DEVELOPMENT OF A FRINGE PROJECTION

METHOD FOR STATIC AND DYNAMIC

MEASUREMENT

WU TAO

(B.Eng. (Hons.))

A THESIS SUBMITTED

FOR THE DEGREE OF MASTER OF ENGINEERING

DEPARTMENT OF MECHANICAL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2003

i

ACKNOWLEDGEMENTS

I will like to express my sincere and deepest appreciation to his supervisors

Assoc. Prof Tay Cho Jui (Department of Mechanical Engineering) and Assist. Prof

Quan Chenggen (Department of Mechanical Engineering) for their invaluable advice

and guidance throughout this project. I would also like to express my gratitude to Mr.

Chiam Tow Jong, Mr. Fu Yu, Ms. Sherrie Han, and Mr. Abdul Malik from the

Experimental Mechanics Laboratory.

I would also like to thank my family. Their financial and spiritual support have

been enabled me to come to Singapore and study at an advanced academic level.

ii

TABLE OF CONTENTS

ACKNOWLEDGEMENTS i

TABLE OF CONTENTS ii

SUMMARY v

NOMENCLATURE vii

LIST OF FIGURES ix

CHAPTER 1 INTRODUCTION 1

1.1 Introduction 1

1.2 Problem 2

1.3 Objectives of the project 3

CHAPTER 2 LITERATURE REVIEW 4

2.1 Application of optical techniques in shape and deformation measurement 4

2.2 Application of optical techniques in dynamic measurement 5

2.3 Development of fringe projection method 7

2.4 Enhancement of dynamic range of optical system 8

2.5 3-D displacement measurement by optical techniques 9

2.6 Digital image correlation (DIC) technique 11

CHAPTER 3 THEORY 12

3.1 Formation of fringe patterns 12

3.1.1 Formation of fringe patterns by interferometry method 12

3.1.2 Formation of fringe patterna by a Liquid Crystal Display (LCD) Projector 14

3.2 Height and phase relationship 15

3.3 Determination of phase value 16

iii

3.4 Phase unwrapping 20

3.5 Enhancement of dynamic range of fringe projection method 21

3.6 Phase shift calibration 24

3.7 Integrated fringe projection and DIC method 24

CHAPTER 4 APPLICATION OF FRINGE PROJECTION METHOD FOR

STATIC MEASUREMENT 36

4.1 Experimental work 36

4.2 Results and discussion 37

CHAPTER 5 APPLICATION OF FRINGE PROJECTION METHOD FOR

DYNAMIC MEASUREMENT 45

5.1 Measurement of dynamic response of a small component 45

5.1.1 Experimental work 45

5.1.2 Results and discussion 46

5.2 Enhancement of dynamic range of the fringe projection method 48

5.2.1 Experimental work 49

5.2.2 Results and discussion 50

CHAPTER 6 INTEGRATED FRINGE PROJECTION AND DIC METHOD 71

6.1 Experimental Work 71

6.2 Results and discussion 72

CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS 94

7.1 Conclusions 94

7.2 Recommendations 97

References 98

APPENDIX A DERIVATION OF TIME-AVERAGE FRINGE PROJECTION

111

iv

APPENDIX B PROCEDURE OF FFT PROCESSING 113

B.1 Fast Fourier Transform 113

B.2 Bandpass Filter for the Fourier Transform Method 113

B.3 Inverse FFT 114

B.4 Unwrapping 115

APPENDIX C PHASE MAPS AT DIFFERENT INTERVALS 121

APPENDIX D LIST OF PUBLICATIONS 126

v

SUMMARY

This thesis is divided into three parts: the first part establishes the theory for a

fringe projection method and its application to shape measurement for both static and

dynamic loading conditions. Experimental verification for both static shape

measurement and dynamic analysis is carried out on a micro membrane, a coin and a

diaphragm of a speaker. The experimental results show excellent agreement with

theoretical values.

Since the exposure period of a camera is reduced with high frame rate, dynamic

measurement with a short exposure is intrinsically light intensity starved. Thus

insufficiency in light intensity often introduces underexposure problem and leads to

poor image quality. To overcome this problem and enhance the dynamic range of the

system, a practical and simple method which involves a white light source (WLS) is

proposed and demonstrated. The theory is presented and an increase in the

measurement range of up to a factor of 6 was achieved.

Since the fringe projection method is mainly based on the height and out-of-

plane displacement of the objects, it is observed that the in-plane displacement has

significantly adverse effects on the results. Therefore, measurement of 3-D

displacement is needed. A novel method combining the fringe projection and digital

image correlation (DIC) into one optical system is developed to simultaneously

measure displacement in three dimensions, where only one camera is used. In this

technique, linear sinusoidal fringes are first projected on an object using a fringe

vi

projector and images of the object�s surface with the fringe pattern are captured by a

CCD camera. With the aid of Fourier transform, the carrier (fringe pattern) in the

images is eliminated while only the background intensity variation is preserved. DIC

is then used to obtain in-plane displacement based on the background images after

carrier elimination. In the mean time, original images are processed by fast Fourier

transform (FFT) technique to deliver information about shapes of the object. Based on

the in-plane displacement vector obtained by DIC, the shapes of the object in different

stages are compared in a reference coordinate to obtain out-of-plane displacement.

Experimental results of 3-D displacement field of a small component are obtained to

confirm the validity of the method.

vii

NOMENCLATURE

*,, CCA Fourier spectra

a Background intensity

Wa White light background introduced

b Variation of fringe pattern

*c Complex conjugate

D Distance from the light source to the screen

d Distance between two virtual points

f Spectra frequency in x-direction

g Distance between LCD and CCD

h Height of the surface

FI Resultant intensity of fringe pattern

'FI Output intensity of fringe pattern

OI Optimal intensity distribution

PI Input intensity of fringe pattern

k Coefficient relating Z to φ

L Length from point A to the edge of the optical wedge

l Distance between reference plane and CCD

0n Refractive index of the optical wedge

P Position of the investigated point

oP Fringes� pitch

Q Centre of the imaging optics

viii

t Thickness of an optical wedge

0u Carrier frequency

W 3-D displacement vector

),,( tyxZ Instant displacement of the vibration surface

),( yxZ Vibration amplitude of one point at ),( yx

α Angle of incidence of the light

β Refractive angle

θ Initial phase angle of vibration

ω Angular velocity of vibration

γ Projection angle

λ Wavelength of a He-Ne laser

),(0 yxϕ Initial phase value

),( yxϕ Phase variable

nϕ Phase shifting interval

δ Phase value

)(tδ Instant displacement

µ , υ Contrast coefficient

x∆ , y∆ , z∆ Displacement in the x -, y - and z - directions

ix

LIST OF FIGURES

Figure 3.1 Schematic diagram of two-point-light-source interferometry with an

optical fiber and an optical wedge. 28

Figure 3.2 Optical geometry for fringe analysis. 29

Figure 3.3 Fourier fringe analysis. 30

(a) spectrum of the interferogram. 30

(b) Processed spectrum after filtering. 30

Figure 3.4 (a) Wrapped phase with 2π jump. 31

(b) Unwrapped phase value. 31

Figure 3.5 Intensity of fringe pattern enhanced by a white light source. 32

Figure 3.6 General scheme of the proposed integrated method. 33

Figure 3.7 Logical model of CRA. 34

Figure 3.8 Schematic diagram of planar deformation process. 35

Figure 4.1 Experimental setup for shape measurement of small components. 39

Figure 4.2 Microphone chip. 40

Figure 4.3 Cross section view of the microphone. 40

Figure 4.4 Image of the fringe pattern on the test surface. 41

Figure 4.5 Relationship between the phase value and height of the test surface. 42

Figure 4.6 3-D plot of the surface of the microphone. 43

Figure 4.7 Cross-section of the micro membrane at x = 125 µ m. 44

Figure 5.1 Experimental setup of fringe projection method for dynamic

measurement. 53

Figure 5.2 (a) Test specimen. 54

x

(b) A close up view of the surface with fringe projection. 54

Figure 5.3 Sinusoidal fringe pattern projected on a small section of the test

specimen. 55

Figure 5.4 Images at 0.002s time interval. 56

Figure 5.5 Calibration of the fringe projection system for the measurement of

vibration in z direction. 57

Figure 5.6 Unwrapped phase maps at 0.002s time interval. 58

Figure 5.7 (a) Vibration amplitude before phase recovery. 59

(b) Vibration amplitude after phase recovery. 59

Figure 5.8 Vibration plots of different regions on the object. 60

Figure 5.9 Comparison of micro values and experimental vibration amplitude. 61

Figure 5.10 Experimental setup with the white light source. 62

Figure 5.11 A small section on the test coin surface. 63

Figure 5.12 Comparison of the images of part of coin recorded at recording rate of

3000 fps. 64

(a) Image recorded with background enhancement. 64

(b) Image recorded without background enhancement. 64

(c) Image processed with an optimal contrast. 64

Figure 5.13 Comparison of the fringe pattern distribution of the cross section of YY.

65

Figure 5.14 (a) 3-D profile of test object before background enhancement. 66

(b) 3-D profile of test object after background enhancement. 66

Figure 5.15 (a) Image of the speaker recorded without background enhancement at

frame rate of 3000 fps. 67

xi

(b) Image of the speaker recorded with white light background

enhancement at frame rate of 3000 fps. 67

Figure 5.16 (a) 3-D profile of speaker diaphragm with background enhancement.68

(b) 3-D profile of the speaker without background enhancement. 68

Figure 5.17 Vibration amplitude of speaker diaphragm. 69

Figure 5.18 Comparison of vibration plots at 3000fps frame rate. 70

Figure 6.1 A close up view of a speaker diaphragm. 76

Figure 6.2 (a) Images of the test surface with fringe pattern with a pitch of 50

pixels. 77

(b) Recovery of intensity distribution of background. 77

Figure 6.3 (a) 3-D representation of the spectrum before filtering. 78

(b) 3-D representation of the spectrum after filtering. 79

Figure 6.4 (a) Images of the test surface with fringe pattern with a pitch of 15

pixels. 80

(b) Recovery of intensity distribution of background. 80

Figure 6.5 (a) Images of the test surface with fringe pattern with a pitch of 1 pixel.

81

(b) Recovery of intensity distribution of background by a 70 pixels ×

70 pixels filtering window. 82

(c) Recovery of intensity distribution of background by a 80 pixels ×

80 pixels filtering window. 83

(d) Recovery of intensity distribution of background by a 100 pixels ×

100 pixels filtering window. 84

(e) Recovery of intensity distribution of background by a 120 pixels ×

120 pixels filtering window. 85

xii

Figure 6.6 (a) A close up view of the test surface with fringe pattern before

displacement (225 µ m, 300 µ m and 140 µ m in x -, y - and z -

directions respectively). 86

(b) A close up view of the test surface with fringe pattern after

displacement (225 µ m, 300 µ m and 140 µ m in x -, y - and z -

directions respectively). 87

Figure 6.7 Phase map of the test surface. 88

Figure 6.8 Images of the background of the test surface after CRA. 89

(a) Image of the background of the test surface before displacement. 89

(b) Image of the background of the test surface after displacement. 90

Figure 6.9 Calibration of the measurement system. 91

(a) Calibration and error analysis of displacement along X direction. 91

(b) Calibration and error analysis of displacement along Y direction. 91

(c) Calibration and error analysis of out-of-plane displacement. 92

Figure 6.10 3-D displacement vector of the test surface. 93

Figure B.1 (a) Real part of the Fourier spectrum, (b) Imaginary part of the Fourier

spectrum, (c) Spectrum. 116

Figure B.2 Dialog box for the bandpass filter for the Fourier transform method.117

Figure B.3 (a) Filtered real part of the Fourier spectrum, (b) Filtered imaginary

part of the Fourier spectrum, (c) Amplitude spectrum of the result. 118

Figure B.4 (a) Back transformed real part of the Fourier spectrum, (b) Back

transformed imaginary part of the Fourier spectrum, (c) Mod π2 -phase.

119

Figure B.5 Unwrapped phase map. 120

Figure C.1 Phase maps at different time intervals 125

1

Chapter 1 Introduction

1.1 Introduction

Optical techniques can be classified into either static or dynamic methods with

respect to the loading conditions. Some examples of the static approach are for shape

measurement and deformation measurement, while vibrational excitation belongs to

the dynamic measurement mode.

Projected fringes for the measurement of surface shape is a non-contact optical

method that has been widely recognized in the contour measurement of various

diffuse objects. The technique is referred to as fringe projection. This method uses

parallel or divergent fringes projected onto the object surface, either by a conventional

imaging system or by coherent light interfererence patterns, in which the projection

and recording directions are different. The resulting phase distribution of the measured

fringe pattern includes information on the surface height variation of the object. An

analysis of the fringe patterns is normally carried out either by the phase shifting

technique or the fast Fourier transform (FFT) method. Both produce wrapped phase

maps, where π2 phase jumps caused by the nature of arctangent must then be

removed by the process known as phase unwrapping to recover the surface heights.

