3D RECONSTRUCTION OF UNSTAINED CELLS FROM A SINGLE DEFOCUSED HOLOGRAM Sunaina Rajora

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3D RECONSTRUCTION OF UNSTAINED CELLS FROM A SINGLE

DEFOCUSED HOLOGRAM

Sunaina Rajora

Department of Physics,

Indian Institute of Technology Delhi,

New Delhi 110016 India.

sunainarajora1511@gmail.com

Mansi Butola

Department of Physics,

Indian Institute of Technology Delhi,

New Delhi 110016 India.

mansibutola83@gmail.com

Kedar Khare

Optics and Photonics Center &

Department of Physics,

Indian Institute of Technology Delhi,

New Delhi 110016 India.

kedark@physics.iitd.ac.in

October 25, 2022

ABSTRACT

We investigate the problem of 3D complex ﬁeld reconstruction corresponding to unstained red blood

cells (RBCs) with a single defocused off-axis digital hologram. We employ recently introduced

mean gradient descent (MGD) optimization framework, to solve the 3D recovery problem. While

investigating volume recovery problem for a continuous phase object like RBC, we came across an

interesting feature of the back-propagated ﬁeld that it does not show clear focusing effect. Therefore

the sparsity enforcement within the iterative optimization framework given the single hologram data

cannot effectively restrict the true object volume. For phase objects, it is known that the amplitude

contrast of the back-propagated object ﬁeld at the focus plane is minimum and it increases at the

defocus planes. We therefore use this information available in the detector ﬁeld data to device weights

as a function of inverse of amplitude contrast. This weight function is employed in the iterative steps

of the optimization algorithm to assist the object volume localization. The experimental illustrations

of 3D volume reconstruction of the healthy as well as the malaria infected RBCs are presented.

The proposed methodology is simple to implement experimentally and provides an approximate

tomographic solution which is axially restricted and is consistent with the object ﬁeld data.

1 Introduction

Holography has historically been associated with 3D imaging capability. In traditional ﬁlm based holography, the

recorded hologram is re-illuminated by conjugate reference beam for replay. The replay ﬁeld, when viewed by a human

observer, provides perception of a 3D image. A critical aspect of this 3D image perception is the ability of human eyes

to concentrate on focused objects while ignoring the defocused and blurred background. Film based holography is

now largely getting replaced by digital holography where the hologram is recorded on a digital array sensor and the

replay process is carried out via numerical computation. The 3D image formation problem in digital holography is

quite different in nature. Mimicking the ﬁlm based holographic replay numerically leads to a reconstructed ﬁeld in the

original object volume, however, now the focusing capability of human eye does not have any role to play in 3D image

perception. If the numerically replayed (or back-propagated) object ﬁeld in the original object volume is carefully

examined, one ﬁnds that the reconstructed ﬁeld is quite different from the original object function. In particular, it

may be shown that the replay process, at least in the weak scattering approximation, is actually a Hermitian transpose

(rather than inverse) operation corresponding to the forward object ﬁeld formation process at the hologram recording

arXiv:2210.12594v1 [eess.IV] 23 Oct 2022

APREPRINT - OCTOBER 25, 2022

plane [

]. Since the replay process is linear in nature, there is also a problem associated with degrees of freedom being

reconstructed. Formation of a 3D image from a single hologram will mean creation of more information (or voxels)

in a 3D volume out of a smaller number of pixels in the hologram data [

]. As suggested in the simulation results

in [

], this degrees of freedom issue may be addressed if the 3D object under consideration is sparse in nature. The

3D image reconstruction problem then may be handled with a sparsity assisted optimization algorithm [

]. The main

aim of the present work is to apply the sparsity based 3D image reconstruction ideas to experimental data consisting

of a single defocused hologram corresponding to an unstained red blood cell (RBC) object. Unstained RBCs may be

considered as "simple" in their structure, so that, the weak scattering approximation and sparse reconstruction concepts

may be applied to this case. In particular it is of interest to know to what extent the 3D object reconstruction by sparse

optimization methodology differs from a traditional back-propagation based 3D image recovery.

