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Chapter
Coded Aperture Correlation
Holography (COACH) - A Research
Journey from 3D Incoherent
Optical Imaging to Quantitative
Phase Imaging
Joseph Rosen, Angika Bulbul, Nathaniel Hai and Mani R. Rai
Abstract
Coded aperture correlation holography (COACH) combines incoherent digital
holography with coded aperture imaging. COACH is also a method to record incoherent digital holograms of three-dimensional object scenes. Still, COACH can be used for
several other incoherent and coherent optical applications. In this chapter, we survey
the prime landmarks on the topic of COACH from two major perspectives: architectures and applications of the various systems. We explore the main configurations of
hologram recorders in the COACH systems. For each design, we describe some of the
recent implementations of these recorders in optical imaging. We conclude the chapter with general ideas on this technology.
Keywords: incoherent holography, digital holography, Fresnel incoherent correlation
holography, digital holographic microscopy, phase-shifting interferometry
1. Introduction
Imaging by optical waves has been known in the technology world for centuries
[1]. For most of this time, imaging has been direct in the sense that images recorded
on the eye retina, photographic film, or electronic sensor have been replicas of the
observed scenes. However, the computing revolution of the second half of the twentieth century has opened many possibilities for indirect rather than direct imaging. In
indirect imaging, a modified version of the observed scene is transferred from the
image sensor to the computer to process and reconstruct the image of the original
scene. One of the indirect imaging methods is coded aperture imaging, proposed in
the sixties for X-ray imaging [2–4] and later adapted to the visible light using coded
phase-masks [5] instead of an array of randomly distributed pinholes used in X-ray
imaging [4].
Digital holography [6–8] can also be classified as indirect imaging, although it is
special in the sense that the pattern recorded by the image sensor is an interference
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Holography - Recent Advances and Applications
pattern between two light beams. At least one of the beams originates from the object.
However, in the case of incoherent digital holography by self-interference, both
interfering beams originate from the object [7]. In 2016, the two different concepts of
coded phase-aperture imaging and incoherent digital holography were combined into
a new indirect imaging method, dubbed coded aperture correlation holography
(COACH) [9]. COACH merges the merits of these two different imaging modalities
and enables three-dimensional (3D) imaging with interesting and unexpected features. More specifically, COACH is an electro-optical technique to record digital holograms of two- and three-dimensional scenes, where at least part of the light from the
object passes through a coded phase-mask. COACH was initially proposed as an
additional method to record incoherent digital holograms without scanning and
evolved in several different directions. The COACH concept was inspired by several
previous methods and systems [2–5, 10–12] and has already stimulated several studies
since then [13–33]; some of them are mentioned in the following. This chapter provides an overview of research activities in the technology of COACH done by several
researchers in the field.
About a year after the invention of COACH, a simpler version of it was proposed.
This version operates without two-wave interference and can demonstrate some
applications, such as 3D imaging. The modified version was called interferenceless
COACH (I-COACH) [15]. Usually, the I-COACH is preferred whenever an application can be performed by both COACH and I-COACH with the same quality. This rule
of thumb is reasonable because the calibration of a single wave system, such as ICOACH, is simpler, and its noise immunity is higher than that of an incoherent
interferometer, such as COACH. However, not all the applications successfully
implemented by COACH can be executed by I-COACH, and examples are given in the
following.
COACH and I-COACH were initially invented for 3D imaging of incoherently
illuminated scenes. Recently, the concept of coherent COACH with and without
two-wave interference has been examined [34–36]. A central application of coherent
digital holography is quantitative phase imaging (QPI) [37], and hence in the
following, we review different ways to implement QPI using COACH [35, 36].
This review consists of six main sections. The development of incoherent COACH
and I-COACH architectures, with different modalities and characteristics, are
reviewed in the following two sections. We describe the coherently illuminated
I-COACH and COACH techniques in the fourth and fifth sections, respectively. The
concluding section summarizes the review.
2. Incoherent COACH
Incoherent COACH belongs to the family of self-interference digital holography
systems [38]. The general optical configuration of these systems is shown in Figure 1.