Phase unwrapping is normally carried out by comparing the phase at neighboring

pixels by adding or subtracting. In application, the fringe projection method has been

proven to be a promising tool for deformation measurement and curvature

measurement purpose.

2

1.2 Problem

However, most research in fringe projection was based mainly on static

measurement with phase-shifting technique. In such application, static loading is

commonly applied to the test specimen to achieve the desired results. Over the years,

although descriptions of the technique were often presented, no formal treatment of

dynamic fringe projection had been dealt with.

Dynamic measurement by fringe projection, which can be coupled with either

static or dynamic measurement of the object, will enable the fringe pattern to be

monitored live as it is produced and as it changes with time under the action of a

varying load. This attribute makes dynamic measurement by fringe projection

particularly useful for monitoring both time-dependent and transient incidents.

If true 3-D displacement analysis is to be performed, the system must monitor

the objects with more than one camera. This will require multiple projection-detection

system, preferably working on the basis of the calibration principle, or with reference

to a precalibrated measurement volume. These methods can effectively monitor 3-D

displacement field, but most often two or more cameras are used to record 3-D

information about the objects. There are some key limitations of multiple camera

systems including 1) ill-suitability for dynamic measurement, 2) mismatch in the

triangulation of corresponding points and 3) a calibration process that is laborious and

time-consuming. Therefore single camera systems are greatly desired. Conventionally,

fringe projection method is mostly used in out-of-plane displacement.

3

1.3 Objectives of the Project

To formulate the fringe projection method for measuring components under

vibrational loading conditions and 3-D translation, the objectives of this project are as

follows:

1) To demonstrate the application of fringe projection method in both static and

dynamic measurement;

2) To enhance the dynamic range of the measurement system; and

3) To develop a novel fringe projection method integrated with digital image

correlation (DIC) technique which enables the determination of shape and 3-D

displacement using only one camera.

In the first chapter of the thesis, the objectives of this project are defined. The

historical development of optical techniques will be presented in chapter 2. Chapter 3

to 6 will cover the main part of the project. These will include the theoretical

derivation, the experimental techniques, followed by a detailed discussion on each of

these topics. Chapter 7 will give the conclusions and lay down some recommendations

for future investigation.

A list of publications arising out of this research is shown in Appendix D.

4

Chapter 2 Literature Review

2.1 Application of Optical Techniques in Shape and Deformation

Measurement

Optical metrology has been developed rapidly since 1960s. Since then the

surface measurement technique is regarded as one of the main components of optical

metrology. In the early days of optics, a laser scanning machine was used as a surface

detection tool. However, because of the time-consuming nature of point-by-point

measurements, it may take a long time to perform the surface measurement. The main

advantages of optical metrology, such as full field measurement, were utilized. Some

techniques, such as shadow moiré method, are still used in surface measurement.

Shadow moiré method [1-3] involves positioning a grating close to an object and its

shadow on the object is observed through the grating. The method is useful for

measuring 3-D shape of a relatively small object, however, the size of the object to be

measured is restricted to the grating size. The sensitivity of the method is from the

order of microns to that of millimeters depending on the frequency and the amount of

relative rotation of the grating.

Holographic method involves generation of a contour fringe pattern by two

reconstructed images of a double-exposure hologram. Thalmann and Dandliker [4]

reported holographic contouring using electronic phase measurement, which is based

on two-illumination source, and the use of a microcomputer for data reduction.

5

ESPI (electronic speckle pattern interferometry), which is based on laser diodes

and single-mode fiber optics is also developed for measuring surface contour [5].

However, poor quality of the contouring images remains the main limitation of this

technique for surface measurement.

Shearography [6, 7] has also been used in surface shape measurement. Unlike

holography, shearography does not require special vibration isolation since a separate

reference beam is not required; hence, it is a practical tool that can be used in an

industrial environment. Optical grating methods have been applied to the

measurement of 3-D shape [8-10]. In this method five separate defocused images

using Ronchi grating are projected onto an object and the deformed grating images are

captured by a CCD camera and evaluated by the phase-shifting technique. The method

is used for relatively large objects.

By the end of 1980s, computer vision with refractive moiré and projection fringe

methods were developed for surface measurement. As classical approaches using

mechanical probes remain inherently slow and ill suited for measurement of curved

surfaces, 3-D sensing by non-contact optical methods are studied extensively for these

applications. In industrial metrology, the non-contacting and non-destructive

automated surface shape measurement technique is a desirable tool for vibration

analysis, quality control, and contour mapping.

2.2 Application of Optical Techniques in Dynamic Measurement

6

The discussion so far has emphasized mainly on the static shape and curvature

measurement of test specimens. In industrial metrology, a non-contact and non-

destructive vibration measurement technique is a desirable tool for contour mapping,

quality control and vibration analysis. Optical techniques for vibration measurement

have been well established and can be traced to the earlier days of optical methods.

The development of laser Doppler vibrometers (LDV) [11, 12] for use in engineering

testing was stimulated by the advances of easily detecting sub-nm amplitudes devices

over a frequency range from static to MHz. However, laser vibrometers are generally

intended to make measurements at a single point on the surface of a test object. Some

solutions for the whole-field measurement of vibration with optical techniques have

been successfully proposed. Hung [13] employed shearography method to vibration

measurement by digitizing speckle images of a deforming object using a high-speed

digital image acquisition system. Moore [14] presented an Electronic Speckle Pattern

Interferometer (ESPI) system that has enabled non-harmonic vibrations to be

measured with micro second temporal resolution. Kokidko [15] developed the shadow

Moiré method to measure deformation of a plastic panel by using high-speed

photography. Nemes [16] presented a system based on grating projection and Fast

Fourier Transform (FFT) technique to measure the transient surface shape in a

polymer membrane inflation test. FFT technique with carrier fringe method [17, 18]

has also been widely employed in dynamic measurement as the technique requires

only one image for phase determination. Other methods [19, 20] based on a high-

speed camera, using FFT technique have also been reported. Tiziani [21-24]

developed pulsed digital holography for measurement of deformations and vibrations

for various objects. Using time-averaged method, holographic method allows

measurement of shape of structures subjected to vibration excitation. Chambard [25]

7

extended the method to include pulsed-TV holography for vibration analysis

applications. Real-time pulsed ESPI [26] (Electronic Speckle Pattern Interferometry)

based on a high precision scheme that synchronizes and fixes an object point during

rotation is used to study out-of-plane vibrations in a noisy environment. Aslan [27]

developed a real time laser interferometry system for measurement of displacement in

hostile environment.

2.3 Development of fringe projection method

Fringe projection method is a suitable method for three-dimensional optical

topometry [28-36]. It is a useful addition to other methods such as confocal

microscopy [37-40] and white-light interferometry [41-43]. Pixel-related devices offer

a much wider range of possibilities, since virtually all desired intensity distributions

can be generated. Triangulation and fringe projection are very appropriate and the

most frequently employed techniques for macroscopic shape analysis. For fringe

projection, a grating, e.g. with a sinusoidal intensity distribution, is imaged onto the

surface for the measurement. The fringe deformation is used for height calculation.

Typically only a few video-frames are need to be recorded in order to obtain a full

field 3D measurement. The image processing based measurement principle enables

very fast measurement. Phase values are determined by calculating Fourier

transformation, filtering in the spatial frequency domain and calculating inverse

Fourier transformation. Compared with the moiré topography, fast Fourier transform

method can accomplish a fully automatic distinction between a depression and an

evaluation of the object shape. It requires no fringe order assignments or fringe center

8

determination and requires no interpolation between fringes because it gives the

height distribution at every pixel over the entire fields.

After Takeda [44], the FFT method has been extensively studied [45]. A two-

dimensional Fourier transform is applied to provide a better separation of the height

information from noise when speckle-like structures and discontinuities exist in the

fringe pattern.

2.4 Enhancement of Dynamic Range of Optical System

An important problem remains in dynamic imaging systems is underexposure of

a CCD sensor in high-speed application. At a high frame rate, the exposure period of a

CCD camera is decreased and hence reduced intensity is absorbed by the photosensors

in the CCD [46]. This will result in insufficient information being recorded. This

problem becomes more serious when the system is used to measure micro-

components with a Long-Distance Microscope (LDM) which has a limited aperture.

Hence acquisition of image becomes an optimization problem of adapting the

dynamic range of the scene to the dynamic range of the camera.

To modulate the intensity in dark and bright areas, Tiziani [47] developed a

method by the use of a three-chip color camera (RGB). The three-color channels are

recorded simultaneously and the combined output of the RGB channels allows the use

of full spatial resolution for each color channel as compared to a one-chip color CCD

camera. To overcome the problem of low intensity Pedrini [48] presented a method

which employs an image intensifier coupled to a CCD sensor. The image intensifier

9

together with an electronic shutter action allow recording of dim test surface with a

short exposure period. To improve the shuttering characteristic Ito [49] suggested a

method which consists of a proximity focused image intensifier with a micro-channel

plate and an external transparent electrode. The method can effectively increase the

intensity on the specimen to fall within the dynamic range of the camera. However,

the apparatus used is somewhat luxurious and data processing is complex.

2.5 3-D Displacement Measurement by Optical Techniques

In many areas of physics and engineering, measurement of three-dimensional

displacement fields for an object that is being translated is of great interest. Within

optical metrology, several techniques that measure all three components of a

deformation simultaneously already exist. Formerly, the most widely used technique

in this regard was speckle photography. This technique essentially consists of

recording incoherent superposition of two or more speckle patterns generated before

and after the motion of object and then analyzing the recorded speckle-gram by

Fourier transform to reveal the object deformation. Some works in this field dealt with

only in-plane displacement or out-of-plane displacement. But, in practice, the two

kinds of motions are often coupled together. So it is always desirable to have a single

technique providing a measure of the total object motion with effective in-plane and

out-of-plane components.

Several speckle-based techniques for 3-D displacement measurement have been

developed. One of them used both He-Ne laser and polychromatic dye laser; some

others obtained results by analyzing the null-speckle displacement ring. These

10

requirements introduce some inconvenience in system architecture or inaccuracy in

measurement. Recently a technique using a photorefractive speckle correlator has

been proposed. In this technique a double- or multi-exposure speckle interferogram is

recorded with the use of a photorefractive crystal, and this interferogram is then

placed as an input in a Fourier transform system to get correlation spots. The position

of the spot is the indication of the object motion. Still this method has some

drawbacks. First, the use of correlation operation increases the complexity and time of

measurement. Second, for general 3-D displacement and tilt, the correlation spot and

the dc term are not in the same plane, so we have to move the observation plane

longitudinally to meet the focus position of correlation term. Since the longitudinal

distribution of the correlation spot changes gradually, it is not always easy to locate

exactly its sharpest spot, consequently this gives rise to another error source.

Two or three-beam holography is suitable when the deformation components

are of equal magnitude and small. Chiang [50, 51] has developed at least two different

techniques. One is based on moiré interferometry and the other is called holospeckle

interferometry where the in-plane components of the deformation are analyzed using

speckle photography and the out-of-plane component by holographic interferometry.

The non-interferometric methods use stereovision to obtain the true deformation field

by capturing the apparent motion of a reference pattern from two cameras in space.

Henao [52] developed a technique where a diffraction grating was used to form

stereovision. During the last decade several researchers [53-55] have presented

systems based on digital correlation algorithms combined with a stereo pair of CCD

cameras. Such systems can handle discontinuities in the deformation and measure

deformations. Pawlowski [56] applied a spatio-temporal approach, in which the

11

temporal analysis of the intensity variation at a given pixel provides information about

out-of-plane displacement. In-plane motion of the object is determined by a

photogrammetry-based marker tracking method.

2.6 Digital image correlation (DIC) technique

Digital image correlation (DIC) is a non-contact optical method for displacement

of strain measurement, which was introduced by Sutton in 1980s [57]. Currently it has

been well developed and applied to many industrial fields [58-66] as a robust

measurement method. The technique is based on the gray level correlation between

the two digital images in the undeformed and deformed states. The natural or artificial

surface patterns in the images are the carrier that records the surface displacement

information of an object. By making use of the correlation algorithm of gray level of

the pixels in the two images, the displacement fields can be obtained.

12

Chapter 3 Theory

3.1 Formation of Fringe Patterns

There are different approaches for generating fringe patterns such as

interferometry, triangulation and spatial light modulating by a liquid crystal modulator.

3.1.1 Formation of Fringe patterns by Interferometry Method

The arrangement that incorporates an optical wedge, enables a fringe pattern

with a fine pitch to be obtained. This technique has the advantage of requiring a

simple experimental setup and optical arrangement. Due to laser interference in a

perfect common path the proposed fringe projector is compact and provides a stable

and highly visible fringe pattern. This is suitable for micro-components measurement.

