A Comparative Study of Ray and Wave Theory for Phase Contrast Tomography

A COMPARATIVE STUDY OF RAY AND WAVE THEORY FOR PHASE CONTRAST TOMOGRAPHY

A Thesis Submitted in Partial Fulﬁllment of the Requirements for the Degree of

Bachelor-Master of Technology(Dual Degree)

by Jyoti Meena

DEPARTMENT OF ELECTRICAL ENGINEERING

INDIAN INSTITUTE OF TECHNOLOGY, KANPUR

September 2012

Abstract

In x-ray imaging, hard x-rays(high energy) characterized by straight path travel, deep penetration and absorption provide a basic tool to estimate the internal structure of the body. However because of high absorption and deep penetration, problems of ionization also come into the picture. Increasing the x-ray dose to get a high resolution image has emerged as a concerning and crucial factor in medical imaging. On the other hand soft x-rays(low energy) are characterized by refraction, diffraction and they are less absorbed. Their amplitude is affected by the absorption coefficient while their phase is affected by the refractive index of the medium. In this thesis, towards an objective of soft-tissue imaging with x-ray phase contrast tomography, we make a novel comparative study of models and reconstructions in ray-theoretic and wave-theoretic(Rytov approximation based) phase-contrast optical tomography. Further, with the objective of enabling greater flexibility in modeling and reconstruction schemes, a local plane wave approximation based phase-retrieval from beam-deflection-data is proposed and evaluated. Reconstruction schemes in ray optical tomography are typically based on beam-deflection and optical path length difference data-types. Further the use of wave-theoretic approaches such as the Rytov linearized approximation enables us to conceptually better address the issues of multifrequency reconstructions, since information observed by wave approximation are frequency dependent while in ray approximation frequency comes in form of scaling factor, in addition the Rytov approximation converts the nonlinear reconstruction problem of OPD-based ray-inversion to a linear problem. Dedicated to my parents... Acknowledgements

First of all, I would like to thank my thesis supervisor, Dr. Naren Naik, for his patience and continuous encouragement throughout this work. I am also grateful for the independence. I would also like to thank Dr. Prabhat Munshi for his support, guidance and most importantly for his blessings. A thanks also goes to all my friends of past ﬁve years for their support and love. Finally, I thank my parents, Lakhan Bai Meena and G. R. Meena, my sisters, Shashi and Annu, and my brothers, Sumer Singh Meena and Shitanshu and Bhabhi for their love, encouragement and unremitting support throughout my years of study. They have made this work possible. Contents

List of Figures 9

1 Introduction 1

2 Ray and wave theory modeling of phase diﬀerence data 11 2.1 Rytov approximation ...... 11 2.1.1 Validity conditions for Rytov approximation ...... 13 2.1.2 The Discretized Forward Model ...... 14 2.2 Optical path length diﬀerence evaluation ...... 15 2.3 Regularized reconstruction via Rytov approximation ...... 16 2.3.1 The discretized forward solution ...... 16 2.3.2 Vector to grid interpolation ...... 17 2.4 Numerical studies ...... 18 2.4.1 Ray tracing in the angular displacement form ...... 18 2.4.2 Rytov approximation ...... 18 2.5 Reconstruction ...... 20 2.6 Conclusions ...... 25

3 Deﬂection angle modeling and phase-retrieval 26 3.1 Ray approximation ...... 26 3.2 Rytov approximation ...... 30 3.3 Phase data estimation from deﬂection angle data ...... 33

7 3.3.1 Signiﬁcance of the phase projection data detection ...... 33 3.3.2 Local plane wave assumption phase retrieval ...... 34 3.4 Numerical Studies ...... 35 3.4.1 Paraxial ray model ...... 35 3.4.2 Rytov approximation ...... 37 3.4.3 Estimated phase ...... 39 3.5 Reconstruction ...... 41 3.5.1 Paraxial ray model ...... 42 3.5.2 Rytov approximation ...... 44 3.5.3 Estimated phase ...... 46 3.6 Conclusions ...... 50

4 Summary and Perspective 51

A Derivation of the phase diﬀerence by Rytov approximation 53 A.1 Scattered Phase ...... 53 A.2 Rytov approximation ...... 54

B Analytical Calculation of Coeﬃcients 56

C Bicubic Interpolation 60

D Mapping from coarse grid to ﬁne grid 63

E Representation of projection data with B-spline Interpolation 67

References 71 List of Figures

1.1 Phase shift and attenuation of a wave in a medium. Inside the medium with refractive index n = 1 − δ − iβ. The wave get attenuated,phase shifted and deflected with respect to the wave propagating in free space indicated by green lines, blue lines and red lines respectively[1] . . . . .5 1.2 Schematic drawing of several phase contrast imaging methods[2] . . . .7 1.3 Basic Moirédeflectometer setup for phase objects. P.O.: Phase object;

G1,G2: Ronchi rulings; S: mat screen [3] ...... 10

2.1 Geometry for beam-deflection tomography for Rytov approximation [4] 12 2.2 Comparison of phase data based on Rytov approximation with OPL model for (a)different resolution, (b)receiver plane at different distances 19 2.3 Original phantom used in forward projection data calculation and reconstruction ...... 20 2.4 Comparison of phase difference projection data based on Rytov approximation which is used in reconstruction with OPL model ...... 21 2.5 Reconstruction from Rytov approximation by phase difference projection data, nrme = 12.95%, (b) Reconstructed phantom ...... 22 2.6 Original parabolic refractive-index phantom used in forward projection data calculation and reconstruction ...... 23

9 2.7 Reconstruction from Rytov approximation by phase diﬀerence projection data for parabolic refractive-index phantom, normalized root-mean- square error(nrme) = 32.63%, (b) Reconstructed phantom ...... 24

3.1 Geometry for beam-deflection tomographic reconstruction [5] ...... 27 3.2 Discrete representation of an object [4] ...... 31 3.3 Comparison of ray theory based beam-deflections from paraxial approximation and eikonal equation for (a) different resolution, (b) receiver plane at different distances ...... 36 3.4 Comparison of deflection angle projection data based on Rytov approximation with the eikonal equation model for (a) different resolution, (b) receiver plane at different distances ...... 38 3.5 Estimated phase difference data from Rytov deflection angle data with and without side lobes ...... 39 3.6 Estimated phase difference data from Rytov deflection angle data with and without side lobes ...... 40 3.7 Deflection angle projection data based on Rytov approximation and paraxial ray model which have been used in reconstruction ...... 41 3.8 Reconstruction from paraxial ray model based deflection angle projection data for selfoc-microlens phantom with nrme = 13.76%, (b) Recon- structed phantom ...... 42 3.9 Reconstruction from paraxial ray model based deflection angle projection data for parabolic refractive index phantom with nrme = 35.51%, (b) Reconstructed phantom ...... 43 3.10 Reconstruction from Rytov approximation based deflection angle projection data for selfoc-microlens phantom with nrme = 13.17%, (b) Re- constructed phantom ...... 44 3.11 Reconstruction from Rytov approximation based deflection angle projection data for parabolic refractive index phantom with nrme = 32.57%, (b) Reconstructed phantom ...... 45 3.12 Reconstruction from estimated phase difference data(Detection plane at L × 0.5), nrme = 50.19%, (b) Reconstructed phantom ...... 46 3.13 Reconstruction from estimated phase difference data (Detection plane at L × 0.5), nrme = 18.39%, (b) Reconstructed phantom ...... 47 3.14 Reconstruction from estimated phase difference data for for parabolic refractive-index phantom(Detection plane at L × 0.5), nrme = 64.77%, (b) Reconstructed phantom ...... 48 3.15 Reconstruction from estimated phase difference data for for parabolic refractive-index phantom without side lobes (Detection plane at L×0.5), nrme = 29.20%, (b) Reconstructed phantom ...... 49

B.1 Coordinate Transformation[4] ...... 57

C.1 For each of the four points in (a), we supplies one function value, two ﬁrst derivatives, and one cross-derivative, a total of quantities[6]. . . . . 61 Chapter 1

Introduction

1.1 Motivation

Evaluation of the internal structure of a material, a component or a system without causing damage to it, is an important issue in various ﬁelds. Tomography is an imaging modality which gives us information about the internal structure of an object via the interaction of electromagnetic impulses with the unknown system[7]. This method is used in radiology, archaeology, biology, geophysics, oceanography, astrophysics, material sci- ence, astrophysics, quantum information, and many other sciences. Various sources like x-rays in CT(Computerized Tomography), gamma rays in SPECT(Single-photon emission computed tomography), radio-frequency waves in MRI(Magnetic resonance imaging), electron-positron annihilation in PET(Positron emission tomography), electrons in TEM(Transmission electron microscopy), ions in atom probe, magnetic particles in magnetic particle imaging etc. are used according to diﬀerent characteristics and applications of the medium.

X-rays have been useful tool in detection of pathology of the skeletal system and for detecting the diseases in soft tissues as well. Computerized tomography, ﬂuoroscopy, radiotherapy etc. are some well-known tools for the detection and diagnose in the 2 medical ﬁeld.

A single CT imaging procedure is suﬃcient to create a 3D model of the desired part of body through the absorption properties. However, ionization of the body cells caused by absorption may lead to some directly or indirectly damage to DNA in case of high energy dose or repeatedly CT scan. The most common cancers caused by radiation exposure are lung cancer, breast cancer, thyroid cancer, stomach cancer and leukemia[8].

An important issue today is how to decrease the radiation dose during CT exam- inations without reducing the image quality. Higher radiation dose would result in higher-resolution image, while lower dose could lead to noise and unsharpness in images. Certainly, increasing the dosage quantity would increase adverse side effects in- cluding risk of radiation induced cancer. In addition, image quality in CT is affected by several artifacts like streak-artifact, partial volume effect, ring-artifact, noise-artifact, motion-artifact, windmill and beam hardening. Beam hardening occurs because of the frequency dependent difference in attenuation derived through position and different energy level of the source.

