Investigations on transform domain techniques in joint-transform correlator architectures for optical pattern recognition

(1)

INVESTIGATIONS ON TRANSFORM DOMAIN

TECHNIQUES IN JOINT-TRANSFORM CORRELATOR ARCHITECTURES FOR

OPTICAL PATTERN RECOGNITION

by

RENU TRIPATHI

Department of Physics

Thesis submitted

in fulfillment of requirements for t'hd degree of Doctor of Philosophy to the

INDIAN INSTITUTE OF TECHNOLOGY, DELHI

NEW DELHI — 110 016 (INDIA)

(2)

CERTIFICATE

This is to certify that the thesis entitled, "Investigations on Transform Domain Techniques in Joint-Transform Correlator Architectures for Optical Pattern Recognition", being submitted by Ms. Rent' Tripathi, to the Indian Institute of Technology Delhi, New Delhi, for the award of Degree of Doctor of Philosophy in Physics is a record of bonaride research work carried out by her under our supervision and guidance. She has fulfilled the requirements for submission of thesis, which to the best of our knowledge, has reached the requisite standard.

Material contained in this thesis has not been submitted in part or full to any other t Iniversity or Institute for the award Of any- degree or diploma.

- V\J

(T.C. (:oel) (Kellar 'ugh)

Professor, Department of Physics Professor & Department of Physics Indian Institute of Technology Delhi, J,pdian Institute of Technology Delhi,

New Delhi 110 016 (India) New Delhi — 110 016 (India)

October 1998

(3)

Acknowledgements

I express my gratitude to Professor Kehar Singh and to Professor T.C. Goel for introducing me to the field of Optical Information Processing. They have been sources of constant inspiration, motivation, and encouragement all through the years of my stay at I.I.T. Delhi.

I also acknowledge the 1.1.1'. Delhi authorities for helping me with excellent research facilities, and for the financial support in terms of the fellowship. Financial assistance from Department of Science & Technology, Government of India, and Council of Scientific and Industrial Research, Government of India is also gratefully acknowledged.

It is impossible to mention the names of everyone who helped me formulate my ideas.

I Inwever, I owe special thanks to my seniors and colleagues Dr. Arvind Kumar, and Mr. G.S. Pan.

Thanks are also due to Dr. Joby Joseph, and Mr. G. Unnikrishnan for the help and support.

I acknowledge Dr. V.P. Pande, Professor, Department of Mathematics and Computer Sciences, Kumaun University Campus, Almora for always being encouraging in my academic endeavors.

Its time to thank my younger brothers and Bhabhi for constant support, and assistance. I also thank Mr. Bharat Shastri & all family members for constant moral support.

With gratitude and love, I dedicate this thesis to my dear parents and to my elder brother who made it possible with their encouragement and constant support over the years.

Eeit.t.t Lh...42,20‘,44 1

(REM.) TRIPATHI)

(4)

Abstract

Research in the field of pattern recognition is marked by significant developments over the years. It deals with generic problem of locating/tracking a target in a single image. Model based techniques, neural networks and correlation based pattern recognition techniques are three most popular approaches to solve pattern recognition problems. Correlation based pattern recognition technique is the technique most commonly adopted in the domain of optics.

Correlation gives a measure of similarity between the objects to be recognized with the stored object of reference. It does not require data specific operations such as segmentation, feature extraction, and on-line model synthesis etc. Therefore, the processors and algorithms used in correlation can be shared in multi-sensor environment.

Optical pattern recognition (OPR) based on correlation is achieved mainly by use of Fourier optical processors, termed as optical correlators. They perform object recognition task by comparing the acquired target information with a reference in the Fourier domain.

Frequency plane optical correlator uses complex matched filter to generate correlation between two signals [Yu and Jutamulia, 1992; Vander Lugt, 1992; Goodman, 1996]. Joint-transform correlator (JTC) is another alternative optical architecture that has attracted large interest among researchers as it avoids synthesis of matched filter [Vander Lugt, 1992; Goodman, 19961. In this architecture, the available input space-bandwidth is shared by a target and a reference signal and a guard band ensures separation of correlation from undesired terms brined in the output plane. Correlation results obtained using these optical processors exhibit shift-invariance and shape-dependent recognition.