As shown in Fig. 3.1(a), the fiber end S of an optical fibre acts as a point light

source and emits a spherical wave front. The wave front is split into two portions,

which correspond to EFGH and 1111 HGFE , respectively. Interference fringes in the

superimposed area GHFE 11 of the two portions are formed from the coherent light of

two point light sources, 1S and 2S , and are equivalent to a pair of pinholes in Young�s

interferometer configuration, as shown in Fig. 3.1(b). Young�s fringes with a

sinusoidal light intensity distribution will thus emerge on the observation screen [67,

68]. The fringes� pitch 0P is given by

13

λ)/(0 dDP = (3.1)

where d is the distance between the two virtual point sources, D is the distance from

the light source to the screen, and λ is the wavelength of a He-Ne laser. According to

Fig. 3.1(a), wedge thickness t at point A is given by

θtanLt = (3.2)

where θ is the wedge angle and L is the length from point A to the edge of the

optical wedge. If the thickness (t) of the wedge is much smaller than D and L , beam

AO is nearly parallel to beam 1CO . Hence the separation distance CD between AO

and 1CO equals the separation d of the two point light sources, 1S and 2S . From

geometry, distance d is given by

αβ sintan2td = (3.3)

where β is the refractive angle at point A , as shown in Fig. 3.1(a), and α is the angle

of incidence of the light. Hence, by Snell�s law,

βα sinsin 0n= (3.4)

where 0n is the refractive index of the optical wedge. Combining Eqs. (3.2)-(3.4), we

can write the separation d of the two point light sources as

14

θαα tan)]sin(tan[arcsinsin2 0nLd = (3.5)

If light source S is shifted in the Z direction only, Eq. (3.5) can be simplified as

follows:

θtanKd = (3.6)

where the constant )]sin(tan[arcsinsin2 0 αα nLK = . Hence the pitch )( 0P of the

fringe pattern is given by

θλ

tan0 KDP = (3.7)

Using optical wedges with different wedge angles )(θ one can obtain fringe

patterns with different pitches.

3.1.2 Formation of Fringe Patterns by a Liquid Crystal Display

(LCD) Projector

For the projection of the fringes a high-resolution spatial light modulator (SLM)

is appropriate. Today, a large number of SLMs are readily available based on twisted-

nematic liquid crystal displays, digital micromirror devices, and reflective LCDs.

Images can be written into the LCD by supplying the driving electronics from a

computer. Brightness and contrast could be set manually on the LCD driver board.

15

The LCD can provide us with grayscale capability, making it possible to use both gray

code and phase-shifting algorithms with sinusoidal fringes. The advantages of this

method are (a) extreme versatility in use, (b) matrix display builds up randomly

configurable patterns for projection, (c) there is an internal memory that can store up

to 32 images and 32 lines, (d) the fringes can be as small as 1 pixel × 1 pixel for

measurement of small objects, which has been of great help to this project.

3.2 Height and Phase relationship

Figure 3.2 shows the optical geometry of the projection and imaging system.

Points P and E are the centers of the exit pupils for the projection and imaging lenses

respectively. Every point on the reference plane is characterized by a unique phase

value, with respect to a reference point such as B , which is stored in the computer

memory as a system characteristic. The detector array is used to measure the phase at

the sampling points. For example, the phase at a point on the reference plane and

phase at a corresponding point on the object surface are measured. The phase mapping

algorithm then searches for a point on the reference plane and based on the similar

triangles, the phase and height relationship can be given by

dACdLAC

yxh+

=1

)(),( (3.8)

16

However, if the distance between the sensor and the reference plane is large

compared to the pitch of the projected fringes, under normal viewing conditions the

phase and profile relationship is given by:

CDCD kfg

lFGglyxh ϕ

πϕ

===2

),( (3.9)

where l is the distance between the sensor and the reference plane, g the distance

between the sensor and the projector, f the spatial frequency of the projected fringes

on the reference plane, )2/( gflk π= is an optical coefficient related to the

configuration of the optical measuring system and CDϕ is a phase angle which

contains the surface height information, k is a constant for a measurement system and

given by fd

Lkπ2⋅

= . The height of the object can then be calculated if the value of k

is known by a calibration process.

3.3 Determination of Phase Value

There are two popular techniques to determine phase value. The first is the

phase shifting method [69-72]. Basically, it employs known phase shifts for one of the

light beams by means of a phase shifter. Therefore the relative phase difference of two

interference waves is changed artificially. The phase value ),( yxδ of a certain point

on the interference field can be calculated from several images with a phase shift

interval of nϕ .

17

]),(cos[),(),(),( nn yxyxbyxayxI ϕδ +⋅+= (3.10)

where ϕϕ )1( −= nn , mn ...2,1= , 3≥m .

For four-step phase shifting method, the phase value of each point in the image

is obtained by,

),(),(),(),(

arctan),(31

24

yxIyxIyxIyxI

yx−−

=δ (3.11)

In general,

∑

∑

−

−

⋅

⋅= m

nnn

m

nnn

yxI

yxIyx

1

1

sin),(

sin),(arctan),(

ϕ

ϕδ (3.12)

The second technique is the Fourier Transform method [73-78], in which the

fringe map is Fourier transformed. The Fourier spectrum is band-pass filtered and

shifted to be centered at zero frequency. This is followed by inverse Fourier transform

to yield a π2 module phase map. The method has the advantage of using only one

interference fringe pattern for processing hence the background intensity variation and

speckle noise are reduced.

The input fringe pattern can be described by:

18

)],(),(2cos[),(),(),( 00 yxyxxuyxbyxayxf ϕϕπ +++= (3.13)

where ),( yxa and ),( yxb are the background and modulation terms respectively, 0u is

a spatial carrier frequency, ),(0 yxϕ is an initial phase, ),( yxϕ is a phase variable

which contains the desired information.

For the simplicity of analysis, the initial phase ),(0 yxϕ has been assumed to be

zero. For the purpose of Fourier fringe analysis, the input fringe pattern can be written

in the following form,

)]2(exp[),()]2(exp[),(),(),( 0*

0 xuiyxcxuiyxcyxayxf ππ −++= (3.14)

where )],(exp[),(21),( yxiyxbyxc ϕ= and the term containing �*� denotes its

complex conjugate. The Fourier transform )(uF of the recorded intensity distribution

),( yxf is given by

),(),(),(),( 0*

0 yuuCyuuCyuAyuF ++−+= (3.15)

In most cases, ),( yxa , ),( yxb and ),( yxϕ vary slowly compared to the carrier

frequency 0u , hence the spectra are separated from each other by the carrier frequency

0u . One of the side lobes is weighted by the Hanning window and translated by 0u

towards the origin to obtain ),( yuC . The central lobe and either of the two spectral

19

side lobes are filtered out by translating one of the side lobes to zero frequency. This

is shown in Figure 3.3.

Taking the inverse Fourier transform of ),( yuC with respect to x yields ),( yxc ,

the phase distribution may then be calculated point wise using the expression

=

)],(Re[)],(Im[arctan),(

yxcyxcyxϕ (3.16)

where )],(Im[ yxc and )],(Re[ yxc denote the imaginary and real parts of ),( yxc

respectively.

The displacement of the object can be evaluated by analyzing the phase value

distribution of the neighborhood points. For a vibrating object, the displacement )(tδ

of a given point at a time t is related to its instantaneous height ),,( tyxh and is given

by

)0,,(),,(),,( yxhtyxhtyxZ −= (3.17)

where )0(h denotes the instantaneous displacement of a reference plane. Using a

function generator, the displacement ),,( tyxZ ( µ m) is given by

)sin(),(),,( ωθ tyxZtyxZ += (3.18)

20

where ),( yxZ is the amplitude of the vibration function, θ the initial phase angle,

and ω the angular velocity. These values may be obtained from the settings of the

function generator.

3.4 Phase Unwrapping

In both phase shifting technique and Fourier transform technique, the phase is

obtained by means of the inverse trigonometric function: arctangent. Due to its nature,

this function returns only principal values, i.e., values in [ ππ,− ], generating a

discontinuous phase map wrapped into a [ ππ,− ] interval. Hence this map should be

unwrapped to the [ ∞∞− , ] interval before the phase values can be converted to

continuous values of the physical variable of interest. Phase unwrapping [79-81] is

essential in optical metrology by phase stepping and spatial altering techniques as

represented in Fig. 3.4. However, determination of the absolute phase from its

principal value can be approached in various ways, including pixel by pixel, block by

block, and frame by frame. Gierloff [82] proposed a phase unwrapping algorithm that

operates by dividing the fringe field into regions of inconsistency and then relating

these areas to one another. Green and Walker [83] presented an algorithm that uses

knowledge of the frequency band limits of a wrapped phase map. A method based on

the identification of discontinuity sources that mark the start or end of a π2 phase

discontinuity was developed by Cusack and Huntley [84]. Some of the methods seem

to be very complicated with the assumption that the wrapped image is very noisy.

However, if the phase map is obtained after applying a noise-suppressing phase

mapping algorithm or with the noises filtered out as described in the previous sections,

phase unwrapping is a relative simple and straightforward task.

21

A simple but robust phase unwrapping algorithm has been applied in this project.

In summary, it seeks phase jumps greater than π and corrects them by addition or

subtraction of a π2 offset until the difference between adjacent pixels is less than π .

This operation is iterated, rightward line by line, until every pixel in the data set has

been unwrapped. This algorithm does not need any pre-processing of the wrapped

image to reduce the noise, nor does it require any effort to choose a special

unwrapping path to avoid the noisy points.

3.5 Enhancement of Dynamic Range of Fringe Projection Method

To solve the underexposure problem encountered in the previous study, it is

necessary to adapt the dynamic range of the scene to the dynamic range of the camera.

In normal application, the sensor in a CCD camera is sensitive to light intensity in 8

bit range where the range of gray value is from 0 to 255. When a sinusoidal fringe

pattern is projected on a diffused test surface by a LCD projector particularly, the light

intensity on the test surface may be lower than the threshold level of the photosensors

in the camera [85] for a high-speed camera with a short exposure period. Hence

underexposure problem is introduced as shown in Fig. 3.5, where the recorded

intensity distribution ( PI ′ ) of the CCD sensor does not correlate with the projected

fringe ( PI ). To overcome this problem a WLS which superimposes a white light

background intensity distribution ( Wa ) is introduced. When the resultant intensity

distribution of the superposed fringe pattern ( FI ) is within the threshold level of the

22

photosensors, the camera would record an intensity distribution ( FI ′ ) correlated with

the input ( FI ).

The instantaneous displacement ),,( tyxZ of a given point on the object surface

may be written as:

tyxZtyxZ ωcos),(),,( = (3.19)

After introducing the reference plane at 0=Z , Eq. (3.13) can be written as:

ktyxZtyx ωϕ cos),(),,( = (3.20)

Substituting Eq. (3.20) into Eq. (3.13),

]cos),(),(2cos[),(),( 00 ktyxZyxxyxbyxaI PPP

ωϕπµ +++= (3.21)

By introducing a white light background ),( yxaW , the fringe intensity

distribution ( )PI is superimposed with the white light background and the resulting

light intensity distribution in the imposed image )( FI can be written as,

)],(cos),(cos2cos[),(),(),( 0 yxk

tyxZp

xyxbyxayxaI PpWF φωαπ ++++= (3.22)

23

Since the overall intensity distribution )( FI is within the threshold of the CCD

sensors as shown in Fig. 3.5, the instantaneous intensity of the imposed image )( FI

would be recorded as ( FI ′ ) during a finite exposure period, T , given by the following:

dtyxk

tyxZp

xyxbTyxaTyxaITt

tPPWF )],(cos),(cos2[cos),(),(),( 0φωαπ ++′+′+′=′ ∫

+

(3.23)

where ),( yxaW′ is the output for ),( yxaW . Now the dynamic range of the input

intensity distribution is adjusted to match the dynamic range of the high-speed camera

and FI ′ is further amplified to obtain a final image with an intensity distribution of 0I

in order to produce an optimal contrast:

})],(cos),(cos2[cos),(),({),( 00 dtyxk

tyxZp

xyxbTyxaTyxaITt

tPPW φωαπυµ ++′+′+′= ∫

+

(3.24)

where µ and υ are contrast coefficients, which are determined from the output of the

WLS and the fringe projector respectively. To obtain an optimal contrast, the gray

values of the background )],(),([ yxayxa PW ′+′µ and the modulation function

),( yxbP′υ are both modulated to a gray level of approximately 128 and the fringe

pattern intensity covers the full range of the output intensity of 0 to 255. Finally, each

optimal fringe image based on Eq. (3.24) can be further processed by FFT image

processing method. With further derivation, the method may have the potential for

time-average measurement (See appendix A).

