Structured transmittance illumination coherence
Aditya Chandra Mandal1,2, Tushar Sarkar1, Zeev Zalevsky3, and Rakesh Kumar Singh1,*
Laboratory of Information Photonics and Optical Metrology, Department of Physics, Indian Institute of Technology
(Banaras Hindu University), Varanasi, 221005, Uttar Pradesh, India
2Department of Mining Engineering, Indian Institute of Technology (Banaras Hindu University), Varanasi, 221005,
Uttar Pradesh, India
3Bar-Ilan University, Faculty of Engineering and Nano Technology Center, Ramat-Gan, Israel
The coherence holography offers an unconventional way to reconstruct the hologram where an incoherent light illumination is
used for reconstruction purposes, and object encoded into the hologram is reconstructed as the distribution of the complex
coherence function. Measurement of the coherence function usually requires an interferometric setup and array detectors.
This paper presents an entirely new idea of reconstruction of the complex coherence function in the coherence holography
without an interferometric setup. This is realized by structured pattern projections on the incoherent source structure and
implementing measurement of the cross-covariance of the intensities by a single-pixel detector. This technique, named
structured transmittance illumination coherence holography (STICH), helps to reconstruct the complex coherence from the
intensity measurement in a single-pixel detector without an interferometric setup and also keeps advantages of the intensity
correlations. A simple experimental setup is presented as a ﬁrst step to realize the technique, and results based on the
computer modeling of the experimental setup are presented to show validation of the idea.
Since its inception more than seventy years ago, holography offers a powerful tool for imaging and light synthesis
availability of high-quality array detectors and reconstruction algorithms has further revolutionized the holography, and optical
reconstruction in the holography is replaced by the digital means, and technique is called digital holography (DH). The DH
keeps inherent advantages of the holography concerning the complex amplitude distribution and additionally offers a simpliﬁed
reconstruction of the hologram by a numerical means
. Conventional DH records and reconstructs the complex waveﬁeld by an
optical ﬁeld distribution itself. The phase information of the waveﬁeld is an important physical parameter of the light. Among
various methods to recover the phase information, the DH is a well-established technique for quantitative phase imaging (QPI).
Various QPI techniques have been developed, and signiﬁcant among them are in-line, off-axis, and phase-shifting holography.
In recent years, attempts have been made to improve the transverse spatial resolution in the DH by random
. Requirements to retrieve information from the self-luminous or incoherent object have also inspired new
trends in the holography12–18.
In a signiﬁcant development, Takeda and co-workers have developed an unconventional holography called coherence
holography (CH), where the information of the complex ﬁeld is reconstructed as a distribution of the spatial coherence
the hologram is reconstructed by an incoherent light illumination rather than by a coherent light. The CH has opened new
research directions on recording and shaping the spatial coherence for applications such as spatial coherence tomography,
proﬁlometry, imaging and coherence current
. Principle of the CH is derived from the van Cittert-Zernike theorem, which
connects the incoherent source structure with a far-ﬁeld spatial coherence of the light. The main task in the CH is to design
an appropriate interferometer for the measurement of spatial coherence. These interferometers mainly employ second-order
correlation or fourth-order correlations measurement by the array detectors. In contrast to the interferometers based on
the second-order correlations, the fourth-order correlation, i.e., intensity interferometers are highly stable for the coherence
. Recently, Naik et al.
made use of the fourth-order correlation in the hologram, and technique is called a
photon correlation holography (PCH). Basic principle of the PCH is derived from a connection between the cross-covariance of
the intensities with the modulus square of the Fourier spectrum of the incoherent structure. However, phase information of
the spectrum is lost in the PCH in contrast to the CH and DH techniques. Recovery of complex ﬁeld parameters in the CH
arXiv:2109.10326v1 [physics.optics] 21 Sep 2021
and DH is possible by employing an interferometer setup which makes the system bulky and prone to external disturbances
. Recently, holographic methods based on interference of the coherence waves have been proposed
to overcome the phase loss issue in the PCH experiments
. However, the interferometric systems bring bulkiness in the
experimental implementation and also require ﬂexible control over the reference ﬁeld to get the interference fringes in the
cross-covariance function at the array detectors plane.
