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NEW SIMPLE DSPI SETUPS AND
IMPROVEMENT OF NOISE TOLERANCE OF DSPI

presented by

Xu Ding

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6/01 c:/CIRC/DateDue.p65-p.15

NEW SIMPLE DSPI SETUPS AND
IMPROVEMENT OF NOISE TOLERANCE OF DSPI

By

Xu Ding

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mechanical Engineering

2004

ABSTRACT

NEW SIMPLE DSPI SETUPS AND
IMPROVEMENT OF NOISE TOLERANCE OF DSPI

Xu Ding

The general objective of this research is to improve the environmental immunity of
the digital speckle pattern interferometry (DSPI) technique and simplify the DSPI
apparatus so that DSPI can be more robust and cost effective, meaning that this technique

can be used for measurement in more realistic industrial environments.

An improved Max-Min scanning method (IMMS) for phase determination has been
developed in this research work, and its application to the determination of phase maps
was successfully demonstrated in some DSPI. Because signal-processing techniques in
the time domain are employed in the new IMMS method, the improved algorithm has

shown good environmental tolerance.

By application of the newly developed IMMS method, a new technique to calibrate
the phase shifting behavior has been derived. Its applications to check the behavior of
piezo electronic transducer (PZT) phase shifters were successfully demonstrated in
experiments. This technique uses the IMMS method to compute the phase values along
the entire intensity waveform to obtain the diagram of phase shifting versus the driving

voltages to the PZT at some pixels, so that the linearity of the phase shifting can be

clearly displayed. By considering more pixel points along chosen directions over the
image, the tilt and non-uniformity of the phase shifting can be shown in a 3-D graph.
This calibration algorithm eliminates the drawbacks caused by the common assumption
of linearity and uniformity of the phase shifter, thus efficiently minimizing the movement
error of the PZT. This new calibration method can be used to evaluate a few properties

of the phase shifter simultaneously, such as non-linearity, non-uniformity and tilt.

Since the IMMS method shows good environment tolerance, particularly
minimizing the effect of errors of phase shifting, both in-plane DSPI and out-of-plane
DSPI could be simplified very much compared to the commonly used setups. These
simplified setups were shown to work quantitatively with the IMMS phase shifting
method. Very good noise immunity of the presented setup was seen. This simplification
makes DSPI more cost effective and broadens the application of the electronic speckle
pattern interferometry (ESPI) technique to noisy service environments outside the

laboratory.

To my parents

iv

ACKNOWLEDGMENTS

The author wishes to offer his sincerest appreciation and gratitude to his advisor,
Dr. Gary Lee Cloud, for his valuable guidance, encouragement and support throughout

this research work.

The author also would like to thank the other members of his committee, Dr.
Dahsin Liu, Dr. Martin A. Crimp, and Dr. Gary J. Burgess for their interest in this

research.

Last but not least, he thanks his wife Xiaoyun Huang for her patience,

understanding, and encouragement.

TABLE OF CONTENTS

 

 

 

 

 

 

 

 

 

 

 

 

LIST OF TABLES ....................................................................................... ix
LIST OF FIGURES ........................................................................................ x
Chapter 1. Introduction 1
1.1 Problem outline ......................................................................................................... 1
1.2 Objectives ................................................................................................................. 3
1.3 Relevant literature review ......................................................................................... 4
Chapter 2. Summary of speckle technique 12
2.1 Introduction ............................................................................................................. 12
2.2 Concept of laser speckles ........................................................................................ 12
2.3 Speckle pattern interferometry ................................................................................ 14
2.4 DSPI technique ....................................................................................................... 17
2.5 Phase shifting technique ......................................................................................... 18
2.5.1 Carré technique 20
2.5.2 Three-step technique 21
2.5.3 Four-step technique 22
2.5.4 Seven-step technique 23
2.5.5 Max-min scanning algorithm 23
2.5.6 Sinusoidal fitting technique 24
2.5.7 Comparison of phase shifting techniques 24
2.6 Image processing .................................................................................................... 25
2.6.1 Phase change map 25
2.6.2 Image filtering 26
2.6.3 Phase unwrapping 28

2.7 Displacement and strain computation ..................................................................... 31

vi

 

 

 

 

 

 

 

 

 

 

 

 

2.8 Sensitivity and spatial resolution of DSPI .............................................................. 33
2.9 Practical DSPI setup ............................................................................................... 34
Chapter 3. IMMS technique 36
3.1 Introduction ............................................................................................................. 36
3.2 The Principle of the IMMS method for phase determination ................................. 38
3.2.1 Mathematical Development 38
3.2.2 Algorithm to determine IMax, IMin and ¢. 42

3.3 Experimental Examples .......................................................................................... 46
3.3.1 Determination of the Crack Tip Position 46
3.3.2 Measurement of In-plane Displacements 50

3.4 The noise tolerance property of the IMMS method ................................................ 52
3.4.1 Numerical simulations 53
3.4.2 The effect of real noise Signals 61
Chapter 4. A technique for calibrating and evaluating phase shifter 68
4.1 Introduction ............................................................................................................. 68
4.2 Scanning method for PZT’S calibration .................................................................. 69
4.3 Calibrations of two phase shifters ........................................................................... 76
4.4 Advantages of the new technique ........................................................................... 85
Chapter 5. Simplified DSPI setups 86
5.1 A simple in-plane DSPI set up ................................................................................ 86
5.1.1 Introduction 86
5.1.2 Setup and Tests 88

5.2 A simple out-of-plane DSPI setup .......................................................................... 94
5.3 Conclusions ................................................................................................. 107
Chapter 6. Summary and discussions 108
Chapter 7. Conclusions and recommendations 112

 

vii

REFERENCES 114

 

viii

LIST OF TABLES

. d1
Table 3.1 Quadrant determination of (15 by Signs of Cos¢ and fig. ............................... 42

Table 3.2 Comparison of the in-plane displacements calculated by the new algorithm and

induced displacements. ...................................................................................................... 51
Table 3.3 Some examples of displacement measurements ............................................... 67
Table 5.1 The comparison between induced and measured results .................................. 93
Table 5.2 The comparison between induced and measured results ................................ 107

ix

LIST OF FIGURES

Figure 2.1. Laser speckle. ................................................................................................. 13
Figure 2.2. A simple in-plane speckle interferometer ....................................................... 14
Figure 2.3. A fringe pattern of speckle pattern. ................................................................ 17
Figure 2.4. Phase change map with errors. ....................................................................... 26
Figure 2.5. The phase map after smoothing of Figure 2.4 map. ....................................... 27
Figure 2.6. Unwrapped phase map of Figure 2.5. ............................................................. 29
Figure 2.7. A typical in-plane sensitive DSPI system. ..................................................... 34
Figure 3.1. Object illuminated by two beams. ................................................................. 38
Figure 3.2. An example of intensity wave obtained by phase shifting. .......................... 44
Figure 3.3. Waveform of Fig. 3.2 after curve fitting. ....................................................... 44
Figure 3.4. Load condition of CT specimen. .................................................................... 47
Figure 3.5. Raw phase change map. ................................................................................. 48
Figure 3.6. Phase change map after smoothing. ............................................................... 48
Figure 3.7. Crack tip position from the new IMMS algorithm. ........................................ 49
Figure 3.8. Crack tip position from the Carré method. ..................................................... 49
Figure 3.9. Calibration plate. ............................................................................................ 50
Figure 3.10. Correlation fringe map obtained by the new algorithm ................................ 51
Figure 3.11. A recorded good intensity changing curve. .................................................. 54
Figure 3.12. Gaussian white noise signals. ....................................................................... 54
Figure 3.13. The intensity waveform with Gaussian noise added on. .............................. 55

Figure 3.14. The fitting curve of the original recorded signals. ....................................... 56

Figure 3.15. The fitting curve of noise-added intensity signals ........................................ 56
Figure 3.16. Gaussian noise with a 6.5 deviation. ............................................................ 57
Figure 3.17. Intensity waveform with the new noise added on. ....................................... 58
Figure 3.18. The fitting curve of an intensity wave with a 6.5 deviation noise added. 58