24

3.6 Phase Shift Calibration

Calibration of the system is carried out by shifting the test object through a

known distance Zδ in the z-axis and the corresponding phase value Vδ on the

specimen is determined. The two sets of images are then processed. Several points

were chosen on the unwrapped maps of the first image and the phase values at these

points are noted. From the phase map of the second image, the same points are chosen

and the phase difference between these points would give the phase difference for the

height difference. Hence the relationship between height and phase difference is found.

The object height relative to the base plane can then be calculated by multiplying the

phase values with the corresponding factor.

3.7 Integrated Fringe Projection and DIC Method

Since DIC and fringe projection provide in-plane and out-of-plane displacement

measurement respectively, the combination of DIC and fringe projection techniques

would provide a 3-D displacement measurement of a planar object.

In digital image correlation, the image intensity acts as an information carrier.

Hence surface illumination should be uniform to ensure that the gray values on a

surface do not change greatly during deformation. However in fringe projection, the

fringe intensity is highly non-uniform. One way to overcome this problem is to filter

out the fringes by Fourier transform. By filtering out a small area of the fringe

frequency in the frequency domain followed by an inverse Fourier transform, the

background intensity would be restored.

25

When the object undergoes 3-D deformation, the deformed and reference

profiles generated by FFT are shifted by a distance equal to its in-plane deformation.

Hence to obtain the out-of-plane displacement accurately, an interpolation process

should be applied to the images being processed to obtain the final profiles.

An approach tailored for the particular requirements of DIC, was developed.

From among a range of possible alternatives, we opted for a simple algorithm where

the only data operations are a Fourier transform followed by a filter convolution. Then

the recovery procedure becomes very simple, consisting solely of an inverse Fourier

transform. Figure 3.7 shows the logical model of carrier removal algorithm (CRA).

Firstly original images are mapped into Fourier spectrums; secondly a low-pass filter

is operated to isolate the Fourier coefficients from zero (DC) up to the cutoff

frequency; thirdly, an inverse FFT of the resulting spectrum is computed and the

intensity distribution of the background is obtained.

In spatial domain, as defined in Eq. (3.10), ),( yxa describes the background

(object�s surface) variation, while ),( yxb represents the variation of fringes. In

frequency domain, as defined by Fourier transform, A , which describes the transform

of the function ),( yxa is preserved by CRA, while C and *C which represent

transforms of the function ),( yxb are eliminated by a low-pass filter. Therefore, the

Fourier transform of ),( yxa is isolated and, by inverse Fourier transform, the term

),( yxa itself can be obtained. It is only necessary then to apply DIC technique to the

resulting image for in-plane displacement vector ),( yxD ∆∆ .

26

Figure 3.8 illustrates schematically the in-plane deformation process of an object.

In order to obtain the in-plane displacement mu and mv of point M in the reference

image, a subset of pixels S around point M is chosen, and it is matched with a

corresponding set 1S in the deformed image. If subset S is sufficiently small, the

coordinates of points in 1S can be approximated by first-order Taylor expansion as

follows:

yyux

xuuxx

MMmmn ∆×

∂∂+∆×

∂∂+++= )1(1 (3.25)

yyvx

xvvyy

MMmmn ∆×

∂∂+∆×

∂∂+++= )1(1 (3.26)

where the coordinates are as shown in Fig. 3.8.

Let ),( yxI and ),( yxI d be the gray value distribution of the undeformed and

deformed image respectively. For a subset S , a correlation coefficient C is defined as:

[ ]

∑∑

∈

∈

−=

SNnn

SNnndnn

yxf

yxfyxfC 2

211

),(

),(),( (3.27)

where ),( nn yx is a point in subset S in the reference image, and ),( 11 nn yx the

corresponding point in subset S1 (defined by Eqs. (3.25) and (3.26)) in the deformed

image. It is clear that if parameters mm vu , are the real displacement and

27

MMMM yv

yu

xv

xu

∂∂

∂∂

∂∂

∂∂ ,,, are the displacement derivatives of point M , the correlation

coefficient C would be zero. Hence minimization of the coefficient C would provide

the best estimates of the parameters. Minimization of the correlation coefficient C is a

non-linear optimization process. Newton-Raphson and Levenburg-Marquardt iteration

methods are usually used in the implementation of algorithm.

To achieve sub-pixel accuracy, interpolation schemes should be implemented to

reconstruct a continuous gray value distribution in the deformed images. Normally

higher order interpolation scheme would provide more accurate result, but with the

limitation of more computation time. The choices of different schemes are depended

on the different requirements, bi-cubic and bi-quintic spline interpolation scheme are

widely used.

28

(a)

(b)

Fig. 3.1 Schematic diagram of two-point-light-source interferometry with an optical fiber and an optical wedge.

Screen

'S

S 2S

'2S

1S

'1S

P

δ

D

d

θS

Optical Fiber

α

E1E

A

B C H

D

1H1R

2R

3R0n

Optical Wedge

G

O

F

1G

1O

1F

Forward Surface

Screen

2S

1S

β

Back Surface

L

Z

X

29

Figure 3.2 Optical geometry for fringe analysis

Q

G B F

h D

3 � D Object

O X

Z

Reference Plane

g

l

E

30

Figure 3.3 Fourier fringe analysis. (a) Spectrum of the interferogram; (b) Processed spectrum after filtering.

I (u, y)

C

-uo u (b)

-uo u

C* C

u

F (u, y)

A

(a)

31

Figure 3.4

(a) Wrapped phase with 2π jump.

(b) Unwrapped phase value.

X (Pixel)

X (Pixel)

Phas

e va

lue

Phas

e va

lue

32

Fig. 3.5 Intensity of fringe pattern enhanced by a white light source.

Output intensity (Gray Value)

Input intensity

Pixel

Threshold of sensors

Output intensity of fringe pattern FI ′

PI ′0I

I/O Curve of CCD Camera Sensors

Resultant intensity of fringe pattern FI

PI

255

Wa

Wa

0

33

Fig. 3.6 General scheme of the proposed integrated method.

Filter (Cutoff the frequency of the fringe)

Inverse Fourier Transform

In plane displacement ),( yxD ∆∆ obtained by

digital correlation

Images with fringe pattern

Fast Fourier Transform

Wrapped Phase map

Pixels renumbered based on ),( yxD ∆∆

Phase unwrapping

Out of plane displacement

3-D displacement field obtained

34

Fig. 3.7 Logical model of CRA.

Representation of the image

after removal of carrier

Fourier transform

(Transform the original images into frequency domain)

Low pass filter

(Preserve the frequency of the background other

frequency set to zero)

Inverse Fourier transform

(Transform the frequency of the background back to

spatial domain)

Original image

35

Fig. 3.8 Schematic diagram of planar deformation process.

X (Pixel)

Y (P

ixel

)

36

Chapter 4 Application of Fringe Projection Method for

Static Measurement

This chapter describes the fringe projection method for static profile

measurement. Linear sinusoidal fringes which are generated by a LCD projector are

projected onto an object surface. The distorted fringe pattern caused by the surface

profile is captured by a CCD camera and stored in a digital frame buffer. With the aid

of FFT technique, the method is applied onto a micro-component

4.1 Experimental Work

The setup of the system is shown schematically in Fig. 4.1. A micromembrane

with an area of 400 µ m× 400 µ m is used as a test specimen. The micromembrane is

part of a microphone bonded in an IC chip as shown in Fig. 4.2. The membrane is

fully clamped at boundaries by a perforated backplate cap. It can be seen in Fig. 4.3

that the wafer and backplate cap keep the membrane rigid in the x - and y -directions

when the membrane is loaded in the z -direction. A CCD camera with a LDM lens is

used to capture the projected fringe patterns. The sensitivity of the system increases

with the angle between the LCD projector and the camera. Increasing the angle would

however create shadow areas on the object and affect the image quality. Hence, an

appropriate angle which depends on the accuracy and size of object should be chosen

for the system. In the experiment an angle of 024 is chosen. Images are recorded by

the CCD camera and are stored in a digital frame storage card for further processing.

37

The LCD projector provides a resolution of 832 × 624 pixels, which can be adjusted

between the relative transparency of 0 and 255. The pitch of the fringe pattern can be

as small as 1 pixel × 1 pixel and thus is suitable for measurement in micro scale.

4.2 Results and Discussion

Figure 4.4 shows the image of a fringe pattern, which is projected on the

unloaded micromembrane by using the LCD projector. The image is subsequently

processed by FFT from which the phase value can be determined. The phase value of

),( yxϕ obtained is wrapped in π2 modules, therefore, phase unwrapping is carried

out in order to convert the discontinuous phase distribution to a continuous one by

adding or subtracting π2 where necessary. The procedure is carried out along each

row and then repeated along the columns. To relate the phase values to the profile of

the test surface, calibration of the system is necessary. It is carried out by shifting the

test object through a known distance Zδ in the z-axis in micro scale and the

corresponding phase value Vδ on the specimen is determined as shown in Fig. 4.5. In

this experiment, the calibrated value for Zδ is given by VZ δδ 65.222= . Figure 4.6

shows a 3-D plot of the membrane. A cross-section of the micro membrane at x =

125 µ m is shown in Fig. 4.7.

There are two issues concerning the resolution of this method: the ),( yx spatial

resolution and out-of-plane resolution. The ),( yx resolution depends on the spatial

resolution of the projector and CCD camera. In this investigation, the camera has a

spatial resolution of 512 pixels × 512 pixels, which means that a total of 250,000

surface points can be monitored. Should hardware with higher spatial resolution be

38

used, the method would be able to simultaneously measure more points. For example,

a resolution of 1000 pixels × 1000 pixels would facilitate measurement of 1 million

points in one image. The Z resolution, however, depends on the gray level resolution

of the recording hardware. In this project, the hardware used has an 8-bit resolution,

i.e. 256 gray levels. The resolution would be increased with the use of higher gray-

level hardware. However, the higher resolution hardware may lead to higher

electronic noise and thus set a limit on the resolution. Note that the profile of the

micro membrane in Fig. 4.6 does not look very smooth. This is due to a limitation of

the conventional CCD sensor (768 pixels × 576 pixels). If a high performance CCD

camera with a higher resolution is used instead, the plot would be smoother.

The setup shown in Fig. 4.1 is based on reflection of a computer-generated

fringe pattern from a specularly reflective surface. The fringe pattern is perturbed in

accordance with the test surface and the fringe phase bears information on the height

of the test surface. Instead of deriving the mathematical relationship between the

fringe phase and the surface height, this relationship is obtained by a calibration

procedure. This fringe projection method makes it easy to achieve shape measurement

of the micro-component. With further development, the method can be implemented

for dynamic measurement, which will be discussed in the next chapter.

39

Fig. 4.1 Experimental setup for shape measurement of small components.

Test Surface

LDM High-speed CCD Camera

LCD Projector

Monitor

Computer Frame Grabber LCD Controller

Three-axis

Translation Stage

Microphone

Z

X

Y

α α

Power Supply

40

Figure 4.2 Microphone chip.

Fig. 4.3 Cross section view of the microphone.

Membrane (thickness 2 mµ )

Acoustic holesmµφ500

Air gap Perforated backplate cap

Si substrate

2000 µ m

650 µ m 400 µ m

Y

Z

41

Fig. 4.4 Image of fringe pattern on test surface of a microphone without loading.

0.1 mm

X=125 µm

X=125 µm

42

y = 222. 65x + 0. 7046R2 = 0. 9999

0

5

10

15

20

25

30

35

40

45

0 0. 02 0. 04 0. 06 0. 08 0. 1 0. 12 0. 14 0. 16 0. 18 0. 2Phase Value

Dis

plac

emen

t Shi

ft ( µ

m)

Fig. 4.5 Relationship between phase value and displacement shift of the test surface.

43

Fig. 4.6 3-D plot of the surface of the microphone.

Y (Pixel)

X (Pixel)

z (P

hase

Val

ue)

44

Profile of the membrane at cross-sectionX=125 µm

-100

0

100

200

300

400

500

600

700

0 25 50 75 100 125 150 175 200 225 250 275

Y (µm)

Hei

ght (

µm)

Fig. 4.7 Cross-section of the micro membrane at x = 125 µ m.

45

Chapter 5 Application of Fringe Projection Method for

Dynamic Measurement

5.1 Measurement of Dynamic Response of A Small Component

In this section, the fringe projection method is used measure the dynamic

response of a small vibrating component. The method enables measurement of

vibrating object point-by-point on its test surface as does the LDV method. The

method utilizes a long working distance microscope, a high-speed CCD camera, a

LCD fringe projector and the FFT technique. Linear fringe patterns are projected on a

vibrating object and the fringe patterns are captured by a high-speed camera and

stored in a digital frame buffer. With the aid of the FFT technique, the amplitude and

vibrating frequency of vibrating rigid objects are obtained.