In conventional absorption-based radiography, the x-ray phase shift information is not used for image reconstruction. Phase shift is sensitive to the variation in refractive index. For soft tissues in which absorbing elements are minimal but refractive index variation is present; phase contrast tomography can play a signiﬁcant role. Conse- quently, phase signal can be obtained with much lower dose deposition than absorption; a very important issue when radiation damage has to be taken care of in living systems.

Information regarding amplitude change of source while interacting with a medium 3 has been a well known aspect for determining the structure of the medium. The change in phase and the change in the direction of traveling wave have emerged of great importance due to the recent development of sensitive projection detectors[9, 10, 11, 12]. In this thesis towards a goal of soft tissues imaging with phase contrast x-ray imaging, we have made a study of models and reconstructions in ray-theoretic[5] and a Rytov approximation based wave-theoretic[4] phase contrast optical tomography. A local plane wave approximation based phase-retrieval from deﬂection data is also proposed to enable ﬂexibility of reconstruction schemes.

1.2 X-ray through soft tissue

X-rays are electromagnetic radiation having wavelength in the range of 0.01 nm to 10 nm. Electromagnetic radiation interaction with medium can be typically modeled based on wavelike or particle like properties of impulses.

Low energy x-rays interact with the whole atoms, moderate energy x-rays interact with the electrons and high energy x-rays interact with the nuclei. There are five kinds of interactions which are defined for x-ray and these are known as classical or coherent-scattering, Compton-effect, photoelectric-effect, pair-production and photo- disintegration. Photoelectric-effect is an absorbing process because x-ray attenuate due to the change in their energy level by interaction with nuclei. When x-ray photons are only partially absorbed, it is called a scattering process such as in the Compton- effect, pair-production, photo-disintegration and classic-scatter.

Of the above five interactions, only two are concerns with absorption based diagnostic applications of x-rays, namely the Compton-effect and photoelectric-effect. If we consider the deflection in the direction of x-rays while interacting with matter, coherent-scattering comes into the picture. 4

Low energy x-rays of about 10keV interaction produce a change in the direction. There is no loss of energy and no ionization. Low energy x-rays are of little importance in absorption based diagnostic techniques, but in the phase contrast tomographic imaging they have been proven of great importance[13].

1.3 Complex Refractive Index

In general, when x-rays pass through a sample, their amplitude is decreased and their phase is shifted as shown in ﬁg. 1.1. This change can be described by a complex form of refractive index of the medium(with real part deviating very slightly from unity),

n = 1 − nδ − iβ (1.1) where,

nδ = refractive index decrements causing phase shifts and β = absorption coeﬃcient,

4πβ µ = (1.2) λρ

where, µ = linear absorption coeﬃcient for absorption imaging, λ = wavelength and ρ = mass density

Related phase change and attenuation on propagating a distance are given by,

2π Z z I 4π Z z φ = nδ(x)dx and − log = β(x)dx (1.3) λ 0 I0 λ 0 where,

I0 = incident intensity and I = ﬁnal intensity 5

To see the eﬀect of refractive index, consider a plane wave propagating through the medium with refractive index n given as,

ink.r i(1−δ)k.r −βk.r ψ(r) = E0e = E0e e (1.4) where, k = wave vector, r = position vector and

E0 = amplitude of the electric ﬁeld

Figure 1.1: Phase shift and attenuation of a wave in a medium. Inside the medium with refractive index n = 1 − δ − iβ. The wave get attenuated,phase shifted and deﬂected with respect to the wave propagating in free space indicated by green lines, blue lines and red lines respectively[1]

The change in amplitude and intensity of a wave traveling through a medium relative to a wave traveling through vacuum(n=1 case) are given by eqn.1.5 and eqn.1.6 respectively.

−βkl ∆E = E0(1 − e ) (1.5) 6

2 −βkl 2 −2βkl −µl ∆I = |E0| − |E0e | = I0 − I0e = I0(1 − e ) (1.6) where, µ = 2kβ, l = length of block of material

The second part of the refractive index is the real part δ. The real part follows the phase diﬀerence of source wave relative to the wave traveling in the vacuum. This change in phase at a point r = (x, y) is,

∆Φ = δk.r (1.7) where,

δk = k − k0 with, k wave-number in medium and k0 wave-number in vacuum.

In general, equation1.7 can be rewritten as, Z ∆Φ = (k cos θ + k sin θ) δ(x, y)dxdy (1.8) where, θ = deﬂection angle at r = (x, y)

The change in phase also results in a change in direction of the x-rays as seen in ﬁg. 1.1. The angular change in the direction with paraxial approximation is given by[5],

Z ∂(δ(x, y)dx) α = (1.9) ∂y

Now, we can see how the real and the imaginary parts of the refractive index aﬀect the wave as they pass through the material. This information may be use to measure the real and imaginary parts of the refractive index, which are corresponding to phase contrast imaging and the conventional attenuation-based imaging. 7

Reconstruction from phase data has certain advantages over reconstruction from deﬂection angle data, which are: 1. In Rytov approximation based solution of beam deﬂection tomography, the system of linear equations contains the data in the measurement operator; the reconstruction thus potentially more susceptible to noise than the phase measurement based ones. 2. The conceptually better utilization of multi-frequency information in the linearized wave-theory type of inversions.

1.4 Methods for sensing the phase variations

We have several ways, in which information regrading the phase can be achieved in the form of forward projection data, shown in ﬁg. 1.2[14].

Figure 1.2: Schematic drawing of several phase contrast imaging methods[2]

Crystal interferometer Crystal interferometers uses a number of crystal reflections to split the x-ray beam and 8 let one of them pass through the sample before they recombine. A sketch of a crystal interferometer set-up is shown in fig. 1.2(a), considered very good for synchrotron use and high resolution studies. It is based on the optical path length difference between the two beams, hence limited by need for stability where small vibrations can change the optical path length enough to disturb the measurements and also limited in the field of view by the size of the crystal optics.

Analyzer based imaging When well collimated x-ray beam passes through a sample, beam is slightly refracted. In analyzer based imaging (ABI), refraction is imaged using the Bragg reflection of one or multiple analyzer crystals. A sketch of an ABI setup is shown in fig. 1.2(b), measures derivative of the phase. This set-up is difficult to extend to tomography as crystals are normally aligned such that derivative of refractive index is measured in direction parallel to tomographic axis. Main limitations for source in laboratory are temporal coherence, which limits available flux. Due to diffraction angles and sizes of analyzer crystals field of view will normally also be limited. Also, the derivative of a quantity would be more susceptible to noise than the quantity itself.

Propagation based imaging A diﬀerent approach to phase imaging is propagation based phase contrast. The propagation based imaging(PBI) is in many senses the simplest kind of phase contrast imaging, as no optical elements are required in beam and constraint on spectral width is relaxed [15]. PBI relies upon on interference fringes arising in free space propagation in the Fresnel regime, as illustrated in ﬁg.1.2(c). The measured intensity fringes are thus not a direct measure of phase like the crystal interferometer, but rather the Laplacian of phase front. 9

In order to achieve interference of propagating beam, a very high degree of spatial coherence is required, and a high resolution detector is needed to observe the fringes. A series of images is then recorded at diﬀerent propagation distances in order to unam- biguously determine the phase of the wave front. This method is particularly good at edge enhancement, and is hence well suited for e.g. ﬁber samples, foam or localization of in-homogeneity in metals also in tomography setup. However, for imaging of soft tissue and small density variations this method is not optimal[16].

Grating based imaging Grating based imaging (GBI) is related to the crystal interferometer in sense that it consists of a beam splitter and a beam analyzer, and GBI is related to ABI by fact that the first derivative of the phase front is measured. The beam splitter grating splits beam by diffraction, but diffraction orders are separated by less than a milli-radian, and diffracted beams are hence not spatially separated, but will interfere to create an intensity pattern downstream of beam-splitter at a distance defined by the Talbot effect, as shown in fig. 1.2(d). Refraction in a sample is measured by detecting transverse shift of interference pattern with a high resolution detector or an analyzer grating. Tomographic reconstruction of differential phase is possible even without initial integration to retrieve the quantitative phase shift, and this kind of tomographic reconstruction has turned out to be an advantageous in local tomography.

Methods based on deflection angle projection data On the basis of projection data, optical computerized tomography can be divided into, phase tomography and deflection tomography. Phase tomography uses optical path lengths of rays passing through test objects as projection data, such as in interferometric tomography or holography, while deflection tomography takes the deflection angles of rays passing through test objects as projection data, such as in Moirédeflectometry, 10 shadowgraphy, or a position-sensitive detector[17].

Figure 1.3: Basic Moir´edeﬂectometer setup for phase objects. P.O.: Phase object;

G1,G2: Ronchi rulings; S: mat screen [3]

Moirédeflectometry shown in fig.1.3, is a noncoherent technique equivalent to interferometry but, instead of measuring differences in optical path length (which are proportional to refractive index), it measures ray deflection of a collimated beam (which is proportional to refractive-index gradient). The accuracy of Moirédeflec- tometry is diffraction-limited like interferometry, but it is superior to interferometry with respect to mechanical stability. The requirement for mechanical stability in Moiré deflectometry is one-tenth of the desired sensitivity, compared with λ/10 in interferometry. The deflection-angle measurement does not require a reference beam, and ap- paratus for beam-deflection tomography is not as complicated as phase-measurement tomography[18]. Chapter 2

Ray and wave theory modeling of phase diﬀerence data

In this chapter we carry out modeling studies using Rytov and ray models with optical path diﬀerence data. Reconstructions are shown for a Rytov approximation based forward model.