Figure 3.20. The fitting curve of the intensity wave with a 6.5 deviation high frequency

noise added ................................................................................................................ 60
Figure 3.22. Intensity change curve with noise added. ..................................................... 62
Figure 3.21. Noise signals under A condition ................................................................... 62
Figure 3.23. The fitting curve obtained by IMMS. ........................................................... 63
Figure 3.24. Noise signal. ................................................................................................. 64
Figure 3.25. Intensity curve with noise added. ................................................................. 65
Figure 3.26. The IMMS result. ......................................................................................... 65
Figure 3.27. Noise sample obtained under case C. ........................................................... 66
Figure 4.1. Phase step with 21: ambiguity. ........................................................................ 70
Figure 4.2. Phase step V.S. driving voltage. .................................................................... 70
Figure 4.3. Two lines are drawn on the image. ................................................................. 71
Figure 4.4. A 3-D plot of the obtained phase shifting results. .......................................... 73
Figure 4.5. Tilt and non- uniformity in x-direction. ......................................................... 74
Figure 4.6. Tilt and non- uniformity in y-direction. ......................................................... 75
Figure 4.7. Recorded intensity waveform. ........................................................................ 77
Figure 4.8. Fitting curve obtained by IMMS. ................................................................... 78
Figure 4.9.The phase shifter .............................................................................................. 79

xi

Figure 4.10. Intensity curve at one pixel point obtained by phase Shifting. ..................... 80

Figure 4.11. The fitting intensity curve of the Figure 4.11. .............................................. 80
Figure 4.12. Phase step versus driving voltages. .............................................................. 81
Figure 4.13. 3-D graph of the phase shifting along X and Y axis .................................... 82
Figure 4.14. Tilt and non-uniformity in x-direction. ........................................................ 83
Figure 4.15. Tilt and non-uniformity in y-direction. ........................................................ 84

Figure 5.1. Two basic practical illumination apparatus of in-plane speckle interferometry.

................................................................................................................................... 88
Figure 5.2. The arrangement used in this work. ............................................................... 88
Figure 5.3. The experimental setup and the PZT mirror ................................................... 89
Figure 5.4. The tilt and non-linearity of the PZT mirror in the horizontal direction. ...... 90
Figure 5.5. The tilt and non-linearity of the PZT mirror in the vertical direction. .......... 91
Figure 5.6. The comparison between two arrangements of DSPI. ................................... 92
Figure 5.7. Schematic of the simple out-of-plane sensitive DSPI system. ....................... 95
Figure 5.8. The Out-of-plane DSPI setup. ........................................................................ 96
Figure 5.9. The phase shifiing plate. ................................................................................. 96
Figure 5.10. A Speckle picture with the PZT driving setup on the right ........................... 98
Figure 5.1 1. The linearity of the phase shifting. ............................................................... 99
Figure 5.12. The 3-D graph of phase Shift. ..................................................................... 100

Figure 5.13. The tilt and non-uniformity of the phase shifting along the x-direction. 101
Figure 5.14. The tilt and non-uniformity of the phase shifting along the y-direction. 102
Figure 5.15. A raw phase difference map. ...................................................................... 103

Figure 5.16. The phase difference map after filtering. ................................................... 104

xii

Figure 5.17. The measured displacement field in gray scale. ......................................... 105

Figure 5.18. The 3-D map of out-of-plane rigid body rotation ....................................... 106

xiii

Chapter 1. Introduction

1.1 Problem outline

Speckle pattern interferometry (SPI) is an accurate, non-contact, full-field technique
for displacement field measurement on naturally rough surfaces. The employment of
phase shifting techniques enhances the precision and the convenience of the SP1
technique for quantitative measurement. With the SP1 technique, there is no requirement
for surface preparations. In addition, the application of electronic detectors and digital
signal processing techniques in computers make SPI a real-time measurement technique.

The technique is alternatively termed as digital Speckle pattern interferometry (DSPI).

The advantages of the DSPI technique are very attractive for many researchers and
industries. Unfortunately, the intrinsic sensitivity of the DSPI technique also makes it
vulnerable to environmental disturbances such as vibration, beam source fluctuation, air
currents, and thermal convection. In addition, the motion of the piezoelectric transducer
commonly used to perform phase shifting in DSPI apparatus is not exact, and the CC D
detector is not truly linear either. These effects introduce errors in the calculation of
measurement results of the DSPI (Phillion 1997). These are the main reasons why the

DSPI technique has not found its way to wide factory applications.

To make the DSPI technique more applicable to practical industrial problems, in the
past years, a number of researchers have been working on improving the environmental

tolerance of the DSPI technique. These research works can be classified into two broad

categories: (1) improvement of the signal-processing algorithm, and, (2) improvement of

hardware arrangements for the setup.

A small amount of vibration can introduce large phase errors and these errors are
difficult to eliminate from the phase data; but they can be minimized with proper
algorithm design. Many of the past works focused on improving signal processing
algorithms, including developing new phase evaluation methods, performing new phase

unwrapping algorithms, and developing new filtering techniques.

. In some situations, where the amount of vibrations is so great that the measurement
is not possible with standard phase shifting interferometers, another solution is necessary.
A range of partial solutions to the noise sensitivity problem has been developed. These
include the use of pulsed lasers to freeze the specimen motion, common-path
configurations such as shearing interferometers, active phase-stabilization systems, and

dynamic speckle interferometers based on high-speed video cameras. (Ruiz, et al. 2001)

However, in measurement systems that result in nonsinusoidal periodic waveforms
(either by choice or by imperfections such as phase-shift errors, multiple interference
beams, non-linearity in detector), the performance of existing algorithms is often
inadequate (Larkin 1992). It is evident from a survey of literature that much of the effort
in the development of phase shifting techniques has been Spent on addressing errors
introduced by the phase shifting itself. Few reports about improvement of the hardware

setup can be found.

l.2 Objectives

In order to make the DSPI setup more robust and cost effective, so that this
technique can be practically used for measurement in realistic industry environments, the
general obj ective of this work is to improve the environmental immunity of the DSPI

technique and simplify the DSPI apparatus.

Development of a new signal-processing algorithm for phase determination that has
an improved noise tolerance is the first choice. It is known that the temporal phase
Shifting technique requires the recording of at least three phase-shifted interferograms,
which must be taken sequentially. The sequential recording of inerferograms can lead to
disturbances by thermal and mechanical fluctuations during the required recording steps.
In addition, fast object deformations cannot be detected (Bothe, Burke, and Helmers,
1997). However, the time-domain phase shifting techniques offer several advantages,
including improved noise immunity, insensitivity to spatial variations in the detector
response, high-Spatial-frequency resolution, and ease of implementation, over spatial-

domain methods of interferogram analysis

In the first part of this research, some digital signal-processing techniques are
employed to process the time-domain signals of phase Shifting interferometers to improve
the noise tolerance of the DSPI technique. Based upon the new algorithm, a Simple in-
plane sensitive DSPI setup and a new phase shifting out-of-plane sensitive DSPI are

implemented to measure displacement fields quantitatively.

By application of the new algorithm, a new calibration method for PZT phase

shifters is developed in this work. The new calibration technique offers a convenient on-

line method to evaluate several characteristics of the phase Shifter simultaneously,

including the non-linearity, tilting, and non-uniformity of the phase shifter motion.

1.3 Relevant literature review

Much development has taken place in digital speckle pattern interferometry
techniques during the recent past. Some literature reviews that are relevant to the current

research work follow:
A. The development of phase extraction techniques.

Phase shifting techniques are commonly used in interferometers for phase
determination. The main idea is to induce some known or unknown phase changes
artificially in the interference system to obtain more information so as to determine the
phase information that is needed. The phase shifting techniques for speckle pattern
interferometry include the three-steps, Carre' method, four-steps, seven-steps, Fourier-

transform, and sinusoidal curve fitting methods.

As summarized by Cloud (1995), the three-step phase-shifting technique, the four-
step phase shifting technique, and the Carre' technique are some basic/common phase
shifting techniques. The three-step and four-step techniques use a known phase Shift
obtained by calibrating the phase shifter. In 1966, The Carré technique was devised. This
technique uses a constant phase Shift step; but it is independent of the amount of the
phase shift, which means that calibration of the phase shifter can be omitted. It is
believed that the Carré technique has a better error immunity than the other two

techniques (Cheng and Wyant 1985).

A single phase-step algorithm was reported by Sesselmann, and Goncalves (1998).
By combining intensity equations obtained before and after displacement, only two
phase-stepped interferograms with a known phase step for each displacement state are
enough to retrieve the phase angle induced by the displacement. However, because this
method relies on the accuracy of the phase step, it is very susceptible to noise such as

vibrations and air currents that result in serious phase step errors.

Surrel (1996) reported a technique to associate a characteristic polynomial with any
phase-shifting algorithm based upon an assumption that the fringe pattern has a sine
profile and constant phase step. He demonstrated that a 2j+2 phase step was necessary to
obtain insensitivity to the jth harmonic content in the presence of a constant phase-

shifting miscalibration. Six-step and ten-step algorithms were derived in his paper.