On the other hand, signiﬁcant attempts have been made to develop computational imaging techniques such as single-pixel
imaging with random and structured ﬁeld illumination over the past few years
. In contrast to using conventional cameras
and two-dimensional array detectors, single-pixel techniques make use of projection of light patterns onto a sample while a
single-pixel detector measures the light intensity collected for each pattern. Therefore, stage of spatial sampling is moved
from the camera to the programmable diffraction element where the structured patterns are loaded. Single-pixel imaging
techniques have brought advantages such as use of a non-visible wavelength or precise time resolution, which can be costly and
practically challenging to realize as a pixilated imaging device. Recently, a combination of optical and computation channels
has been developed for the reconstruction of the three-dimensional (3D) amplitude object from a single-pixel detector, and
technique is called hybrid correlation holography (HCH)
. This technique makes use of cross-covariance of the intensity and
is derived from the connection between complex coherence function and intensity correlation for Gaussian random ﬁeld. A new
scheme based on the recovery of the complex-valued object in a modiﬁed HCH scheme with an interferometric setup has been
In this paper, we present a new technique for the reconstruction of the complex ﬁeld within the framework of the PCH and
present a new theoretical basis for the reconstruction in correlation holography. This approach equips correlation imaging with
a complete wavefront reconstruction without an interferometric setup but keeping the advantage of the intensity correlation. For
this purpose, a structured light illumination is projected on the incoherent structure, and a far-ﬁeld spectrum is measured by a
single-pixel detector. A complex Fourier spectrum from the intensities is successfully obtained from the four-step phase shifting
in the structured illumination. Although the Fourier spectrum measurement in the CH is based on the Hanbury Brown-Twiss
(HBT) approach with a single-pixel detector but active illumination strategy in the proposed technique helps to overcome the
phase loss problem of the typical HBT approach. Applying two-dimensional (2D) Fourier transform (IFT) to the obtained
spectrum yields the desired DH. The phase-shifting illumination approach also brings the elimination of noise that is statistically
the same. A detailed theoretical foundation and implementation of the proposed technique in comparison to the CH and PCH
are discussed below.
Basic principles of the CH and PCH have been discussed in detail in Ref.
. However, for the sake of continuity and to
connect with basic principle of the proposed technique, we brieﬂy describe the CH and PCH. Fig. 1(a) represents a coherent
recording of the complex ﬁeld of an object in the Fourier hologram
. The hologram
is read out with the incoherent
light, as shown in Fig. 1(b). To describe the reconstruction process of the hologram in Fig. 1(b), consider the complex ﬁeld of
light immediately behind the hologram as
E(r) = H(r)exp[iφ(r)] (1)
denotes the imaginary unit and
H(r) = |H(r)|exp[iδ(r)]
being the amplitude transmittance of the
hologram and deterministic phase of the readout light, respectively. The spatial vector at the source is
. The random
phase inserted in the light path to destroy spatial coherence by the rotating ground glass (RGG) is represented by
at a ﬁxed
. A lens in Fig. 1(b) with focal distance
is used to Fourier transform the randomly scatted light ﬁeld from the source,
and the complex ﬁeld on the observation plane becomes
E(k) = ZE(r)exp[−i k ·r]dr =ZH(r)exp[iφ(r)]exp[−i k ·r]dr (2)
is spatial frequency coordinate at the observation point. Two-point correlation of the random ﬁeld is
W(k1,k2) = hE∗(k1)E(k2)i
=ZZ H∗(k1)H(k2)hexp[i(φ(k2)−φ(k1))]iexp[−i(k2·r2−k1·r1)]dr2d r1
represents the ensemble average which will be replaced by the temporal average in the experiment. The rotating
ground glass is considered to produce an incoherent source, i.e.