2.1 Rytov approximation

The Rytov approximation is derived by considering the total ﬁeld represented by a complex phase [19].

u(r) = e(iφ(r)) (2.1)

The in-homogeneous wave equation would be given by

(∇2 + k2)u = 0 (2.2)

We can express the total complex phase,φ, as the sum of an incident phase function

φ0 and a scattered complex phase φs as

φ(r) = φ0(r) + φs(r) (2.3) 2.1 Rytov approximation 12

where

(iφ0(r)) u0(r) = e (2.4) is the incident ﬁeld. We can obtain the equation for the scattered phase as[7]

2 2 2 (∇ + k0)u0φs = −u0[(∇φs) + o(r)] (2.5)

Solving for φs, we obtain Z 1 0 2 0 0 φs = g(r − r )u0[(∇φs) + o(r )]dr (2.6) u0(r) The Rytov approximation assumes that the term in the square brackets in the above equation can be approximated by

2 0 ∼ 0 (∇φs) + o(r ) = o(r ) (2.7) thus, the ﬁrst-order Rytov approximation to the scattered phase φs becomes Z 1 0 0 0 0 φs = g(r − r )u0(r )o(r )dr (2.8) u0(r)

Figure 2.1: Geometry for beam-deﬂection tomography for Rytov approximation [4]

We can write the phase diﬀerence φ(r) as(derived in Appendix A) k2 ZZ φ(r) = ks .r − d2r0O(r0)[N (kR)sin(ks .R) + J (kR)cos(ks .R)] (2.9) 0 4 0 0 0 0 2.1 Rytov approximation 13

where,

0 n2(r0) O(r ) = 1 − 2 = the object refractive index function and n0 R = r − r0 = the vector from detector point to sample point

th th Ni is the i order Neuman function, and Ji is the i order Bessel function. The two components s0x and s0y of the unit propagation vector s0 are equal to 1 and 0, respectively.

A couple of aspects in our problem are: 1. The geometrical dimensions of the variation in the distribution of the refractive index n(r0) are much greater than the wavelength of the incident wave, i.e. one side of element grid in which the refractive index is constant is much greater than the wavelength. 2. The magnitude of the variation in the refractive index is small, i.e.the derivative of refractive index has low value.

2.1.1 Validity conditions for Rytov approximation

In deriving the Rytov approximation we made the assumption that

2 0 ∼ 0 (∇φs) + o(r ) = o(r ) (2.10)

It is true only when

0 2 o(r ) (∇φs) (2.11)

If we write o(r0) in terms of the refractive index

0 2 2 o(r ) = k0[n (r) − 1] (2.12)

2 2 = k0[(1 + nδ(r)) − 1] (2.13)

2 2 = k0[(1 + 2nδ(r) + nδ(r)) − 1] (2.14)

2 2 = k0[2nδ(r) + nδ(r)] (2.15) 2.1 Rytov approximation 14

For small nδ, the object function could be written linearly related to the refractive index derivation as,

0 2 o(r ) ' 2k0nδ(r) (2.16)

So, the condition in eqn. (2.11) could be as,

2 (∆φs) nδ 2 (2.17) k0 We can observe from here that the size of the object is not a factor in the Rytov approximation. Putting expression for k0, we could ﬁnd a necessary condition for the validity of the Rytov approximation given as

∆φ λ2 n s (2.18) δ 2π

The change in scattered phase over one wavelength is important. In this sense the Rytov approximation is valid when the phase change over a single wavelength is small.

We can see that forward projection data based on phase difference differs for different wavelength as inside term integration terms depend upon the wave-number. This proves that diffraction gives deep insight into the wave-material interaction than refraction.

2.1.2 The Discretized Forward Model

We can write the total phase φ(r) obtained from Rytov approximation(given in appendix A)as,

2 Z xmax Z ymax k 2 0 0 φ(r) = ks0.r − d (r )O(r )[N0(kR)sin(ks0.R) + J0(kR)cos(ks0.R)] 4 xmin ymin (2.19) where,

(xmin, ymin) = initial point in object plane, 2.2 Optical path length diﬀerence evaluation 15

(xmax, ymax) = ﬁnal point in object plane

Phase diﬀerence could be written in a vector form as

∆φ = Ao, A ∈

where,

Z xmax Z ymax Aij = [N0(kR)sin(ks0.R) + J0(kR)cos(ks0.R)]dxdy (2.21) xmin ymin

o = Object array in vector form(∈

2.2 Optical path length diﬀerence evaluation

Considering tomography based on axis-symmetric object and obtaining optical path length measurements from that gives more general insight into the problem. Data are obtained by interferometry, given section 1. The unknown refractive index field is given by n(x, y). One ray travels a curved path through the object plane because of refraction due to n(x, y), and the reference ray travels along a straight path through a medium which has uniform refractive index n0. We have taken our field of interest in 2D circular plane. The optical path length difference is defined as[20],

Z rmax OPL = ∆φ = (n(x, y) − n0)ds (2.22) rmin where, ds = diﬀerential length of the ray,

rmin = initial point of interaction of ray with object plane,

rmax = leaving point of ray from object plane and 2.3 Regularized reconstruction via Rytov approximation 16

Ray path which is curved due to the refraction is governed by the ray equation[1]. Equation(2.22) gives us the phase difference obtained by the ray through the propagation from refractive index plane. We are considering only the refraction effect and neglecting the diffraction affect here with assuming size of in-homogeneity in soft tissues is more than 100 times of the source radiation wavelength.

2.3 Regularized reconstruction via Rytov approximation

In this section, the linearized Rytov approximation is inverted to obtain reconstructions of the unknown refractive index. Finding a solutions to a linear problems would involve inversion of the forward model. Common types of solutions are the least squares, minimum norm, and weighted minimum norm solutions. A typical solution should depend continuously on data to be stable. In our problem since typically the measurement operator is ill-posed, we use regularization to produce stable reconstruction.

Typically two techniques dominate this problem, namely, Tikhonov regularization[21] and truncated singular value decomposition(TSVD)[22]. A regularization method re- places the original forward operator with a well-conditioned approximation that is close to it.

2.3.1 The discretized forward solution

Phase diﬀerence given in a matrix form in eqn.(2.20). The linear least-square problem can be expressed as,

2 minxkAo − ∆φk2, M > N (2.23)

The above problem can be seen to be a discrete ill-posed problem since, 1. The singular values of A decay gradually to zero 2.3 Regularized reconstruction via Rytov approximation 17

2. The ratio between the largest and the smallest nonzero singular values is large.

Considering this, Tikhonov regularization has been used to solve the above linear discrete ill-posed problem. It deﬁned the regularized solution oλ as the minimizer of the following weighted combination of the residual norm and the side constraint

2 2 ∗ oλ = arg min{kAo − ∆φk2 + λ kL(o − o )k} (2.24)

Here, the matrix L is typically either the identity matrix In or a p × n discrete approximation of the (n − p) − th derivative operator, in which case L is a banded matrix with full row rank. Where regularization parameter λ controls the weight given to minimization of the side constraint relative to minimization of the residual norm.

Numerical observation for oλ are composed by Tikhonov regularization given the singular values of matrix A were decaying for the projection data.

2.3.2 Vector to grid interpolation

We came up with an interpolation scheme from 1D h vector to 2D phantom, since our phantom is a function of vector h.

o = Ch (2.25) where C = interpolation matrix from h vector to 2D refractive index phantom(∈

We have used bilinear interpolation to create interpolation matrix C. We can use other basis to basis(2D to 2D) interpolation from coarse grid to ﬁne grid in case of non-symmetric object(given in appendixD). 2.4 Numerical studies 18

2.4 Numerical studies

2.4.1 Ray tracing in the angular displacement form

For an accurate forward data-set we need to know the exact ray path through the object plane. Angular displacement form of eikonal equation has been used to trace the ray at each step. Let θ to be ray angle with respect to x-axis, then this angle should satisfy the following first-order differential equation[1], dθ 1 ∂n ∂n = (cos θ − sin θ ) (2.26) ds n ∂y ∂x Where ds is the differential length of the ray. Fourth-order Runge-Kutta method has been used for updating θ based on eqn.(2.26).

2.4.2 Rytov approximation

A comparative analysis has been done for different resolution cases to show the resolution effect on the stability of the wave model. Phase data is much more stable comparative to deflection angle data. Grid size d taken more than λ × 20 started showing instability. Projection angle has been taken equal to 0.0 radians. For a stable projection data case, N = 50 with d = λ × 32, we check the semi-near field and far field approximation and their effect on image reconstruction. Phase data changes it’s form although giving the accurate reconstruction.