Tomographic 3D phase imaging ﬁnds applications in number of areas like bio-medical imaging [

], cryo-electron

microscopy [

], X-ray computed tomography [

]. The problem of tomographic image reconstruction was ﬁrst introduced

in the early work of Wolf [

], where it was showed that the 3D refractive index distribution of an object can be retrieved

by solving an inverse scattering problem. This work considered weak scattering approximations to establish a relation

between the Fourier transform of the 3D object function and the 2D complex-valued scattered ﬁeld [

] as may be

recorded using a holographic imaging system. The Fourier relation is known by the name Fourier diffraction theorem

[

] and the imaging modality is also sometimes refereed to as optical diffraction tomography (ODT). In a typical

ODT setup, multiple holograms of a 3D object are captured at different viewing angles in order to ﬁll the 3D Fourier

space of the object as per the Fourier diffraction theorem. Advance hardware developments try to record maximum

possible number of views either by changing the illumination angle of the incident laser beam on to the sample or by

tilting the sample stage itself or by employing both [

]. In the ODT community more attention has been paid to the

problem formulation based on 3D refractive index recovery which inherently arises in the formulation of the Fourier

diffraction theorem [

]. In the present work, we attempt this problem in a slightly different manner where the

unknown quantity is the 3D object ﬁeld function rather than the 3D refractive index distribution, which may make the

formulation somewhat simpler. Further we use the numerically recovered complex-valued object ﬁeld in the hologram

plane (rather than hologram intensity) as our starting data. The 2D complex object ﬁeld can be obtained from numerical

processing of digital hologram(s) recorded on a digital sensor (CCD or CMOS). The 3D object ﬁeld to be recovered is

considered to be composed of thin slices separated by axial sampling distance

∆z

. The phases may then be related to

the material present in

∆z

thickness of respective slices. In prior literature, different scattering approximations have

been suggested like Born or Rytov, based on the phase structure and size of the object [

]. For objects with complex

structures where multiple scattering effects can not be neglected, more sophisticated methods like beam propagation

method[

], split-step non-paraxial method [

] have been suggested. For simplistic objects like human RBCs, ﬁrst

order scattering approximation may be considered sufﬁcient to model the forward problem [

]. In ﬁrst order Born

approximation, each slice of the 3D object volume is propagated independently and the individual propagated ﬁelds at

the detector plane may be simply added to make the 2D object ﬁeld.

Our aim in this work is to investigate the 3D object recovery problem for simple objects from a single defocused

hologram recorded on a 2D sensor array. We have studied this problem for simulated discrete objects in [

], which

we extend here for the real biological samples like RBCs that have continuous structure. Imaging of both healthy and

malaria infected RBCs is illustrated. The problem of 3D reconstruction with single hologram is studied in literature

for objects like particle ﬁelds in in-line holographic conﬁguration [

]. In [

], the beam propagation method is

employed to model the forward problem and the 3D refractive index distribution of the particle-ﬁeld like object function

is determined by using a regularized optimization algorithm wherein the measured data is the single in-line hologram

intensity. Note that here the role of regularization is to handle both the object twin and defocus components. The

work in [17, 16] also uses single in-line hologram intensity and iteratively estimates the 3D complex ﬁeld distribution

of discrete particles or text-like objects by application of 3D de-convolution and iterative reﬁnement. We ﬁnd that

for continuous nature of the 3D object under study, a careful examination of the 3D recovery problem is required.

For example, if the characteristic of the back-propagated ﬁeld corresponding to the RBCs is observed, we ﬁnd that

it is quite different from the particle ﬁelds. The amplitude of the back-propagated ﬁeld corresponding to RBCs does

not show signiﬁcant defocusing effect on moving a small distance away from the RBC image plane, as is generally

observed for particle ﬁelds. With such back-propagated complex ﬁelds, the localization of the object volume is thus

not a straightforward problem even if the sparsity enforcing priors like total variation are employed in optimization

framework. This volume localization problem simpliﬁes for the ODT problem when multiple views of the same sample

are captured [

]. As our present work focuses on the 3D reconstruction with single 2D complex ﬁeld data at the

detector plane, we provide some discussion on the object volume localization in the current problem framework. In

particular, we use the inverse amplitude contrast as a weighting factor to improve the conﬁnement of a cell object in the

reconstruction volume as we will discuss later in the paper. For iterative reconstruction, we employ the optimization

methodology of mean gradient descent which was introduced in an earlier work [

] for single shot interferogram

analysis and 2D image de-convolution [

]. The advantage of this methodology is that it does not require regularization

APREPRINT - OCTOBER 25, 2022

Figure 1: System schematic of reconstruction problem. The planes shown with red solid lines correspond to the location

of the RBC and its defocused image. Blue dotted lines refer to the perfect focus plane and its conjugate at the detector.

Focal lengths f1and f2are not shown to scale.

parameter by design and hence the tedious task of empirical tuning is avoided as in standard optimization approaches

[21, 22, 23].