hexp[i(φ(r2)−φ(r1))]i ≡ δ(r2−r1)
Figure 1. (a) Represents coherent recording of complex ﬁeld of an object located at Z=0 , Lis a lens with focal distance f
(b) Represents the reconstruction process of the hologram with incoherent light, where
is a lens with focal distance
, RGG is
rotating ground glass,
is hologram ; (c) Represents the PCH setup: intensity correlation holography scheme; (d) Represents
the STICH setup, where SL is structured light pattern, His hologram, Dis single-pixel detector
Therefore, Eq. 3transforms into the van Cittert Zernike theorem as
F(k) = ZI(r)exp[−i k ·r]dr (4)
I(r) = H∗(r)H(r)
is the source hologram placed at the RGG plane and
represents the Fourier spectrum of the
incoherent source at the far-ﬁeld. The basic principle of the CH is described by Eq. 4and therefore provides reconstruction of
the object as the distribution of the complex coherence function. The random ﬁeld intensity at the observation plane, at a ﬁxed
time tcorresponding to one rotation state of the RGG, is represented as
I(k) = |E(k)|2(5)
The random intensity pattern
is having no direct resemblance to reconstruction of the object. The cross-covariance of
the intensities of the Gaussian random ﬁeld is given as
∆I(k) = I(k)− h∆I(k)i
is the ﬂuctuation of the intensities with respect to its average mean value. Eq. 6highlights the
basic principle of the PCH and is sketched in Fig. 1(c), wherein the phase part of the coherence function is lost. To circumvent
the above-mentioned issue on recovery of complex phase with the only measurement of the cross-covariance of the intensities,
we present a new technique called STICH. Basic principle of the STICH is represented in Fig. 1(d) and described as follows. A
two dimensional (2-D) structured illumination with its spatial frequency
and initial phase
is projected on the RGG.
This structured illumination is a sinusoidal pattern and is represented as
θ(x,y;kx,ky) = a+b·cos(kxx+kyy+θ)(7)
is an un-modulated term of the illumination pattern, and
represents the contrast. The light coming out of the structured
transparency propagates through the RGG, which is used to mimic an incoherent light source. A hologram
is placed next to
the RGG, as shown in Fig. 1(d). Therefore, the instantaneous complex ﬁeld immediately after the His expressed as:
Eθ(x,y;kx,ky) = H(x,y)exp(iφ(x,y))(a+b·cos(kxx+kyy+θ)) (8)
The instantaneous complex ﬁeld at the single-pixel detector is represented as
Eθ(kx,ky) = En+wZZΩ
represents the illuminated area,
is a scale factor whose value depends on the size and the location of the detector,
represents the response of background illumination. The instantaneous random intensity at the single-pixel detector is given as
Iθ(kx,ky) = |Eθ(kx,ky)|2(10)
The random intensity variation from its mean intensity is calculated as
∆Iθ(kx,ky) = Iθ(kx,ky)− hIθ(kx,ky)i(11)
where the angular bracket
denotes the ensemble average and
is mean intensity. The cross-covariance of the
Dθ(kx,ky) = h∆Iθ(kx,ky)∆Iθ(kx,ky)i(12)
The 4-step phase-shifting approach allows each complex Fourier coefﬁcient
to be obtained by every four responses
corresponding to the illumination patterns, i.e.
. The response
are used to obtain
the Fourier spectrum of H, i.e. F(kx,ky)as
[D0(kx,ky)−Dπ(kx,ky)] + i·Dπ/2(kx,ky)−D3π/2(kx,ky)
=2bwRRΩH(x,y)·exp[−i(kxx+kyy)] d xdy (13)
The Fourier coefﬁcient is expressed as
F(kx,ky) = RRΩH(x,y)·exp[−i(kxx+kyy)] dxdy
2bw [D0(kx,ky)−Dπ(kx,ky)] + i·Dπ/2(kx,ky)−D3π/2(kx,ky)(14)
By computing Fourier coefﬁcients
(i.e., the Fourier spectrum) using Eq. 14 for a complete set of
, the desired complex
ﬁeld distribution is reconstructed. A 4-step phase-shifting sinusoid illumination plays an essential role in the proposed technique.