Gaussian quadrature integration method [6] has been used to perform the 2D integration over the image element region with side length d for calculating the phase diﬀerence with node points 5. For the reconstruction regularization tool has been used from Per Christian Hansen[21] Matlab package. Phase data is very much stable and give good reconstruction result for large range with respect to resolution and receiver distances, thus we could say it is much more computation friendly in comparison to the deﬂection angle data. 2.4 Numerical studies 19

Phase data stability

1.2 ray wave with N=25 wave with N=32 1 wave with N=38 wave with N=40 wave with N=50

0.8

0.6

(radians) 0.4 φ

0.2

−0.2 −4 −3 −2 −1 0 1 2 3 4 −4 Receiver line(m) x 10

1.2 ray wave with d=0.2L wave with d=0.5L 1 wave with d=L wave with d=3L wave with d=5L

0.8

0.6

(radians) 0.4 φ

0.2

−0.2 −4 −3 −2 −1 0 1 2 3 4 −4 Receiver line(m) x 10

Figure 2.2: Comparison of phase data based on Rytov approximation with OPL model for (a)diﬀerent resolution, (b)receiver plane at diﬀerent distances 2.5 Reconstruction 20

2.5 Reconstruction

Selfoc-microlens phantom

−4 x 10

−3 x 10 7 1

0.8 6

0.6 5

0.4 4

0.2

Object function(O(r)) 3

0 2 4

2 4 1 −4 0 2 x 10 0 −2 −4 −2 x 10 −4 −4 y−axis(m) x−axis(m)

Figure 2.3: Original phantom used in forward projection data calculation and reconstruction

Wave approximation has been used in beam-deﬂection optical tomography measuring the 2D densities. This algorithm measures 2D refractive index properties of the selfoc-microlens phantom given by

2 n(r) = nc(1 − βr /2), r ≤ D/2, (2.27) where, r = distance from lens center,

nc = 1.558, β1/2 = 0.225mm−1 The diameter of the phantom has been taken 0.25mm with total length of object plane is taken to be L = 1mm. 2.5 Reconstruction 21

Phase data used in reconstruction for selfoc microlens phantom

1.6 ray wave 1.4

1.2

0.8

0.6 (radians) φ

0.4

0.2

−0.2 −4 −3 −2 −1 0 1 2 3 4 −4 Receiver line(m) x 10

Figure 2.4: Comparison of phase diﬀerence projection data based on Rytov approximation which is used in reconstruction with OPL model 2.5 Reconstruction 22

Reconstruction from phase data for selfoc microlens phantom

−3 x 10

) 1 0 Original 0.8 Reconstructed

0.6

0.4

0.2

Refractive index(n(r)−n 0 0 0.2 0.4 0.6 0.8 1 1.2 −4 Distance from object−center(m) x 10

−4 x 10 −3 x 10 8 1 7 6 5 0.5 4 3 0 2 4

Object function(O(r)) 2 4 1 0 2 0 −4 −2 −2 x 10 −4 −4 −4 x 10 y−axis(m) x−axis(m)

Figure 2.5: Reconstruction from Rytov approximation by phase diﬀerence projection data, nrme = 12.95%, (b) Reconstructed phantom

Reconstruction from phase difference data gives much more accurate results shown by fig. 2.5 in comparative to the deflection angle data which gives us the property of computation compatibility of phase data. 2.5 Reconstruction 23

Parabolic refractive-index phantom

−4 x 10

5 −4 x 10

6 4.5

5 4

4 3.5

3 3

2 2.5

2 Object function(O(r)) 1

0 1.5

4 1 2 4

−4 0 2 x 10 0.5 0 −2 −4 −2 x 10 −4 −4 0 y−axis(m) x−axis(m)

Figure 2.6: Original parabolic refractive-index phantom used in forward projection data calculation and reconstruction

A parabolic refractive-index phantom could be given by [23]

n 1 − 2∆( r )g1/2 0 ≤ r ≤ a n(r) = 1 a (2.28) n2 , r ≥ a with

2 2 n1 − h2 ∆ = 2 (2.29) 2n1 where,

n1 = refractive index of core center,

n2 = refractive index of cladding, a = core radius and g = a parameter signifies the difference of the refractive index profile (typically ≈ 2) 2.5 Reconstruction 24

Reconstruction for parabolic refractive-index phantom from phase data

−4 x 10

) 8 0 Original 6 Reconstructed

Refractive index(n(r)−n −2 0 0.2 0.4 0.6 0.8 1 1.2 −4 Distance from object−center(m) x 10

−4 x 10 −4 x 10 5 6 4 4 3 2 2 0 4 1

Object function(O(r)) 2 4 0 2 0 −4 −2 −2 x 10 −4 −4 0 −4 x 10 y−axis(m) x−axis(m)

Figure 2.7: Reconstruction from Rytov approximation by phase diﬀerence projection data for parabolic refractive-index phantom, normalized root-mean-square error(nrme) = 32.63%, (b) Reconstructed phantom 2.6 Conclusions 25

2.6 Conclusions

In this chapter, we ﬁrst compared optical path length diﬀerence data obtained by solving the eikonal equation and a linearized(Rytov) approximation to the Helmholtz equation. Subsequently, we demonstrate reconstructions for the Rytov-approximation modelled data.

The phase data obtained from the two models is seen to be comparable; the results obtained here need to be compared to experimental data to see which yeilds a close ﬁt. The results demonstrate the viability of Rytov-approximation based reconstructions for essentially ray-domain problems. Chapter 3

Deﬂection angle modeling and phase-retrieval

In this chapter, two mathematical models have been discussed to interpret the linear approximation of a forward model based deflection angle projection data. One is ray approximation beam-deflection model [5] which assumes almost straight ray path due to slight refractive index variation. Other is Rytov approximation based beam-deflection which takes diffraction effects into account [4]. Both give forward projection data in the form of deflection angle based on Moirédeflectometry(beam-deflection) set-up.

3.1 Ray approximation

We have discussed in chapter 1 that x-rays are affected by refraction, diffraction and absorption phenomenon based on the properties of the medium and wavelength of the source radiation. In ray model, the diffraction effect has been neglected to concentrate on the refraction effect. With refraction effects, geometrical propagation based concept could be used. For the slight variation in refractive index we can apply the linear approximation based on beam-deflection [5]. 3.1 Ray approximation 27

Consider an index-of-refraction distribution given by n(x, y) as shown in ﬁg.3.1. The deﬂection angle θ of a ray projected at angle α at distance y0 from the x0 axis is given by [5]

Figure 3.1: Geometry for beam-deﬂection tomographic reconstruction [5]

0 Z x max 0 0 0 −2 ∂n(x , y ) 0 θ(y , α) = 2 0 dx (3.1) 0 n0 x min ∂y where, θ(y0, α) = ﬁnal deﬂection angle for one projection,

0 x min = initial interaction point of ray, 0 x max = ﬁnal interaction point of ray

The above equation has been derived by following assumptions: (1) In-plane approximation: Electromagnetic radiation propagates in 3D space, by in-plane approximation only 2D propagation in x − y plane has been considered in above equation. (2) Paraxial approximation: Projection rays are parallel to the x0-axis, refraction should in principle aﬀect wave propagation along both x0and y0 axis; we are assuming 3.1 Ray approximation 28 variations along y0-axis dominate variations along x0-axis; hence taking partial derivative along the y0-axis and integrating along the x0-axis.

Discretization of the forward problem

The diﬀerentiation along y0 in eqn.(3.1) at projection view angle α could be written for y0 + ∆y0 → y0 with ∆y0 → 0 as

∂n(x0, y0) n(x0, y0 + ∆y0) − n(x0, y0) = (3.2) ∂y0 ∆y0

In eqn.(3.2), we need to know the refractive index values at (x0, y0 + ∆y0) and at (x0, y0). We have used 2nd order bicubic local piece-wise polynomial basis interpolation function [24] to ﬁnd the refractive index values at these points.

a (x0, y0 + ∆y0)n − b (x0, y0)n = ij j ij j (3.3) ∆y0

c (x0, y0)n = ij j (3.4) ∆y0 where,

0 0 0 th th aij(x , y +∆y ) = the coeﬃcients of interpolation function for i ray and j object element,

0 0 th th bij(x , y ) = the coeﬃcients of interpolation function for i ray and j object element and

0 0 0 0 0 0 0 cij(x , y ) , aij(x , y +∆y )−bij(x , y ) = the coeﬃcients of interpolated diﬀerential function at (x0, y0)

We can write eqn.(3.1) in the form of quadrature sum as

N 0 Z x max 0 0 −2 X ∂n(x , y ) 0 θ = 2 0 dx (3.5) n 0 ∂y 0 j=1 x min 3.1 Ray approximation 29

Plugin eqn.(3.4) into eqn.(3.5) in form of

N 0 Z x max 0 0 −2 X cij(x , y )nj 0 θ = 2 0 dx (3.6) n 0 ∆y 0 j=1 x min Using Simpson’s (1/3)rd rule [6] for integration with respect to x0-axis could be express for grid size(along x0-axis) x0 to x0 + ∆x0 as, 0 0 0 0 0 0 0 ∆x cij(x , y )nj cij(x , y )nj cij(x , y )nj 0 0 = 0 |x0 + 0 |x0+∆x0 + 4 0 | (x +∆x ) (3.7) 6 ∆y ∆y ∆y 2

0 0 = Bij(x , y )nj (3.8) where, 0 0 0 ∆x 0 0 0 0 0 0 0 0 Bij(x , y ) = 0 cij(x , y )|x0 + cij(x , y )|x0+∆x0 + 4cij(x , y )| (x +∆x ) (3.9) 6∆y 2 Finally eqn.(3.1) in the following form could be used for numerical application. 0 Z x max 0 0 −2 ∂n(x , y ) 0 θ = 2 0 dx (3.10) 0 n0 x min ∂y N −2 X = B (x0, y0)n (3.11) n2 ij j 0 j=1 N X = Aijoj (3.12) j=1 = Ao (3.13) where,

−2 M×N Aij = 2 Bij = forward weighting matrix for paraxial ray model(∈ < ), n0 2 nj N×1 oj = (1 − 2 ) = Object array in vector form(∈ < ), n0 M = Total number of projections, N = Total number of object elements(=J2)

We should keep in mind that while we are calculating the weighting matrix A to ﬁnd the deﬂection angle forward projection data, we don’t know the actual ray path. Using a straight ray path assumption is works because we dealing with only slightly varying refractive index distribution in our problem. 3.2 Rytov approximation 30

3.2 Rytov approximation

Using the equations of the geometrical optics [25], we obtain the unity propagation vector s of the scattered ﬁeld U(r)

∇φ(r) s = (3.14) n0 The deflection angle corresponding to the scattered field on receiver line, given in fig. 2.1 is thus written as

∂φ(r) ! −1 ∂y θ(r) = tan ∂φ(r) (3.15) ∂x Under the assumptions that are necessary for the eqn.(3.15), the Rytov approximation holds for the scattered ﬁeld. The partial diﬀerentiation of the phase φ(r) with respect to x could be obtained as given in appendix A.