The flow of information starts from the light emitted from each object point in the
upper part of Figure 1. The light propagates toward a beam splitting unit and is split
into two waves. Each wave is modulated differently by a modulation component. The
two waves originate from the same object point and hence are mutually coherent,
although the light emitted, or reflected, from the object is spatially incoherent. Therefore because of the mutual coherence, the two waves with different wavefronts
interfere at the sensor plane. The image sensor accumulates the entire interference
patterns of all the input points to an incoherent hologram. A single hologram, or
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Figure 1.
Recording and reconstruction of holograms in a general self-interference digital holography system.
several acquired holograms, are introduced into a digital computer, where the various
operations of the digital processor are schematically shown in the lower part of
Figure 1. In the case of several holograms, they are superposed into a single digital
hologram. Finally, the image of the object is reconstructed from the processed hologram by an appropriate numerical algorithm.
COACH was proposed as a generalized case of Fresnel incoherent correlation
holography (FINCH) [10–12], a well-known technique of recording holograms, which
also belongs to the self-interference systems. In FINCH, a quadratic phase-mask
modulates at least one of the two waves. In COACH, on the other hand, the quadratic
phase-mask of FINCH is substituted by a diffractive chaotic phase-aperture. The
initial goal of COACH was like FINCH, that is, to acquire a hologram of the 3D
observed scene illuminated by quasi-monochromatic spatially incoherent light.
COACH’s optical scheme is depicted in Figure 2. The light from an object is split into
two beams, and only one of the object beams is modulated by the chaotic mask termed
coded phase-mask (CPM). The modulated beam is coherently interfered with the
unmodulated object beam due to their common origin. Because COACH is on-axis
system, it needs a phase-shifting procedure and complex hologram synthesis [39].
That means that three holograms are recorded, each of which with the CPM multiplied by a different phase-constant. The three holograms are superposed digitally in
the computer such that the result is a complex-valued hologram. This digital hologram
is reconstructed into a single image without the twin image and the bias term.
Unlike other well-known incoherent hologram recorders, such as FINCH
[10–12, 40] and Michelson-interferometer-based incoherent holographic systems
[41–44], COACH does not have a defined image plane where the wavefront can
numerically propagate from the hologram to the reconstruction plane. Hence, COACH
has different recording and reconstruction procedures. In other words, COACH consists
of a two-step recording procedure: a one-time calibration and then imaging. In the
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Holography - Recent Advances and Applications
Figure 2.
Schematic diagram of coded aperture correlation holography (COACH). SLM - spatial light modulator.
initial stage of the calibration, one illuminates a moving pinhole along the optical axis,
and the image sensor records a point spread hologram (PSH) for every axial location of
the pinhole. The set of PSHs is accumulated in a library for later use in the imaging
stage. Following the calibration process, an object hologram is recorded under the same
restrictions and with the coded apertures as the PSH acquisition. The 3D image of the
observed scene is reconstructed by a two-dimensional (2D) cross-correlation between
the object hologram and the corresponding elements of the PSH library.
Although FINCH influenced the COACH structure, COACH has different features
than FINCH. The image reconstruction has been modified to 2D cross-correlations
with guidestar responses instead of the Fresnel back-propagation of FINCH
[10–12, 40]. Compared to FINCH, COACH has better axial resolution but worse
lateral resolution [9, 45]. However, the main difference is that COACH can do the
same holographic 3D imaging without two-wave interference [15]. Nevertheless,
several applications can only be performed by a version of the original COACH with
two-wave interference. One of such applications is a one-channel-at-a-time incoherent synthetic aperture imager [46], summarized next.
2.1 One-channel-at-time incoherent synthetic aperture
An interesting application for COACH is incoherent imaging with synthetic aperture (SA). SA is a familiar super-resolution method and a conventional technique in
astronomy to accomplish image resolution beyond the diffraction limit dictated by the
physical aperture [47] of the telescope. Since its invention a century ago [48], incoherent SA imaging was usually realized by at least two optical channels operating
simultaneously. The wave interference between two incoming light beams, both originated from the same object, was recorded over time from several viewpoints within
the SA region. Then, the interference intensity patterns were processed to produce an
image of the object with a resolution equivalent to complete SA [22, 48]. A singlechannel SA is possible for cases of imaging systems with coherent light [49], but
astronomical imaging is usually done with incoherent light sources. A solution to this
double-channel problem of SA incoherent imaging is the lately proposed incoherent
single-channel SA technique termed one-channel-at-time incoherent synthetic aperture imager (OCTISAI) [46].