Several seven-step techniques were reported by Zhang (1998), Larkin and Oreb
(1992), De Groot (1995), Hibino, Oreb, F arrant, and Larkin (1995). A Fourier description
method was used to expand the intensity equation of the speckle pattern interference.
Seven phase steps were used to acquire speckle pattern signals to compute the Fourier
coefficients. The differences between these seven step techniques are that different step
values were used, and different data processing equations were employed. They were all

claimed to be insensitive to the second-harmonic component to some degree.

Farrel and Player (1992) developed a variable step algorithm for phase
determination. The Lissajous figure technique and ellipse fitting were employed to detect
and quantify phase change between frames with accuracy around 4 degrees. Furthermore,
two algorithms for N different steps were presented. Numerical Simulations Showed the

algorithms had poor performance under some specific phase step values. In 1994, an

inter-pixel algorithm and a dual-step algorithm based on the same Lissajous figure and
ellipse fitting techniques were reported. The authors pointed out that the computational

effort required in these algorithms is much greater than for fixed step algorithms.

An algorithm that iS immune to tilt phase-Shifting error for phase-Shifting
interferometers was presented by Chen, Guo, and Wei (2000). It is assumed that the tilt
errors are linearly distributed throughout each speckle pattern interferogram so that all
pixels remained on the same phase plane after tilt occurs. The first-order Taylor series
expansion of the phase shifting error equations was used to determine the actual phase
shift plane. Numerical simulations showed that this algorithm could compensate for
phase-measurement errors caused by both translation and tilt-shift errors, based upon the

assumption that all shifted phases are kept on the same phase-shift plane.

A Fourier-transform method for phase determination was presented by Goldberg
and Bokor (2001). Based upon the assumption that the phase increment was uniform
throughout all the domain points, a F ourier-transform technique was performed to
determine the N global phase positions introduced during the phase shifting steps. Then,
the phase positions were substituted into the least-square method as input information to
compute the phase angles. The Fourier-transform method relies highly on the fact that the
phase Shifting process does not change the spatially varying components of the optical
path difference, which means the spatial carrier frequency must be high enough to
adequately separate the first order signal from the zeroth order components in the Fourier
domain. If a large portion of the interferometric data is available, this technique can be

very robust in the presence of noise.

The Max-Min scanning method was reported by Vikhagen (1990), and applied by
Wang, Grant, (1995), and Chen, Gramaglia, and Yeazell, (2000). In the Max-Min
method, the basic idea is that a set of recorded intensity signals is sorted to find the
maximum and minimum intensity values at each pixel, and the phase angle is calculated

from the values by the following equation.

 

 

 

(I _ [Max +1Min}
2
¢ = ArcCos
IMax _ [Min
\ 2 /

 

 

After the above computations, a small constant step phase shift is performed to
obtain additional sign images to determine the Sign of the calculated phase value both

before and after displacement occurrence.

A sinusoid least square fitting method was reported by Macy and Bokor (1983), and
by Ransom and Kokal (1986). In the sinusoid fitting method, an assortment of intensity
data was recorded during the phase shifting procedure. The intensity wave was assumed
to be of sinusoidal waveform. The sinusoidal fitting model, I(v)=a(v)+b(v)Cos(cp(v)), was
used to fit the recorded signals. After solving a series of equations, fitting parameters,
a(v), b(v) and (p(v) were determined. Okada, Dato, and Tsujiuchi (1991) modified the
least-square method. Both phase shift and phase angle were counted as unknowns
Simultaneously. Two steps of linear least—square calculations were performed, one to
obtain the phase distribution and the other one to obtain the phase shift. The steps were

repeated until the results converge.

An active phase-shifting method was reported by Yamaguchi, Liu, Kato (1996). A
spatial filtering detector was placed in the setup to detect the fringe movements caused by
external perturbations. The detected information was fed back to the PZT phase shifter to

stabilize and adjust the phase step.

The spatial phase shifting method (SPSM) has been designed to avoid problems
such as fluctuation of beam light and non-linearity of phase shifting motion that arise due
to the acquisition of different patterns at different time as is the norm in the above
techniques. By combining rotational polarizing components or diffraction optical

elements, a set of phase-shifted interferograms can be acquired simultaneously.

Koliopoulos (1996) presented a simultaneous phase shift interferometer(SPSI). The
setup of the SPSI differs from the traditional method of obtaining phase shifted
interferograms, which use PZT mirrors to physically create computer-controlled phase
shifts over a short time period. The SPSI creates four phase-shifted interferograms using
polarization optics to simultaneously produce four phase-Shifted interferograms. Four
CCD cameras are placed in the setup to acquire four interferograms simultaneously.
These four cameras are co-aligned on a sub-pixel basis so that any point on the object
being tested is seen at the equivalent pixel in each of the four cameras. This phase
algorithm is relatively restricted. Additional errors appear because the patterns are
obtained by different cameras or from different parts of the same camera, which
introduce variations of intensity. Moreover, precise alignment of the optical elements
with sub pixel accuracy is required. (Dorrio, Fernandez, 1999, Bothe, Burke, and

Helmers 1997).

Some methods of freezing the fringe map have been reported also, such as using a
pulse laser or a high speed camera (Melozzi, Pezzati, and Mazzoni 1995). They are not

discussed here.

If there are no means to perform phase Shifting, the Fourier transform technique, the
convolution algorithm, the sinusoidal fitting method, and the spatial synchronous
detection methods can be some alternatives to evaluate the phase information. It is often
believed that the phase shifting methods are more precise than these methods that rely on
a single pattern image. These techniques will not be discussed in detail. Corresponding

references can be found in Massing and Heppner (2001).

B. Calibration techniques.

The accuracy of most phase shifting techniques relies on the accuracy of the phase
step. Even those methods claiming to be insensitive to miscalibration of the phase shifter
will benefit from knowing precisely the behavior of the phase Shifter. So, the accurate

calibration of the phase Shifter is still highly desired in many interferometry techniques.

Few methods have been reported for the calibration of phase shifters used in

interferometry techniques.

The simplest one is the phase-lock method as mentioned by Cheng and Wyant

(1985). Once the two images of fringes, before and after phase stepping, are observed to
be identical and all fringes are shifted to the position of their neighbor fringes, the 21:-

phase step is determined. This phase-lock method has the advantage that it is not

sensitive to nonlinearity of the phase shifter. However, this phase-lock method can only

be used to calibrate the 21t-phase step. Once the Zn phase step has been determined, the
intermediate phase steps can be computed by a simple interpolation. The accuracy of the

interpolation results strictly depends on the linearity of the phase step schemes.

The Carre’ phase shifting method was used by Cheng and Wyant (1985) to calibrate
the phase shifter. Four phase-Shifted interferograms are used in the Carré phase Shifting
algorithm to determine the phase step. The Carre’ algorithm can also be used to obtain
information on all four unknowns in the intensity equation of speckle interference. This

calibration method relies on the assumption of good linear movement of the phase shifter.

A two-image method was reported by Brug (1999). Two intensity images were
obtained by a phase step. The correlations between these two phase-stepped intensity
images were computed, and then the phase step was retrieved from the correlation result.
With this method, the phase changes must be ensured to be linear in the area where the

method is applied.

All these methods have some obvious drawbacks and limitations. In addition, none
of them can check the detailed behavior of the shifter, i.e., the linearity, the tilt, and the
non-uniformity of the phase shifter. A convenient real-time calibration method is desired

to thoroughly characterize phase shifters.

C. Simple ESPI setups.

Relatively little research have been published about simplifying the system of
apparatus of DSPI. A compact DSPI sensor was introduced by Siebert, Wegner and

Ettemeyer (2001), but the construction of that DSPI sensor is still a trade secret.

10

Several variants of a simple out-of-plane sensitive DSPI setup were reported and
patented by Cloud (2000, 2003). To make these Simple setups work quantitatively, a
method to perform phase shifting is desired. The newly Improved Max-Min Scanning
(IMMS) phase shifiing algorithm has a good immunity to tilt, non-linearity and non-
uniforrnity of phase shifting as well as vibration noise, as reported by Ding (2002) and by
Ding and Cloud (2004). This approach is applicable to other types of phase-shifting

interferometry, but it is used here to greatly expand the capability of DSPI.

11

Chapter 2. Summary of Speckle technique

2.1 Introduction

This chapter briefly introduces the digital speckle pattern interferometry technique
(DSPI), including the basic theory of speckle interferometry, phase shifting techniques,
image processing techniques, and typical DSPI setups. The main sources of the

information are Cloud (1995), Hong (1997), and Lanza di Scalea (1996).

2.2 Concept of laser speckles

Illumination of a rough surface (roughness of the order of the wavelength of the
illumination light) by a coherent light creates a grainy intensity distribution in space
termed a speckle pattern as shown in Figure 2.1. For some years, it was considered a
special kind of noise, and it limited the usefulness of lasers. However, with understanding
that the speckle is a result of the coherent addition of waves scattered from a rough
surface, its information-carrying property was discovered. This discovery led to a new
class of optical measurement techniques known as speckle methods.