Use of frequency plane correlators and JTCs has been traditionally described for a variety of pattern-matching problems. The reliance on the performance of an optical correlator primarily depends on its ability to detect targets of interest, discriminate against undesired object, robustness to the noise in the input scene, and tolerance to distortions or deformations in the target. Techniques, architectures, and algorithms have been developed to construct efficient optical correlators to meet the challenges in pattern recognition applications. Design and

(5)

optimization of correlation filters [Flannery and Horner, 1989; Chatwin and Wang, 1996] and use of nonlinear processing techniques [Javidi, 1989; Kuo, 1994; Yu and Gregory, 1996]

amenable to optical implementation have spurred a flurry of research activity in the field. To extend the applicability of optical processors in distortion invariant object recognition [Flannery and Homer, 1989; Goodman, 1996], various kinds of geometric transforms [Casasent and Psaltis, 1977; Goodman, 1996] (e.g., Mellin, Fourier-Mellin, log-polar, coordinate transforms, and circular harmonic expansion) have been proposed without incurring severe loss in signal-to-noise (SNR) ratio.

Wavelet transform (WT) is a new family of time-frequency transform [Chui, 1992] that is well suited for analysis of transients and non-stationary signals. Two-dimensional WT has been successfully applied to multi-resolution image analysis, image compression, reconstruction, and enhancement etc. [Young, 1993]. One of the most common interpretation of WI' is that it processes the signal through a bank of band-pass/wavelet filters [Szu and C'aullield, 1992]. This implies that the high and low frequency information present in the signal is sampled at the same rate, Therefore, WT proves to be more efficient for analyzing local features of the image compared to the use of conventional Fourier transform in pattern recognition.

Optical wavelet correlators [Sheng etal, 1993; Yang etal, 1994] have been proposed recently. Instead of computing the correlation between the target and reference, optical wavelet correlators compute correlation between wavelet coefficients of target and reference patterns.

Hy virtue of choosing the wavelet functions and their dilation factors, these correlators feature efficient target recognition and discrimination in the presence of unknown background clutter and noise. Use of extracted wavelet features at different resolution forms a set of rotationally distorted training images which can be used in invariant pattern recognition. A filter based on polynomial expansion constructed out of a scaled bank of wavelet filters multiplied by 1-D wavelet weight function has also been proposed [Shabtay et al, 1997] for achieving invariant pattern recognition.

(6)

Spectral transforms are widely used in signal and image processing. Hartley transform (HT) [Bracewell, 1986] is a Fourier related spectral transform which provides transform domain information in a purely real form. HT has been proposed with definite advantages for signal processing including nonlinear filtering and noise rejection, image reconstruction, and computation of image moments in image classification applications. Use of HT has also been suggested in optical correlation systems [Cottrell et al, 1987] to construct real-matched filters alleviating the need for phase encoding as required in case of a FT. It gives superior false correlation rejection and uniformly sized correlation signals.

Space-variant object recognition forms another class of pattern recognition problem.

There are several approaches for obtaining space-variant detection using optical techniques.

Use of angle-multiplexed holographic filters, phase-encoded reference beams and Fresnel transform in optical correlators enables space-variant object detection. Fractional Fourier transform (FRT) has recently been developed [Lohmann, 1993; Mendlovic and Ozaktas, 1993]

as a generalized form of conventional Fourier transform. Unlike the ordinary FT, it combines both spatial

-

domain and frequency

-

domain characteristics of the original image. Fractional correlator (Mendlovic et al, 1995] based on FRT operation has recently been formulated as a powerful approach to obtain space-variant object recognition. The performance of such a correlator has been analyzed according to the standard criteria of SNR, peak-to-correlation energy, and flamer efficiency. Fractional correlator that yields an application to build filters based on FRI' and insensitive to scale change of' the object have been experimentally demonstrated. A planar-optic configuration [Song et al, 1997] with fractional correlation filter and Fresnel lenses fabricated on a glass substrate has been implemented recently.

The basic

yrc

architecture has been used to propose various applications other than correlation of two signals. The system uses two laterally displaced signals presented together in the input plane of the FT lens. Based on this, a deconvolution scheme for image restoration [Javidi et al, 1989], a simple technique for image enhancement [Javidi et al, 1990] and implementation of wavelet transform [Szu and Caulfield, 1992] have been proposed. With the availability of high contrast and high resolution optically addressed SLM and sqaurc-law CCD

iii

(7)

array, the architectural simplicity of a nonlinear JTC promises several other futuristic applications such as in associative retrieval, sensing, and displacement measurement.

From an implementation stand-point, the architectures for optical correlators have matured considerably due to the technological advancements made in various interfacing and recording devices such as spatial light modulators (SLMs) and nonlinear optical materials.