As in many other COACH systems, the CPM of OCTISAI is synthesized using a
modified version of the Gerchberg-Saxton algorithm (GSA) [50]. Then, the CPM is
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divided into N (in the following example N = 64) equal parts for the SA implementation. The optical setup of the OCTISAI experiment is shown in Figure 3 and described
next. The system is first calibrated by collimating the light diffracted from a pinhole,
where the collimating lens L1 mimics the far-field imaging condition. A polarizer P1
polarizes the collimated light to be oriented at 450 regarding the active orientation of a
spatial light modulator (SLM). The SLM is used as the display on which the CPMs of
OCTISAI and all other systems in this chapter are displayed. Only a partial area of the
SLM is used at a time, and all other parts are activated in a raster scan mode. Because
of the polarization angle, the light is split into two orthogonal linear polarizations
beyond the SLM. The CPM modulates one polarized wave, and the other wave passes
the SLM without any change. Beyond the polarizer P2, also oriented at 450 to the
SLM’s active axis, both beams have the same orientation enabling to record a pattern
of interference between the two beams. The interference pattern between the modulated and unmodulated beams is captured by the image sensor. Three phase-shifted
PSHs for the input point object (pinhole) are recorded for every partial aperture at
each position in the SA region. Then, three phase-shifted object holograms are captured for the input object with the same phase-apertures as before in the calibration.
Next, using the digital computation capabilities, the entire PSH parts are stitched
together into one synthetic PSH. The parts of the object hologram are also processed
into one synthetic object hologram by a similar procedure. The final image with the
enhanced resolution is obtained by a 2D cross-correlation between the two synthetic
holograms.
The complete experiment of OCTISAI is extensively described in [46], and here
we briefly describe only the main results. In the experiment, a collection of PSHs was
produced using three CPMs, each having a phase-shift exp(iθj), where θ1,2,3 = 0o, 120o,
and 240o. A pinhole of 25 μm diameter was positioned in the input. After the PSH
creation, group 3, element 1 of the United States Air Force (USAF) negative resolution
chart, replaced the pinhole. We recorded the three object holograms with the same
three CPMs used for the PSHs. The synthetic object holograms and PSHs were produced by stitching respective partial holograms and superimposing corresponding
Figure 3.
The tabletop experimental setup for one-channel-at-time incoherent synthetic aperture imager (OCTISAI) inside the
blue rectangle, BS1 and BS2 – beamsplitters, CMOS camera - Complementary metal-oxide-semiconductor camera,
L01, L02, and L1 - refractive lenses, LED1 and LED2 - identical light-emitting diodes, P1 and P2 – polarizers, SLM spatial light modulator, and USAF - United States Air Force resolution target. Adapted from [46].
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Holography - Recent Advances and Applications
Figure 4.
(a1, a2) COACH reconstructed images and (a3, a4) direct images of limited aperture (a1, a3) and full aperture
(a2, a4), magnitude and phase of (b1, b2) PSH and (b3, b4) object holograms of the complete SA, (c1-c6)
reconstructed images after stitching of (c1) 8 central horizontal sub-holograms, (c2) eight central vertical subholograms, (c3) 2 2, (c4) 4 4, (c5) 6 6 central sub-holograms, and (c6) full 64 (8 8) sub-holograms.
Adapted from [46].
synthetic three-intensity responses. Finally, the object hologram was cross-correlated
with the phase-only filtered version of the synthetic PSH. The outcome of this crosscorrelation is the final reconstructed image. The COACH images related to the partial
and complete apertures are shown in Figures 4(a1) and (a2), respectively. For comparison, Figures 4(a3) and (a4) show the corresponding images of direct imaging
with a setup of a single lens and similar numerical apertures. The stitched holograms
after the superposition are shown in Figure 4(b). Figure 4(c) presents the
reconstructed images for OCTISAI with various area sizes of the SA holograms.