2 Z xmax Z ymax ∂φ(r) k 2 0 0 k 0 = ks0x − d r O(r ){[( )(x − x )J1(kR) − (ks0x)N0(kR)]cos(ks0.R) ∂x 4 xmin ymin R k − [( )(x − x0)N (kR) − (ks )J (kR)]sin(ks .R)} R 1 0x 0 0 where,

0 n2(r0) O(r ) = 1 − 2 = the object refractive index function and n0 R = r − r0 = the vector from detector point to sample point

th th Ni is the i order Neuman function, and Ji is the i order Bessel function. The two components s0x and s0y of the unit propagation vector s0 are equal to 1 and 0, 0 0 respectively. We could obtain dφ(r)/dy by replacing x, x , and s0xwith y, y , and s0y, respectively, in above equation.

Assuming that the object O(r0) is constant in a small square region whose sides are of length d is called the image element and its region denoted by vector rj. The object is represented by a square region that contains J × J image elements, and we 3.2 Rytov approximation 31

0 set oj = O(r ). A deﬂection angle θ(r) is detected at point r = ri, and the number of detecting points is I. The value of ∂φ(r)/∂x at r = ri is denoted by ∂φi/∂xi.

Figure 3.2: Discrete representation of an object [4]

Then we can rewrite the partial phase derivatives as

J2 k2 X ∂φ /∂x = ks + ( ) o C (3.16) i i 0x 4 j ij j=1

J2 k2 X ∂φ /∂y = ks + ( ) o D (3.17) i i 0y 4 j ij j=1 where,

Z xmax Z ymax 2 k Cij = d rj{[( )(xi − xj)J1(kRij) − (ks0x)N0(kRij)]cos(ks0.Rij) xmin ymin Rij k − [( )(x − x )N (kR ) − (ks )J (kR )]sin(ks .R )} R i j 1 ij 0x 0 ij 0 ij

Rij = ri − rj (3.18)

and Dij is obtained by replacing xi, xj and s0x with yi, yj ands0y, respectively, in above equation. 3.2 Rytov approximation 32

We can also rewrite eqn. (3.15) by using eqn. (3.16) as follows

J2 X k(tan θiCij − Dij)Oj = −4s0x tan θi + 4s0y, θi = θ(ri) (3.19) j=1

Discretization of the forward problem

Linear system of equation given in eqn.(3.19) could be written in matrix form as,

Ao = b (3.20) where,

Aij = k(tan θiCij − Dij) 2 nj oj = (1 − 2 ), n0

bi = −4s0x tan θi + 4s0y, M = Total number of projections and N = Total number of object elements(=J2)

The distribution of deflection angle θ(ri) on the receiver line is considered as projection data at one view angle. We could rotate the object and obtain projection data at different view angles. The values of s0x and s0y are always 1 and 0, respectively, and the vector ri is fixed or we can rotate the source and detector while holding object still, which we can take s0x = cos θi, s0y = sin θi with ri values varying accordingly.

An analytical calculation to compute the coeﬃcients Cij and Dij has been shown in appendix B.

We could reconstruct the object plane by solving eqn.(3.20). Our weighting matrix

A have element in the form (tan θiCij − Dij), which depend upon the projection data itself. In practical application, projection data always contains some amount of noise. So, there is noise term sitting in the expression of the weighting matrix itself. This will 3.3 Phase data estimation from deﬂection angle data 33 typically create computational issues in the reconstructions. However, in the present work, we have considered noiseless data since our preliminary focus was on model comparison.

3.3 Phase data estimation from deﬂection angle data

3.3.1 Signiﬁcance of the phase projection data detection

Phase contrast x-ray computerized tomography is a newly emerging technique because of its advantage of low energy x-ray dose over high energy x-ray dose and easy detection in soft tissues. Related to two kinds of projection data, phase-contrast tomography can be divided into phase tomography and deﬂection tomography. The projection data obtained from the phase tomography gives us the direct summation of the refractive index known as optical path length of ray such as in interferometric tomography.

Z rmax 0 Pphase = n(r)d(r ) (3.21) rmin On the other hand projection obtained from deflection tomography gives us the summation of the derivative of the refractive index in the form of deflection angle of the ray such in moire deflectometry. With paraxial approximation the deflection tomography can be expressed by

Z rmax ∂n(r) 0 Pdeflection = d(r ) (3.22) rmin ∂y

Deflection tomography could be used in much practical condition, but projection data(deflection angle) obtained from it is not very computation friendly in terms of reconstruction process. Based upon local plane wave approximation, a phase-retrieval could be obtained from deflection angle data. Reasonable reconstructions obtained from this data using Rytov approximation based inversion are demonstrated. 3.3 Phase data estimation from deflection angle data 34

3.3.2 Local plane wave assumption phase retrieval

Considering a local plane wave approximation of the phase front at the receiver, we can write for a deﬂection angle θ,

φ(x, y) = k(s.r) = k(x cos θ + y sin θ) (3.23)

where,

k = k0n = wave-number with k0 is the wave-number in vacuum and n is the medium refractive index, (x, y) = receiver vector points

Note that this approximation satisﬁes the relation equation(3.15). Initial phase is given by

φ(x0, y0) = k(s0.r0) = k(s0xx0 + s0yy0) (3.24) where,

(x0, y0) = initial vector point for plane wave in object plane

Phase diﬀerence can be written as

∆φ(r) = φ(x, y) − φ(x0, y0) (3.25)

The eqn.(3.25) shows an important way to reconstruction by deflection tomography. Deflection projection can be converted to phase projection by means of eqn.(3.25). Then practical algorithms used for phase tomography, such as FBP and ART, can be applied to deflection tomography[17]. 3.4 Numerical Studies 35

3.4 Numerical Studies

3.4.1 Paraxial ray model

Beam-deflection based on paraxial ray model derived deflection angle data has been compared to the eikonal based angular displacement model for different resolutions cases(N = 50, 40, 38, 32, 25).

Source with optical wavelength 0.63 µm has been used at 0.0 radians projection angle. Beam-deflection data is not affected by changing the receiver plane distance, because it is not specified based on far field or near field as described by fig. 3.3.

Reconstruction from estimated phase data has been done through the non-negative least square toolbox given in matlab. Finally an interpolation for nonzero values has been done to improve the reconstruction quality. 3.4 Numerical Studies 36

Paraxial ray model based deﬂection data stability

0.1 ray paraxial with N=25 0.08 paraxial with N=32 paraxial with N=38 paraxial with N=40 0.06 paraxial with N=50

0.04

0.02

0 (radians) θ −0.02

−0.04

−0.06

−0.08

−0.1 −4 −3 −2 −1 0 1 2 3 4 −4 Receiver line(m) x 10

0.1 ray paraxial with d=0.2L 0.08 paraxial with d=0.5L paraxial with d=L paraxial with d=3L 0.06 paraxial with d=5L

0.04

0.02

0 (radians) θ −0.02

−0.04

−0.06

−0.08

−0.1 −4 −3 −2 −1 0 1 2 3 4 −4 Receiver line(m) x 10

Figure 3.3: Comparison of ray theory based beam-deflections from paraxial approximation and eikonal equation for (a) different resolution, (b) receiver plane at different distances 3.4 Numerical Studies 37

3.4.2 Rytov approximation

Rytov model based deﬂection angle projection data has been compared with the eikonal equation solution for diﬀerent resolutions cases(N = 50, 40, 38, 32, 25).

To calculate the coeﬃcient Cij and Dij in 2D, Gaussian quadrature integration method [6] has been used over the image element region with side length d with 5 node points.

The grid size has also to be set-up for accurate reconstruction. Grid size has to be greater than 25.2µm. Deﬂection angle forward projection data start to show instability for d ≤ 25.2µm and so do the reconstruction images derived from it. We took grid size d=λ × 32 with N = 50 for further analysis.

Observations have been taken for semi-near field(L × 0.5) to describe the receiver distance effect on forward projection data and reconstruction. Forward projection data shows the significance changes with respect to the distance of the receiver plane due to the effect of evanescent waves. In practice, these evanescent waves always decays rapidly far from the boundary, and can be ignored at a distance more than 10×λ from an in-homogeneity. 3.4 Numerical Studies 38

Rytov approximation based deﬂection angle data stability

0.1 ray wave with N=25 0.08 wave with N=32 wave with N=38 wave with N=40 0.06 wave with N=50

0.04

0.02

0 (radians) θ −0.02

−0.04

−0.06

−0.08

−0.1 −4 −3 −2 −1 0 1 2 3 4 −4 Receiver line(m) x 10

0.06 ray wave with d=0.2L wave with d=0.5L wave with d=L 0.04 wave with d=3L wave with d=5L

0.02

0 (radians) θ

−0.02

−0.04

−0.06 −4 −3 −2 −1 0 1 2 3 4 −4 Receiver line(m) x 10

Figure 3.4: Comparison of deflection angle projection data based on Rytov approximation with the eikonal equation model for (a) different resolution, (b) receiver plane at different distances 3.4 Numerical Studies 39

3.4.3 Estimated phase

Numerical Results have been obtained for the phase data obtained by above local plane wave assumption and than compared to Rytov phase data. In the far field, detected phase data would not get accurately reconstructed image, but when we decreased the distance between image plane and the receiver to make it in semi-near field, the image has improved. Estimated phase difference from deflection angle data for selfoc microlens phantom

1.2 original wave φ 1 estimated φ 0.8

0.6

0.4 (radians)

φ 0.2

−0.2 −4 −3 −2 −1 0 1 2 3 4 −4 Receiver line(m) x 10

0.8 original wave φ 0.6 estimated φ (without side lobes)

0.4

0.2 (radians) φ 0

−0.2 −4 −3 −2 −1 0 1 2 3 4 −4 Receiver line(m) x 10

Figure 3.5: Estimated phase diﬀerence data from Rytov deﬂection angle data with and without side lobes 3.4 Numerical Studies 40

Side lobes in estimated phase data casing error in refractive index. We remove the estimated phase data from homogeneous region which has no refractive index variation to improve the accuracy in estimated phase data. Estimated phase diﬀerence for parabolic refractive-index phantom

