Quantitative phase imaging and reconstruction for biological applications

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Quantitative phase imaging andreconstructions for biological

applications

Kou Shan Shan

Optical Bioimaging Laboratory

NUS Graduate School for Integrative Sciences and Engineering

A thesis submitted for the degree ofDoctor of Philosophy

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Abstract

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Acknowledgements

I would like to express sincere appreciation and gratitude towards mythesis supervisor, Professor Colin Sheppard for his sparkling wisdomthat enlightens me all the time His generous guidance and continuoussupport has encouraged me to complete this thesis work From thevery beginning of the first discussion, Colin presented me a whole newworld of fantasy in optics and microscopy with his signature drawingsin Fourier optics , and like Alice in Wonderland, I was so intriguedand absorbed with optics I thank him for not only introducing meto the field and teach me as a mentor, but also for his enthusiasmand creativity that constantly inspires me along the PhD journey.Discussions with him have always been interesting, and he has anamazing ability in answering my most primitive questions with clarityand simplicity.

I would also like to thank my thesis advisory committee members Prof.Michael Raghunath and Dr Chen Nanguang for their continuoussupport Their valuable feedback during the Ph.D qualifying examhelps me to steer more smoothly towards the completion of this thesiswork.

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PhD journey These short attachments not only broaden the hori-zons of my work, but also set up interesting research links and usefulcollaborations In chronological order, I have visited the followingprofessors and their group.

• Professor David Sampson from University of Western Australia(UWA, OBEL group), where I had a great deal of hands-on ex-periences, particularly on Digital Fourier Holography This iswhere I started with holography.The discussions with his stu-dents, Tim Hillman, Thomas Gutzler, and Dirk Schneiderheinzewere also invaluable.

• Professor George Barbastathesis from Massachusetts Institute ofTechnology (MIT, 3D Optics group), where closer collaborativework has been established George’s advice and mind-bendingdiscussions are gratefully acknowledged In particular, his stu-dent Laura Waller has become such a great colleague and friend.• Professor Christian Depeursinge and Dr Pierre Marquet fromEcole Polytechnique F´ed´erale de Lausanne (EFPL, MVD group),where understandings of holography was deepened I am alsograteful for the extended discussions with their lab membersJohnas Kăuhn, Nicolas Pavillon and Yann Cotte.

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With these friends, we not only talk about research, but also work to-gether on community projects We have set up a student chapter ofthe Optical Society of America (OSA), and I am immensely gratefulfor their kind support during my presidentship of this chapter in theinaugural period I am also particularly indebted to my colleague andgood friend, Tang Wai Teng for his numerical recipes which helpedme on several occasions in the MATLAB programing.

Preparing biological samples is a tedious and cumbersome job thatI did not have proper training of I am thankful to have some goodfriends to help me prepare all the samples used throughout this thesis.Koh Hui Shan spent three weeks of her personal time on teaching meA to Z on how to culture cells Sun Jia from Pharmacology, EdwinLiu from Biochemistry, Ping Yuan from Division of Bioengineeringsupplied me various types of cells from their own experiments.A bunch of happy and cheerful friends in the bioimaging laboratoryalways put smiles on my face, and because of them, the social life ofPh.D is never boring “Du Li Fan Tuan” which means “IndependentGroup for Rice” is the code for us Besides regular gatherings forgood food (essential why it is called the rice group), we enjoyed allkinds of sports and our traveling footprints were left all cross trans-Singapore-Malaysia.