Figures 4(c1) and (c2) are produced using the central eight, horizontally [4(c1)] and
vertically, [4(c2)] stitched partial holograms, respectively. Figures 4(c3)-(c6) show
the reconstruction results with 2 2, 4 4, and 6 6 central sub-holograms, and the
entire 64 sub-holograms. The resolution enhancement by raising the number of
stitched partial holograms is demonstrated. Comparing Figure 4(c6) with Figures 4
(a1) and (a3), one can conclude that OCTISAI’s images have higher resolution
than the images taken with a limited aperture in both techniques of COACH and
direct imaging.
3. Interferenceless incoherent COACH
As mentioned above, interferenceless coded aperture correlation holography
(I-COACH) was published in 2017 [15] as a simpler configuration of the earlier
proposed COACH [9]. Both systems spatially modify incoherent light by chaotic
phase-masks. However, unlike COACH, I-COACH records holograms without twobeam interference. I-COACH is an incoherent 3D imaging method in which the image
is digitally obtained by numerical 2D cross-correlation between the hologram of the
object and the library of PSHs. The PSHs are recorded once in the calibration mode of
the system, before the imaging stage, as shown in Figure 5. The same chaotic CPMs
modulate the light waves in both the calibration and imaging stages. The modulated
light is recorded by a digital camera after propagating in the free space. I-COACH
system without two-beam interference can produce similar results as COACH because
the intensity point-response of I-COACH on the sensor plane is highly sensitive to the
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Figure 5.
Schematic diagram of interferenceless coded aperture correlation holography (I-COACH). The upper scheme refers
to the calibration mode, whereas the lower describes the imaging mode.
axial location of the input point. Mathematically, the high sensitivity means that the
cross-correlation between two intensity responses for two points located at two different axial positions is much smaller than the autocorrelation of each response [15].
Thus, the entire object points can be reconstructed in the 3D image space using 2D
cross-correlations between a multi-point object and the library PSHs. The early configuration of I-COACH [15] has been developed into different systems with various
architectures and with a variety of algorithms [16–33], each with strengths and weaknesses. The typical design of I-COACH is shown in Figure 5, where the same physical
setup is schematically depicted in two modes of operation. The upper scheme shows
the calibration process, in which the system collects a library of PSHs acquired for an
object point positioned at different axial locations. When the library is completed, the
same setup works in the imaging mode shown in the lower part of Figure 5. An
incoherently illuminated 3D object replaces the single point in the system’s input. The
object intensity response recorded by the sensor is 2D cross-correlated with each PSH
of the library. The assembly of cross-correlation results is the desired reconstructed
3D image. This general scheme describes most I-COACH types and has been the basis
for developments that have evolved since 2017 [16–31]; one example of an I-COACH
application is depth-of-field engineering, briefly described next.
3.1 Depth-of-field engineering
Long depth-of-field (DOF) in imaging systems has been important for many
applications [51]. Generally, the DOF is dictated by the numerical aperture of the
optical system. Reducing the numerical aperture extends the DOF, but it also unfavorably decreases the lateral image resolution of the system. Several methods have
been advanced to extend the DOF of the optical system [51–59] with a minimal
resolution decrease. Still, the complicated experimental and computational requirements have stimulated a search for simpler methods. This subsection reviews a new
technique proposed first in [60] to engineer the DOF of imaging systems. DOF
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Holography - Recent Advances and Applications
engineering is done by integrating radial quartic phase-functions (RQPFs) [61, 62]
into the incoherent I-COACH shown schematically in Figure 6. The phase-mask
displayed on the SLM of Figure 6 is a fusion of three separate phase-masks. The first is
the chaotic CPM generated by GSA with the constraints of sparse dots on the camera
plane [63] and a constant magnitude on the SLM plane. The second element is a
positive diffractive lens used to fulfill the 2D Fourier relations of the GSA between the
planes of the SLM and camera. The focal length of the diffractive lens f is determined
such that each object is imaged on the camera. In other words, for the distance objectSLM (dOS) and SLM-camera (dSC), the three lengths satisfy the imaging equation
1=f ¼ ð1=dOS Þ þ ð1=dSC Þ. The distance object-SLM is chosen as the distance from the
center of the object space to the SLM. The third mask is the above-mentioned RQPF
implemented
to
h
i extend the DOF as desired. The RQPF with the phase-function
exp i2π ðr=pÞ4 stretches the DOF of the sparse dots created by the CPM, where p is the
modulation parameter controlling the length of the DOF, and r is the radial coordinate
on the SLM plane. Near the back focal point of the diffractive lens on the camera plane,
the RQPF generates sword beams with an almost constant intensity along a controlled
propagation distance and a relatively narrow beam-like shape in any transverse plane
[61, 62]. The 3D location and the length of the DOF can be determined by changing the
parameters of the RQPF and the focal length of the lens. Multiplexing various threesomes of phase-masks (diffractive lens, CPM, and RQPF) with different modulation
parameters can create various focusing curves. For instance, an imaging system that
can image objects in two non-connected sub-volumes in the object space. In this
example, the entire objects inside these sub-volumes remain in focus, while the images
outside these sub-volumes are blurred and seen out-of-focus. The unusual DOF enables
to image targets in specific sub-volumes simultaneously (or successively), whereas
objects in other sub-volumes are blurred. Moreover, the engineered DOF allows to
transversely shift an image from one volume relative to another image from another
volume. Mutual transverse shifts of sub-volumes can avoid overlap between images
when one object is behind or in front of another object.