Eq. 13 can not only assemble the Fourier spectrum of the desired hologram image but also eliminate undesired direct current
Experimental design and algorithm
A possible experimental design for the proposed technique is shown in Fig. 2. A monochromatic collimated laser light is folded
by a beam splitter (BS) and incident on a spatial light modulator (SLM). The SLM in Fig. 2 is considered to be a reﬂective type
and loaded with the
number of sinusoidal gratings in a sequence. The sinusoidal pattern displayed to the SLM is inserted
into the incident beam, and subsequently, this structured light transmits through the BS and illuminates the RGG. The RGG
introduces randomness in the incident structured light. As shown in Fig. 1(a), a computer-generated hologram of an off-axis
object is used as transparency and placed adjacent to the RGG. The structured pattern embedded in the stochastic ﬁeld due to
the RGG illuminates a hologram
. Scattering of the light through the RGG generates a stochastic ﬁeld with the Gaussian
statistics. The scattered light further propagates and is Fourier transformed by a lens
at the single-pixel detector plane
Corresponding to the spatial frequency and initial phase of the loaded structured pattern, an instantaneous random ﬁeld at
the single-pixel detector is represented by
. The instantaneous signal at the detector is represented as
After the single-pixel measurement for a particular random phase mask, we stored the value in our personal computer (PC)
for post-processing. Due to the Gaussian statistics, the cross-covariance of the intensities is estimated at the single-pixel
corresponding to different sets of random phase masks introduced by the RGG. The cross-covariance of the intensities at the
single-pixel detector for a given frequency pair
and initial phase
is represented as
. For a complete set of
Fourier coefﬁcients, we illuminate the hologram
by the structured patterns with full sets of spatial frequency
Each complex Fourier coefﬁcient corresponding to that spatial frequency is extracted by using a 4-step phase-shifting approach.
The number of Fourier coefﬁcients in the Fourier domain is the same as the number of pixels in the spatial domain.
The algorithm proposed in this paper is an iterative heuristic that aims to reconstruct complex object encoded into the
hologram as the distribution of the complex coherence function. In contrast to the previously reported CH, here we report the
use of the structured illumination at the RGG plane. This strategy makes reconstruction procedure completely different from
previously developed reconstruction approach and also equips us to extract the complex ﬁeld even from a single-pixel detector.
This is a unique feature of our proposed technique that helps to recover the complex coherence without interferometry. The
algorithm in our work is implemented using MATLAB and simulated on a personal computer. Fig. 3 shows the steps of the
algorithm, which are:
1. A hologram of size N×Npixels is taken as transparency.
Figure 2. Experimental conﬁguration: BS- beam splitter, SLM- spatial light modulator, RGG- rotating ground glass, T-
transparency, L- lens with focal distance f, D- single-pixel detector, PC- personal computer.
2. Construction of sinusoidal patterns and random phase masks :
(a) Total Mnumber of different random phase masks of same size of hologram are generated.
The 2-D sinusoidal patterns of size
pixels with initial phase
are constructed by
considering discretized spatial frequency space −k<kx,ky<k.
3. Iterative steps:
First, a 2-D sinusoidal pattern for that particular frequency pair
is taken. A single-pixel detector is used to
sense the random light ﬁeld ( matrix multiplication of hologram and sinusoidal pattern and random phase mask ),
and corresponding random intensity is represented using Eq. 10.
In such a way, other intensity patterns at single-pixel detector corresponding to different sets of random phase
masks introduced by the RGG are obtained. Random intensity variations from its mean intensity are calculated
using Eq. 11. Cross-covariance of the intensity for the taken spatial frequency pair
is calculated using Eq.
Calculation of each complex valued Fourier coefﬁcient
corresponding to the spatial frequency pair is
simulated from four responses Dθ(θ=0,π/2,π,3π/2)( see Eq. 14).