Laser speckle is categorized into two types: objective speckle and subjective
speckle. When an optically rough surface whose roughness is of the order of the
wavelength of laser beam is illuminated by laser light, the intensity of the scattered light
varies randomly with position. This effect is known as objective speckle. When a lens is
used to form an image of the illuminated rough surface, the image shows a similar

random intensity variation that is referred to as subjective speckle. The size of the

12

speckles is dependent on the aperture, focal length of the lens, and the wavelength of the
illuminating light. The speckle size on the object can be written as:

_1.22x(1+M)/1F

S subject - M (2_1)

 

Where Ssubject is the size of the subjective speckles at the object plane;
M is the magnification of the lens;
it is the wave length of the illumination beam;
F is the aperture ratio (focal length/aperture).

The real speckle size is usually in the range of 5 to 50 pm.

The speckle size at the image is computed by

S = 1.22 x (1+ M )4F (22)

subject

 

Figure 2.1. Laser speckle.

l3

From this equation, it is known that the resolution of the detector must be greater than the
speckle size to fully track the behavior of the speckle. The maximum resolution required

of the setup is determined by equation 2.2.

2.3 Speckle pattern interferometry

When a laser speckle pattern is mixed with a second coherent beam, the intensity
image of the interference, which has a random Speckle appearance, is known as speckle
interference. In the following parts, an in-plane sensitive interferometer, as Shown in

Figure 2.2, will be used to explain the fringe formation of Speckle interference.

Illumination Beam IA

imaging

syste j—

fll ,
lll ’

illumination Beam l3 imaging
plane

Q

 

object

 

 

 

 

.__l

//

Figure 2.2. A simple in—plane speckle interferometer.

The intensity at any point of the speckle pattern can be written as:

I=IA+IB+ZMC0S¢ (2.3)

where I is the resultant intensity at the given point;

1,, is the intensity of beam 1;

13 is the intensity of beam 2;

¢ is the phase difference between beam 1 and beam 2.
AS the speckle moves along the x-axis together with the object surface, the path length
difference between the two illumination beams changes. Therefore, the relative phase
difference (15 between the two illumination beams is varied, and consequently the

resultant intensity of the speckle varies. The speckle variation contains the information
about the local displacement of the object surface.

Once a displacement occurs, the speckle intensity changes to:

1' = I, + I, + 241,1, Cos(¢ + M) (2...

where A (1) is the phase change induced by the movement of the speckle.

Let dx represent the surface displacement in the x direction shown in Figure 2.2, the

phase change is given as (Cloud, 1995):

A¢= —£xd sin6

(2.5)

where (9 is the incidence angle of the illumination beams; xi is the wave length of the

light.

15

In equation 2.4, when A ¢ =2nrt, (n=0,1, 2, 3...), maximum correlation happens between
I and l'. The correlation becomes zero when A¢ =(2n+1)1t, (n=0,1, 2, 3...). On the whole

field of the image, the speckle pattern is random, but the phase change (A¢ ) will be
smoothly varying.

Considering equation 2.5, the maximum correlation is along the lines where

_ 112.
x 23in6? (2:6)

 

and zero correlation is along the lines where

_ (n + 1)].
2 Sin 6 (2'7)

Variations of the speckle pattern correlation between I and I' over the entire image appear

 

as a fringe pattern. If I' is subtracted from I, which is commonly done, dark fringes occur

mi.

along the lines where x Z 2 sin 6 . Figure 2.3 presents an example of a fringe

pattern from Speckle pattern correlation interferometry.

Equation 2.5 indicates that the local displacement information of the object is
contained in the phase change indicated by the interference at each speckle pattern.
Therefore, speckle pattern interferometry can be used to measure displacement point by
point, meaning that the formation of the fringe pattern of the speckle correlation is not

necessary to deduce the displacement information of the object.

16

:-‘IL_II (1'3

\'

._-"|]§>$I,.)§qi.1TE-.

 

Figure 2.3. A fringe pattern of speckle pattern.

2.4 DSPI technique

Equation 2.1 shows that the size of speckles can be controlled by the F number of
image lens. The minimum speckle size is generally between 5 and 100m (Cloud 1996,
Hong, S. S. 1997). It was soon realized that speckle pattern interferometry could be
performed by electronic means such as video camera, electronic signal processing, and
computer techniques, thereby increasing its accuracy and speed of measurement. This
technique came to be known as electronic speckle pattern interferometry (ESPI). The
major advantage of ESPI is that it enables real-time measurement of a surface
displacement field without photographic processing. This advantage makes ESPI a good

technique to form correlation fringe maps and perform non-destructive evaluation

qualitatively. However, the requirements to locate the center of fringes and number the

fringe order make it unsatisfactory for quantitative measurement.

When a phase-shifting technique is introduced into speckle pattern interferometry,
the phase and displacement information can be retrieved point by point without forming
an interference fringe pattern. The application of phase shifting techniques makes ESPI a

convenient quantitative technique.

With the advantages of digital techniques, various data processing steps can be done
through digital methods inside a computer rather than in analog devices. When the
computer takes over the role of processing all signals in the digital domain, The ESPI

technique is alternatively termed digital speckle pattern interferometry (DSPI).

2.5 Phase Shifting technique
If the relative phase (,6 can be computed both before and after deformation, the
phase change [8475 at the speckle can be calculated by the subtraction between mm, and

(balm; firrthermore, the displacement can be determined at this point quantitatively. That

is the objective of the phase Shifting technique: to quantitatively determine phase value
¢.
There are three unknowns in the intensity equation of speckle:

I = I A + [B + 2./I .41 B COS¢ , namely the beam one intensity IA, beam two intensity [3,

and the relative phase ¢ between these two illumination beams. Obviously, a minimum
of three equations is needed to determine the unknowns. Well-known phase shifting

techniques can be used to quantitatively determine the phase value 45. Several phase

18

shifting algorithms have been developed by researchers, including the three-Step
technique, Carré technique, max-min scanning technique, etc. The main idea is to
introduce some known or unknown phase changes artificially to get several additional
intensity equations in order to derive the ¢ parameter. The equations may be generally

expressed as

In =IA +IB +2./IAIBCOS(¢+an)

where IA is the intensity of beam one;

13 is the intensity of beam two;
(15 is the initial relative phase between the beams one and two;
a" is the introduced phase shift;

n is the index of the phase shift step which is determined by the phase
shifting algorithm.

The final computation algorithm of phase value depends on the specific phase shifting

parameters including the size and number of steps.

Commonly, the phase shifting technique uses some device, usually a mirror
mounted on a piezoelectric transducer (PZT), to change the path length of one beam in an
interferometer so as to induce some known phase changes in the interference. The
corresponding intensities are recorded to derive the phase information point by point.
Several different phase shifting algorithms were developed in the past years for DSPI.
Dorrio, and Fernandez (1999) and Surrel (2000) categorized and reviewed these

techniques in detail. Some algorithms are briefly introduced in the following sections.

19

2.5.1 Carré technique

In 1966, Carre' developed a phase evaluation algorithm. Constant phase shift steps

an=-%a, —%a,%a,%a are introduced into equation 2.3 to obtain four speckle
patterns:
3
11ZIA+IB+2‘/IAIBC0S(¢—‘2—a) (23)
1
I2 =IA +13 +2./IAIBCOS(¢—Ea) (29)
{—— 1
I3:IA+IB+2 IAIBCOS(¢+§a) (210)
[—— 3
I4=IA+IB+2 IAIBC0S(¢+§'a) (2.11)

For this case, the phase can be computed as

 

J[3(12 -13)-(11-14)l[(12 -13)+(1.-14)]
(12 +13)—(11+I4)

 

¢ = arctan (2.12)

when a rs near 5 , the rntensrty modulatron, Wthh rs defined as the vrsrbrlrty of

speckles, is

 

 

_ 1 l/[([2_I3)+(Il_I4)]2+[(12+I3)_(11+14)l2
7-21 2 (2.13)

where 10 =IA+IB.

20

With this Carré technique, it is not necessary to calibrate the phase shifter to perform the
above computations to determine phase value. However, since a constant phase step is

assumed, the non-linearity of the phase shifter will affect the accuracy. Secondly, the a

7r . . . . . .
should be near 2 1n order to use the Intensrty modulat1on equatron 2.13 to determme

invalid pixels. Therefore, understanding of the properties of the phase shifter is still

necessary to apply the Carré technique in reality.