SI.M forms the key element in optical correlation systems. The correlation performance is largely governed by the SLM properties [Lee, 1995] such as resolution, frame-rate, and fidelity of complex amplitude modulation. SLM is used in both the input and filter plane of an optical correlator. With respect to the filter plane, the pixel pitch and the space-bandwidth product have a direct influence on the performance of the correlator. Future research in high-speed optical correlators will mature with the development of two constituent technologies, silicon VI .51 and liquid crystal displays.

Photorefractive (PR) crystals are used as real-time recording medium in optical correlators. They outperform the liquid crystal SLMs in terms of resolution, sensitivity and dynamic range. Over the past few years, a large variety of optical correlators [Solymar et al, 1996] have been demonstrated using PR crystals in the Fourier plane. Various wave-mixing phenomena associated with PR crystals enable all-optical nonlinear processing in the recording of holographic filter in a frequency plane correlator or joint power spectrum (JPS) in a JTC.

These have been exploited to propose a class of all-optical nonlinear PR crystal based correlators [Khoury et al, 1994]. Owing to the nature of Bragg diffraction, the PR crystal based optical correlators lack shift-invariance and pose difficulty in off-Bragg read-out of the volume hologram. Use of thin PR crystal plates and long focal length telescopic Fourier transforming systems have been proposed to surmount the above mentioned difficulties.

Many real-time correlation systems based on frequency plane and JTC architectures have been demonstrated recently. It has also been recognized that a correlator is a powerful interconnection system that can employ space and frequency multiplexed weights. Therefore, these systems find applications in optical implementation of neural nets and associative memories. They also form subsystem in other system architectures proposed for programmable

(8)

general-purpose multi-functional optical image processors. A challenge for present day optical correlators is to improve their fidelity in favor of greater dynamic range, and to design optimum algorithms with the limitations of the currently available devices. Research in the field of OPR addresses problems both in the fronts of fundamental and applications of correlation. Concepts from the field of signals and systems, Fourier optics and statistical pattern recognition arc borrowed to envision systems with optimum configuration and high performance. Hardware and analytical advances in OPR research continue in the rapid evolution of correlation architectures to find potential applications in pattern recognition problems.

The present study undertakes implementation of various mathematical transforms in correlation architectures to attain improved con-elation performance. It describes the use of WT in an all-optical PR crystal-based JTC architecture and a hybrid JTC architecture for high performance target identification and discrimination. Implementation of a Hartley transform based JTC has been described as an alternative to a conventional JTC architecture, and advantageous in terms of target tracking alongwith discrimination. A shift-variant scale- invariant JTC has been proposed based on the concept of fractional-Fourier transform in the input plane. This type of JTC has applications when input object has to be correlated only when it is situated in a particular pre-defined area in the input plane, e.g., photograph in an I-card.

Use of incoherent erasure in PR wave-mixing has been proposed to implement nonlinear PR JTC whose performance limits can be tuned from matched filter to inverse filter limit. Concept of correlation has been extended to object displacement measurement application using speckle photography. In-plane, three-dimensional, and tilt motions of an object have been measured with high degree of accuracy. Chirp modulation has been used to decouple the out-of-plane components of object motion from that of in-plane during measurement.

Chapter 1 contains an introduction to, and overview of research and development in the area of correlation based OPR. It discusses some of the proposed homomorphic processing techniques used in JTC architecture and the implementation issues pertaining to them. It also includes a discussion on mathematical transforms (e.g, Wavelet, and fractional-Fourier, and Hartley transforms) used in efficient feature detection and optical transforms used in invariant pattern recognition. Various invariant mathematical transforms are discussed in brief. The

(9)

chapter presents a brief discussion on various PR phenomena with reference to their applications in optical con-elators. Complex modulation characteristics of SLM and its properties also form a section of the present chapter. Charge-coupled device (CCD) detectors and their characteristics are also discussed.

Chapter 2 describes a wavelet filter

-

bank based JTC for pattern discrimination. Use is made of the band-pass characteristics of the wavelet filters to extract features helpful in discriminating between look-alike objects. A series of wavelet filters is generated prior to the correlation operation and stored in a PC as a filter-hank. These filters arc then applied to the JPS. The mother wavelet (Mexican-Hat) chosen for the study has a cut-off frequency of 3dB.