Back to Figure 6, the incoherent light source critically illuminates the observed 3D
scene using a lens L0. In this scheme, an object volume is defined as the volume along
z for which the DOF is extended, such that each object inside the volume produces an
in-focus image in the output. Off-axis sub-volumes in Figure 6 indicate images of onaxis objects that are reconstructed out of the z-axis in the output due to an additional
linear phase-mask attached to the other three-phase-masks (diffractive lens, CPM,
Figure 6.
Optical scheme of the depth-of-field engineering system. DL - diffractive lens, CPM - coded phase-mask, RQPF radial quartic phase-function. Adapted from [60].
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Figure 7.
(a, b) Direct images of the objects with two different lenses and (c) Reconstructed images from a single hologram
using depth-of-field engineering. Adapted from [60].
and RQPF). The light emitted from the object scene is modulated by the combination
of the phase-masks displayed on the SLM. For any point inside the object volume, the
intensity recorded by the camera is distributed in the form of chaotically sparse dots.
Like other I-COACH schemes, this dot pattern is used as the PSH, which reconstructs
the image of any object by cross-correlation with the object hologram. In previous
demonstrations of I-COACH [15, 16], the PSH has been recorded experimentally by
illuminating a pinhole positioned at the system input. However, in [60], the PSH is
digitally computed based on the known experimental parameters. The object reconstruction is done by a nonlinear cross-correlation [21] between the computed PSH and
the object hologram.
Next, we show results of only a single object volume. In other words, the following
I-COACH has extended DOF compared to direct imaging with the same numerical
aperture. More complicated examples of DOF engineering can be found in Ref. [60].
The proposed technique is verified by an experimental setup like the scheme in
Figure 6. Unlike typical I-COACH systems, two targets are positioned in two separate
channels of the experimental setup such that they are located at each end of the object
volume [60]. Two LEDs with refractive lenses separately illuminated the object in
each channel. Both objects are from the USAF transmission resolution chart. In one of
the channels, the object is element 6 of Group 2, and in the other channel, element 1 of
Group 3 is used as the object. The targets are located at distances of 24 and 26 cm from
the SLM, respectively. Light from the two targets was combined by a beamsplitter and
projected on the SLM. The gap between the SLM and the camera was 22 cm. Direct
images of the objects [shown in Figures 7(a) and (b)] on the camera plane were
achieved by displaying only a single diffractive lens on the SLM with the focal length
that satisfies the imaging equation, each for a different object in its own depth. For the
I-COACH system, the PSH was computed using the optimal CPM that yielded ten
randomly distributed dots on the camera plane. The reconstructed images of ICOACH are shown in Figure 7(c). In the case of direct imaging, it is clear from
Figures 7(a) and (b) that the axial gap between the targets was too large to focus both
targets at the same time. On the other hand, in the case of the I-COACH with the
engineered DOF of 3 cm, the reconstructed images of Figure 7(c) show that both
targets are in focus without any resolution decrease.
4. Interferenceless coherent COACH
Optical recording of digital holograms with coherent light traditionally involves
interference between object and reference waves, complicating the image acquisition
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Holography - Recent Advances and Applications
[39]. With the coherent I-COACH, the concept of the coded aperture is adapted from
the area of incoherent holography to record digital holograms of three-dimensional
coherently illuminated scenes without two-wave interference or any kind of scanning.