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Contents

List of Figures xi

List of Tables xv

Acronyms, abbreviations and conventions xviiPublications and presentations xix

1 Introduction 1

1.1 Motivation 1

1.2 Objectives 3

1.3 Major contributions of the thesis 4

1.4 Organization of the thesis 6

2 Overview & Background 92.1 Phase contrast imaging 9

2.1.1 Overview 9

2.1.2 Digital holography and holographic tomography 12

2.1.3 Differential Interference Contrast (DIC) Microscope 14

2.1.4 Non-interferometric methods for phase retrieval 17

2.2 Transfer function analysis for an optical system 18

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CONTENTS3 Image formation in digital holographic microscope (DHM) 253.1 Introduction 253.2 Basic DHM set-up 263.3 DHM V.S interference microscope 263.4 3D CTF for DHM 303.5 Broadband DHM 34

3.6 Discussion and Conclusion 36

4 Image formation in holographic tomography 394.1 Introduction 39

4.2 Optical diffraction tomography with two set-ups 41

4.3 Image formation under paraxial approximations 44

4.3.1 Defocused and in-focus transfer functions 44

4.3.2 3D CTF of transmission holographic tomography with scan-ning of illumination in one direction 46

4.3.3 3D amplitude spread function (APSF) under paraxial treat-ment 48

4.4 Assumption of the Ewald Sphere 49

4.5 Image formation under high-aperture conditions 52

4.5.1 CTF for object rotation in a single direction 52

4.5.2 CTF for illumination rotation in a single direction 55

4.5.3 Refection tomography under high aperture case 60

4.5.4 Amplitude point spread function for 4π case 62

5 Realizing quantitative phase imaging in transmission DHM 655.1 Introduction 65

5.2 Set-up of a holographic transmission microscope 65

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CONTENTS

6 Linearizing DIC for quantitative phase imaging 776.1 Introduction 77

6.2 2D modeling and characterization of DIC microscope 79

6.2.1 Measurement of the shear value 80

6.2.2 The de S´enarmont Compensator 90

6.2.3 A wavefront image model for a pure phase object 91

6.3 Obtaining phase gradient information from DIC 92

6.3.1 Phase shifting (stepping) in DIC 94

6.3.2 Image registration in DIC phase stepping 96

6.4 Reconstruction using phase gradient information 99

6.4.1 Direct integration 100

6.4.2 Hilbert Transform 100

6.4.3 Fourier integration with two orthogonal phase gradients 102

6.4.4 Inverse Abel transform - a special case of fiber refractiveindex reconstruction 107

7 Transport equations and defocused images for quantitative phaserecovery 1137.1 Introduction 113

7.2 Solutions to TIE 115

7.3 Quantitative phase imaging through TIE 122

7.4 Application in DIC microscope 124

7.4.1 Introduction 124

7.4.2 TI-DIC 126

7.4.3 Applications of TI-DIC images with image enhancements 1337.5 Phase retrieval using defocused images 137

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List of Figures

2.1 A schematic for DIC setup 15

2.2 DIC image of HeLa cells 16

2.3 Mind map for Fourier Optics -relationship between optical transferfunctions and point spread functions 21

2.4 Typical cellular components of a procaryote cell 22

2.5 Typical cellular components of an animal (eukaryotic) cell 1 nu-cleolus 2 nucleus 3 ribosome 4 vesicle 5 rough endoplasmicreticulum (ER) 6 Golgi apparatus 7 Cytoskeleton 8 smoothendoplasmic reticulum 9 mitochondria 10 vacuole 11 cytoplasm12 lysosome 13 centrioles within centrosome 23

3.1 DHM set-up 27

3.2 Image content of DHM 28

3.3 Coherence Probe Microscope 31

3.4 Frequency cutoff of interference microscope 32

3.5 Frequency cutoff of DHM 32

3.6 quasi-monochromatic CTF 35

3.7 Spatial content in low-coherence DHM 36

3.8 3D CTF for low-coherence DHM 37

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LIST OF FIGURES

4.3 Defocus CTF under paraxial treatment 47

4.4 3D CTF spatial cutoff - paraxial peanut 49

4.5 3D APSF of paraxial tomographic system using 1D scanning 50

4.6 3D IPSF of paraxial tomographic system using 1D scanning 51

4.7 Analytical geometry for deriving high-aperture object rotation case 534.8 The 3D transfer function cutoff for object rotation in transmission 544.9 Analytical geometry for deriving high-aperture holographic tomog-raphy 57