2.5.2 Three-step technique

In equation 2.3, there are three unknowns IA , 13 , and g). Therefore, a minimum of

three equations is needed to compute the unknowns. An example of three-step algorithm

. 7r .
usrng 3 step rs,

1
I]:IA+IB+2‘VAIAIBC0S(¢+-4—lfl) (2.14)

3
12:IA+IB+21/IAIBC0S(¢+Zfl-) (2.15)

5
I3:IA+IB+2‘\lIAIBC0S(¢+Z-7r) (2.16)

Solving these three equations gives,

I —I
¢ = arctan( 4:71) (2.17)
l 2

The intensity modulation is

21

 

 

7 = \/(I1—12)2 +(Iz —I3)2
210

where 10 =1, +13.
2.5.3 Four-step technique

A commonly used method is the four-step phase Shifting technique. Equal % phase

shifting steps are used in this algorithm.

II = 1, + I, + ZWCOS (¢) (2.1s)
12=IA+IB+2\/—I:I:C0s(¢+%7r) (2.19)
13=IA+IB+ZMC0S(¢+7I) (2.20)
I4=II4+IB+ZMC0S(¢+-:—fl') (2.21)

The phase value is calculated by

I —1
¢ -- arctan( ——4 2 )
Il — 13 (2'22)

The intensity modulation is

 

 

_,/(1,—1,)2+(1,_1,)2
7 _ 210 (2.23)

22

2.5.4 Seven-step technique

Several seven-step techniques were deve10ped in the past decade (Zhang 1998,
Larkin and Oreb 1992, De Groot 1995, Hibino, Oreb, Farrant, and Larkin 1995, Surrel
1996). The authors claimed these seven-step phase shifting techniques to be more

insensitive to linear phase-shift miscalibration. One example among them uses
a = (i — 1); (i=1,...,7).

The phase is computed by

—I1 —412 +13 +814 +15 —416 —I7
1,—212 —713 +715 +216 —17

 

¢ = arctaIll l (2.24)

More detail about its derivation can be found in the references.

2.5.5 Max-min scanning algorithm

The Max-Min scanning method, was reported by Vikhagen (1990), Chen,
Gramaglia, and Yeazell (2000), and Wang, Grant (1995). In this method, a set of
recorded intensity signals is sorted to find the maximum and minimum intensity values at
each pixel, and the phase angle is calculated from the obtained maximum and minimum

intensities by the following equation.

 

 

 

 

 

(I'— IMax +IMin\
2
¢ = ArcCos
I Max — I Min (2.25)
K 2 1

where I is the original intensity at one pixel, IMax and 1M,n are the maximum and

minimum values of the intensity at the same pixel position as I.

23

Additional Sign images are recorded by a small phase step to determine the sign of
the calculated phase value. However, because of the discontinuity of the signal recording,
this algorithm may miss the real maximum and minimum intensity values, causing .
uncertainties in the calculated results. Furthermore, to obtain the sign images, additional

phase steps are needed, which is time consuming and requires accurate small phase steps.

2.5.6 Sinusoidal fitting technique

A sinusoid least square fitting method was reported by Ransom and Kokal (1986),
Macy, and Bokor (1983). In the sinusoid fitting method, a number of intensity data are

recorded during the phase shifting procedure. The intensity wave is assumed to be a

sinusoidal waveform. The fitting model [(1’) = a(V) 'l' b(v) COS¢(V) is used to fit the
recorded signals. After solving a series of equations, parameters, a(v), b(v) and (b(v) are

determined, where ¢(v) is the phase value. This method was generally used to process 2-

D fiinge maps, which can be classed into spatial techniques for phase determination.

2.5.7 Comparison of phase shifting techniques

The Carré, three-step, and four-step techniques are simpler to implement than the
others. But non-linearity and miscalibration of the phase shifter movement will cause
errors in the computed results. Compared with the others, the seven-step, Max-min
scanning, and Sinusoidal techniques have been claimed to present better tolerance to the
above errors. However, these algorithms require more complex mathematics work and
computing time. There are some other techniques to retrieve phase information. The

author could not list them all.

24

2.6 Image processing

Once the phase maps are determined before deformation and after deformation, the
phase maps contain some noise. Therefore, some digital processing techniques are

needed to remove errors and smooth the maps.

2.6.1 Phase change map

Equation 2.5 yields

2A¢

dx = .
47: sm 6

 

(2.26)

where A¢ is the phase change caused by the displacement of the object surface.

Therefore, once the A¢ is determined, the corresponding in-plane displacement can be
computed.
In section 2.5, it is shown that phase values can be determined point by point

through phase shifting techniques. After performing the phase shifting technique, both

before deformation and after deformation, phase values are determined as ¢before and

¢after -

Phase change is

A¢ : ¢after — ¢before (2-27)

25

Once the above procedure is employed pixel by pixel over the entire region of
interest on the image, a phase change map is obtained. Figure 2.4 presents an example of
a phase change map corresponding to some in-plane displacement of a rotated rigid body

surface. Pepper-salt noise can be seen on this picture.

 

Figure 2.4. Phase change map with errors.

2.6.2 Image filtering

When the phase shifting technique is performed in electronic speckle pattern
interferometry, there are two main sources for noisy data points: slight decorrelation of
speckles after deformation, and low intensity modulation as the phase is shifted. During

signal processing, noisy points are generally marked as invalid pixels. To remove most of

26

the invalid pixels, the modulation of intensity at each pixel is checked to determine
whether the point is good. If the calculated modulation is smaller than a given threshold
during the phase shifting, the pixel is marked as an invalid pixel. The threshold value can
be determined by trying different numbers until the best phase map is obtained. This type
of work is very experience- related. These invalid pixels usually show up as pepper-salt
noise in a phase change map, as seen in Figure 2.4. It is well known that a median
window can effectively fill in these pepper-salt points (Hong 1997). To smooth pixel
points where slight decorrelation occurs, a smoothing window can be employed which
uses neighbor pixels to correct the center one in the window. Hence, a good phase change

map can be generated, as seen in Figure 2.5.

 

Figure 2.5. The phase map after smoothing of Figure 2.4 map.

27

2.6.3 Phase unwrapping

With the phase computation from the phase shifting technique, the phase can be
determined to modulo 21:. After subtraction between the phase maps determined before
and after deformation to obtain the A¢ , it is obvious that the phase change map contains
21: ambiguities. The 21: ambiguities present 21: edges in a phase change map, which can

be seen in Figure 2.5.

In order to determine the real displacement of each pixel from the phase change
map, the 21: ambiguities must first be removed from the phase change map. Various
phase unwrapping techniques have been developed to remove the 21: jumps that occur in
the computed phase change map, so that a continuous phase change map can be obtained.
These methods can be classified as: (1) Local phase unwrapping techniques, and (2)

global phase unwrapping techniques.
The following is a brief summarization of some possible options that can be used.

The Simplest phase unwrapping method is just an application of the classical one-
dimensional phase unwrapping idea. A reference point is chosen to begin the unwrapping
procedure. A horizontal line (or a vertical line) of pixels are scanned. Whenever a change
more than a threshold 1: occurs between adjacent pixels, a 21: offset is added to (or
subtracted from) the second pixel until there is no more than a 1: difference between all
adjacent pixels along the line. Using the unwrapped line as the reference, the one-
dimensional unwrapping procedure is then performed along all the vertical lines

(horizontal lines). Finally, the phase ambiguities are removed from the entire image.

28

Figure 2.6 shows an example of an unwrapped phase map, which was obtained by

performing this method on Figure 2.5.

 

Figure 2.6. Unwrapped phase map of Figure 2.5.

When a noise-free phase map is available, this method is very ideal to perform
phase unwrapping. However, noise pixels inevitably exist on phase map. Any noise in the
image will generate an error that propagates along the unwrapping direction. Hence, a
localized noise can affect a whole line of pixels, which typically turns out to cause a path

dependent, global error.

A local phase unwrapping method was developed by Hong (1997). In this method,
an n><n window is used to pick regions on which to perform phase unwrapping. In this
window, the numbers of pixels that are above a given upper-threshold and those that are

below a lower-threshold are counted separately. A 2m: offset is added to those low-

29

threshold pixels to remove the edge when the upper-threshold number is greater than the
low-threshold number. On the other hand, a 2m: offset is subtracted from those upper-
threshold pixels when the opposite condition is true. Also, the number of noise pixels is
counted. If this number exceeds half of the entire pixels over a window that is n>< n, the
pixel elements remain invalid. The advantage of this algorithm is that it is a path-
independent local technique. So, the local noise does not expand to other regions. It must
be pointed out that this local phase unwrapping method can be used to compute the strain

field from phase change maps directly.