1.2 original wave φ 1 estimated φ 0.8

0.6

0.4 (radians)

φ 0.2

−0.2 −4 −3 −2 −1 0 1 2 3 4 −4 Receiver line(m) x 10

0.6 original wave φ 0.5 estimated φ (without side lobes) 0.4

0.3

0.2 (radians)

φ 0.1

−0.1 −4 −3 −2 −1 0 1 2 3 4 −4 Receiver line(m) x 10

Figure 3.6: Estimated phase diﬀerence data from Rytov deﬂection angle data with and without side lobes 3.5 Reconstruction 41

3.5 Reconstruction

Deﬂection data used in reconstruction

0.1 eikonal wave 0.08 paraxial

0.06

0.04

0.02

0 (radians) θ −0.02

−0.04

−0.06

−0.08

−0.1 −4 −3 −2 −1 0 1 2 3 4 −4 Receiver line(m) x 10

Figure 3.7: Deﬂection angle projection data based on Rytov approximation and paraxial ray model which have been used in reconstruction

Rytov approximation has been applied on N = 50 number of points with grid size of d=λ × 32. L denotes the total length of the object plane. Both, paraxial ray model based and Rytov approximation based deﬂection data used in reconstruction has been shown in ﬁg.3.7 in compare to standard Runge-Kutta method. 3.5 Reconstruction 42

3.5.1 Paraxial ray model

Reconstruction from paraxial ray model for selfoc microlens phantom

−3 x 10

) 1 0 Original 0.8 Reconstructed

0.6

0.4

0.2

Refractive index(n(r)−n 0 0 0.2 0.4 0.6 0.8 1 1.2 −4 Distance from object−center(m) x 10

−4 x 10 −3 x 10 8

1 7 6 5 0.5 4 3 0 2 4

Object function(O(r)) 2 4 1 0 2 0 −4 −2 −2 x 10 −4 −4 −4 x 10 y−axis(m) x−axis(m)

Figure 3.8: Reconstruction from paraxial ray model based deﬂection angle projection data for selfoc-microlens phantom with nrme = 13.76%, (b) Reconstructed phantom . 3.5 Reconstruction 43

Reconstruction from paraxial ray model for parabolic refractive index phantom

−4 x 10

) 8 0 Original 6 Reconstructed

Refractive index(n(r)−n −2 0 0.2 0.4 0.6 0.8 1 1.2 −4 Distance from object−center(m) x 10

−4 x 10 −4 x 10 5 6 4 4 3 2 2 0 4 1

Object function(O(r)) 2 4 0 2 0 −4 −2 −2 x 10 −4 −4 0 −4 x 10 y−axis(m) x−axis(m)

Figure 3.9: Reconstruction from paraxial ray model based deﬂection angle projection data for parabolic refractive index phantom with nrme = 35.51%, (b) Reconstructed phantom . 3.5 Reconstruction 44

3.5.2 Rytov approximation

Reconstruction from Rytov approximation for selfoc microlens phantom

−3 x 10

) 1 0 Original 0.8 Reconstructed

0.6

0.4

0.2

Refractive index(n(r)−n 0 0 0.2 0.4 0.6 0.8 1 1.2 −4 Distance from object−center(m) x 10

−4 x 10 −3 x 10 8 1 7 6 5 0.5 4 3 0 2 4

Object function(O(r)) 2 4 1 0 2 0 −4 −2 −2 x 10 −4 −4 −4 x 10 y−axis(m) x−axis(m)

Figure 3.10: Reconstruction from Rytov approximation based deﬂection angle projection data for selfoc-microlens phantom with nrme = 13.17%, (b) Reconstructed phantom . 3.5 Reconstruction 45

Reconstruction from Rytov approximation for parabolic refractive index phantom

−4 x 10

) 8 0 Original 6 Reconstructed

Refractive index(n(r)−n −2 0 0.2 0.4 0.6 0.8 1 1.2 −4 Distance from object−center(m) x 10

−4 x 10 −4 x 10 5 6 4 4 3 2 2 0 4 1

Object function(O(r)) 2 4 0 2 0 −4 −2 −2 x 10 −4 −4 −4 x 10 y−axis(m) x−axis(m)

Figure 3.11: Reconstruction from Rytov approximation based deﬂection angle projection data for parabolic refractive index phantom with nrme = 32.57%, (b) Recon- structed phantom . 3.5 Reconstruction 46

3.5.3 Estimated phase

Reconstruction from estimated phase data has been done using non-negative con- straints in least square method. After reconstruction, data needs to be interpolated for non-zero values which provide more accuracy as shown in ﬁg. 3.13. Normalized root-mean-square error(nrme) are given with each ﬁgure. Reconstruction from estimated phase data for selfoc microlens phantom

−3 x 10

) 2 0 Original Reconstructed 1.5

0.5

Refractive index(n(r)−n 0 0 0.2 0.4 0.6 0.8 1 1.2 −4 Distance from object−center(m) x 10

−4 x 10 −3 x 10 16 2 14 12 10 1 8 6 0 4 4

Object function(O(r)) 2 4 2 0 2 0 −4 −2 −2 x 10 −4 −4 −4 x 10 y−axis(m) x−axis(m)

Figure 3.12: Reconstruction from estimated phase diﬀerence data(Detection plane at L × 0.5), nrme = 50.19%, (b) Reconstructed phantom 3.5 Reconstruction 47

Reconstruction from estimated phase data without side lobes for selfoc microlens phantom

−3 x 10

) 1 0 Original 0.8 Reconstructed

0.6

0.4

0.2

Refractive index(n(r)−n 0 0 0.2 0.4 0.6 0.8 1 1.2 −4 Distance from object−center(m) x 10

−4 x 10 −3 x 10 6

1 5

4 0.5 3

2 0 4 1

Object function(O(r)) 2 4 0 2 0 −4 −2 −2 x 10 −4 −4 −4 x 10 y−axis(m) x−axis(m)

Figure 3.13: Reconstruction from estimated phase diﬀerence data (Detection plane at L × 0.5), nrme = 18.39%, (b) Reconstructed phantom 3.5 Reconstruction 48

Reconstruction for parabolic refractive-index phantom

−3 x 10

) 1.4 0 Original 1.2 Reconstructed 1

0.8

0.6

0.4

0.2

Refractive index(n(r)−n 0 0 0.2 0.4 0.6 0.8 1 1.2 −4 Distance from object−center(m) x 10

−3 x 10 −3 x 10 1.2 1.5 1

1 0.8

0.5 0.6

0.4 0 4 0.2

Object function(O(r)) 2 4 0 2 0 −4 −2 −2 x 10 −4 −4 0 −4 x 10 y−axis(m) x−axis(m)

Figure 3.14: Reconstruction from estimated phase diﬀerence data for for parabolic refractive-index phantom(Detection plane at L × 0.5), nrme = 64.77%, (b) Recon- structed phantom 3.5 Reconstruction 49

Reconstruction for parabolic refractive-index phantom without side lobes

−4 x 10

) 8 0 Original Reconstructed 6

Refractive index(n(r)−n 0 0 0.2 0.4 0.6 0.8 1 1.2 −4 Distance from object−center(m) x 10

−4 x 10 −3 x 10 6

1 5

4 0.5 3

2 0 4 1

Object function(O(r)) 2 4 0 2 0 −4 −2 −2 x 10 −4 −4 −4 x 10 y−axis(m) x−axis(m)

Figure 3.15: Reconstruction from estimated phase diﬀerence data for for parabolic refractive-index phantom without side lobes (Detection plane at L × 0.5), nrme = 29.20%, (b) Reconstructed phantom 3.6 Conclusions 50

3.6 Conclusions

In this chapter, we have compared beam deﬂection data obtained from forward models based on the ray-equation(paraxial approximation and the eikonal-equation) and the Rytov approximation.

We observe that the deﬂection data and reconstruction obtained from the Rytov approximation is quite sensitive to the location of the detector plane. Best reconstruction were seen to be obtained fro receiver distances about half the object size. This observation needs to be further analyzed.

The local plane wave assumption based phase-retrieval gives phase data that needs incorporation a prior support information to achieve acceptable levels of data and reconstruction correlation with the ground truth. This is an encouraging result gives a direction to look into for further development of more sophisticated schemes. Chapter 4

Summary and Perspective

Comparison of ray and wave model

Low dose x-ray imaging of phase objects is an emerging modality in x-ray tomography. Most of the algorithms typically use ray theory reconstructions based on beam deflection data since it is easier to obtain than phase difference data. Keeping in mind that x-ray source are typically not monochromatic, we need to investigate the use of wave theoretic model for reconstructions from beam deflection and phase difference data, since they better use multifrequency information.

In this thesis, we made a novel comparative study of models and reconstructions in ray theoretic and wave theoretic(Rytov approximation based) phase-contrast optical tomography. The present comparative studies between ray and wave theory based models have been carried out at optical frequencies to compare results in known benchmark cases.

The Ray approximation is derived with refraction effect domination which happens when the in-homogeneity is much larger than the wavelength of the source radiation. When size of homogeneity becomes comparable to the wavelength of the source radi- 52 ation, diffraction effects dominate, which we model on the basis of wave theory in the Rytov approximation. Comparison of these two techniques to derive the forward projection data (deflection and phase) gives the important insight in terms of limitations of the respective approximation.

Phase data is much more stable than deflection data both in the sense of resolutions, receiver distance and quality of reconstructions. However, the physical instrumentation set-up for phase-data is very sensitive and has difficulty with large specimen. Deflection angle data is very sensitive to receiver distance and resolution. The error in reconstructed images changes quickly when we alter any of the two above. Ray modeled beam-deflection however sustains stability with respect to far field and near field, but obtained reconstructed image quality was not very good.