Simulations are performed to verify the validity of the proposed scheme with synthetic and real-life images. Quantitative measurements such as normalized peak intensity, discrimination ratio, de

-

to

-

correlation peak intensity ratio, and full-width at half-maximum intensity arc calculated to gain a better insight into the correlation performance or the system. Experiments are performed in two ways: (i) using a photorcfractive BaTia; crystal for recording the WS, and reading it with wavelet-filter-modulated read-out beam, (ii) using a single lens geometry in a hybrid setup using an amplitude-modulated spatial light modulator.

While the first set-up makes the process real-time due to real-time recording/processing capability of the photorefractive crystal, the second setup combines the speed of optical processing with the flexibility of digital processing to realize such a technique in real-time.

Experimental results obtained show high discrimination, and sharp correlation peaks for perfectly matching templates and highly reduced correlation peaks for unmatched patterns.

Since the systems operate in real-time, these may find applications in correlation systems where shift, scale, and rotation invariance is required along with target discrimination.

Hartley transform (HT) is of particular interest in analysis and processing of images since it can be implemented optically without introducing phase ambiguities unlike the intensity-only observations of Fourier transform (FT). Chapter 3 presents a simple polarization-based interferometric technique to implement the Hartley transform based JTC.

(11.1TC). Use of HT in correlation yields information related to absolute position of the object

(10)

present in the input plane. However, inverse FT of Hartley joint power spectrum (HJPS) produces various intensity distributions alongwith the desired correlation output. Some of these distributions when present simultaneously with correlation can create ambiguity in the detection process. The use of chirp modulation in HJTC enables to extract information regarding object correlation and absolute object position contained in the output plane by delineating them through the action of a Fresnel zone plate. An optimum threshold function has been used to implement binary HJTC and binary chirp modulated HJTC in order to attain improved light efficiency and target discrimination in correlation. Theoretical analysis describing the principle of operation has been discussed. Simplicity of the experimental set-up and realization of HJTC suggest that the use of HT can serve as an alternative to using FT in a JTC.

Fractional Fourier transform (FRT) has been recently identified as a generalized form of conventional FT and has been used to introduce fractional correlation (FC) operation that serves as a powerful approach for space-variant recognition. Chapter 4 presents a space- variant pattern recognition technique based on modified definition of FRT and fractional correlation. Computer simulation studies have been performed to show space-variance and scale sensitivity properties of modified FC. Results obtained for FC also reveal narrowing of the correlation profile. A phase-only FC has also been proposed for attaining input light efficiency and pattern discrimination.

PR crystals show nonlinear properties during wave-mixing which can be tailor made and utilized in Fourier domain optical processing. Chapter 5 describes the use of an incoherent erasure (IE) beam in PR four wave-mixing (FWM) and two wave-mixing (TWM) geometries to demonstrate nonlinear optical correlation. Simple device theories based on incoherent-to- coherent conversion and coupled-wave equations governing TWM have been used to explain the principle of operation of PR IEJTCs. Presence of IE beam influences nonlinear transformation on JPS from soft-clipping limit to hard-clipping limit. Computer simulation results show that the degree of nonlinearity can be controlled by varying the beam ratios associated with the respective wave-mixing geometries. The performance of the correlator can he switched from classical to inverse filter limit. A single experimental setup has been used to

vii

(11)

demonstrate both TWM and FWM based PR IEJTCs using a PR BaTiO1 crystal. Experimental results show pattern discrimination and noise immunity of PR IEJTCs. These JTCs form a new family of PR JTCs. PR crystals with fast writing capabilities and possessing good sensitivity at erasure wavelength can help in realizing nonlinear PR IEJTCs operating with high correlation speed. Since a FWM geometry does not involve beam-coupling or energy transfer, the IEJTC using FWM geometry can be implemented in non-photorefractive media. Our study asserts the importance of IEJTCs and encourages further investigations using suitable PR materials.

Speckle photography is a non-contact, simple, and elegant method of assessing small displacements, tilts, contours, and surface characteristics of an object. Speckle analysis involves measurement of spacing and inclination of interference fringes recorded in a photographic medium. There have been few attempts to apply real-time optical processing techniques to speckle metrology using SLMs and PR crystals. Although the introduction of PR materials and SLMs has made the process real-time, yet the analysis of speckle interference patterns is cumbersome as it has to be done manually or by using micro-densitometers. In the subsequent chapters, correlation based techniques have been tried out in displacement measurement. The correlation of specklegrams gives a direct measure of the degree of match between two displaced states of an object. PR crystal has been used to generate speckle interference pattern for various kinds of object motion. A correlator geometry has been used to analyze the speckle interference pattern through diffraction.

