# Quantitative analysis of numerically focused red blood cells using
subdivided two-beam interference (STBI) based lateral-shearing digital
holographic microscope

Shital Devinder  Instrument Design and Development Center, IIT Delhi  Ashish
Lal  Tathagato Rai Dastidar  SigTuple Technologies Pvt Ltd. Bangalore, India.
Satish Kumar Dubey  Instrument Design and Development Center, IIT Delhi
http://web.iitd.ac.in/~satishdubey/<EMAIL_ADDRESS>
###### Abstract

A lateral shearing interferometer based digital holographic microscopy has
been realized to study the morphology and dynamics of the biological samples
quantitatively. Here a lateral shearing interferometer is embedded with a
conventional microscope and a CMOS sensor to form a lateral shearing digital
holographic microscope. It enables recording of the image plane holograms of
the object that can be numerically reconstructed to estimate its 3-D profile.
This yield the depth information of the sample in addition to its bright field
image. To overcome the duplicate image problem, a method called subdivided
two-beam interference (STBI) has been used that enables the recording of the
hologram equivalent to the two-beam holographic microscope. Since a small
region of the smear is sufficient to study its cellular morphology. Here
region of interest is isolated from the rest of the sample and STBI method is
applied to analyze the peripheral blood smear quantitatively. In addition,
focusing on the cells is ensured using a computational approach called total
variation (TV) method. It is planned to automate the process to get the best
focused 3D profile of the object (cells). In this study, STBI and TV
approaches were applied to record and reconstruct the holograms corresponding
to the human blood smear.

subdivided two-beam interference, Digital holographic microscopy, Resolution,
Total Variation.

††preprint: AIP/123-QED

##  I Introduction:

Digital holographic microscopy using two beam configuration is a common method
used for the phase imaging of microscopic objects like RBCs, WBCs, platelets
etc 6255748  article1  Mann:05 . Phase and amplitude information of the object
are encoded on a charged couple device(CCD) or on a complementary metal-oxide-
semiconductor (CMOS) camera in the form of interference fringes called the
hologram. Phase and amplitude information is retrieved from the digitally
recorded holograms using numerical reconstruction methods Schnars_2002 .Two
beam configuration enables recording of wide field of view (FOV) of the sample
under investigation. Also, it is possible to have wider angular tilt between
object and reference beams. The angular tilt need to be higher than a certain
level to avoid the overlap of dc and cross terms in the Fourier domain.
However there are techniques for high quality image recovery even when the dc
and the cross terms in the hologram overlap in the Fourier domain. Khare:13
Zhang:04 . One such configuration is Machzender Interferometer based digital
holographic microscope (DHM). In this configuration, a coherent beam is split
into two, one illuminates the object and other is kept unaltered. These
wavefronts follow different paths in space and are so directed to produce
interference fringes on CCD. The separation of beams in two arms of setup
leads to the reduced temporal phase stability which results in the
fluctuations of phase of object over time. This types of configuration also
requires more number of optical components, making the setup difficult to
align, bulky and expensive. This problem has been addressed using a common
path configuration called lateral shearing digital holographic microscope(LS-
DHM) Singh:12  article  Singh:18  having a high temporal phase stability. This
configuration uses almost half of the optical components that are used in
standard two beam interferometer based DHM making it easier to align, compact
and cost effective. However this configuration suffers from duplicate image
problem which emerges due to the self referencing geometry. Self overlap of
the object beam also leads to the loss of the object information. Object
information remains available only in the region where there is a overlap of
altered region (having object information) with that of unaltered (no object
information) region of sheared wavefront. The problems of object information
overlap and the duplicate image have been addressed in the present manuscript.
In this regard, following two approaches have been incorporated, (i) scanning
the peripheral region of the sample under investigation and (ii) maintaining
the lateral shear distance approximately equal to the half of the field of
view. Moreover, accurate focusing of the field of view is essential for the
quantitative analysis of the blood smear because even a small defocus alters
the phase of the object considerably. The defocus may result in the alteration
of the features like cell volume, surface area etc. by great deal which is
used for the identification and classification of different type of cellular
objects. In two beam interference based DHM, sample is focused by blocking the
reference beam and recording a bright field image of highest contrast.
Unblocking the reference beam results in a well focused hologram. However,
this is not possible in LS-DHM, as interfering beams are always overlapping.
In this manuscript, a computational approach has been proposed for finding the
best focus hologram out of a stack of holograms recorded around the plane of
the best focus.The best focused hologram is decided on the basis of the total
variation (TV) method. Hologram with the highest TV value is considered to be
the best focused hologram.