4.10 Cross-sectional view of the transmission CTF 59

4.11 The 3D CTF spatial cutoff for transmission case - high N A peanut 614.12 The 3D CTF spatial cutoff for reflection 61

4.13 The 3D intensity point spread function (IPSF) for 4π tomography 635.1 Schematic of the off-axis holographic microscopy set-up 66

5.2 Infinity-corrected optical system 67

5.3 Actual off-axis holographic microscopy set-up 68

5.4 Digital reconstruction of image holography 71

5.5 Digital reconstruction of a single hologram for USAF target at 3xmagnification 72

5.6 Digital reconstruction of a single hologram for USAF target at 10xmagnification 73

5.7 Digital reconstruction of a single hologram for phase target withcrab pattern at 10x magnification 75

5.8 A comparison between coherent phase reconstruction and partiallycoherent phase reconstruction methods 76

6.1 Schematic diagram of DIC setup and hardware modification 81

6.2 Image-based measurement of the shear at 45 degree of bias 82

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LIST OF FIGURES

6.4 Image-based measurement of the shear at 0 degree of bias 84

6.5 3D mesh plot of the shear PSF at 0 degree of bias 85

6.6 BFP image of DIC shear using 40x objective (condenser side) 87

6.7 Cosine fitting of BFP image using 40x objective (condenser side) 886.8 BFP image of DIC shear using 40x objective (objective side) 89

6.9 Cosine fitting of BFP image using 40x objective (objective side) 896.10 Essential components for constructing the de S´enarmont compen-sator 91

6.11 2D crab DIC image model for a pure phase object 93

6.12 DIC phase stepping for cheek cell 97

6.13 Phase gradient information from DIC phase stepping for cheek cell(Image registration incorporated) 97

6.14 Phase gradient information from DIC phase stepping for cheek cellfrom unregistered images 99

6.15 1D Integration from phase gradient 100

6.16 Hilbert Transform of phase gradient 101

6.17 Phase gradient images along two orthogonal directions 104

6.18 Phase reconstruction from two orthogonal phase gradient images 1056.19 Mis-aligned phase reconstruction from two orthogonal phase gra-dient images that are not registered 106

6.20 Comparison of phase reconstruction from orthogonal gradients andTIE based approach The cell line is HepG2 liver cancer cells 107

6.21 Inverse Abel transform of a mock fiber in simulation 109

6.22 Comparison between a single DIC image and normalized line profile1106.23 Inverse Abel transform for a real fiber and its reconstruction fromsingle DIC images 110

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LIST OF FIGURES

7.2 A stack of defocused images for solving TIE in bright field

mi-croscopy setup 117

7.3 Frequency response of common central difference differentiator 120

7.4 Frequency response of smooth noise-robust differentiator 120

7.5 Noise-robust Differentiator for axial intensity difference 121

7.6 TIE and DHM reconstruction comparison 123

7.7 AFM section measurement of the crab phase object 125

7.8 Simulated phase reconstruction from TI-DIC 128

7.9 Original DIC images of cheek cell and TI-DIC reconstruction images1307.10 Validating the TI-DIC process with phase gradient information 131

7.11 TI-DIC Fiber reconstruction 132

7.12 TI-DIC on live cells 132

7.13 Binary phase image from TI-DIC images 134

7.14 Segmented image from TI-DIC images 135

7.15 PSF for estimating the shear 135

7.16 Deconvolution image on TI-DIC cheek cells 136

7.17 Deconvolution image on TI-DIC mphage cells 137

7.18 TIE and WOTF reconstruction for cheek cell 141

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List of Tables

2.1 List of phase imaging methods 122.2 List of common cellular components and their refractive indices 245.1 Key optical components for DHM set-up 676.1 List of parameters for BFP cosine model fitting for prism at

con-denser side 876.2 List of parameters for BFP cosine model fitting for prism at

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Acronyms, abbreviations andconventions

DHM: Digital Holographic Microscope

DIC: Differential Interference Contrast microscopeEM: electron microscopy

FBP: filtered back propagationFT: Fourier transfomr

HeLa: immortal human cervical cancer cell line used in scientific researchLED: light-emitting diode

MO: microscope objectiven: refractive index

N.A: numerical aperture

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0 ACRONYMS, ABBREVIATIONS AND CONVENTIONS

OTF: optical transfer function or incoherent optical transfer function (dependingon context)

PSF: point spread function

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Publications and presentations

Chapters 3-5 are based on the work presented in the following journal publica-tions.