Estimation of phase data

Computationally phase data is more beneficial. The deflection angle data model obtained from Rytov approximation contains measurements in it’s measurements matrix, however deflection data can be obtained in much more practical and applicable scenar- ios than phase data making it potentially susceptible to noise.

In the present work, we have utilized a local plane wave approximation to estimate the phase from the beam deﬂections. The reconstruction results obtained are encouraging enough to motivate the use of deﬂection data in subsequent phase retrieval schemes. Appendix A

Derivation of the phase diﬀerence by Rytov approximation

A.1 Scattered Phase

Scattered phase for in-homogeneous wave from [7] is given by Z 1 0 2 0 0 φs = g(r − r )u0[(∇φs) + o(r )]dr (A.1) u0(r) where,

φs= Scattered phase for in-homogeneous wave,

u0 = A exp (jk0s.r) = incident ﬁeld for plane wave, 0 i (1) 0 g(r − r ) = 4 H0 (k0R) = Green’s function in 2D with R = |r − r | 0 2 2 o(r ) = k0[n (r) − 1] = Object-plane function in terms of refractive index

where n(r) is the electromagnetic refractive index of the media and is given by s µ(r)(r) n(r) = (A.2) µ00

Here we have used µ and to represent the magnetic permeability and dielectric constant and the subscript zero to indicate their average values. k0 indicates the wave- A.2 Rytov approximation 54 number in vacuum. s is unit propagation vector and r shows receiver plane vector, whereas r0 shows the position of a point in object plane.

A.2 Rytov approximation

Using the Rytov approximation we assume that the term in brackets in the above equation can be approximated by

2 0 ∼ 0 (∇φs) + o(r ) = o(r ) (A.3) applying this, the ﬁrst-order Rytov approximation to the function of scattered phase

φs becomes Z 1 0 0 0 0 φs = g(r − r )u0(r )o(r )dr (A.4) u0(r)

Z 1 0 0 0 0 φs(r) = g(r − r )u0(r )o(r )dr (A.5) u0(r) where, the Green’s function is given by i g(r − r0) = H(1)(k R) (A.6) 4 0 0 with the ﬁrst kind Hankel function of zero-order(Bessel function of third kind). Hankel function is given by,

(1) H0 (k0R) = J0(k0R) + iN0(k0R) (A.7) where,

J0(k0R) = Bessel function of ﬁrst kind and

N0(k0R) = Bessel function of second kind (Neumann function)

Putting eqn.(A.6) and (A.7) into the eqn.(A.5), we get Z 1 i 0 0 0 φs(r) = (J0(kR) + iN0(kR))u0(r )o(r )dr (A.8) u0(r) 4 A.2 Rytov approximation 55

now, putting the expression for incident ﬁeld and the object plane in above equation gives, Z 1 i 0 2 2 0 0 φs(r) = (J0(kR) + iN0(kR))A exp (ik0s.r )k (n (r ) − 1)dr A exp (ik0s.r) 4 Z ik2 = − (J (kR) + iN (kR)) exp (ik s.(r0 − r))(1 − n2(r0))dr0 4 0 0 0 k2 Z = − i(J (kR) + iN (kR)) exp (−ik s.(r − r0))(1 − n2(r0))dr0 4 0 0 0

Using Euler’s formula and putting O(r0) = (1 − n2(r0)) with R = r − r0 in above equation we get,

k2 Z φ (r) = − i(J (kR) + iN (kR)) × (cos (k s.R) − i sin (k s.R))O(r0)dr0 s 4 0 0 0 0 k2 Z = − i(J (kR) × cos (k s.R) + N (kR) × sin (k s.R)) 4 0 0 0 0 0 0 (J0(kR) × sin (k0s.R) − N0(kR) × cos (k0s.R))O(r )dr

The second term in the integration is real and is accountable for amplitude change.

For scattered phase term we would have the expression

k2 Z φ (r) = − (J (kR) × cos (k s.R) + N (kR) × sin (k s.R))O(r0)dr0 (A.9) s 4 0 0 0 0

Initial phase is given by φ0(r) = ks0.r, then the total scattered phase would be

∆φ(r) = φ0(r) − φs(r) (A.10) k2 Z = ks .r − (J (kR) × cos (k s.R) + N (kR) × sin (k s.R))O(r0)dr0 0 4 0 0 0 0 (A.11) as given in the eqn.(2.9). Appendix B

Analytical Calculation of Coeﬃcients

We can reduce the double integral to calculate the coefficients given in Rytov approximation to a single integral. We eliminate the suffix ij of R and R for simplicity. We take the assumption of R is a few millimeters here, so the value of kR can be considered as infinite. Then Bessel and Neuman functions are approximated in their asymptotic form as below

2 1/2 3π J (kR) ∼ cos kR − (B.1) 1 πkR 4

2 1/2 3π N (kR) ∼ sin kR − (B.2) 1 πkR 4

2 1/2 π J (kR) ∼ cos kR − (B.3) 0 πkR 4

2 1/2 π N (kR) ∼ sin kR − (B.4) 0 πkR 4 Second assumption is the length d of the sides of an image element is so small that

3/2 1/2 the values of (kR) (xi − xj) and (1/R) in coeﬃcients estimation do not change 57

greatly when the vector rj moves in the region of an image element, then these factors can be moved out of the integral. Then integral eqn. could be rewritten as

2k 1/2 ZZ 3π C = (x − x ) d2r cos kR − cos(ks .R) ij πR3 i j j 4 0

2k 1/2 ZZ π + s d2r sin kR − cos(ks .R) πR 0x j 4 0

2k 1/2 ZZ 3π − (x − x ) d2r sin kR − sin(ks .R) πR3 i j j 4 0

2k 1/2 ZZ π − s d2r cos kR − sin(ks .R) (B.5) πR 0x j 4 0

r r

Figure B.1: Coordinate Transformation[4]

To reduce the double integral to a single integral in eqn. B.5, we introduce a new variable γ deﬁned by the angle between the vector R and the direction of the x axis, as shown in ﬁg. B.1. Using the variable γ, we have

ks0.R = k(s0xcosγ + s0ysinγ)R = KγR (B.6) 58

2 The integral element d rj is replaced with RdRdγ, and R changes from R1γ to R2γ at the angle γ, which changes from γ1 to γ2 in an image element, as shown in ﬁg. B.1. So, we can rewrite eqn. B.5 as

2k 1/2 Z γ2 Z R2γ 3π Cij = 3 (xi − xj) cos kR − cos(KγR)RdRdγ πR γ1 R1γ 4

2k 1/2 Z γ2 Z R2γ π + s0x sin kR − cos(KγR)RdRdγ πR γ1 R1γ 4

2k 1/2 Z γ2 Z R2γ 3π − 3 (xi − xj) sin kR − sin(KγR)RdRdγ πR γ1 R1γ 4

2k 1/2 Z γ2 Z R2γ π − s0x cos kR − sin(KγR)RdRdγ (B.7) πR γ1 R1γ 4

The ﬁrst integral term F1 in eqn. B.7 can be expressed as

Z γ2 Z R2γ 3π 3π F1 = cos (k + Kγ)R − + cos (k − Kγ)R − RdRdγ γ1 R1γ 4 4

Z γ2 Z R2γ 3π 3π = Rcos (k + Kγ)R − + Rcos (k − Kγ)R − dRdγ γ1 R1γ 4 4

! Z γ2 Z R2γ 3π Z R2γ dRZ R2γ 3π = [ R cos (k + Kγ)R − dR + cos (k + Kγ)R − dR γ1 R1γ 4 R1γ dR R1γ 4

! Z R2γ 3π Z R2γ dRZ R2γ 3π + R cos (k + Kγ)R − dR + cos (k + Kγ)R − dR ]dγ R1γ 4 R1γ dR R1γ 4

! Z γ2 Rsin (k + K )R − 3π Z R2γ sin (k + K )R − 3π dR = [ γ 4 |R2γ − γ 4 dR R1γ γ1 (k + Kγ) R1γ (k + Kγ) 59

! Rsin (k − K )R − 3π Z R2γ sin (k − K )R − 3π dR + γ 4 |R2γ − γ 4 dR ]dγ R1γ (k − Kγ) R1γ (k − Kγ)

! Z γ2 3π 3π Rsin (k + Kγ)R − 4 cos (k + Kγ)R − 4 = [ + 2 γ1 (k + Kγ) (k + Kγ)

! Rsin (k − K )R − 3π cos (k − K )R − 3π + γ 4 + γ 4 ]|R2γ ]dγ (B.8) 2 R1γ (k − Kγ) (k − Kγ)

By above two assumptions, we can neglect term with 1/(k − K2 in eqn. B.8 Iˆ±) ! Z γ2 Rsin (k + K )R − 3π Rsin (k − K )R − 3π F = [ γ 4 + γ 4 |R2γ ]dγ (B.9) 1 R1γ γ1 (k + Kγ) (k − Kγ)

So, eqn. B.7 becomes

1/2 γ 3π 3π ! 2k Z 2 Rsin (k + K )R − Rsin (k − K )R − C = (x − x ) [ γ 4 + γ 4 |R2γ ]dγ ij 3 i j R1γ πR γ1 (k + Kγ) (k − Kγ)

1/2 γ π π ! 2k Z 2 Rcos (k + K )R − Rcos (k − K )R − + s [ − γ 4 − γ 4 |R2γ ]dγ 0x R1γ πR γ1 (k + Kγ) (k − Kγ)

1/2 γ 3π 3π ! 2k Z 2 Rsin (k + K )R − Rsin (k − K )R − − (x − x ) [ γ 4 − γ 4 |R2γ ]dγ 3 i j R1γ πR γ1 (k + Kγ) (k − Kγ)

1/2 γ π π ! 2k Z 2 Rcos (k + K )R − Rcos (k − K )R − − s [ − γ 4 − γ 4 |R2γ ]dγ 0x R1γ πR γ1 (k + Kγ) (k − Kγ)

1/2 1/2 ! Z γ2 2k 2k 3 (x − x ) 3π s 3π C = 2 πR i j Rsin (k + K )R − − πR 0x Rsin (k − K )R − |R2γ ]dγ ij γ γ R1γ γ1 (k + Kγ)R 4 (k − Kγ)R 4 (B.10)

Dij could be obtained similarly for y axis. Appendix C

Bicubic Interpolation

We represented the object in the square region that contains J × J image elements. The length of the one image element in kept so small, so that it could satisfy the assumptions which were necessary in the Rytov approximation. Hence, the number of J 2 may become large, we reduce the number of unknown by using only the unknowns

Og whose element are located at intervals of distance L × d. The number g may be given by

g = 1 + (u − 1)L + (v − 1)L × J, (u, v) = 1 ∼ N, (C.1) where the number of unknowns is N 2 and J = (N − 1)L + 1. After reconstructing the object Og which is obtained with the reduced dimension, original object can be interpolated by bi-cubic Interpolation [6]. The interpolated surface can then be written as

X3 X3 i j p(x1, x2) = aij(x1 − x1l) (x2 − x2l) (C.2) i=0 j=0 x1l, x2l are the coordinates of the lower left corner of the grid square corresponding to

(x1, x2)[1].