Figure 1: Schematic of LS-DHM used for the study of blood cells.

##  II Methods:

###  II.1 Lab setup:

The LS-DHM setup consists of a coherent light source, a microscopic objective
for magnification, a wedged glass plate and a detector (CCD/CMOS camera). The
schematic of the proposed setup is as shown in Fig.1. The used light source is
a He-Ne laser ($\lambda=633nm,5mW,unpolarised$). Beam is spatially filtered
using a $40X$ M.O. and a $10um$ pinhole. A $100mm$ focal length lens is used
for collimation . The collimated beam is directed on the sample which is
subsequently magnified by a ($10X$) microscopic objective. This magnified
version of the object wavefront is made to fall on a thick ($10mm$) shear
plate. The object beam is reflected from front and back surfaces of the shear
plate. These two reflected object beams make an interference pattern, which is
recorded on a CCD/CMOS camera.

###  II.2 Subdivided two beam interference:

As a consequence of the conventional lateral shearing interferometer system,
recorded hologram contains duplicate images and information overlap due to the
interference between two sheared object beams. This overlap degrades the
reconstruction results and makes it difficult to extract the phase of the
object. To avoid overlap of object information, periphery of the sample is
scanned which essentially yields the object wavefront as two sub-divided beams
article  with and without object information as shown in Fig.(2).

Figure 2: Diagram showing STBI at the periphery of the blood smear.

In such a case if the lateral shear distance is made equal to half of the FOV,
we get a hologram which is equivalent to that of the two beam interference
DHM. Assuming the interference between altered part and the unaltered part of
the object wavefront only, the recorded fringe pattern can be mathematically
expressed as

$\displaystyle I_{recorded}(x,y)=|O_{1}(x,y)+O_{2}(x,y)|^{2}$ (1)
$\displaystyle I_{recorded}(x,y)=O_{1}(x,y)^{2}+O_{2}(x,y)^{2}+O_{1}^{*}(x,y)$
$\displaystyle O_{2}(x,y)+O_{1}(x,y)O_{2}^{*}(x,y)$ (2)

where $O_{1}(x,y)$ refers to the altered part of the object wavefront,
reflected from the front surface of the shear plate and $O_{2}(x,y)$ refers to
the unaltered part of the object wavefront, reflected from the back surface of
the shear plate. $O_{2}(x,y)$ is reflected from the back surface of the shear
plate. It travels additional optical path, which adds a universal phase to the
wavefront. This unwanted phase is removed by subtracting the background phase
from the phase calculated when the object is present in the FOV. The
subtraction leaves only the object phase.

####  II.2.1 Focusing:

For the extraction of accurate phase of microscopic samples, focusing of the
sample is crucial. A small defocus leads to a significant phase error. In two
beam interference microscope, sample can be brought to focus by blocking one
of the beams (reference) and looking for the highest contrast image. The plane
of highest contrast can be considered as the plane of focus. However, two
beams always overlap in shearing interference microscope. It is difficult to
first isolate and then look for the sharp bright field image. We propose a
method to find the focused hologram out of a number of recorded holograms. As
phase information of the object is encoded in terms of the modulated fringes,
the maximum modulation of fringes correspond to the phase of the object.
Therefore, a well focused hologram is one where in fringe modulations are
maximum. Modulation in fringes can be calculated in terms of the total
gradient of the image. Which is called as the total variation (TV) of the
image. Total variation is the gradient of the image summed over all the
pixels. Therefore, for the best focused hologram, total variation is maximum.
Stack of holograms were recorded across the focus plane and one with the
maximum TV was labeled as focused hologram.

##  III Recording a hologram

For recording a hologram using the proposed system, region of interest must be
chosen very carefully, taking into account the object location, thickness and
refractive index of the shear plate and the angle of incidence at the shear
plate. In the proposed microscope, hologram is produced by interference of two
sheared object beams which are reflected from the front and back surfaces of
the shear plate. In conventional systems, the duplicate image of the object
could not be removed due to small lateral shear distance. This is because more
than half of the object beam overlaps with itself. This problem is taken care
by maintaining the lateral shearing distance less than or equal to half of the
FOV in the proposed setup.