• S S Kou and C J R Sheppard, ”Imaging in digital holographic mi-croscopy,” Opt Express 15, 13640-13648 (2007) (Virtual Journal of Biomed-ical Optics)

• S S Kou and C J R Sheppard, ”Image formation in holographic tomog-raphy,” Opt Lett 33, 2362-2364 (2008) (Virtual Journal of BiomedicalOptics)

• S S Kou and C J R Sheppard, ”Image formation in transmission holo-graphic tomography: High aperture imaging conditions,” Appl Opt 34,H168-H175 (2009) (Virtual Journal of Biomedical Optics)

Part of Chapter 7 talks about work presented in the following journal publi-cations.

• S S Kou, Laura Waller, George Barbastathis, and C J R Sheppard, ”ATransport-of-Intensity approach to DIC microscope for quantitative phaseimaging in live cells,” Opt Lett 35, 447-449 (2010).

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0 PUBLICATIONS AND PRESENTATIONS

Work related to Chapters 3-5 has been presented in the following conferences.

• S S Kou and C J R Sheppard, ”Transfer function analysis for holo-graphic tomography,” Focus on Microscopy Krakow Poland (2009).• S S Kou and C J R Sheppard, Comparison of three dimensional

trans-fer function analysis of alternative phase imaging methods, Proceedings ofSPIE, Vol 6443, 64430Q (2007).

Work in Chapter 6 results in the following conferences presentations.

• S S Kou and C J R Sheppard, ”Linear phase recovery from DIC mi-croscope,” in International Conference on Advanced Phase MeasurementMethods in Optics and Imaging, AIP Proceedings, Locarno, Switzerland(2010).

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1

Introduction

1.1Motivation

The Specialized microscopic technique called phase contrast imaging is oftenused to visualize living and unstained biological specimens (Hoffman & Davidson,2007) These specimens are sometimes called phase objects because they barelyabsorb or change the intensity when light passes through and give very weak con-trast in conventional bright-field microscopy (Pluta, 1989) What is introducedis a phase delay in the speed of light due to the intrinsic optical property of re-fractive index The contrast enhancement mechanism then converts these phaseshifts into intensity variations for human eyes to perceive Minute structural andmorphological information, therefore, can be embedded in such images Since noexogenous contrast agents are used, the technique is able to provide nondestruc-tive information about the dynamic states of living cells and organisms.

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1 INTRODUCTION

a change in the distribution of scattered light field afterwards For this reason,measurement of the phase distribution properties of cells and tissues providesmeaningful detection of cellular and sub-cellular scattering processes On theother hand, to date, capabilities for obtaining quantitative metrology in phaseimaging, especially three-dimensional data, are rather limited There have beenremarkably few quantitative studies of image formation in different types of phasecontrast microscopy (Cogswell et al., 1997) This thesis sets out to analyze thecommonly used phase imaging modality of digital holography first and points outits pros and cons for phase reconstruction In particular, the focus is tuned onits limitations in 3D imaging capacity Ways of improving this include tomog-raphy and partially coherent methods The thesis follows on to analyze some ofthese examples, among which, optical diffraction tomography (ODT) (alternativename: holographic tomography) (Dăandliker & Weiss, 1970; Devaney, 1982a; Wolf,1969) and Quantitative Phase Microscopy (QPM) (Paganin & Nugent, 1998)based on the principle of the Transport of Intensity Equation (TIE) (Teague,1983) are most promising to give 3D quantitative phase data For TIE, effects ofpartially-coherent imaging, nonlinearity in large phase gradients, and combina-tion of amplitude and phase informacombina-tion are among the key issues to tackle Forholographic tomography, diffraction and inverse scattering, appropriate approx-imations of the form of the object in either Born or Kirchhoff approxapprox-imations(Born & Wolf, 2005), digital interpolations and inverse filtering are all among thecomplications that render the reconstruction difficult to deal with For all thesereasons, it is very meaningful to develop valid models of the image formation pro-cess which analyze the behavior of the systems and offer further advancementsand more accurate phase reconstruction algorithms.