In addition to the obvious advantages of combining interferenceless holographic systems with coherent light, the proposed method enables relatively rapid image acquisition made possible by its inherent high signal-to-noise ratio (SNR). In [34], the ICOACH method was implemented for generating coherent holograms without interference between reference and object waves. The technique, called interferenceless
coherent coded aperture correlation holography (IC-COACH), creates a bi-polar digital hologram of a 3D scene from two camera shots where the scene is illuminated by
coherent laser light. The 3D image of the observed scene is reconstructed from the
hologram by a deconvolution-like process.
To understand the evolution from incoherent to coherent I-COACH, we briefly
summarize the principles of incoherent I-COACH first. Generally, an incoherent ICOACH hologram denotes a 2D function containing an image of a 3D scene, such that
the image can be digitally reconstructed from the 2D function. Mathematically, the 2D
digital hologram of a 3D object is given by,
ð
HOBJ ðrÞ ¼ IOBJ ðr; zÞ ∗ pðr; zÞdz,
(1)
where ∗ is 2D convolution at each z plane, r ¼ ðx, yÞ are the transverse coordinates,
and pðr; zÞ is the PSH of the recording system, which can be a general complex [15] or
bi-polar real [16] function. The library ofPSHs
is a priori acquired in a calibration
process with a guidestar, in which each p r; zj from the PSH library is computed as a
response to an object point at zj. Once the library is ready, and an object hologram is
recorded, 2D cross-correlations between the object hologram and each PSH from the
library reconstruct each zj plane of the 3D image. This computation process is based on
the linearity of incoherent optical systems with 2D intensity signals expressed by the
following familiar convolution,
IOut ðrÞ ¼ IIn ðrÞ ∗ jhðrÞj2 ,
(2)
where hðrÞ is the coherent point spread function of the optical system. IIn ðrÞ and
IOut ðrÞ are the system input and output intensities, respectively. In contrast to incoherent, coherent optical systems are linear in corresponding to 2D complex amplitudes, and they obey the relation,
IOut ðrÞ ¼ jAIn ðrÞ ∗ hðrÞj2 ,
(3)
where AIn ðrÞ is the input 2D complex amplitude fulfilling the equation IIn ðrÞ ¼
jAIn ðrÞj2 : Because of the nonlinearity of Eq. (3), the implementation of the I-COACH
concept in the coherent system is possible only for special cases. Hence, the coherent
processor should be adapted in such a way that can satisfy the relation,
jAIn ðrÞ ∗ hðrÞj2 ≈ jAIn ðrÞj2 ∗ qðrÞ,
(4)
where AIn ðrÞ represents a broad set of input objects, and we assume that hðrÞ and qðrÞ
are nontrivial functions. Eq. (4) is satisfied if hðrÞ is a set of points distributed over the
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camera plane such that the gap between any two points is wider than the size of AIn ðrÞ
[34]. The pattern of the random points on the camera plane is considered the system’s
PSH. Hence, the CPM replicates the object to a set of the same images chaotically
distributed over part of the camera plane. Such CPMs are created by a modified
version of GSA [50], in which iterative transformations between the CPM’s plane and
the spectral plane are done with suitable constraints at each plane. The constraint at
the CPM’s plane is a constant magnitude distribution because the CPM is displayed on
a phase-only SLM. In the spectral plane, which is also the camera plane, the intensity
is constrained to be in a shape of randomly distributed dots over all or part of the
plane.
The optical configuration of the IC-COACH system of [34] shown in Figure 8 is
based on the classical 4-f spatial filtering system, with the SLM positioned at the
Fourier domain and the camera at the image plane. In this setup, the spatial spectrum
of the object is modulated by the CPM displayed on the SLM. The CPM was produced
by the GSA to duplicate the input object over an ensemble of points randomly distributed at the camera plane. Two different chaotic CPMs are sequentially displayed
on the SLM to create two different random sets of replications of the object. These two
sets are subtracted from each other to produce a bi-polar object hologram. The ability
of IC-COACH to image multi-plane objects is accomplished by multiplexing on the
SLM, a few independent CPMs, each of which yields an in-focus different set of dots
on a different transverse plane. Each ensemble of out-of-focus dots becomes focused
on the camera plane for a point object positioned at the corresponding transverse
plane. Lastly, the desired transverse image of the observed 3D scene is reconstructed
by cross-correlation between the object hologram and the corresponding PSH.