This paper is organized as follows: In Section 2, we describe the problem of 3D volume reconstruction of RBC from

a single defocused hologram and further discuss about the peculiar nature of the back-propagated ﬁeld of the RBC.

Section 3 describes the methodology of mean gradient descent optimization and show its application to the problem of

3D reconstruction. In Section 4, we describe the inverse amplitude contrast as a new weighting criteria employed in the

iterative optimization algorithm to restrict the object volume. Further we show 3D volume reconstruction results for

two examples of a healthy and malaria infected RBCs. Finally in Section 5, we highlight the conclusions and future

directions of our work.

2 Problem overview

The problem we wish to address is illustrated in Fig. 1. A red blood cell (RBC) object is placed at a defocus plane of an

inﬁnity corrected 40x microscopic imaging system which forms one arm of a balanced digital holographic microscope

system (Make: Holmarc Product: HO-DHM-UT01-FA). The other arm of the interferometer brings in a tilted plane

reference beam. A digital hologram of the defocused cell is recorded on an array sensor (Make: IDS GmBH, uEye

3070, pixel size 3.45

µm

) leading to an off-axis Fresnel zone hologram of the RBC as shown in Fig. 2(a). The DHM

system uses a 650 nm diode laser for illumination purposes in the quantitative phase mode. The system is additionally

equipped with a switchable white light illumination which makes the system work as an inﬁnity corrected brightﬁeld

microscope when required by the users. To model the problem of 3D reconstruction as in [

], the hologram of a single

RBC is recorded at a defocus from the perfect focus plane. We identify the focus plane directly in the hologram domain

by visually inspecting the fringe pattern. As RBC can be considered essentially as a nearly pure phase object, the image

plane hologram predominantly shows phase modulation of fringes (in the form of fringe bending) [

]. When the cell is

displaced from the focus plane, the phase information is transferred to amplitude and as a result fringes additionally

show amplitude modulation in a defocus plane. In the experiment, the defocus is approximately set to

10 µm

using a

motorized z-stage. The problem of interest now is to use the complex object ﬁeld recovered in the detector plane and ﬁt

it to a 3D object ﬁeld in a box centered on the location of the image plane of the RBC. The 2D complex-valued ﬁeld

at the detector plane is reconstructed from the off-axis hologram, shown in Fig. 2 (a) using Fourier domain ﬁltering

of one of the cross-terms in the hologram. Under weak scattering approximation, this complex-valued object ﬁeld

recovered from the recorded hologram can be considered as a sum of the background illumination and the scattered

ﬁeld from the RBC. We remove the effect of background by mean subtraction and further add a Gaussian window over

the mean subtracted object ﬁeld to remove any residual ﬁeld. The Fourier ﬁltering approach uses the entire hologram,

however, we are displaying the resultant phase and amplitude maps of the background subtracted object ﬁeld in the

region of interest (ROI) of

280 ×280

pixels (or

24µm ×24µm

after accounting for 40x magniﬁcation) in Figs. 2(b),

V(x, y)

and is treated as a starting data for our

3D reconstruction problem. Once we acquired the data

V(x, y)

, now the ﬁrst step is to locate the defocus distance

numerically, which we do by observing the amplitude contrast of the RBC ﬁeld obtained by back-propagating

V(x, y)

at different z-distances from the detector plane [

]. To ﬁnd the focus distance, we back-propagate the ﬁeld

V(x, y)

via angular spectrum propagation and calculate the standard deviation (

) of its amplitude values within a central

circular window of radius

pixels. Figure 2(d) shows the plot of amplitude contrast

√σ

against various z-distances

from the detector ranging from

0µm

18µm

. We ﬁnd that the amplitude contrast is minimum at a z-distance of

9µm

which is indicated by blue arrow in Fig. 2(d). It can be seen that the numerically estimated defocus distance is quite

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3DRECONSTRUCTIONOFUNSTAINEDCELLSFROMASINGLEDEFOCUSEDHOLOGRAMSunainaRajoraDepartmentofPhysics,IndianInstituteofTechnologyDelhi,NewDelhi110016India.sunainarajora1511@gmail.comMansiButolaDepartmentofPhysics,IndianInstituteofTechnologyDelhi,NewDelhi110016India.mansibutola83@gmail.comKedarKhareOpticsandP...

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3D RECONSTRUCTION OF UNSTAINED CELLS FROM A SINGLE DEFOCUSED HOLOGRAM Sunaina Rajora

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