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1.2 Objectives

to analyze shapes of living cells, microorganisms, thin tissue slides, fibers, glassfragments, lithographic patterns and even sub-cellular particles In medical diag-nosis and treatment areas such as laser-induced thermo-therapy, understandingthe alteration of optical properties of tissues such as refractive index changesduring diseases is crucial and is a major challenge for current development ofthe technology Quantitative phase imaging methods developed in this thesiscould have potential applications in improving the current status of such medicaldiagnosis area.

1.2Objectives

The main goal of this thesis project is to develop algorithms and design instru-mental set-ups that retrieve quantitative phase information from various trans-mission microscopic modalities and to investigate the characteristics and possibleapplications of these phase imaging methods in biological applications Digitalholographic microscope (DHM), Differential Interference Contrast (DIC) micro-scope and the Transport of Intensity Equations (TIE) are among the opticalmodalities explored for such investigations The research includes the followingaims:

1 To identify the problems in each specific microscopic instrumentation thathinders the recovery of quantitative phase.

2 To devise methods that solve these problems, firstly through analyticalmeans The process involves investigations using physical optics modeling andcoding of algorithms using mathematical tools.

3 To study the characteristics of the proposed algorithms and set-ups withboth simulations and experimental verifications.

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1 INTRODUCTION

1.3Major contributions of the thesis

This thesis makes four major contributions to the research field of quantitativephase imaging and measurements.

Model of image formation in DHM and holographic tomographyAlthough experimentally implemented, digital holographic microscopy and to-mography had not been studied for their behaviours using imaging formationtheories before Therefore the imaging process was not fully understood and thesubsequent results of reconstructions were often not optimized Using the co-herent transfer function (CTF) concept from Fourier optics, the spatial cutoffsof three-dimensional (3D) CTFs for DHM and holographic tomography were forthe first time derived analytically Especially with the case of 1D illuminationrotation for holographic tomography, the visualization of the 3D CTF cutoff inboth paraxial treatment and high N.A cases were presented, such results beingunprecedented The theoretical formalism presented could provide invaluable in-sights into ways of improving these systems for 3D phase imaging capacity Itwill not only guide the reconstruction process, but also offer links between theFourier space and the spatial space that generate enhanced filtering techniquesaccounting for diffraction effects that were often previously ignored.

Framework of DIC phase retrieval

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1.3 Major contributions of the thesis

further developments in improved phase imaging with DIC.

TIE implementation and a study of defocusing for weak phase ob-jects

It is known that TIE is a Poisson type second order differential equation thatcan be used to solve for the two-dimensional (2D) phase information from irra-diance measurements only TIE is also robust to work with partially coherentlight source, which promises better resolution than interferometric phase recov-ery methods One way of improved phase imaging was achieved with a Fourierimplementation of the TIE equations In particular, a mathematical relation-ship of DIC adapted TIE equations was found This unconventional approachgreatly enhances opportunities for quantitative phase imaging with live cells incommercial DIC microscopes For the specific case of weak phase objects that areoften valid with biological specimens, the fundamental basis for phase retrievalusing defocusing was discussed Novel phase recovery using defocused weak ob-ject transfer functions (WOTFs) was demonstrated using real experimental datafor the first time Such simple yet robust inversion technique could have manypotential applications in real-time quantitative phase imaging.