Figure 9 shows the reconstructed images for two different planes and two different
gaps between the object planes, forming two different multi-plane scenes.
Figure 8.
Experimental setup of interferenceless coherent coded aperture correlation holography (IC-COACH) with two
independent illumination channels. BS1,2,3: Beamsplitters, M1,2: mirrors, and SLM: spatial light modulator.
Adapted from [34].
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Holography - Recent Advances and Applications
Figure 9.
Image reconstructions of different object planes obtained by a correlation with the corresponding PSH in ICCOACH. Adapted from [34].
Experimental demonstration for imaging diffusely reflective objects also appears in
[34], making the IC-COACH system suitable for processing speckle images obtained
by coherent illumination.
5. Coherent COACH with two-wave interference
IC-COACH described in the previous section is an adaptation of the incoherent
I-COACH to the case of coherent illumination. This system is capable of imaging 3D
scenes holographically, but it cannot do phase imaging of any kind. To enable QPI of
transparent objects, COACH has been integrated with a Mach-Zehnder interferometer
[35, 36]. QPI, in general, is done by capturing the wavefront passing through thin
transparent objects and converting it to an optical thickness map of the examined
objects. This method is useful for many applications, including label-free biological
cell imaging [64, 65] and nondestructive quality tests [66, 67].
Like the previous demonstrations of I-COACH [34, 63], the image of the observed
object is projected to randomly and sparsely distributed replications over the camera
plane. As before, the replications are obtained by a pseudorandom CPM synthesized
by modified GSA [50]. The CPM is displayed on a phase SLM in the configuration of
the coherent sparse COACH (CS-COACH) shown in Figure 10. The image sensor
records the interference pattern between the waves of the image replications and of a
reference tilted plane wave as follows:
12
Coded Aperture Correlation Holography (COACH) - A Research Journey from 3D Incoherent…
DOI: http://dx.doi.org/10.5772/intechopen.105962
Figure 10.
Optical configuration of coherent sparse COACH. MO: Microscope objective, BS1,2,3: Beamsplitters, M1: Mirror,
and SLM: Spatial light modulator. Adapted from [36].
N
X
Iðx, yÞ ¼ Oðx, yÞ exp ½jϕðx, yÞ ∗
δ x
i¼1
xi , y
2π
x sin θx þ y sin θy
yi þR exp j
λ
2
,
(5)
where O(x, y) is the object amplitude, and ϕ(x, y) is its phase, R is the reference
wave amplitude, λ is the illumination wavelength, N is the number of image replications, (xi, yi) are the displacement values of the i-th replica from the camera origin,
and (θx, θy) are the angles between the object and reference waves in the x-z and y-z
planes, respectively. It should be noted that off-axis holography is used to acquire
holograms by a single camera shot. A digital filtering process in the spatial frequency
domain eliminates the bias term and the twin image from the recorded intensity
pattern. The processed object hologram is:
HOBJ ðx, yÞ ¼ R ∗ Oðx, yÞ exp ½jϕðx, yÞ ∗
N
X
δ x
i¼1
xi , y
yi :
(6)
This hologram includes several randomly distributed replications of the object over
the image plane. Like the procedure explained in [34, 63], the reconstruction of the
object’s complex amplitude is performed by 2D cross-correlation between the object
hologram HOBJ and the PSHs. Figures 11(a) and (b) show the phase-image of polystyrene microspheres (FocalCheck, 6 μm diameter) with the proposed CS-COACH
method. For comparison purposes, the phase-images extracted from a regular
Mach-Zehnder interferometer using conventional off-axis holography are shown in
Figures 11(c) and (d). It is apparent that the image of CS-COACH has higher SNR
than the conventional technique. This advantage is attributed to the averaging procedure over several replications accompanied by the reconstruction using crosscorrelation. Noise reduction is one of the several advantages of CS-COACH in comparison to open-aperture equivalent systems. Another advantage presented in Ref.