We can ﬁnd the values of aij by given function values and their derivative grouping 61

Figure C.1: For each of the four points in (a), we supplies one function value, two ﬁrst derivatives, and one cross-derivative, a total of quantities[6].

the unknown parameters aij in a vector,

T Γ = [a00 a10 a20 a30 a01 a11 a21 a31 a02 a12 a22 a32 a03 a13 a23 a33] (C.3)

and

X = [f(0, 0) f(1, 0) f(0, 1) f(1, 1) fx1(0, 0) fx1(1, 0) fx1(0, 1) fx1(1, 1) ... T fx2(0, 0) fx2(1, 0) fx2(0, 1) fx1(1, 1) fx1x2(0, 0) fx1x2(1, 0) fx1x2(0, 1) fx1x2(1, 1)] We can written above two vectors in the form of linear equation

CΓ = X (C.4) where, 62

C−1 =   1000000000000000      1111000000000000       1000100010001000       1111111111111111       0100000000000000       0123000000000000       0100010001000100       0123012301230123       0000100000000000       0000111100000000       0000100020003000       0000111122223333       0000010000000000       0000012300000000       0000010002000300    0000012302460369

Γ = C−1X, (C.5) Appendix D

Mapping from coarse grid to ﬁne grid

Often in practical scenario for non-symmetric object we can fulfill the condition given in eqn.(2.23) by having multiple projections at different angle. In this case we need to to a 2D to 2D mapping from coarse grid to fine grid. We create a projection vector to map from a coarse grid base to fine grid base. Practically,the fine grids for reconstruction are hard to achieve, so interpolation from coarse grid to fine grid is necessary to carry out backward process with good basis. In model-based image reconstruction problems, a parameter may have a different basis representation in the forward and inverse model and must be mapped whenever the forward solver presented. We use bicubic piecewise polynomial basis for this purpose. It could be done simply mapping the coarse grid to fine grid through vector projection. The projection matrix for this mapping can be derived as follows, Let {bP } and {bB} be two orthonormal basis expansions of domain Ω, so that the continuous function f(r) is represented as

NP P X P P f(r) ≈ f (r) = Fk bk (r) (D.1) k=1 where, 64

P bk (r) = pixel basis,

NP = number of points in ﬁne grid

NB B X B B f(r) ≈ f (r) = Fk bk (r) (D.2) k=1 where,

B bk (r) = bicubic piece-wise polynomial basis

NP = number of points in coarse grid

To look into bicubic piece-wise polynomial basis, let us ﬁrst assume a rectangular shaped domain Ω = {(x, y)|0≤x≤dx and 0≤y≤dy}. Let δx be the grid spacing in the ij ij x axis and δy be the grid spacing in y axis, so the center c of basis function b (r) is located at

ij c = (iδx, jδy) (D.3)

Then bij(r) is given by

ij ij ij b (x, y) = bx (x) × by (y), (D.4) and ij 1 − x if dx ≤ 1, |x − iδx| bx (x) = , dx = (D.5) 0 if dx > 1 δx and ij 1 − y if dy] ≤ 1, |y − iδy| by (y) = , dy = (D.6) 0 if dy > 1 δy where, i and j denotes the location of the co-ordinates of lower left corner in respective grid[1] with k = (i − 1)N + j. This approach can be extended to piece-wise nth order polynomials as an approximation to the sinc function kernel of the box ﬁlter. We can also consider a tricubic 65 regular grid, with basis functions given by 1 3 3 3 3 3 ij 12 [|dx − 2| − 4|dx − 1| + 6|dx| − 4|dx + 1| + |dx| + 2 ] if dx ≤ 1, bx (x) = , 0 if dx > 1 (D.7) and 1 3 3 3 3 3 ij 12 [|dy − 2| − 4|dy − 1| + 6|dy| − 4|dy + 1| + |dy| + 2 ] if dy ≤ 1, by (y) = , 0 if dy > 1 (D.8)

Piece-wise polynomial basis satisfy the normalization condition

X bij(r) = 1 (D.9) i,j

The discrete vectors F P ∈ RNP and F B ∈ RNB are given by Z P P Fk = f(r)bk (r)dr, (D.10) Ω

Z B B Fk = f(r)bk (r)dr (D.11) Ω The map{bB} → {bP } is thus obtained from

Z NB Z P B P X B P B Fk = f (r)bk (r)dr = Fq bk (r)bq (r)dr (D.12) Ω q=1 Ω or

F P ≈ PF B (D.13) where,

P ∈ RNP NB = the projection matrix for the map {bB} → {bP },

PT ∈ RNB NP = the map {bP } → {bB}

If both bP and bB are local basis expansions then the projection matrices will be sparse. 66

The forward model for ﬁne grid can be represent in pixel basis form given by,

AF P = t (D.14) where, t = projection data, A = forward matrix and F P = ﬁne grid discretization

We apply the reconstruction algorithms on matrix A for eqn.(D.14). This equation could be convert in the form of coarse grid representation with bicubic piece-wise polynomial basis through projection matrix P by putting eqn.(D.13 into the eqn.(D.14)) given as,

A(PF B) = t (D.15)

A0F B = t (D.16) where, A0 = AP = forward matrix and F B = coarse grid discretization

Now, we apply the reconstruction algorithms on matrix A0 for eqn.(D.16), which is typically works on given corase grid and give the solution in ﬁne grid.

Coarse grid has been taken with d=λ×128, which gives N = 13 point to interpolate. Fine grid has been obtained through projection matrix, which is working ﬁne for 1 → 4 interpolation. Number of points in ﬁne grid are N = 50. Appendix E

Representation of projection data with B-spline Interpolation

Deﬂection angle projection data given by Rytov approximation is

∂φ(r) ! ∂y tan θ(r) = ∂φ(r) (E.1) ∂x

We can always represent phase as function of (x, y) plane given by [26]

3 3 1 X X φ(r) = φ(x, y) = P (u)P (v)q(x , y ) (E.2) 36 m n i+m−1 j+n−1 m=0 n=0 where, i and j denotes the location of the co-ordinates of lower left corner in respective grid[1].

The above equation, phase is given in form of Interpolation through B-spline function. The polynomial P(u) and P(v) are deﬁned by 68

      3 P0(u) −1 3 −3 1 u            2   P1(u)   3 −6 0 4   u    =            P2(u)   −3 3 3 1   u        P3(u) 1 0 0 0 1

q denotes the control point vector on the (x, y) plane and (u, v) are given by

x−xi u = ∆ , xi−1 = xi − ∆ < x < xi + ∆ = xi+1

y−yi v = ∆ , yi−1 = yi − ∆ < y < yi + ∆ = yi+1

and ∆ is the mesh size.

The polynomial Pm(u) can also be deﬁned recursively for m order in terms of x co-ordinates of control point vector q as follows

m x − xk m−1 xk+m+1 − x m−1 Pk (x) = Pk (x) + Pk+1 (x) (E.3) xk+m − xk xk+m+1 − xk+1 In our case m = 3, so we have

3 x − xk 2 xk+4 − x 2 Pk(x) = Pk(x) + Pk+1(x) (E.4) xk+3 − xk xk+4 − xk+1 The zeroth order polynomial P is deﬁned as

0 Pk(x) = U(x − xk)U(xk+1 − x) (E.5)

where, U = unit step function deﬁned on x k = number of control points

The partial diﬀerentiation of φ(x, y) could be given in terms of diﬀerentiation of B-spline polynomial like 69

3 3 ∂φ(x, y) 1 X X = φ (x, y) = P (u)P (v)q(x , y ) (E.6) ∂x x 36 mx n i+m−1 j+n−1 m=0 n=0 where, Pmx(u) is given by

      2 P0x(u) −3 6 −3 0 u              P1x(u)  1  9 −12 0 0   u    =       ∆      P2x(u)   −9 6 3 0   1        P3x(u) 3 0 0 0 0

φx(x, y) could also be obtained by Pn(v) respectively.

Putting these expressions in the eqn.(E.1), we get

φ (x, y) tan θ(r) = y (E.7) φx(x, y)

φx(x, y) tan θ(x, y) = φy(x, y) (E.8)

In the matrix form, the above equation could be represented by,

(Ax × tan θ)q = Ayq (E.9)

(Ax × tan θ − Ay)q = 0 (E.10)

Aq = 0 (E.11) where,

A = Ax × tan θ − Ay, 1 P3 P3 Ax = φx(x, y) = ∆36 m=0 n=0 Pmx(u)Pn(v), 70

1 P3 P3 Ay = φy(x, y) = ∆36 m=0 n=0 Pm(u)Pny(v),

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