanida J
,
Stern A
,
Javidi B
2012
“Multidimensional imaging using compressive Fresnel holography,”
Opt. Lett.
37
2013 
2015
DOI : 10.1364/OL.37.002013
Rivenson Y
,
Rot A
,
Balber S
,
Stern A
,
Rosen J
2012
“Recovery of partially occluded objects by applying compressive Fresnel holography,”
Opt. Lett.
37
1757 
1759
DOI : 10.1364/OL.37.001757
Shaked N. T
,
Katz B
,
Rosen J
2009
“Review of threedimentional holographic imaging by multipleviewpointprojection based methods,”
Appl. Opt.
48
H120 
H136
Rivenson Y
,
Stern A
,
Rosen J
2011
“Compressive multiple view projection incoherent holography,”
Opt. Express
19
6109 
6118
DOI : 10.1364/OE.19.006109
Lim S
,
Marks D
,
Brady D
2011
“Sampling and processing for compressive holography,”
Appl. Opt.
50
H75 
H86
Li J
,
Wang Y
,
Li R
,
Li Y
2013
“Singlepixel holographic 3D imaging system based on compressive sensing,”
Digital Holography and 3D Imaging Technical Digest ©, OSA
Li J
,
Wang Y
,
Li R
,
Li Y
2013
“Coherent singledetector 3D imaging system,”
Proc. SPIE
8913
891303 
Clemente P
,
Duran V
,
Tajahuerce E
,
Andres E
,
Climent V
,
Lancis J
2013
“Compressive holography with a singlepixel detector,”
Opt. Lett.
38
2524 
2527
DOI : 10.1364/OL.38.002524
Meng X. F
,
Cai L. Z
,
Xu X. F
,
Yang X. L
,
Shen X. X
,
Dong G. Y
,
Wang Y. R
2006
“Twostep phaseshifting interferometry and its application in image encryption,”
Opt. Lett.
31
1414 
1416
DOI : 10.1364/OL.31.001414
Liu J.P
,
Poon T.C
2009
“Twosteponly quadrature phaseshifting digital holography,”
Opt. Lett.
34
250 
252
DOI : 10.1364/OL.34.000250
Li J
,
Zheng T
,
Liu Q.Z
,
Li R
2012
“Doubleimage encryption on joint transform correlator using twosteponly quadrature phaseshifting digital holography,”
Opt. Commun.
285
1704 
1709
DOI : 10.1016/j.optcom.2011.11.115