Based on the well known least-square method, an efficient phase unwrapping
algorithm, the minimum Lp-norm algorithm, was reported by Ghiglia and Romero (1996).
This minimum Lp-norm algorithm determines the unwrapped phase field by solving the
discrete function of the wrapped and unwrapped phase fields. For more detailed
information, refer to Ghiglia and Romero (1996). Furthermore, as reported by Kaufmann
et al (1998), some numerical simulations showed that this algorithm can handle high
densities of inconsistent pixels, objects with edges and holes when the order p=0. With
this algorithm, some complex equations need to be solved and several weighting

parameters need to be determined, which results in this being a time-consuming method.

Strand and Taxt (1999) evaluated the performance of several two-dimensional
phase unwrapping algorithms with respect to correct unwrapped phase results and
execution time. Comparisons of experimental results indicated that, in diverse noise
conditions, there is always a trade-off between the efficiency and the amount of time

needed to choose the most appropriate method.

30

In the past several years, new phase unwrapping algorithms have appeared at a

rapid rate (e.g. He 2002, Herraez 2002, Huang, and Lai 2002, Gens 2003).

2.7 Displacement and strain computation

Once the phase change map is computed through the previous techniques, the in-

plane displacement field can be determined by

_ 2A1
x 471 Sin 0

 

(2.28)

where (1,2 is the displacement;

2: is the wave length of illumination;
0 is the angle of illumination;

A¢ is the phase change caused by the displacement of the object
surface

The illumination angle will affect the sensitivity of the measurement. As will be
discussed in the next section.

If only the displacement field is desired, no further processing is necessary.
However, a strain field is often demanded in mechanics analysis. It is obvious that the
strain field can be determined by the first derivative of the displacement map. The least
square fitting method is commonly used to compute the derivatives of the discrete data

SCI.

The idea is to find a polynomial fitting function of the displacement field by the

least square technique and to compute the derivatives of the fitting plane.

31

For an n><n window, let UiJ represent the displacement value in this window
obtained from the previous steps, where i,j=1...n. A linear fitting plane for a 2-D field

can be written as

(1059)”) = ax + by + C (2.29)

where the a, b, and c are three fitting parameters. The least square method
minimizes the difference between the fitting data and the test results by the following

equations.

8 : 22(U(xi,j1yi,j) — unj)2

i=1 j=l

_ n n 2
‘ X Z (“xiv + by :31 + c — ”1.1) (2.30)

i=1 j=1

The partial derivatives with respect to a, b, and c are set to zero to minimize the 8 .

After derivations, the following equations are obtained

n n
2 2 xnjuw.

i=1 j=l

 

 

a =
” ” 2 (2.31)
22...,
i=1 j=1
n n
22%.“...-
_ i=1 j=1
b _ n n
2 (2.32)
yr)-
1=1 j=l

32

 

c = ' 2 (2.33)

From equation 2.29, the a is the spatial derivative of the displacement map in the x

direction, meaning it is the strain in x direction. The b is the strain in the y direction.

2.8 Sensitivity and spatial resolution of DSPI

The sensitivity is a key factor for the choice of an experimental measurement
technique. The sensitivity of the DSPI technique depends on the spatial resolution limit of

the image system and the illumination angle.

The sensitivity of the correlation fringes can be computed by

_ /l
S _ 2sint9 (2'34)

where d3 is the sensitivity of the fringes;

6 is the illumination angle;
2 is the wave length of the beam, which is 633nm for Helium-Neon laser.

For 8-bit, 256 gray levels image processing, the measurement sensitivity of DSPI is

dS/256 in theory (Hong 1997). The bigger the illumination angle, the higher the

sensitivity.

33

The Size of the speckle and the resolution of the imaging system determine the

maximum spatial resolution of DSPI setups (Equation 2.2).

The actual sensitivity of the measurement requires some experimental calibrations.

2.9 Practical DSPI setup

There are two typical applications of DSPI technique: to measure in-plane

displacement field and to measure out-of-plane displacement field.

Figure 2.7 presents a commonly used in-plane-sensitive arrangement of DSPI.

 

 

 

M /é_____| Laser generator

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

BS PZT ' =1:
l
M
MO d MO
_ _ M

M o
L ' Object

Z

X

Figure 2.7. A typical in-plane sensitive DSPI system.

(M=mirror, BS=Beam splitter, MO=microscope object)

34

In figure 2.7, after the Helium-Neon laser generator, a beam splitter is used to
obtain two coherent beams so that they can interfere with each other. A phase shifter,
which is a mirror mounted on a piezoelectric transducer, is placed in one beam path. The
driving voltage is controlled by the computer to vary the path length of the beam to
perform the phase shifting. Spatial filters and microscope objectives are used to filter and
expand the two beams to illuminate the object’s surface. A CCD camera is used to record
the speckle patterns. Recorded images are sent to the computer to perform signal

processing.

If the setup is arranged so that one illuminating beam is directed into the camera
and combined with the other object beam as a reference beam, this in-plane sensitive

DSPI is changed into an out-of-plane sensitive DSPI setup.

Within the computer, proper algorithms, image capture, phase shifting, image
filtering, phase unwrapping, displacement and strain field computation techniques are

implemented to obtain the final quantitative results as needed.

35

Chapter 3. IMMS technique

3.1 Introduction

The phase shifting technique is a powerful tool to determine the phase angle for
interferometric optical measurements such as moire, speckle interferometry and speckle
shearing. As mentioned in Chapter 1, many phase shifting algorithms have been
developed by Creath (1985), Sesselmann et al. (1998), Thesing (1998), Zhang, et a1.
(1998), Koliopoulos (1996), and Hibino (1999), among others. Complete and detailed
classifications and descriptions of phase shifting algorithms were presented by Dorrio
and Fernandez (1999) and Surrel (2000). In Chapter 2, some phase shifting techniques
were briefly introduced. One of them, the Max-Min scanning method, was proposed a
decade ago by Vikhagen (1990). In this method, a set of recorded intensity signals is
sorted to find the maximum and minimum intensity values at each pixel, and the phase
angle is calculated from the values obtained. Additional Sign images are recorded both
before and after deformation by use of tiny constant phase steps to determine the Sign of
the calculated phase value (Vikhagen 1990, Chen, et a1 2000, Wang, et a1 1995).
However, because of the discontinuity of the signal record, the sorting of the recorded
signal sequences may miss the real maximum and minimum intensity values, causing
uncertainties in the calculated results. F urtherrnore, to obtain the Sign images, additional
phase steps are needed both before and after deformation, which requires accurate phase

steps and is time consuming.

36

This chapter develops a new signal-processing algorithm for the Max-Min scanning
method for phase determination. This new method will be named the Improved Max-Min
Scanning phase determination method (IMMS). In this algorithm, the acquisition and
processing of data are improved by employing a curve fitting technique to establish the
waveform of the light intensity obtained from recorded signals. From this curve, the
maximum and minimum intensity values are determined and the phase angle is
computed. The Sign of the phase can also be determined directly from this curve.
Advantages of this approach include improved data processing efficiency and accuracy of
the result. There is no need to acquire additional images to determine the Sign of the
computed phase angle. In addition, calibration of the phase shifter is not required, and, in

certain instances, the phase shifting device can be eliminated entirely from the setup.

Phase shifting is a key step in optical measurement. There is always a big concern
about the environmental tolerance of all kinds of phase determination techniques, because
the noise tolerance determines the requirements for the working conditions of the
techniques. In other words, the noise tolerance establishes limitations for the application
of the phase shifting techniques. The majority of current techniques require a vibration-
insulated table in order to work. In this chapter, employment of low-pass filtering and
curve fitting techniques to the recorded signals in the time domain is shown to improve
the noise tolerance of the developed IMMS method. Numerical simulations and tests are

presented.

37

3.2 The Principle of the IMMS method for phase determination

3.2.1 Mathematical Development

In this chapter, the in-plane-sensitive speckle pattern interferometer is used to
explain the derivation of the new signal-processing algorithm. The final result of this
chapter can be applied to the out-of-plane sensitive interferometer as well as other
interferometers that require a phase shifting technique. To illustrate the theoretical
derivation of the new algorithm, suppose that an object is illuminated by two coherent
laser beams 11 and 12, as shown in Figure 3.1. A speckle pattern is constructed both on the

surface of the object and on the sensor part of the imaging system.

Imaging system

 

 

_J

sensor
|2

Figure 3.1. Object illuminated by two beams.