5.1.1 Experimental Work

Figure 5.1 shows the experimental setup. A small coin with a diffuse surface is

used as a test object on which fringes are projected through a LDM lens and a LCD

projector. A high-speed camera with a LDM lens is used to capture the projected

fringe patterns. The coin was attached on a shaker, which has a sinusoidal output. The

shaker was mounted on a test rig consisting of a micrometer-driven translation stage

which enables out-of-plane translation. In the experiment an angle of 026 is chosen.

46

Images which can be displayed on a monitor are recorded by the high-speed camera

with a video frame rate of 500 frames/s.

5.1.2 Results and Discussion

Figure 5.2 shows part of the coin with a projected fringe pattern on the number

�2�. A close-up view of the fringe pattern with a projected fringe area of 1 mm × 0.5

mm is shown in Fig. 5.3. The coin was subjected to a vibrating frequency of 50 Hz

and images (see Fig. 5.4) were recorded at intervals of 0.002s. The images were

subsequently processed by FFT method and phase maps were obtained (See appendix

B). Calibration of the system is carried out by shifting the test object through a known

distance Zδ in the z-axis in micro scale to determine the relationship of Zδ and

Vδ (phase values of different points on the test surface to represent their heights) as

shown in Fig. 5.5. In this experiment, the calibrated value is given by VZ δδ 73.26= .

Figure 5.6 shows the phase maps at different time intervals after fringe processing,

where the gray values of the images represent the height of the test surface at different

points.

It should be noted that a surface profile variation (such as letters on a coin) is not

necessary for dynamic measurement. Measurement of a vibrating plane (no variation

in surface profile) can be easily obtained by this method. However, this method is also

suitable for measuring objects with profile variation surface. The surface profile of the

object can be obtained together with vibration frequency and amplitude.

47

Such profile variation on the surface may introduce �shadow effects�. This

problem was reduced in principle by adjustment of experimental setup with an optimal

angle for projection to minimize the shadow effect. Furthermore, since the object

undergoes rigid-body vibration, investigation of a small number of points on the

surface would represent the whole surface. Therefore, in the data analysis procedure,

an area (a small number of points in this area) on the surface was deliberately chosen.

The vibration plots of the test surface before and after temporal phase recovery

is shown in Figs. 5.7a & 5.7b respectively. For real time measurement of dynamic

events, provided the sampling rate is not so high that the phase change is more than

π± between two consecutive recordings, temporal phase recovery is needed.

Temporal phase recovery is carried out by the addition or subtraction of a 2π phase

value and is determined by whether a neighboring (in time axis) point is moving

towards or away from a reference point on the Z-axis. For movement towards the

reference point, a 2π phase angle would be added while for movement away from the

reference point, a 2π phase angle would be subtracted. The experimental values of

parameters A, ω and θ in Eq. (3.18) can be obtained from the phase maps as follows:

A certain region (marked with a square in Fig. 5.6) on the phase map at a particular

time (t=0.002s in this case) is chosen and their instantaneous displacements are

obtained from the calibrated phase values. Eight different regions are investigated to

minimize the errors as shown in Fig. 5.8. The process is repeated for a second

(t=0.004s) and third phase maps (t=0.006s) (See appendix C). With these known

values Eq. (3.18) can be rewritten as:

)68.314284.33sin(63.57)( tt +−=δ (5.1)

48

To further validate the method, the measurement results are compared with the

frequency output of the function generator and the vibration amplitude obtained by a

photogrammetry-based marker tracking test. In this test, the object surface is shifted

by the micrometer (resolution 0.5µ m) between the extreme positions of the vibration.

The sequential stages of movement are saved frame by frame and subsequently

compared with the images captured when the object is vibrating. By monitoring the

position of markers attached to the object, the vibration amplitude can be obtained. A

comparison of the results obtained is shown in Fig. 5.9. It is seen that the present

results show excellent agreement with that obtained using the micrometer and the

maximum discrepancy is less than 1%. It is also seen that the present results agree

closely with the input sinusoidal function. The proposed method has the advantage of

requiring a relatively simple experimental set-up and optical arrangement and hence

utilizes fewer optical components.

5.2 Enhancement of dynamic range of the system

The method described above is further applied on the vibrating object using a

background enhancement technique to increase the temporal resolution of the system.

With the proposed technique, the high-speed camera is able to function at a higher

frame rate of up to 3000 fps and thus an increase in the measurement range of up to a

factor of 6 (compared to the original frame rate limit of 500 fps) with enhancement in

the temporal resolution in the system is achieved. In the use of high-speed camera, a

problem often encountered is poor fringe image quality due mainly to insufficient

exposure of the CCD sensors. This problem is especially serious when the system

49

incorporates LDM lens which are used to measure small components. To solve this

problem, we developed a simple method to enable the method to capture good quality

images at a high frame rate. It employs a LCD projector, a high-speed CCD camera

and a WLS. Measurements are conducted on a vibrating speaker diaphragm. The

proposed method not only enables harmonic vibration measurement to be studied, but

also shows the potential application for measurement of non-repeatable transient

events.

5.2.1 Experimental Work

Figure 5.10 shows the experimental setup which consists of two LDMs, a high-

speed camera, a LCD fringe projector, a WLS (component W) and image processing

hardware. Figure 5.11 shows a number �2� on the test surface of the coin. A LDM

lens (magnification ×6 ) is focused on the fringes and another LDM lens is mounted

on a high-speed camera to capture the image. The WLS used to modulate the

background intensity is an Optem Model 29-60-74 Fibre Light Source, which has a

maximum output of 150 Watts. To ensure the light intensity of the test surface is

increased uniformly, the WLS is arranged along the same axis as the projector. Values

of µ and υ in Eq. (3.24) are adjusted by changing the power of the WLS and settings

of the fringe projector according to the output of the sensor (i.e. the gray value

distribution in 0I ). To Images of the test surface are recorded by the high-speed

camera at a frame rate of 3000 frames per second (fps).

Tests were also conducted on a speaker diaphragm subjected to free vibration.

The speaker, 5 cm in diameter, is mounted on a 3-D stage excited by a sinusoidal

50

voltage. Various stages of vibration of the speaker diaphragm are captured frame by

frame and stored for further processing.

5.2.2 Results and Discussion

Figure 5.12a and 5.12b show respectively the images of the test coin before and

after the introduction of a white light background. It is seen in Fig. 5.12b that the

image quality improved significantly and the recorded image intensity ( 0I ) shows

improved fringe quality (Fig. 5.12c). Figure 5.13 shows the gray value distribution of

cross-section YY as indicated in Fig.5.12. Values of )],(),([ yxayxa PW +µ and

),( yxbPυ are both modulated around 128 to make the fringe pattern displayed in a

wide range of gray value and thus produce optimal contrast. Figure 5.12c is further

processed by FFT technique to obtain the profile of the vibrating surface and a 3-D

plot of the surface profile is shown in Fig. 5.14a. Comparing with a 3-D image

obtained without background enhancement (in Fig. 5.14b), it is seen that the 3-D

profile obtained using the proposed method shows much better results.

Fringe pattern on the speaker superimposed with background was captured at a

frame rate of 3000 fps as shown in Fig. 5.15b, where the exciting frequency for the

speaker is 50 Hz. For comparison, the WLS was subsequently switched off and

images of the fringe pattern without the background were recorded at the same frame

rate as shown in Fig. 5.15a,. As can be seen, the images in Fig. 5.15a is underexposed

and in poor quality with insufficient light intensity. Obviously, the images with the

background are improved significantly as shown in Fig. 5.15b.

51

Image in Fig. 5.15b is subsequently processed by FFT technique to obtain

unwrapped phase maps. A 3-D profile of the vibrating test surface is shown in Fig.

5.16b compared with the one (Fig. 5.16a) which is obtained from Fig. 5.15a. As can

be seen, the result has been improved by this method. The method was further applied

to dynamic measurement of the speaker vibrating at a frequency of 100Hz. A

sequence of images of the speaker was recorded at 3000 fps, giving about 30 frames

per vibration period. Hence the profile of the diaphragm at a specific time can be

obtained. Subsequently whole-field displacement of the test surface at a specific time

and the vibration frequency and amplitude are determined. Figure 5.17 shows a plot of

the vibration amplitude obtained by comparing two phase maps at the extreme

positions of vibration. The mode shape of the vibrating diaphragm is determined by

considering a point on the phase map at a particular time (t=3.3ms in this case) and its

instantaneous displacement is obtained (from the calibrated phase values). The

process is repeated for 30 points within the vibration period of the object. With these

values known, the vibration can thus be plotted in Fig. 5.18. It can be seen that the

accuracy of the experimental results with the proposed method is better than the

original one without using white light background indicated by the black dot symbol.

Comparing with micro values, it is seen that the proposed method shows excellent

agreement. The micro values were obtained from the function generator and the

photogrammetry-based marker tracking test, described in section 5.1.2. The results

show a maximum discrepancy of less than 2%.

It is note worthy that without the white light background enhancement, the high-

speed camera can only capture image at a frame rate of up to 500 fps. However, with

52

the enhanced white light background, images of good quality can be captured at frame

speed of up to 3000 fps.

53

Fig. 5.1 Experimental setup of fringe projection method for dynamic measurement.

Function Generator

Object

LDM Lens High-Speed CCD Camera

LCD Projector

Monitor

Computer Frame Grabber LCD Controller

A

Three-axis Translation

Shaker X

Z

Y

54

(a) Test specimen.

(b) A close up view of the surface with fringe projection.

Fig. 5.2

1 mm

0.5 mm

55

Fig. 5.3 Sinusoidal fringe pattern projected on a small section of the test specimen.

0.05 mm

56

(a) (b)

(c) (d)

Images at different time intervals (a) 0s (b) 0.002s (c) 0.004s (d) 0.006s

Fig. 5.4 Images at 0.002s time interval.

57

y = 267. 25x + 0. 0412R2 = 0. 9999

0

5

10

15

20

25

30

35

40

45

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Phase Shift

Dis

plac

emen

t Shi

ft ( µ

m)

Fig. 5.5 Calibration of the fringe projection system for the measurement of vibration in z direction.

58

t= 0

.006

s t=

0.0

04s

t= 0

.002

s

Fi

g. 5

.6 U

nwra

pped

pha

se m

aps a

t 0.0

02s t

ime

inte

rval

.

45

40

35

30

25

M

icro

ns

20

15

10

5 0

59

-5

-4

-3

-2

-1

0

1

2

3

4

5

0 2 4 6 8 10 12 14 16 18 20 22

Sampling Number

Phas

e Va

lue

(a) Vibration amplitude before phase recovery.

0

1

2

3

4

5

0 2 4 6 8 10 12 14 16 18 20 22

Sampling Number

Phas

e Va

lue

(b) Vibration amplitude after phase recovery.

Fig. 5.7

60

-80

-60

-40

-20

0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Time (ms)

Ampl

itude

(mm

)

Fig. 5.8 Vibration plots of different regions on the object.

61

-40

-20

0

20

40

60

80

100

0 5 10 15 20 25 30 35 40

Time (ms)

Am

plitu

de ( µ

m)

Experimental Results by Fringe ProjectionExperimental Results by MicrometerFunction Generator Plot

Fig. 5.9 Comparison of micro values and experimental vibration amplitude.

62

Fig. 5.10 Experimental setup with the white light source.

Computer

Video Signal Processor

Monitor

High-speed

CCD camera

LCD Projector

LDM Lens

LDM Lens

Reference Plane

Object

3-Axis Translator

Z

X Y

Function Generator

W

63

Fig. 5.11 A small section on the test coin surface.

1 mm

64

(c)

(a )

Y

Y

Fig.

5.1

2 C

ompa

rison

of t

he im

ages

of p

art o

f coi

n re

cord

ed a

t rec

ordi

ng ra

te o

f 300

0 fp

s. (a

) Im

age

reco

rded

with

bac

kgro

und

enha

ncem

ent.

(b) I

mag

e re

cord

ed w

ithou

t bac

kgro

und

enha

ncem

ent.

(c) I

mag

e pr

oces

sed

with

an

optim

al c

ontra

st.

(b)

65

Fig. 5.13 Comparison of fringe pattern distribution of the cross section along YY.

0

50

100

150

200

250

0 10 20 30 40 50 60 70 80 90 100

Pixel

Gra

y Va

lue

Gray level distribution with enhanced backgroundGray level distribution without enhanced backgroundGray level distribution of matched fringe image

)( FI ′ )( PI ′

)( 0I

66

(a) 3-D profile of test object before background enhancement.

(b) 3-D profile of test object after background enhancement.

Fig. 5.14

67

(a) Image of the speaker recorded without white light background enhancement at frame rate of 3000 fps.

(b) Image of the speaker recorded with background enhancement at frame rate of 3000 fps.

Fig.5.15

68

.