Chapter 6 describes a correlation technique for measurement of in-plane object motion.

A PR BaTiO3 crystal has been utilized for recording double and multiple exposure speckle patterns in a PR TWM geometry. The FT of the speckled object beams generates interference fringes in real-time. These are subsequently grabbed by a CCD camera and displayed on an electrically addressed SLM. The FT of the displayed speckle interference pattern yields correlation output. Positions of the peaks in the correlation plane vary linearly with the in-plane displacement. The position and intensity of the correlation peak give a measure of in-plane object displacement and degree of correlation/decorrelation. The speckle fringes are binarized to produce sharp and well-defined con-elation peaks which enable precise measurement of object displacement compared to the fringe analysis method. The correlation output obtained in

(12)

a multiple-exposure case contains multiple peaks which helps in a more accurate measurement of object displacement by averaging the distance measured between consecutive adjacent peaks. The method promises high degree of accuracy and increased range for displacement measurement in real-time.

Chapter 7 reports the use of a PR correlator for measurement of three-dimensional rigid body motion. The effective three-dimensional displacement comprises of in-plane and out-of-plane components of object motion. Since it is not possible to decouple and analyze both components of motion separately, it is desirable to devise a technique that could give a measure of total object motion. In the proposed technique, the out-of-plane component of the object motion produces a chirp modulation in the speckle interference pattern. Resulting speckle fringes are curved and form an amplitude encoded Fresnel zone plate. This leads to the fbcusing of correlation peaks (formed by in-plane component of object motion) along different planes in the longitudinal direction. While the separation between the correlation peak positions gives a measure of the object's in-plane motion, the distance between the longitudinal planes where the correlation peaks come into focus, gives a measure of the object's out-of-plane motion. Both simulation and experimental studies have been performed to establish the proposed technique.

Tilt motion of an object is considered to be a complex motion as it poses difficulty in fringe analysis. The techniques described so far for measurement of object tilt suffer from being less accurate and non-real-time. Chapter 8 proposes a real-time speckle correlation technique for the measurement of object tilt. Tilt can occur either with the center of axis of object rotation lying on optical axis or away from it. A general theory has been developed taking into account both the situations. When the center of axis of rotation of the object lies on the optical axis, the speckle fringes formed in the observation plane due to the object tilt are found to be straight. The sensitivity of correlation based tilt measurement has been studied for different positions of the diffused object along the longitudinal direction. When the center of axis of rotation of the diffused object is shifted away from the optical axis along the transverse direction, the fringes formed due to object tilt are curved. The sensitivity of tilt measurement in this case is found to be higher for different object positions along the longitudinal direction. A

(13)

wide range of simulation and experimental studies are performed to establish the sensitivity and accuracy of the correlation based measurement to object tilt.

Chapter 9 contains a summary of future trends and scope for future work in the area of optical pattern recognition. Some of these include use of transformations and algorithms for invariant pattern recognition and use of nonlinear correlator architectures to realize Hough processor, associative memory, and neural networks.

(14)

TABLE OF CONTENTS Abstract

List of Figures vii

Chapter 1

AN OVERVIEW OF VARIOUS MATHEMATICAL TRANSFORMATIONS USED IN PATTERN ANALYSIS, AND A DISCUSSION ON PHYSICAL

CHARACTERISTICS OF MATERIALS AND DEVICES USED IN OPTICAL PATTERN RECOGNITION.

1.1 Introduction

1.2 Pattern Recognition using Coherent Optical Systems

1.2.1 Vander Lugt or matched filter based optical correlator 1.2.2 Joint-transform correlator

1.3 Mathematical Transformations used in Pattern Analysis

1 2 4 6 10

1.3.1 Utility of mathematical transforms 10

1.3.2 Wavelet transform 11

1.3.2.1 Definition and properties 11

1.3.2.2 Optical implementation 17

1.3.2.3 Use of wavelet transform in optical correlation 21 1.3.2.4 Use of wavelet transfdrm in invariant pattern recognition 24 1.3.3 Fractional Fourier transform: a new tool for signal analysis 26