38

According to the well known optical theory as summarized in section 2.3 of the
previous chapter, at each point of the image area, the intensity of the Speckle interference

can be expressed by the following equation

1 = [1+ I2 + 21/1112 Cos ¢ (3.1)

where I is the resultant intensity;
IA is the intensity of beam 1;
13 is the intensity of beam 2;
¢ is the phase difference between beam 1 and beam 2.
In this equation, there are three unknowns, 11 , 12 and ¢. Among them, the phase (15
is the key parameter that is the goal of the phase determination techniques, because the
change of this phase angle during deformation is the container of the displacement

information at each speckle point.

From equation (3.1), it is known that the intensity I is a cosine function of the

relative phase difference between the two interfering beams, which means that the

intensity I is a cosinusoidal waveform. Furthermore, the I I + I 2 represents the
background of the speckle, and the 2./I,I2 represents the visibility of the Speckle point.

In this equation, when ¢ = 2n1t, n = O, 1, 2, , the resultant intensity I reaches the

maximum value:
[Max:Il+12+2 1112 (3.2)

When (I) = (2n+1)1:, n= 0, l, 2, ..., the intensity 1 reaches the minimum value:

39

[Min:Il+12_2 1112 (3.3)

Adding (3.2) and (3.3) gives

 

[WU/11... =2(11+12) (3.4)
$0
IMax +1Min
11 +12 2 2 (3.5)

which is the average intensity value or the background of the intensity wave. Subtracting

(3.2) and (3.3) gives

IM —IMin 24V1112 (3-6)

 

 

 

 

 

 

and
2 II _ IMax —IMin
l 2 —
2 (3.7)
This is the amplitude or the visibility of the intensity wave.
Substitute equations. (3.5) and (3.7) into equation (3.1) to obtain
1 ___ IMax +[Min + IMax —IMin COS¢
2 2 (3.8)
which yields
I _ 1 Max + 1 Min
Cos¢ = 2
1 Max _ 1 Min (3.9)
2

40

Therefore, the phase can be computed as

 

 

 

(I IMax +1Min \
2
¢ = ArcCos
[Max ‘IM... (3.10)
K 2 /

 

 

where I is the original intensity at one pixel, and 1M,x and 1M;n are the maximum and
minimum values of the intensity at the same pixel position as I. Once IMax and 1M,n are
determined, then, from equation (3.10), ¢ can be determined in the range between 0 and
1:.

A second key step is to determine in which quadrant to place the phase.

Taking the derivative of equation (3.1) yields

d1 -
fi=—2,/1112Sm¢ (3.11)

Then

d1

. _ 9%
SIMS—-2 1112 (3.12)

The Sign of 3:; , which is opposite to the Sign of Sin(¢) , can be used to determine the

quadrant in which the phase ¢ is located, as shown in Table 3.1.

41

. d1
Table 3.1. Quadrant determination of g) by Signs of C030) and —.

 

 

 

 

 

d¢
fl
Cos¢ d¢ Quadrant of ¢
+ + Quadrant 4
+ - Quadrant 1
- + Quadrant 3
- - Quadrant 2

 

 

 

 

 

3.2.2 Algorithm to determine 1M“, [Min and ¢.

The question of how to find IMax and IMin still remains. As mentioned before, the
intensity I has the cosinusoidal waveform shape. Therefore, if several cycles of the
intensity wave can be obtained at one pixel, the IMax and 1M)n are easily determined from
the waveform at this point. In our experimental setup, a PZT phase shifter is used to
artificially change the path length of one beam so that the phase difference between the
two interfering beams is changed. A computer is used to record digital pictures of the
light intensity of the speckle image for each mirror location. The Nyquist theorem must

be fulfilled to sample the intensity waveform. To minimize the chance that the intensity

42

background and the fringe visibility will change, the signal recording should be finished
in a short time, as soon as possible. After recording, at each image point, the light
intensity values are plotted against the driving voltages of the PZT, and a low-pass filter
and least-square curve fitting method are used to process the test values to obtain a good
intensity changing waveform. For low pass filter and curve fitting techniques, a large
number of complex mathematical equations need to be solved. Fortunately, many
commercial software packages are capable of doing this processing, making it easy to
develop an efficient program. Since only the intensity waveform instead of the fitting
parameters is needed to determine 1Max and IMin, there are more options for commercial
software to do curve fitting. In the LabVIEW environment used here, a linear low pass
Butterworth filter is used to filter noise (National Instrument Corporation, 1998).
Because the intensity waveform is not purely cosinusoidal, a general polynomial curve fit
of 10th —order is used to fit the recorded signals at each image pixel. A smooth intensity

' waveform is obtained. Figures 3.2 and 3.3 Show an example of such an intensity wave at
one point on a test Specimen before and after curve fitting. On the fitting curve, the
derivative at each point is computed to find the local maximum and minimum intensity
values. The average of all local maximum intensities is counted as the 1M... , and the

average of all local minimtun intensities is counted as the 1M)n . By substituting these

parameters into equation (3.10), the phase ¢ can be calculated.

43

 

Figure 3.2. An example of intensity wave obtained by phase
shifting.

 

Figure 3.3. Waveform of Fig. 3.2 after curve fitting.

44

In order to determine the quadrant of the obtained phase, the derivative of the fitting
d1 . . . . .
curve d—V at the pornt where the drrvrng voltage equals zero 1S computed. Obvrously,

i has the same Sign as g. From table 3.1, if i is positive, the sign of phase 05
dV d¢ dV

obtained from equation. (3.10) is reversed to locate the phase angle in quadrant 3 or 4. If
23% is negative, for example, then 05 should be placed in quadrant 1 or 2. The sign of

Cos( ¢) , already determined, establishes which of these two quadrants is the correct one.

After completing the above procedure at each point across the entire image field, the
phase map is determined modulo 21:. The 21: ambiguities can be removed by phase
unwrapping techniques discussed in Chapter 2 (Creath 1985, Sesselmann et al. 1998,
Kaufmann, et a1. 1998).

This algorithm, which scans the intensity, has two potential difficulties: (1) if the
amplitude of the intensity wave is too small, the waveform will be blurred by noisy
signals at a certain point; (2) if the intensity value before phase Shifting is not between the
1Max and IMin , which results in the value of the cosine equation being greater than 1, the

phase angle can not be calculated.

[Mar — Ill/fin

To solve the first problem, the amplitude of the waveform can be

checked. If the amplitude at a certain point is smaller than a given threshold value
determined by the user’s experience, the point is marked as an invalid pixel. A big
threshold, which implies a high percentage of invalid pixels in the whole image, makes it

difficult, even impossible, to obtain a smooth phase map. A too-small threshold results in

45

a very rough phase map. The threshold can be determined by trying different numbers
until the best phase map is obtained. In this work, amplitude 4 gray levels were employed
as the threshold. To solve the second problem, the intensity value can be approximated
by the nearest value that lies within the range of the intensity wave. However, if the

value exceeds the intensity range considerably, the point is marked as an invalid pixel.

3.3 Experimental Examples

To verify the phase shifting algorithm presented above, the method was first applied
in a digital speckle pattern interferometry (DSPI) setup to measure the crack tip position
in a compact tension (CT) specimen (Cloud, Ding, and Raju 2002). Second, the in-plane
displacement of a disk that was given a controlled small rotation was measured (Cloud
1995, 1979). For this purpose, a LabVIEW program was designed (National Instrument
Corporation, 1998). This program acquires image data from a CCD camera pixel-by-
pixel through a frame grabbing board (National Instruments, PCI-IMAG-1402), controls
the PZT to implement phase shifting, calculates the phase at each pixel, and finally

computes the displacement or the coordinates of the crack tip.

3.3.1 Determination of the Crack Tip Position

In this experiment, the setup that was used by Cloud, Ding, and Raju (2002) was
used. The results obtained from the new algorithm were compared to those obtained by
the Carré phase shifting technique in which 1: /2 phase shifts were used. Less than one
minute on a PHI-500 PC served to finish signal processing of a 200x200 pixel region, and

forty recorded images were used in the new algorithm. Figure 3.4 is a diagram of the

specimen.

46

 

+—O

Figure 3.4. Load condition of CT specimen.

 

 

 

 

Figures 3.5 to 3.7 were obtained by the new algorithm. Figure 3.5 shows the raw
phase change map with salt-and pepper noise. A 5x5 smoothing filter process (National
Instrument Corporation 1998) was used to remove the noise and to obtain a smooth phase
change map as seen in Figure 3.6. After performing phase unwrapping to remove 2 1:
ambiguities on phase change map, a continuous smooth phase map is seen in Figure 3.7.
Figure 3.8 is the phase map after unwrapping obtained by using the Carré phase shifting
method. The coordinates of the crack tip are indicated on the images (Figure3.7 and
Figure 3.8). Through these coordinates, it was established that the crack tip positions
determined by the two methods were in a good agreement. The new IMMS method

works as well as the Carre’ phase shifting method in this application.