(a) 3-D profile of the speaker without background enhancement.

(b) 3-D profile of speaker diaphragm with background enhancement.

Fig. 5.16

69

Fig. 5.17 Vibration amplitude of speaker diaphragm.

70

-1.5

-1

-0.5

0

0.5

1

1.5

0 1 2 3 4 5 6 7 8 9 10

Time (ms)

Am

plitu

de (m

m)

With background enhancementWithout background enhancementMicro values

Fig. 5.18 Comparison of vibration plots at 3000fps frame rate.

71

Chapter 6 Integrated Fringe Projection and DIC Method

After developing the dynamic fringe projection method for vibration

measurement, we try to look at an integrated fringe projection method to measure the

3-D displacement using only one camera. The method which uses only one camera

and is different from the conventional multi-camera based systems for 3-D

displacement measurement, shows a significant improvement over conventional

method.

6.1 Experimental Work

Part of a speaker (3.6 cm × 3.6 cm) with a diffuse surface is used as a test object

as shown in Fig. 6.1. The speaker was mounted on a test rig consisting of a

micrometer-driven translation stage enabling rigid-body translations in the out-of-

plane and in-plane directions. Distortions in the fringe pattern on the test surface are

recorded on the image plane of a CCD camera with a LDM lens. The CCD camera is

located perpendicular to the base on which the object is mounted, and the angle α

between the optical axis of the fringe projector and the optical axis of the CCD camera

is o8.32 . The object was subjected to a rigid-body motion which is defined by a 3-D

displacement vector ),,( ,,, yxyxyx zyxW ∆∆∆ by using a 3-axis translation stage with a

resolution of 0.5 µ m.

The in-plane displacement ),( ,, yxyx yxD ∆∆ of the object is determined by a

carrier removal operation and DIC according to Eq. (3.27). The value of

72

),( ,, yxyx yxD ∆∆ is subsequently used for modulating the reference coordinate

according to Eq. (3.27). After that, the shape and out-of-plane movement )( ,yxzD ∆

are determined by fringe projection.

6.2 Results and Discussion

Figure 6.2(a) shows the image of the test surface with a projected fringe pattern,

the pitch of which is 50 pixels. The Fourier transform of the fringe pattern is taken and

three peaks are obtained: the Fourier transform of the term ),( yxa is located in the

center of the spectrum and Fourier transform of ),( yxc and ),(* yxc is symmetric

with respect to the center and located at a distance that is determined by the frequency

of the fringe pattern. To obtain the contour of the test surface, the Fourier transform of

),( yxc is usually isolated, and by inverse Fourier transforming the peak, the value of

),( yxc itself is obtained, while the frequency outside the range of the phase are set to

zero. It is then necessary to separate the imaginary and the real parts and perform an

arctan operation to obtain the phase value of the fringe pattern.

However, to obtain the in-plane displacement of the object, the windowing

operation is placed at the center of the spectrum, and the Fourier transform of ),( yxa

is isolated. By inverse Fourier transforming the peak, the value of ),( yxa , which is

the intensity distribution of the background in the image, is obtained. The size of the

window is determined by the position of the Fourier transform of ),( yxc and ),(* yxc

in the spectrum, which is dependent on the number of fringes in the x direction on the

73

original image. Therefore, to retain more of the background data, denser fringe

patterns are preferred and thus a bigger filtering window size is preferred.

A 3-D representation of the spectrum is illustrated in Fig. 6.3(a). A low-pass

filter is applied to the spectrum to isolate the Fourier transform of ),( yxa , while

getting rid of the rest. Provided that we have chosen the cutoff frequency (directly

proportional to the size of the filtering window) well, the filter will eliminate the

fringe pattern completely, but still preserves enough background information. The

quality level of the resulting images must be such that the background information is

discernable and sharp, while the fringe pattern is removed thoroughly. Therefore, the

cutoff frequency value needs to be chosen carefully. Usually, a higher cutoff

frequency, i.e. big filtering window can preserve more coefficients of frequency area

and improve image sharpness, but it runs the risk of incomplete removal of the fringe

pattern, and if part of the Fourier transforms of ),( yxc and ),(* yxc are maintained.

In this experiment, a filtering window of 10 pixels × 10 pixels is applied to the

spectrum to isolate ),( yxa as shown in Fig. 6.3(b). Inverse Fourier transform is then

performed onto Fig. 6.3(b) to obtain ),( yxa itself as shown in Fig. 6.2(b). As can be

seen in Fig. 6.2(b), a large amount of data of ),( yxa is disregarded and the image

looks very blur. This is due to the small size of the filtering window. Figure 6.4(b)

shows the intensity of ),( yxa by applying a 20 pixels × 20 pixels window to the

fringe pattern with a pitch of 15 pixels (see Fig. 6.4a). Compared to Fig. 6.2(b), the

image is slightly sharper. The procedure is repeated on a fringe pattern with a pitch of

1 pixel. The result is shown in Fig. 6.5. It can be seen that the quality of the images

are improved significantly and ),( yxa is recovered with higher frequency.

74

A smaller part (1.4 cm × 1.2 cm) of the speaker was investigated for 3-D

displacement measurement. The images displayed in Fig. 6.6(a) and 6.6(b) show the

test surface being translated in the x -, y - and z - directions by a 3 - axis micrometer

driven translator with a resolution of 0.5 µ m. The magnitude of the displacement

from the micrometer reading are 225 µ m, 300 µ m and 140 µ m in x -, y - and z -

directions, respectively. A 3-D plot of the test surface obtained by FFT technique is

shown in Fig. 6.7. Figure 6.8(a) shows the image after the carrier removal algorithm

(CRA) procedure. It appears similar when compared with an image without a

projected fringe pattern (see Fig. 6.8(b)). In-plane displacement is subsequently

computed by applying a DIC algorithm to the image. Since the test surface has

undergone a rigid body translation, the displacement of the whole surface is

theoretically similar and thus an average operation was taken on the whole test surface.

The displacement in the x - and y - directions are 6.2 pixels and 8.4 pixels

respectively. A two � dimensional ( x and y directions) calibration procedure was

carried out. Figure 6.9 (a) and 6.9 (b) show the calibration plot, where the pixels were

converted into global coordinate. In this experiment, the relationship between pixel

and length is 1 pixel = 36.0 µ m and 1 pixel = 38.8 µ m for the x - and y - directions,

respectively.

Based on the value of the in-plane displacement, the shape data of the deformed

test surface was shifted and compared with the undeformed test surface in a 3-D

coordinate system. The displacement of the object can be evaluated by analyzing the

phase value distribution of the neighborhood points. The out-of-plane displacement

75

),( yxZδ of each point ( yx, ) is related to its height ),( yxh′ - before translation and

),( yxh - after translation and given by the equation

),(),(),( yxhyxhyxZ −′=δ (6.1)

The value of phase variation Zδ for the rigid body translation was 3.209. The

setup was subsequently calibrated in the z - direction by shifting the test object

through a known distance Zδ and the corresponding phase value Vδ on the specimen

is determined. In this experiment, the calibrated value for Zδ is given by VZ δδ 9.42=

as shown in Fig. 6.9(c). Figure 6.10 shows the 3-D displacement vector with the value

of 222.0 µ m, 297.4 µ m and 137.3 µ m in the x -, y - and z - directions. The

measurement results are compared with the values obtained from the micrometer and

show excellent agreement with discrepancies of 1.28%, 1.01% and 1.93% in x , y

and z directions, respectively.

76

Fig. 6.1 A close up view of a speaker diaphragm.

77

(a) Images of the test surface with fringe pattern with a pitch of 50 pixels.

(b) Recovery of intensity distribution of background.

Fig. 6.2

78

(a) 3-D representation of the spectrum before filtering.

Fig. 6.3

79

(b) 3-D representation of the spectrum after filtering.

Fig. 6.3

80

(a) Images of the test surface with fringe pattern with a pitch of 15 pixels.

(b) Recovery of intensity distribution of background.

Fig. 6.4

81

(a) Images of the test surface with fringe pattern with a pitch of 1 pixel.

Fig. 6.5

82

(b) Recovery of intensity distribution of background by a 70 pixels × 70 pixels filtering window.

Fig. 6.5

83

(c) Recovery of intensity distribution of background by an 80 pixels × 80 pixels filtering window.

Fig. 6.5

84

(d) Recovery of intensity distribution of background by a 100 pixels × 100 pixels filtering window.

Fig. 6.5

85

(e) Recovery of intensity distribution of background by a 120 pixels × 120 pixels filtering window.

Fig. 6.5

86

(a) A close up view of the test surface with fringe pattern before displacement (225 µ m, 300 µ m and 140 µ m in x -, y - and z - directions respectively).

Fig. 6.6

87

(b) A close up view of the test surface with fringe pattern after displacement (225 µ m, 300 µ m and 140 µ m in x -, y - and z - directions respectively).

Fig. 6.6

88

Fig. 6.7 Phase map of the test surface.

89

(a) Image of the background of the test surface before displacement.

Fig. 6.8 Images of the background of the test surface after CRA.

90

(b) Image of the background of the test surface after displacement.

Fig. 6.8 Images of the background of the test surface after CRA.

91

y = 0.0278xR2 = 0.9997

0

2

4

6

8

10

12

14

16

0 100 200 300 400 500 600

Translation in X direction by Micrometer (µm)

Dis

plac

emen

t in

X di

rect

ion

(Pix

el)

Theoretical Value of displacement along X directionExperimental Value of displacement along X direction by CRALeast-Squares FitLeast-Squares Fit

(a) Calibration and error analysis of displacement along X direction.

y = 0.0258xR2 = 0.9999

0

2

4

6

8

10

12

14

16

0 100 200 300 400 500 600

Translation in Y direction by Micrometer (µm)

Dis

plac

emen

t in

Y di

rect

ion

(Pix

el)

Theoretical Value of displacement along Y directionExperimental Value of displacement along Y direction by CRALeast-Squares FitLeast-Squares Fit

(b) Calibration and error analysis of displacement along Y direction

Fig. 6.9 Calibration of the measurement system.

92

y = 0.0233xR2 = 0.9998

0

1

2

3

4

5

6

7

8

0 50 100 150 200 250 300 350

Translation in Z direction by Micrometer (µm)

Dis

plac

emen

t in

Z di

rect

ion

(Pha

seva

lue)

Experimental Value of displacement along Z directionLeast-Squares Fit

(c) Calibration and error analysis of out-of-plane displacement.

Fig. 6.9 Calibration of the measurement system.

93

Fig. 6.10 3-D displacement vector of the test surface.

X

Y

Z

),,( zyxD ∆∆∆

x∆ = 222.0 µ m

y∆ = 297.4 µ m

z∆ = 137.3 µ m

94

Chapter 7 Conclusions and Recommendations

7.1 Conclusions

This thesis has concerned research work in: 1. Development of a fringe

projection system, 2. Application of the method to dynamic measurement and

enhancement of the dynamic range of the system and 3. Development of the integrated

fringe projection and DIC method for 3-D displacement measurement.

A dynamic fringe projection method for measurement of small components was

presented. Based on the principle of fringe projection, the system containing LCD

fringe projector, LDM lens, shaker, high-speed camera and controlling device was

designed. Quantitative analysis of each surface point of the specimen can be realized

with the wrapped phase maps extracted from a group of images captured by the high-

speed camera. From the wrapped map, unwrapping method restored the

discontinuities between every two adjacent points caused by the nature of arctangent.

It is proven that the system performs well in the various applications and

environments. Experimental results of vibrational frequency and amplitude are

obtained and compared favorably with theoretical values. The method has also been

successfully applied to the static measurement of micro-components.

To overcome the underexposure problem in dynamic measurement, the

background enhancement method was introduced and the dynamic range of the

system was increased with a factor of 6. The vibration plot of a vibrating diaphragm

95

was also successfully obtained. This method not only enables harmonic vibration

measurement to be studied, but also shows the potential application for measurement

of non-repeatable transient events.

In this project, integration of fringe projection and DIC techniques was also

introduced. The method enabled 3-D displacement of a small component using only

one CCD camera. In the integrated method, DIC was used to measure in-plane

displacement vector, while fringe projection was used to obtain 3-D shape and out-of-

plane displacement. The use of only one camera is a significant improvement to

previous multi-camera systems. The results obtained show excellent agreement with

theoretical values.

The main advantage of this method is that only one camera is needed to retrieve

the 3-D displacement vector. With the aid of the high-speed imaging system, the

proposed method is ready for studying dynamic measurement in both in-plane and

out-of-plane. It is noted the out-of-plane displacement may introduce variations in

magnification rate of the system. This can be avoided by using a telecentric lens to

project fringe pattern and capture images.