Parallel study of coherent and partially-coherent phase imagingtechniques

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char-1 INTRODUCTION

acteristics of the designated phase imaging technique Demonstrations of imagingresults between techniques that are based on coherent light and others that arebased on partially coherent light are given for clear differentiation.

1.4Organization of the thesis

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1.4 Organization of the thesis

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2

Overview & Background

The discussions and findings in this thesis touch various aspects of phase imagingtechniques used in microscopy The purpose of this chapter serves as an intro-duction to the relevant topics that are about to be presented and extended in thefollowing chapters The context of this thesis project is outlined with historicalorigins of each problem The first part of this introduction gives a quick overviewof the scope of the phase imaging methods, with a specific summary of each of therelevant concept The analytical tool called the optical transfer function, whichis frequently used for Fourier optics modeling in this thesis, is subsequently de-scribed Its relationship with the standard point spread function (PSF) analysisfor imaging systems is examined Finally, the physiological meaning of refractiveindex in biological cells is elaborated in the last part of this introduction, forbetter understanding of the potential applications of this thesis project.

2.1Phase contrast imaging

2.1.1Overview

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di-2 OVERVIEW & BACKGROUND

aphragm or defocus the image Unfortunately these maneuvers are usually arbi-trary and often lead to degradation of resolution More advanced techniques havebeen developed over the years Zernike invented Phase Contrast Microscopy andreceived the Nobel prize for it (Zernike, 1955) Although still widely used, theZernike Phase Contrast microscope has significant drawbacks such as halo andshading-off artifacts (Murphy et al., 2009) In addition, for achieving linear phaseresponse in the image intensity, the sample thickness must be no more than 1/10of a wavelength (λ), and this is rather restricted, as biological samples can easilyreach the dimensions of hundreds of λ when the average λ value in visible lightis about 550 nm.

Mathematically, we can express the 1D simplified case of a pure phase objecttransmission as (Born & Wolf, 2005),

T (x) = exp (iφ(x)) (2.1)

Then in a perfect imaging system, the image intensity is directly |T (x)|2 = 1,where the phase information φ is lost! If assuming φ is weak, we can approximatethe above as

T (x) ' 1 + iφ(x) (2.2)The underlining principle of Zernike Phase Contrast relies on pupil modification.The undiffracted (DC) component of the transmitted light is either retardedor advanced by a quarter of a λ through a phase plate placed at the back focalplane (BFP) of the objective The modified light distribution in the image plane isT0(x) = ±i + iφ(x) So that when diffracted light and undiffracted light interferesto form the image,

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2.1 Phase contrast imaging

Using various schemes of pupil modification at the BFP, phase imaging isobtained in other methods such as dark field, Schlieren phase imaging (Born &Wolf, 2005), Hoffman modulation contrast (Pluta, 1989) and offset illumination(Kachar, 1985).

Based on a totally different principle using a shearing interferometer and po-larization, Differential Interference Contrast (DIC) microscope stands out fromthe above mentioned phase imaging methods without introducing any direct pupilmodification, although the DIC prism creates cosine fringes at the BFP It isnowadays much preferred to Zernike Phase Contrast method because of its supe-rior transverse resolution at full illuminating aperture and the three-dimensionalimage sectioning capacity (Inoue & Spring, 1997) Its dependence on polarizationoptics, however, limits its applications for birefringent samples Differential PhaseContrast (DPC) (Amos et al., 2003; Hamilton & Sheppard, 1984) was proposedas an alternative to the full-field DIC, which also uses differential operations toacquire phase gradients, except that in the latter case at least two images areneeded to produce one differential image Due to reciprocity, one can use twoside-by-side CCD image detectors or two semi-disk apertures for the subtractionand shearing operation along one chosen lateral direction DPC can be useful inapplications involving birefringent samples where DIC fails.