[36] is extending the field-of-view (FOV) of the imaging system. Extended FOV
realized with the same focal length of the microscope objective and without sacrificing
the image resolution is an important advantage in microscopy.
13
Holography - Recent Advances and Applications
Figure 11.
Reconstructed phase images of polystyrene microspheres and the phase cross-sections using (a)-(b) CS-COACH
and (c)-(d) conventional off-axis holography. Units of the left panel color bars are radian. Adapted from [36].
6. Discussion and summary
For all its forms, COACH is a rapidly evolving technology because of the desire to
enhance the resulting images and due to the new applications supported by the
method. Any technology of imaging is expected to be as quickly as possible with the
least camera shots. While the early version of I-COACH [15] operated with three
camera shots taken under three independent CPMs, the number of shots and CPMs
was decreased to two in [16]. By multiplexing two CPMs in space instead of time as
before [15, 16], a single-camera shot was applied in Ref. [18]. I-COACH [19] and CSCOACH [36] with extended FOV were demonstrated by calibrating the systems with
extended PSHs beyond the conventional FOV. The numerical reconstruction procedure was changed in [21] by substituting the ordinary linear cross-correlation with
new nonlinear cross-correlation optimized to yield a correlation distribution with the
lowest entropy. A different nonlinear cross-correlation with other cost-function in the
optimization process was employed in [26, 30]. Some of the noise on the resulting
images in the early versions [15, 16] appeared because of the low-intensity level per
pixel of the PSH on the sensor plane. This difficulty was treated in [63] by imposing a
PSH with the structure of sparse dots of light intensity distributed chaotically inside a
limited region. The same problem was differently solved in [30] with PSHs of a ring
shape. The electro-optical calibration in the upper part of Figure 5 was changed by a
14
Coded Aperture Correlation Holography (COACH) - A Research Journey from 3D Incoherent…
DOI: http://dx.doi.org/10.5772/intechopen.105962
pure digital technique of synthesizing the library of PSHs in the computer [68].
Lateral resolution can be considered one of the holy grails of optical imaging. Improving the lateral resolution by I-COACH has been treated in [23, 45, 69, 70] by different
approaches. Usually, I-COACH’s lateral and axial resolutions are the same as those of
lens-based imaging systems with the same numerical aperture. The methods of
[23, 45, 69, 70] improve the lateral resolution beyond the diffraction limit enforced by
the finite numerical aperture of optical systems. In [23], resolution-enhanced images
of the observed objects are reconstructed by a nonlinear cross-correlation between
object holograms and PSHs. In [69, 70], a CPM displayed on the SLM was introduced
between the object and the input aperture of a regular lens-based imager. Thus, the
effective numerical aperture was increased beyond the characteristic numerical aperture of the imaging system. The effective numerical aperture and the improved
resolution limits can be tuned by altering the scattering degree of CPMs [69, 70].
Other applications of COACH and I-COACH and their context in a frame of systems
with dynamic diffractive phase-apertures are reviewed in [17, 71, 72].
To conclude this review, we note that COACH for all its modes is based on the
extension of the resources available for imaging in a few ways. First, the real-valued
aperture function of ordinary direct imaging is replaced with the complex-valued
aperture function of COACH. Second, the COACH aperture is modified over time in
the multiple-shot versions. Finally, an additional stage of digital processing is integrated with the optical system. These additional resources add to the COACH system
new capabilities and unique features. Even though I-COACH is a simpler form of
COACH and thus is preferred for many 3D imaging projects, there are some unusual
applications in which COACH with two-beam interference is required. Incoherent
synthetic aperture imagers [20, 22], the hybrid FINCH-COACH system [45], and
quantitative phase-imagers [35, 36] are characteristic examples of systems that twobeam interference is necessary for their operations. However, other applications can
be implemented successfully on I-COACH; some are presented herein others might be
proposed in the future.
Author details
Joseph Rosen*, Angika Bulbul, Nathaniel Hai and Mani R. Rai
School of Electrical and Computer Engineering, Ben-Gurion University of the Negev,
Beer-Sheva, Israel
*Address all correspondence to: rosenj@bgu.ac.il
© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of
the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided
the original work is properly cited.
15
Holography - Recent Advances and Applications
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