In the measurement of full-field displacements in dynamic situations, the

proposed method may encounter a few difficulties: 1.) Compared to the use of

transducers, the proposed method has the advantages of non-contact and non-

destructive. However, the temporal resolution of the system will be limited by the

frame rate of the high-speed camera, for measurement of a fast event. 2.) The

underexposure problem described in section 5.2 may occur, if the high-speed camera

96

is used at a high frame rate. 3.) The surface of the object must be painted white before

measurement can be carried out. 4.) To achieve measurement of non-harmonic

dynamic displacement, the current system needs to synchronize with the object due to

the short operation duration of the high-speed camera (approx. 4 seconds at 3000 fps).

5.) In dynamic situations, a specially designed calibration procedure may also be

required to relate the displacement to the phase values.

97

7.2 Recommendations

The power and stability of a LCD projector is one of the most significant factors

in the dynamic fringe projection system and the intensity of a projected fringe pattern

may be very low, at a high frame rate. Hence the quality of an image may be greatly

affected due to inefficiency in light intensity. It was also observed that fringe patterns

projected by a LCD projector vibrate at lower frequency and this introduces errors in

experimental results. Hence more powerful and stable projectors would improve test

results.

The measurement presented in this thesis is limited to 3-D rigid-body

displacement. However, with further development the proposed method could be

extended to non rigid-body whole-field displacement measurement. In addition, since

the method requires only one image for each displaced state, it would also have the

potential to be developed for dynamic and in-situ measurement.

Image size recorded by a high-speed camera decreases with high frame rate (128

pixels × 128 pixels at 10000 frames per second). The small size of the image

introduces difficulty for full-field measurement difficult. Hence a more powerful high-

speed camera is recommended.

98

References

1. R. Henao, A. Tagliaferri, and R. Torroba. A contouring approach using single

grating digital shadow Moiré with a phase stepping technique, Optik, 110 (4),

pp.199-201. 1999.

2. A. K. Maji, and M. A. Starnes. Shape measurement and control of deployable

membrane structures, Experimental Mechanics, 40 (2), pp.154-159. 2000.

3. M. M. Ratnam, P. L. Chee, and A. N. Khor. Three-dimensional object

classification using shadow moire and neutral network, Opt. Eng., 40 (9),

pp.2036-2040. 2001.

4. R. Thalmann, and R. Dandliker. Holographic contouring using electronic phase

measurement, Opt. Eng., 24 (6), pp.930-935. 1985.

5. C. P. Lopez, F. M. Santoyoa, R. R. Vera, and M. Funes-Gallanzi. Separation of

vibration fringe data from rotating object fringes using pulsed ESPI, Opt. Lasers

Eng., 38, pp.145-152. 2002.

6. C. T. Griffen, Y. Y. Hung, and F. Chen. Three dimensional shape measurement

using digital shearography, Proc. Of the SPIE � The International Society for Opt.

Eng., 2545, pp.214-220. 1995.

99

7. Y. Y. Hung. Applications of digital shearography for testing of composite

structures, Composites Part B (Engineering), 30B (7), pp.765-773. 1999.

8. Y. M. He, C. J. Tay, and H. M. Shang. Deformation and profile measurement

using the digital projection grating method, Opt. Lasers Eng., 30(5), pp.367-377.

1998.

9. Y. Morimoto, M. Fujigaki, and S. Yoneyama. Shape, stress and strain

measurement using phase analysis of grating or fringe patterns, Proc. of the SPIE

� The International Society for Optical Engineering, 4537, pp.47-52. 2001.

10. Y. Han, L. Ma, and S. P. He. Parameters calibration in topography measurement,

Proc. Of the SPIE � The International Society for Optical Engineering, 4537,

pp.297-300. 2002.

11. L. Scalise and N. Paone. Laser Doppler vibrometry based on self-mixing effect,

Opt. Laser Eng., 38, pp.173-184. 2002.

12. L. Scalise, G. Stavrakakis, A. Pouliezos. Fault detection for quality control of

household appliances by non-invasive laser Doppler technique and likelihood

classifier, Measurement, 25, pp.237�247. 2000.

13. Y. Y. Hung and J. D. Hovanesian. High speed shearography system for measuring

dynamic deformation, Proc. of the SEM Spring Conference on Experimental

Mechanics and Applied Mechanics, Houston, Texas, June 1-3, pp.49-50. 1998.

100

14. A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones. Transient deformation

measurement with electronic speckle pattern interferometry and a high-speed

camera, Appl. Opt., 38, pp.1159-1162. 1999.

15. D. Kokidko, L. Gee, S. C. Chou, F. P. Chiang. Method for measuring transient

out-of-plane deformation during impact, Int J Impact Eng., 19(2), pp.127-33.

1997.

16. Y. Li, A. Nemes, A. Derdouri. Optical 3-D dynamic measurement system and its

application to polymer membrane inflation tests, Opt. Laser Eng., 33, pp.261-276.

2000.

17. D. R. Burton, M. J. Labor. Multichannel Fourier fringe analysis as an aid to

automatic phase unwrapping, Appl. Opt,. 33, pp.2939-2947. 1994.

18. D. J. Bone, H. A. Bachor, and R. J. Sandeman. �Fringe-pattern analysis using 2-D

Fourier transform, Appl. Opt., 25(10), pp.1653-1660. 1986.

19. C. Buckberry, M. Reeves, A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C.

Jones. The application of high-speed TV-holography to time-resolved vibration

measurements, Opt. Laser Eng., 32, pp.387-394. 1999.

20. P. J. Rae, H. T. Goldrein, N. K. Bourne, W. G. Praud, W. G. Forde, L. C. Forde,

and M. Liljekvist. Measurement of dynamic large-strain deformation maps using

an automated fine grid technique, Opt. Laser Eng., 31, pp.113-122. 1999.

101

21. G. Pedrini, S. Schedin, and H.J. Tiziani. Pulsed digital holography combined with

laser vibrometry for 3D measurements of vibrating objects, Opt. Laser Eng. 38,

pp.117-129. 2002.

22. C. Perez-Lopez, F. Mendoza Santoyo, G. Pedrini, S. Schedin, and H. J. Tiziani.

Pulsed digital holographic interferometry for dynamic measurement of rotating

objects with an optical derotator, Appl. Opt. 40(28), pp.5106-5110. 2001.

23. S. Schedin, G. Pedrini, H. J. Tiziani, A. K. Aggarwal, and M. E. Gusev. Highly

sensitive pulsed digital holography for built-in defect analysis with a laser

excitation, Appl. Opt. 40(1), pp.100-103. 2001.

24. S. Schedin, G. Pedrini, H. J. Tiziani, and F. M. Santoyo. Simultaneous three-

dimensional dynamic deformation measurements with pulsed digital holography,

Appl. Opt., 38(34), pp.7056-7062. 1999.

25. J. P. Chambard, V. Chalvidan, X. Carniel, and J. C. Pascal. Pulsed-TV

holography recording for vibration analysis applications, Opt. Laser Eng., 38,

pp.131-143. 2002.

26. C. P. Lopez, F. M. Santoyoa, R. R. Vera, and M. Funes-Gallanzi. Separation of

vibration fringe data from rotating object fringes using pulsed ESPI, Opt. Laser

Eng., 38, pp.145-152. 2002.

102

27. M. Aslan, and B. R. Tittmann. Laser interferometry for measurement of

displacement in hostile environment and in real time, Proc. of the SPIE � The

International Society for Optical Engineering, 3993, pp.68-77. 2000.

28. R. Windecker, M. Fleischer, K. Korner, and H. J. Tiziani. Testing micro devices

with fringe priojection and white-light interferometry, Opt. Lasers Eng., 36(2),

pp.141-154. 2001.

29. X. Y. He, X. Kang, C. Quan, C. J. Tay, S. H. Wang, and H. M. Shang. Optical

methods for the measurement of MEMS materials and structures, Proc. Of the

SPIE � The International Society for Optical Engineering, 4537, pp.63-68. 2002.

30. G. S. Spagnolo, and D. Ambrosini. Diffractive optical element based sensor for

roughness measurement, Sensors and Actuators, A: Physical, 100(2-3), pp.180-

186. 2002.

31. C. Quan, C. J. Tay, X. Y. He, X. Kang, and H. M. Shang. Microscopic surface

contouring by fringe projection method, Opt. Laser Tech., 34(7), pp.547-552.

2002.

32. C. Zhang, P. S. Huang, and F. P. Chiang. Microscopic phase-shifting profilometry

based digital micromirror device technology, App. Opt., 41(28), pp.5896-5904.

2002.

103

33. K. Leonhardt, U. Droste, and H. J. Tiziani. Microshape and rough-surface

analysis by fringe projection, Appl. Opt., 33, pp.7477-7488. 1994.

34. R. Windecker, M. Fleischer, and H. J. Tiziani. Three-dimensional topometry with

stereo microscopes, Opt. Eng., 36, pp.3372-2277. 1997.

35. R. Windecker, S. Franz, and H. J. Tiziani. Optical roughness measurements with

fringe projection, Appl. Opt., 38, pp.2837-2842. 1999.

36. R. Kowarschik, P. Kuhmstedt, J. Gerber, W. Schreiber, and G. Notni. Adaptive

optical three-dimensional measurement with structured light, Opt. Eng., 39,

pp.150-158. 2000.

37. F. Laguarta, R. Artigas, A. Pinto, and I. Al-Khatib. Micromeasurements on

smooth surfaces with a new confocal optical profiler, Proc. of the SPIE � The

International Society for Optical Engineering, 3520, pp.149-160. 1998.

38. P. Pavlicek, and J. Soubusta. Theoretical measurement uncertainty of white-light

interferometry on rough surfaces, App. Opt., 42(10), pp.1809-1813. 2003.

39. T. Wilson, ed., Confocal Microscopy. Chaps. 1-3, London: Academic Press. 1990.

40. D. Steudle, M. Wegner, and H. J. Tiziani. Confocal principle for macro- and

microscopic surface and defect analysis, Opt. Eng., 39, pp.32-39. 2000.

104

41. P. Hlubina. White-light spectral interferometry to measure intermodal dispersion

in two-mode elliptical-core optical fibres, Opt. Commun., 218 (4-6), pp.283-289.

2003.

42. G. S. Kino and S. Chim. Mirau correlation microscope, Appl. Opt., 29, pp.3775-

3783. 1990.

43. P. de Groot and L. Deck. Surface profiling by analysis of white-light

interferograms in the spatial frequency domain, J. Mod. Opt., 42, pp.389-401.

1995.

44. M. Takeda, and K. Motoh. Fourier transform profilometry for the automatic

measurement of 3-D object shapes, Appl. Opt., 22 (24), pp.3977-3982. 1983.

45. J. Li, X. Y. Su, and L. R. Gou. An improved Fourier transform profilometry for

automatic measurement of 3-D object shapes, Opt. Eng., 29 (12), pp.1439-1444.

1990.

46. P. K. Rastogi, Digital speckle pattern interferometry and related techniques. John

Wiley & Sons Ltd, 2001.

47. K. P. Proll, J. M. Nivet, C. Voland, and H. J. Tiziani. Enhancement of the

dynamic range of the detected intensity in an optical measurement system by a

three channel technique, Appl. Opt., 41, pp.130-135. 2002.

105

48. G. Pedrini, H. J. Tiziani, and I. Alexeenko. Digital-holographic interferometry

with an image-intensifier system, Appl. Opt., 41 pp.648-653. 2002.

49. Y. Ito, Y. Katoh, M. Kagata, S. Tomioka, and T. Enoto. Analysis for

improvement of simultaneity of shuttering in an ultra high-speed framing camera,

IEEE Transactions on Magnetics., 36, pp.1774-1778. 2002.

50. Y. Wang and F. P. Chiang. New moire interferometry for measuring three-

dimensional displacements, Opt. Eng., 33, pp.2654-2658. 1994.

51. F. P. Chiang and D. W. Li. Random Speckle patterns for displacement and strain

measurements: some recent advances, Opt. Eng., 24, pp.936-943. 1985.

52. R. Henao, F. Medina, H. J. Rabal, and M. Trivi. Three-dimensional speckle

measurement with a diffraction grating, Appl. Opt., 32, pp.726-729. 1985.

53. P. F. Luo, Y. J. Chao, and M. A. Sutton. Application of stereo vision to three-

dimensional deformations anlyses in fracture experiments, Opt. Eng., 33, pp.981-

980. 1994.

54. J. D. Helm, S. R. McNeill, and M. A. Sutton. Improved three-dimensional image

correlation for surface displacement measurements, Opt. Eng., 35, pp.1911-1920.

1996.

106

55. . Synnergren. Measurements of three-dimensional displacement fields and shape

using electronic speckle photography, Opt. Eng., 36, pp.2302-2310. 1997.

56. M. E. Pawlowski, M. Kujawinska, and M. G. Wgiel. Shape and motion

measurement of time-varying three-dimensional objects based on spatiotemporal

fringe-pattern analysis, Opt. Eng., 41 (2), pp.450-459. 2002.