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2 OVERVIEW & BACKGROUND

Phase Contrast MechanismsDark fieldZernike phase contrastDefocusTransport of Intensity (TIE) EquationOffset illumination

Hoffmann modulation contrastDifferential phase contrast (DPC)

Interference microscopy

Differential Interference Contrast (DIC) microscopyPupil filtering

Shack-Hartmann wavefront sensorDigital holographic microscopy (DHM)

Iterative phase retrieval

Table 2.1: List of phase imaging methods

have received revived interests lately, and particularly in biological applications.Thus they are most promising for a thorough study which might lead to true 3Dperformance A brief introduction of each is introduced in the next few sections,whereas the detailed background information for individual technique is attachedin later chapters, respectively.

2.1.2Digital holography and holographic tomography

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and biomedical studies (Cuche et al., 1999a; Kemper et al., 2006; Marquet et al.,2005; Stern & Javidi, 2007).

DHM is a coherent imaging technique, and its advantage lies in the instanta-neous and quantitative acquisition of both amplitude and the phase informationfrom the reconstruction of the wave-front (Cuche et al., 1999a) Imaging of phasedistributions with high spatial resolution can be used to determine refractive in-dex variations as well as the thickness of the specimen The ability to detect veryminute phase variations also allows quantitative phase imaging to reveal struc-tural characteristics of cells and tissues which, in turn, may have many potentialimplications in medical diagnosis.

Various experimental set-ups have been made in modern DHM systems toextract amplitude, as well as phase information, from different types of phaseobjects In order to overcome the limitation of the CCD pixel size, phase-shiftingtechnique of interferometry was applied to holography (Yamaguchi & Zhang,1997), where a minimum of three holograms need to be captured while the rel-ative phase difference between reference and object beam are changed stepwise.Recent advances in DHM research focuses both on off-axis (Cuche et al., 1999a;Kemper et al., 2006; Marquet et al., 2005) and on-axis (Stern & Javidi, 2007)interference configurations for detection of refractive index, shape, and other in-formation that is useful in biomedical applications Through management ofquantitative phase information, the wave front curvature inherent in the DHMsystem can be compensated for (Ferraro et al., 2003), and it is also feasible toachieve high resolutions using a lens of small numerical aperture (NA) throughFourier holography and synthetic aperture techniques (Alexandrov et al., 2006;Turpin et al., 1995).

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frequency response can be much improved by holographic tomography (Kou &Sheppard, 2007) This is a promising technique to map the 3D complex opticalrefractive indices distribution within an object to its 3D spatial frequency sup-port, thereby generating a real-time 3D image of the object (Charri`ere et al.,2006; Choi et al., 2007; Lauer, 2002; Sharpe et al., 2002) This technique hasbeen variously named optical diffraction tomography (Lauer, 2002), optical pro-jection tomography (Sharpe et al., 2002), and tomographic phase microscopy(Choi et al., 2007) The technique is especially useful for viewing live unstainedbiological samples However, use of a reconstruction algorithm based on theRadon transform or back-projection, adapted from x-ray computed tomography(CT), ignores diffraction Applying it in the regime of optical microscopy maycreate artifacts as here diffraction is not negligible Taking account of diffrac-tion effects promises more precise restoradiffrac-tion of the object In such situadiffrac-tion, amore accurate reconstruction algorithm, for example, filtered back-propagation(FBP) (Devaney, 1982a), is necessary It is known that this latter approach isalso equivalent to filtering in 3D Fourier space.

2.1.3Differential Interference Contrast (DIC) Microscope

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tenths of a micrometer (Preza et al., 1999) The sheared beam is subsequentlycombined in the objective Wollaston prism and then passes through the analyzerbefore viewing An additional parameter called the bias (phase difference be-tween two sheared wavefronts) can be adjusted by sliding the Wollaston prismperpendicular to the optical axis or introducing a de S´enarmont Compensator(Hariharan, 1993).

Figure 2.1: A schematic for DIC setup - The figure shows standard setup inDIC microscope, taken from Olympus Microscope Primer.

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Figure 2.2: DIC image of HeLa cells - The figure shows DIC microscopicimage of live human HeLa cells with magnification factor of 40x.