For the complete set of desired
size Fourier coefﬁcients, we need to iterate the above steps over the discretized
spatial frequency space unless the last iteration is obtained. Total
number of sinusoidal patterns need
to be projected, including the four-step phase-shifting by an SLM shown in Fig. 2, on which the patterns are controlled
by a personal computer (PC) directly. A Fourier coefﬁcients
is obtained with 4 measurements. So, basically,
for fully sampling
size Fourier coefﬁcients consumes
measurements of single-pixel
detector to reconstruct the complex object.
Figure 3. Proposed STICH algorithm steps.
Here, we present computational results processed using MATLAB for the validation of our proposed experimental setup.
The phase information of the waveﬁeld is a critical parameter to examine a complex ﬁeld of an object. Fig 4.(a),(c) and
(b),(d) show both amplitude and phase distributions of objects, letter “P” and number “3” respectively, which are directly
reconstructed from DH. Applying two-dimensional Fourier transform on the DH brings out three spectra: a non-modulating
central DC term, the desired spectrum, and its off-axis local and unit conjugate. Location of off-axis spectrum is governed by
carrier frequency as shown in recording of DH in Fig. 1(a). The unwanted DC terms containing high-frequency content are
digitally suppressed in Fig 4 to highlight the objects located in off-axis position. Fig 5 and Fig 6 represent the reconstructed
complex ﬁelds from holograms using the STICH technique at different numbers of random phase masks (
) and the quality of
reconstruction depends on it. In order to examine the effect of
on reconstruction quality of STICH, we evaluate visibility
) and reconstruction efﬁciency (
for three different numbers of
and results are given in Table 1.
Amplitude and phase distributions of number “3” are shown in Fig 5. (a)-(c) and (d)-(f) for
for letter “P”, Fig 6. (a)-(c) and (d)-(f) show the amplitude and phase distributions for
. In both object’s
reconstruction central DC terms are digitally suppressed as shown in Fig 5 and 6. The central DC in the reconstruction appears
because use of off-axis hologram as the transparency. The visibility of a target reconstruction is deﬁned as the degree to which it
can be distinguished from background noise. It is calculated as the ratio of the average image intensity level in the signal region
to the average background intensity level. Here Otsu’s method44 is used as a global threshold to identify the signal region.
In order to reconstruct complex ﬁelds of size
from hologram of size
using STICH, structured
illumination patterns are generated according to Eq. 7, where
, spatial frequencies range is
at steps of 0.0505.
Figure 4. Recovery of complex ﬁelds without RGG from holograms of objects, letter “P” and number “3” (a) Amplitude
distribution of letter “P” (b) Phase distribution of letter “P” (c) Amplitude distribution of number “3” (d) Phase distribution of
The calculated visibility value for ﬁg. 4.(a) and (c) are 127.6 and 64.7. The calculated visibility and reconstruction efﬁciency
value for ﬁg. 5. (a)-(c) and for ﬁg. 6.(a)-(c) are given in Table 1. (I) and (II) respectively. The upper part of conjugate phase
distributions in Fig. 5 and Fig. 6 are highlighted with white color annular rings. From Table 1 and Fig 5 and Fig 6, it can be
seen that reconstruction quality improves with increase of value of M.
(a) (b) (c)
(d) (e) (f)
Figure 5. Recovery of the complex ﬁelds using STICH for three different Mvalues: an off-axis hologram of number “3” is
used as transparency (a), (d) Amplitude and phase distribution of object for
; (b), (e) Amplitude and phase distribution
of object for M=500; (c), (f) Amplitude and phase distribution of object for M=1000.
(a) (b) (c)
(d) (e) (f)
Recovery of the complex ﬁelds using STICH for three different
values: an off-axis hologram of letter “P” is used
as transparency (a), (d) Amplitude and phase distribution of object for M=200; (b), (e) Amplitude and phase distribution of
object for M=500; (c), (f) Amplitude and phase distribution of object for M=1000.