1.3.3.2 Application of fractional-Fourier transform:

space-variant filtering and object recognition

30

1.3.3.3 Digital implementation 33

1.3.4 Hartley transform and its use in spectral analysis 33 1.3.4.1 Applications of Hartley transform 34

1.3.5 Transform techniques used in distortion-invariant pattern recognition 37

1.3.5.1 Mellin transform 37

1.3.5.2 Log-polar transform 38

1.3.5.3 Circular-harmonic decpmposition 38

1.3.5.4 Mometits 40

1.3.5.5 Hough transform 41

(15)

1.4 Optical Light Modulation using Liquid Crystals 42 1.4.1 Principle of complex amplitude modulation in 42

twisted nematic crystals

1.4.2 Measurement of phase-retardation of SLMs 46 1.4.3 Ferroelectric liquid crystal SLM for binary amplitude and 48

phase-modulation

1.4.4 Characteristics of SLMs and their influence on optical correlation 50 1.5 Charge Coupled Device (CCD) and its Performance Parameters 5 I 1.6 Photorefractive Materials & Their Use in Optical Correlators 56

1.6.1 Performance and related issues 58

1.7 Speckle Photography and Measurement of Object Displacement 60

Chapter 2

PATTERN DISCRIMINATION USING A BANK OF WAVELET FILTERS IN A JOINT-TRANSFORM CORRELATOR

2.1 Introduction 65

2.2 Theoretical Background 67

2.3 Simulation 71

2.4 Experiment 81

2.4,1 Experiments using PR BaTiO3 crystal 81 2.4.2 Experiments using a single lens geometry 87

2.5. Discussion of results 89

2.6 Conclusion 94

Chapter 3

PATTERN DISCRIMINATION AND OBJECT TRACKING USING CONVENTIONAL AND CHIRP-MODULATED HARTLEY

TRANSFORM BASED JOINT-TRANSFORM CORRELATOR

3.1 Introduction 95

3.2 Theory 97

(16)

3.2A Hartley transform 97

3.2B Hartley transform based JTC 97

3.2C Binary HJTC 103

3.3 Computer Simulation 105

3.4 Experiment 110

3.5 Conclusion 118

Chapter 4

REALIZATION OF SHIFT VARIANT FRACTIONAL CORRELATION BASED ON A MODIFIED DEFINITION OF FRACTIONAL FOURIER TRANSFORM AND ITS SENSITIVITY TO OBJECT SCALE VARIATION

4.1 Introduction 119

4.2 Theoretical Background 120

4.2.1 Modified definition of fractional Fourier transform 122 and fractional correlation

4.2.2 Phase-only fractional correlation 123

4.2.3 Shift-variance property 124

4.2.4 Effect of fractional order on correlation profile 125 4.2.5 Effect of object scale variation on FRT 127

4.4 Conclusion 147

Chapter 5

PATTERN DISCRIMINATION USING AN ALL OPTICAL PHOTOREFRACTIVE JOINT-TRANSFORM CORRELATOR EMPLOYING INCOHERENT ERASURE IN TWO- AND FOUR- WAVE MIXING GEOMETRIES IN A BATIO3 CRYSTAL

5.2 (a) FWM Case 152

5.2 (b) TWM Case 154

5.4 Experiment 165

iii

(17)

5.5 Conclusion 172

Chapter 6

IN-PLANE DISPLACEMENT MEASUREMENT USING A PHOTOREFRACTIVE SPECKLE CORRELATOR

6.2 A: Double exposure 177

6.2 B: Multiple exposure 177

6.2.1 Speckle correlation calculation 178

6.2.1 A: Double exposure case 178

6.2.1 B: Multiple exposure case 178

6.2.2 Correlation using fixed thresholding 179

6.3 Simulation 181

6.4 Experiment 185

6.5 Conclusion 194

Chapter 7

TH RE M ENSIONAL DISPLACEMENT MEASUREMENT USING CHIRP MODULATION IN A PHOTOREFRACTIVE CORRELATOR

7.2 Theory 198

7.3 Simulation 205

7.4 Experiment 211

7.5 Conclusion 215

(18)

Chapter 8

OBJECT TILT MEASUREMENT USING A PHOTOREFRACTIVE SPECKLE CORRELATOR: THEORETICAL AND

EXPERIMENTAL ANALYSIS

8A Introduction 217

8.2 Theory 219

8.3 Simulation 228

8.5 Experiment 236

8.6 Conclusion 241

Chapter 9

CONCLUSIONS AND SCOPE FOR FUTURE STUDIES

9.1 Conclusions 243

9.2 Scope for Future Studies 245

REFERENCES 249

AUTHOR'S BIOGRAPHY 267

& LIST OF PUBLICATIONS