##  IV Experiments:

####  IV.0.1 Imaging of human red blood cells:

In certain cases, diseased RBCs can be differentiated from healthy ones on the
basis of phase. Phase of RBCs present in human blood sample is imaged using
the proposed setup. The unstained blood smear was prepared in the lab and the
periphery of the blood smear was scanned using a $10X$ M.O. (for wide FOV).
The plate was kept at $25cm$ from the microscopic objective at an angle of
$45degrees$ (approx.) to the object beam. The camera was kept at $20cm$ from
the shear plate. A raw video of holographic view was recorded on a $24bit$
CMOS camera (frame rate $150f/s$, pixel size of $4.8um$) for approximately
$10s$ while the sample was gradually moved in and out of the focus. Out of the
large no of recorded frames, the frame with maximum TV value was considered
for reconstruction and phase extraction. The feasibility of the TV method has
been established to find out the best focused hologram. The procedure was
repeated multiple times and the TV approach was found to be stable for the
estimation of the focus position.

####  IV.0.2 Reconstruction:

In 20 seconds more than $3000$ frames were recorded. Total variation was
calculated for $1^{st}$,$30^{th}$,$60^{th}$ frames and so on and the hologram
frame with highest TV was considered for reconstruction. The numerical
reconstruction of the hologram was carried out using Fourier transform method
with accurate carrier removal technique Singh:16 . The retrieved phase was
unwrapped using the transport of intensity method Pandey2018 .

Figure 3: (a) TV plot for various holograms recorded along microscope axis.
(b) Phase map for frame no. 40. (c) Phase map for frame no. 58. (d) Phase map
for frame no. 50.  Figure 4: Mean phase variation for stack of 30 holograms in
the neighbourhood of the focused one  Figure 5: (a) Background (Hologram
recorded with no sample present). (b) Hologram recorded for the human blood
smear. (c) Phase map for the recorded hologram. (d) Phase map 3D view of the
recorded hologram.

##  V Results:

Focusing: $Fig.3(a)$ shows the variation of TV for the various frames recorded
along the optical axis. X-axis stands for the $n^{t}h$ no. of hologram and
Y-axis shows the TV values for corresponding holograms. $Fig.3(b)$,$fig.3(c)$
and $fig.3(d)$ show the phase map for the $40^{th}$, $58^{th}$ and $50^{th}$
frame. The average phase for these frames is $0.5632$ , $0.6204$ and $0.6465$
respectively. $Fig.(4)$ Shows the average phase variation in the vicinity of
focus. When in defocus, whether in positive or negative direction, the average
phase is relatively smaller. Average phase is the maximum for the best focused
hologram.

Phase results for Blood cells $Fig.5(a)$ shows background hologram i.e.
without sample being present and $fig.5(b)$ hologram with sample (blood
smear). Holograms for RBCs are recorded at the peripheral smear without
information overlap. Moreover RBCs at the boundary are least effected by the
mechanical procedure of smear preparation and so are best suited for
evaluation. $Fig.5(c)-(d)$ shows the phase map of the RBCs.$Fig.6(a)$ shows
the phase of single RBCs form a healthy sample and $Fig.6(b)$ shows the phase
for an unhealthy sample.

Figure 6: (a) Shows the phase for a healthy RBC, (b) Shows the phase for an
unhealthy RBC.

##  VI Conclusion:

A laboratory breadboard set up has been realized for the phase imaging of the
cellular objects present in the blood sample. It is a compact setup that uses
only few optical components as compared to two beam interference holographic
microscope. The components used in the present set up are a laser diode,
microscopic objective, shear plate and a CMOS digital array. The holograms
were recorded using the altered and unaltered part of the same FOV of the
sample slide by specially focusing the peripheral part of the blood smear to
avoid the duplicate (overlap) image problem. The reflected beams from the two
surfaces of the shear plate had almost same intensity (4% approx for both
interfaces), leading to the best contrast fringes. This helps in retrieving
the object phase with good accuracy. The setup gives high temporal stability
and phase results comparable to those of the two beam interference holographic
microscope. This works well for the sparse samples as it avoids overlapping.
However, the background in such a case may not be uniform leading to small
distortions in the object phase. The FOV can also be increased by filtering
one of the beams. This type of geometry requires an additional lens and a
pinhole of very small diameter. The method may lead to the loss of phase
information due to the finite extent of the pin hole. Because of the common
path geometry, pin hole may also block the high frequencies components of the
object beam. The proposed geometry and methods are free from such issues,
however the overall throughput is relatively small, which can be increased by
using a smear prepared with extended boundaries. This will involve preparation
of multiple smears on the same slide. This is being worked upon and will be
presented in the next manuscript.

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