In conclusion, a new technique entitled STICH is presented to reconstruct the complex coherence from the intensity measure-
ment with a single-pixel detector and without an interferometric setup. This brings the advantages of compatibility in the
reconstruction of complex ﬁelds in the correlation-based imaging system. A experimental conﬁguration and computational
model of it is described to validate our idea. We have demonstrated the reconstructions of complex ﬁelds of objects at different
random phase masks and the quality of reconstruction depends on the value of number of random phase masks used to realize
the thermal light source. The proposed technique is expected to provide new direction on the coherence holography and imaging
through scattering medium.
1. Gabor, D. A new microscopic principle. Nature 161, 777–778 (1948).
2. Leith, E. N. & Upatnieks, J. Reconstructed wavefronts and communication theory. JOSA 52, 1123–1130 (1962).
Yamaguchi, I. Phase-shifting digital holography. In Digital Holography and Three-Dimensional Display, 145–171
Hendry, D. Digital holography: Digital hologram recording, numerical reconstruction and related techniques, u. schnars, w.
jueptner (eds.), springer, berlin (2005),(164 pp.+ ix.,$ 69.95), isbn: 354021934-x (2006).
5. Park, Y. et al. Speckle-ﬁeld digital holographic microscopy. Opt. express 17, 12285–12292 (2009).
Vinu, R., Chen, Z., Pu, J., Otani, Y. & Singh, R. K. Speckle-ﬁeld digital polarization holographic microscopy. Opt. letters
44, 5711–5714 (2019).
7. Choi, Y. et al. Dynamic speckle illumination wide-ﬁeld reﬂection phase microscopy. Opt. letters 39, 6062–6065 (2014).
Gao, P., Pedrini, G. & Osten, W. Structured illumination for resolution enhancement and autofocusing in digital holographic
microscopy. Opt. letters 38, 1328–1330 (2013).
Ma, J., Yuan, C., Situ, G., Pedrini, G. & Osten, W. Resolution enhancement in digital holographic microscopy with
structured illumination. Chin. Opt. Lett. 11, 090901 (2013).
Micó, V., Zheng, J., Garcia, J., Zalevsky, Z. & Gao, P. Resolution enhancement in quantitative phase microscopy. Adv. Opt.
Photonics 11, 135–214 (2019).
11. Gao, P. et al. Recent advances in structured illumination microscopy. J. Physics: Photonics (2021).
12. Rosen, J. & Brooker, G. Digital spatially incoherent fresnel holography. Opt. letters 32, 912–914 (2007).
Tahara, T., Kanno, T., Arai, Y. & Ozawa, T. Single-shot phase-shifting incoherent digital holography. J. Opt.
Bouchal, P., Kapitán, J., Chmelík, R. & Bouchal, Z. Point spread function and two-point resolution in fresnel incoherent
correlation holography. Opt. express 19, 15603–15620 (2011).
15. Rosen, J. & Brooker, G. Fluorescence incoherent color holography. Opt. Express 15, 2244–2250 (2007).
Quan, X., Matoba, O. & Awatsuji, Y. Single-shot incoherent digital holography using a dual-focusing lens with diffraction
gratings. Opt. letters 42, 383–386 (2017).
Watanabe, K. & Nomura, T. Recording spatially incoherent fourier hologram using dual channel rotational shearing
interferometer. Appl. optics 54, A18–A22 (2015).
Rosen, J., Brooker, G., Indebetouw, G. & Shaked, N. T. A review of incoherent digital fresnel holography. J. Hologr.
speckle 5, 124–140 (2009).
19. Takeda, M., Wang, W., Duan, Z. & Miyamoto, Y. Coherence holography. Opt. express 13, 9629–9635 (2005).
Duan, Z., Miyamoto, Y. & Takeda, M. Dispersion-free optical coherence depth sensing with a spatial frequency comb
generated by an angular spectrum modulator. Opt. express 14, 12109–12121 (2006).
Takeda, M., Wang, W., Naik, D. N. & Singh, R. K. Spatial statistical optics and spatial correlation holography: a review.
Opt. Rev. 21, 849–861 (2014).