57. M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson and S. R. McNeill.

Determination of displacements using an improved digital correlation method,

Image Vis. Comput. 1, pp.133-139. 1983.

58. M. A. Sutton, M. Cheng, W. H. Peters, Y. J. Chao, and S. R. McNeill.

Application of an optimized digital correlation method to planar deformation

analysis, Image Vis. Comput. 4, pp.143-150. 1986.

59. H. A. Bruck, S. R. McNeill, M. A. Sutton and W. H. Peters. Digital image

correlation using Newton-Raphson method for partial differential correlation, Exp.

Mech. 29, pp.261-267. 1989.

60. M. A. Sutton, S. R. McNeill, J. Jang, and M. Babai. Effects of subpixel image

restoration on digital correlation error estimates, Opt. Eng. 27, pp.870-877. 1988.

61. G. Vendroux and W. G. Knauss. Submicron deformation field measurements:

Part 2. Improved digital image correlation, Exp. Mech. 38, pp.86-92. 1998.

107

62. H. Lu and P. D. Cary. Deformation measurements by digital image correlation:

implementation of a second-order displacement gradient, Exp. Mech. 40, pp.393-

400. 2000

63. P. Cheng, M. A. Sutton, H. W. Schreier and S. R. McNeill. Full-field speckle

pattern image correlation with B-Spline deformation function, Exp. Mech. 42,

pp.344-352. 2002.

64. T. C. Chu, W. F. Ranson, M. S. Sutton, and W. H. Peters. Applications of digital

image correlation techniques to experimental mechanics, Exp. Mech. 25, pp.232-

244. 1985.

65. W. H. Peters, H Zheng-Hui, M. A. Sutton, and W. F. Ranson. Two-dimensional

fluid velocity measurement by use of digital speckle correlation techniques, Exp.

Mech. 24, pp.117-121. 1984.

66. Y. J. Chao and M. A. Sutton. Measurement of strains in paper tensile specimen

using computer vision and digital image correlation, Part2: Tensile specimen test,

Tappi J. 70, pp.153-156. 1998.

67. S. H. Wang, C. J. Tay, C. G. Quan, and H. M. Shang. An optical fiber fringe

projector for micro-component, Optik, pp.419-422. 2000.

108

68. J. H. Yi, S. H. Kim, and Y. K. Kwak. A nanometric displacement measurement

method using the detection of fringe peak movement, Meas. Sci. Technol.,

pp.1353-1358. 2000.

69. P. S. Huang, Q. J. Hu, and F. P. Chiang. Double three-step phase-shifting

algorithm, Appl. Opt., 41 (1), pp.4503-4509. 2002.

70. C. Quan, C. J. Tay, X. Y. He, X. Kang, and H. M. Shang. Microscopic surface

contouring by fringe projection method, Opt. Laser Tech., 34 (7), pp.547-552.

2002.

71. C.P. Zhang, P. S. Huang, and F. P. Chiang. Microscopic phase-shifting

profilometry based on digital micromirror device technology, Appl. Opt., 41 (28),

pp.5896-5904. 2002.

72. C. Quan, C. J. Tay, X. Kang, X. Y. He, and H. M. Shang. Shape measurement by

use of liquid-crystal display fringe projection with two-step phase shifting, Appl.

Opt., 42 (13), pp.2329-2335. 2003.

73. D. R. Burton and M. J. Labor. Multichannel Fourier fringe analysis as an aid to

automatic phase unwrapping, Appl. Opt., 33, pp.2939-2947. 1994.

74. D. J. Bone, H. A. Bachor, and R. J. Sandeman. Fringe-pattern analysis using a 2-

D Fourier transform, Appl. Opt., 25 (10), pp.1653-1660. 1986.

109

75. M. Takeda, K Mutoh, H. Ina, and S. Kobayashi. Fourier-transform method of

fringe- pattern analysis for computer-based topography and interferometry, J. Opt.

Soc. Am., 72 (1), pp.156-160. 1982.

76. W. W. Macy. Two-dimensional fringe-pattern analysis, Appl. Opt., 22(23),

pp.3698-3901. 1983.

77. C. Quan, C. J. Tay, H. M. Shang, and P. J. Bryanston-Cross. Contour

measurement by fibre optics fringe projection and Fourier transform analysis, Opt.

Commun., 119, pp.479-483. 1995.

78. P. J. Bryanston-Cross, C. Quan, and T. R. Judge. Application of the FFT method

for the quantitative extraction of information from high-resolution interferometric

and photoelastic data, Opt. Laser Technol., 26, pp.147-155. 1994.

79. M. A. Herraez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat. Fast two-

dimensional phase-unwrapping algorithm based on sorting by reliability

following a noncontinuous path, Appl. Opt., 41 (35), pp.7437-7444. 2002.

80. X. Y. He, X. Kang, C. J. Tay, C. Quan, and H. M. Shang. Proposed algorithm for

phase unwrapping, Appl. Opt., 41 (35), pp.7422-7428. 2002.

81. M. A. Herraez, M. A. Gdeisat, D. R. Burton, and M. J. Lalor. Robust, fast, and

effective two-dimensional automatic phase unwrapping algorithm based on image

decomposition, Appl. Opt., 41 (35), pp.7445-7455. 2002.

110

82. J. J. Gierloff. Phase unwrapping by regions, current developments in optical

engineering, Proc. Of the SPIE � The International Society for Optical

Engineering, 818, pp.2-9. 1987.

83. R. J. Green, J. G. Walker. Phase unwrapping using a priori knowledge about the

band limits of a function, industrial inspection, Proc. Of the SPIE � The

International Society for Optical Engineering, 1010, pp.36-43. 1988.

84. R. Cusack, J. M. Huntley, and H. T. Goldrein. Improved noise-immune phase

unwrapping algorithm, Appl. Opt., 34 (5), pp.781-789. 1995.

85. A. J. Moore, J. D. C. Jones, and J. D. R. Valera, Dynamic measurements, In: P. K.

Rastogi, editor. Digital speckle pattern interferometry and related techniques.

John Wiley & Sons, LTD, 2001.

86. W. Osten, F. Elandaloussi, and M. Stache, An introduction to the Fringe

Processor 2.0, Bremer Institut fur Angewandte Strahltechnik, 1998.

111

APPENDIX A Time-average Fringe Projection

Equation (3.23) can be rewritten as following,

}cos),(sin)],(cos2[sin

cos),(cos)],(cos2[cos){,(

),(),(

0

0

dtk

tyxZyxk

x

dtk

tyxZyxp

xyxb

yxayxaI

Tt

t

Tt

tpP

WPF

ωϕαπ

ωφαπ

+−

⋅+

++=′

∫

∫+

+

(A.1)

Subsequently,

dtk

tyxZyxk

xyxb

dtk

tyxZyxp

xyxb

tyxatyxaI

Tt

tP

Tt

tP

WPF

]cos),([sin)],(cos2sin[),(

]cos),([cos)],(cos2cos[),(

),,(),,(

0

0

ωϕαπ

ωφαπ

∫

∫+

+

⋅+⋅−

⋅+⋅

++=′

(A.2)

The integration expression of zero-order first-kind of Bessel�s function is

∫∫ =Γ

=TT

dttAT

dttAT

AJ00

0 )coscos(7724.1

1)coscos()

21(

1)( (A.3)

The first part ),(),( yxayxa WP + of Eq. (A.2) is constant, which depends on the

spatial intensity variation of the background. The second part

112

dtk

tyxZyxp

xyxbTt

tP ]cos),([cos)],(cos2cos[),( 0

ωφαπ∫+

⋅+⋅ can be transformed into

Bessel�s function as shown in Eq. (A.3). The third part

dtk

tyxZyxk

xyxbTt

tP ]cos),([sin)],(cos2sin[),( 0

ωϕαπ∫+

⋅+⋅− cannot be expressed by

Bessel�s function. However, for ]cos),(sin[k

tyxZ ω is an odd function, when it is

integrated within a long period, value of which will approach zero or a constant. Thus

the third part in Eq. (A.2) can be regarded as zero and Eq. (A.2) can be expressed by

the sum of a constant value and a Bessel�s function.

113

APPENDIX B Procedure of FFT Processing

B.1 Fast Fourier Transform

The FFT can be applied only to gray value images which numbers of columns

and rows are a power of 2 [86]. In the opposite case, we must resize the images to

meet the requirement.

As a result of the applied FFT, we will get three floating point images:

• Real part of the Fourier spectrum (See Fig. B.1a),

• Imaginary part of the Fourier spectrum (See Fig. B.1b),

• Spectrum (See Fig. B.1c).

B.2 Bandpass Filter for the Fourier Transform Method

The aim of bandpass filtering is to get a smoothed phase by taking only one part

of the spectrum by translating it to the origin.

The bandpass filter is applied to both the real part and imaginary part of the

Fourier spectrum after FFT. Six co-ordinates are necessary for the determination of

the filter function which must be chosen carefully. They should be determined by

analyzing the amplitude spectrum before calling the bandpass filter. A rectangular

region of the spectrum is extracted by the first point in the left upper corner and the

114

last point in the right bottom. The size of the spectrum around the center to be deleted

should also be given.

After selecting the real part and imaginary part of the Fourier spectrum to be

filtered, the values of the appropriate co-ordinates for the bandpass filter must be

keyed in (See Fig. B.2):

• Start: x and y , co-ordinates of the first pixel (left on the top) of the region to

be filtered,

• End: x and y , co-ordinates of the last pixel (right on the bottom) of the region

to be filtered,

• Zero: x and y , size of the spectrum around the center to be eliminated.

After applying the bandpass filter to the Fourier spectrum, three floating point

images are given:

• Filtered real part of the Fourier spectrum (See Fig. B.3a),

• Filtered imaginary part of the Fourier spectrum (See Fig. B.3b),

• Amplitude spectrum of the result (See Fig. B.3c).

B.3 Inverse FFT

The inverse is applied to the filtered real and imaginary parts of the Fourier

spectrum to deliver three floating point images:

• Back transformed real part of the Fourier spectrum (See Fig. B.4a),,

• Back transformed imaginary part of the Fourier spectrum (See Fig. B.4b),,

• Mod π2 -phase (See Fig. B.4c),.

115

B.4 Unwrapping

Since all phase measuring techniques deliver the phase only mod π2 because of

the sinusoidal nature of the intensity distribution, unwrapping the saw-tooth images to

reconstruct the continuous phase distribution is necessary:

• Select the wrapped phase map,

• Define the start point carefully to activate the unwrapping procedure,

• Unwrapped phase map (See Fig. B.5).

116

(a) (b)

(c)

Fig. B.1 (a) Real part of the Fourier spectrum, (b) Imaginary part of the Fourier spectrum, (c) Spectrum.

117

Fig. B.2 Dialog box for the bandpass filter for the Fourier transform method.

118

(a) (b)

(c)

Fig. B.3 (a) Filtered real part of the Fourier spectrum, (b) Filtered imaginary part of the Fourier spectrum, (c) Amplitude spectrum of the result.

119

(a) (b)

(c)

Fig. B.4 (a) Back transformed real part of the Fourier spectrum, (b) Back transformed imaginary part of the Fourier spectrum, (c) Mod π2 -phase.

120

Fig. B.5 Unwrapped phase map.

121

APPENDIX C Phase Maps of the Test Surface at Different

Intervals

(a) (b)

(c) (d)

122

(e) (f)

(g) (h)

123

(i) (j)

(k) (l)

124

(m) (n)

(o) (p)

125

(q) (r)

(s) (t)

Fig. C.1 Phase maps at different time intervals (a) 0.002s (b) 0.004s (c) 0.006s (d) 0.008s (e) 0.010s (f) 0.012s (g) 0.014s (h) 0.016s (i) 0.018s (j) 0.020s (k) 0.022s (l)

0.024s (m) 0.026s (n) 0.028s (o) 0.030s (p) 0.032s (q) 0.034s (r) 0.036s (s) 0.038s (t) 0.040s

126

APPENDIX D List of Publications

1. C. J. Tay, C. Quan, H. M. Shang, T. Wu, and S. H. Wang. New method for

measuring dynamic response of small components by fringe projection, Optical

Engineering, 42(6), pp.1715-1720. 2003.

2. T. Wu, C. J. Tay, C. Quan, H. M. Shang, and S. H. Wang. Vibration measurement

of micro-components by fringe projection method, Proceedings of the SPIE, 5116,

pp.912-923. 2003.

3. C. Quan, C. J. Tay, T. Wu, S. H. Wang, and H. M. Shang. Extending the dynamic

range of a white-light video measurement system, Submitted to Optics

Communications.

4. C. J. Tay, C. Quan, T. Wu, and Y. H. Huang. An integrated method for 3-D shape

and in-plane displacement measurement using fringe projection, Submitted to

Optical Engineering.