Naik, D. N., Ezawa, T., Singh, R. K., Miyamoto, Y. & Takeda, M. Coherence holography by achromatic 3-d ﬁeld correlation
of generic thermal light with an imaging sagnac shearing interferometer. Opt. express 20, 19658–19669 (2012).
Soni, N. K., Vinu, R. & Singh, R. K. Polarization modulation for imaging behind the scattering medium. Opt. letters
Wang, W. & Takeda, M. Coherence current, coherence vortex, and the conservation law of coherence. Phys. review letters
96, 223904 (2006).
25. Goodman, J. W. Statistical optics (John Wiley & Sons, 2015).
Naik, D. N., Singh, R. K., Ezawa, T., Miyamoto, Y. & Takeda, M. Photon correlation holography. Opt. express
Singh, R. K. & Sharma M, A. Recovery of complex valued objects from two-point intensity correlation measurement.
Appl. Phys. Lett. 104, 111108 (2014).
Singh, R. K., Vyas, S. & Miyamoto, Y. Lensless fourier transform holography for coherence waves. J. Opt.
Kim, K., Somkuwar, A. S., Park, Y., Singh, R. K. et al. Imaging through scattering media using digital holography. Opt.
Commun. 439, 218–223 (2019).
Somkuwar, A. S., Das, B., Vinu, R., Park, Y. & Singh, R. K. Holographic imaging through a scattering layer using speckle
interferometry. JOSA A 34, 1392–1399 (2017).
Chen, L., Singh, R. K., Chen, Z. & Pu, J. Phase shifting digital holography with the hanbury brown–twiss approach. Opt.
Lett. 45, 212–215 (2020).
Chen, L., Chen, Z., Singh, R. K., Vinu, R. & Pu, J. Increasing ﬁeld of view and signal to noise ratio in the quantitative
phase imaging with phase shifting holography based on the hanbury brown-twiss approach. Opt. Lasers Eng.
Zhang, Z., Ma, X. & Zhong, J. Single-pixel imaging by means of fourier spectrum acquisition. Nat. communications
Martínez-León, L. et al. Single-pixel digital holography with phase-encoded illumination. Opt. express
Horisaki, R., Matsui, H., Egami, R. & Tanida, J. Single-pixel compressive diffractive imaging. Appl. Opt.
Edgar, M. P., Gibson, G. M. & Padgett, M. J. Principles and prospects for single-pixel imaging. Nat. photonics
Shin, S., Lee, K., Baek, Y. & Park, Y. Reference-free single-point holographic imaging and realization of an optical
bidirectional transducer. Phys. Rev. Appl. 9, 044042 (2018).
38. Hu, X. et al. Single-pixel phase imaging by fourier spectrum sampling. Appl. Phys. Lett. 114, 051102 (2019).
39. Ota, K. & Hayasaki, Y. Complex-amplitude single-pixel imaging. Opt. letters 43, 3682–3685 (2018).
40. Singh, R. K. Hybrid correlation holography with a single pixel detector. Opt. letters 42, 2515–2518 (2017).
Gibson, G. M., Johnson, S. D. & Padgett, M. J. Single-pixel imaging 12 years on: a review. Opt. Express
Chen, Z., Singh, D., Singh, R. K. & Pu, J. Complex ﬁeld measurement in a single pixel hybrid correlation holography. J.
Phys. Commun. 4, 045009 (2020).
Hillman, T. R. et al. Digital optical phase conjugation for delivering two-dimensional images through turbid media. Sci.
reports 3, 1–5 (2013).
Otsu, N. A threshold selection method from gray-level histograms. IEEE transactions on systems, man, cybernetics
The work is supported by the Science and Engineering Research Board (SERB) India- CORE/2019/000026. T.S acknowledges
support from University Grant Commission (UGC), India for his scholarship.
Author contributions statement
A.C.M conceived of idea and build the theoretical basis, experimental design, and completed simulation and preparation of
manuscript. T.S involved in experimental design, simulation, preparation of manuscript. Z.Z provided advice and assistance,
reviewing and editing the work. R.K.S involved in supervision, formulation of research goals and aims, funding acquisition,
reviewing, and editing.