47

 

Figure 3.5. Raw phase change map.

 

Figure 3.6. Phase change map after smoothing.

48

 

Figure 3.8. Crack tip position from the Carré method.

49

3.3.2 Measurement of In-plane Displacements

Known in-plane displacements on an object were introduced and compared with the

results obtained by applying the developed method in a DSPI setup. A calibration device

 

Figure 3.9. Calibration plate.

that uses a rigid plate that can be given known small in-plane rotation was used (Cloud
1995, 1979). The plate is rotated by using a micrometer that engages a lever arm that is
118 mm long as shown in Figure 3.9. Least reading of the micrometer is 0.0001 in
(0.00254 mm). During the experiment, with small rotations of the plate, the expected
horizontal correlation fringes were observed (Figure 3.10). Three random points on the
surface of the rigid plate were chosen to compare the displacements obtained by this new

algorithm with the known induced displacements. The comparative values and errors are

50

given in Table 3.2. The difference is within 4%. It can be said that the measured

displacement results are in good agreement with induced displacement.

 

Figure 3.10. Correlation fiinge map obtained by the new algoritlnn.

Table 3.2. Comparison of the in-plane displacements calculated by the new algorithm

and induced displacements.

 

 

 

 

 

 

 

 

Rotating Induced Measured
Angle(rad) Displacement(um) Displacement(p.m) Difference
2.0532e—4 3.475 3.51 1.00%
2.7517e-4 5.866 5.64 3.85%
2.540064 6.388 6.16 3.57%

 

51

 

The results of these two examples of applications of the newly developed IMMS
algorithm indicate that the developed IMMS method for phase determination can work
well to perform surface measurements in DSPI apparatus. However these two tests were
performed with the DSPI setup and specimen mounted on vibration-insulated table that
can minimize noise effects significantly. Therefore, the test results only mean that the
new technique can work as well as other existing phase Shifting techniques in a
laboratory environment. The next concern is what advantages the new IMMS method has
compared with other techniques for phase determination. The purpose in developing this
new IMMS method is to improve the noise tolerance of the DSPI setup so that this non-
contact technique for surface measurement is more practical for real application
environments. In the next section, in order to verify the general objectives of this research
work, the noise tolerance of the new method is tested by performing numerical

simulations and real applications.

3.4 The noise tolerance property of the IMMS method

Since some numerical filters, for instance a low pass filter, can be employed in the
algorithm to process the recorded intensity signals in the time domain, a good noise
tolerance is expected for this new IMMS method. To verify this assumption, numerical
simulations are the first simple choice to perform the test work. A second choice is to
apply the new algorithm in some noisy work conditions. In this section, some
experiments are presented to explore the noise tolerance properties of the IMMS method.
The calibration plate shown in Figure 3.9 was used in this test. At ten points chosen
randomly over the entire image, good intensity variation curves are recorded by driving

the PZT mirror in an in-plane sensitive DSPI setup where the whole setup and specimen

52

are placed on a vibration insulated table. Numerical simulations and some real noise
signals are used to test the noise tolerance property of this IMMS method at these ten

points.

3.4.1 Numerical Simulations

A white noise signal that includes wide band of frequency of noise is commonly used
to perform environmental noise simulations. In this research work, a numerical Gaussian
white noise generator that has capability to control the standard deviation of generated
noise signals is used to generate some white noise Signals. These noise signals are added
to the recorded intensity signals. The IMMS method is employed to both the originally
recorded intensity signals and to the noise-added intensity Signals to determine phase
values. Results are compared as in the following.

Figure 3.11 is a recorded good intensity waveform after removing the background
intensity. Figure 3.12 is an example of Gaussian white noise whose standard deviation is

2.0. Figure 3.13 presents the intensity waveform with the Gaussian noise added on.

53

 

Figure 3.12. Gaussian white noise signals.

54

 

Figure 3.13. The intensity waveform with Gaussian noise added on.

In this example, the RMS signal-to-noise ratio is 6.3. After performing the IMMS
method to the Figures 3.11 and 3.13 individually, two final fitting intensity waveforms
were obtained. Figure 3.14 presents the final fitting curve of the originally recorded
intensity wave. Figure 3.15 shows the final result curve of noise-added intensity signals
(Figure 3.13) obtained from IMMS. The results indicate that the effect of this Gaussian

noise could be removed completely by the IMMS method.

55

:15
:3
Figure 3.15. The fitting curve of noise-added intensity signals.

3'

 

56

The deviation of the noise signals was increased gradually. One more example from
among these extensive tests is reported. A Gaussian noise signal with higher deviation is
shown in Figure 3.16. The deviation of this noise is 6.5, more than three times of the
previous one that was at 2.0. The RMS signal-to-noise ratio is decreased to 0.96 in this
example, meaning that the noise is nearly as large as the signal. Adding this noise signal
to the intensity waveform (Figure 3.11) yields Figure 3.17, which is more chaotic than
the signal in Figure 3.13. Figure 3.18 shows the IMMS result from Figure 3.17.
Compared to Figure 3.14, this result indicates that the IMMS method cannot process this

noisy signal efficiently.

By adjusting the deviation of the white noise signals, it is shown that the IMMS
method can remove noise with a small deviation efficiently. Quantitatively, a noise that

has an RMS signal-to-noise ratio larger than 1.8 can be processed successfully.

 

    

HQ. . 'h» ' 4. rm 1‘ 1“?" ’ 3" .

Figure 3.16. Gaussian noise with a 6.5 deviation.

57

 

Extact Fnitti .

,4
.1.

    

‘w

Figure 3.18. The fitting curve of an intensity wave with a 6.5 deviation noise
added.

58

In the next step, a high pass filter removes the low frequency components from the
same very noisy signal (Figure 3.16), which has a 6.5 deviation. Therefore, only the high
frequency components of the noise are left and added to the original signal. Figure 3.19
presents the resultant intensity waveform. The RMS Signal-to-noise ratio is 1.21 in this
example. Figure 3.20 shows that good results are obtained from the IMMS method. This
result indicates that high frequency noise even with big amplitude can be removed by the

IMMS efficiently.

Repetition of the simulation tests showed that the IMMS method gives very good
immunity to both high frequency noise and small amplitude low frequency white noise.
Large amplitude low frequency noise can damage the efficiency of the noise immunity of

the IMMS method.

 

Figure 3.19. The intensity waveform with high frequency component of the
noise added on.

59

 

Figure 3.20. The fitting curve of the intensity wave with a 6.5 deviation high
frequency noise added.

60

3.4.2 The effect of real noise signals

At various points on the Specimen, intensity changes induced by environmental
noise (such as air turbulence and vibration) were recorded under three different situations
by DSPI setup and added to good intensity curves. The IMMS method was employed to

process these noise-added signals.

A. The specimen was placed on a vibration-insulated table:

In this case, both the DSPI equipment and specimen were placed on a vibration-
insulated table. At some pixels, the intensity changes induced by environmental noise
were recorded. A noise Signal example is shown in Figure 3.21. This noise was added to
a good intensity waveform as Shown in Figure 3.22. In this example, the RMS signal-to-
noise ratio is 7.22. On this table, the vibration is minute. So, the RMS-SNR was bigger
than for the other cases. The IMMS method was employed to process the data. Figure
3.23 gives the fitting result from the IMMS method. The phase value determined by the
IMMS is —3. 14 Rad, which exactly matches the phase obtained from the original signals.
The figures and the match of the phase values indicate that the IMMS method works very

well in this Situation, as expected.

61

 

Figure 3.21. Noise signals under A condition.

 

Figure 3.22. Intensity change curve with noise added.

62

 

Extract Ettin'

    

Figure 3.23. The fitting curve obtained by IMMS.

63

B. The specimen was put on the same table without air support vibration isolation:

In this case, the DSPI setup and the specimen were still on the vibration-insulated
table, but the air was released from the supporting structure of the table. Figure 3.24 is an
example of recorded noise signals under this situation. The amplitude of this noise was
bigger than that of case A. The original intensity curve was distorted much more than in
the previous case, as seen by comparing the sample photo of Figure 3.25 and Figure 3.22.
In this situation, the specimen vibration was very serious. The RMS-SNR is 1.81, which
is much smaller than in case A. Figure 3.26 shows the IMMS processing result. The
phase results still match exactly at —2.29Rad. The IMMS method still works well in this

case.

 

 

V31“ ._ , ... it

Figure 3.24. Noise signal.

64

 

ise added.

thno

Intensrty curve WI

.25

3

igure

F

m

Extract Fi

 

Figure 3.26. The IMMS result.

65

C. Noise signals were recorded with the specimen placed on a cart.

In this case, the specimen was placed on an ordinary laboratory cart that is much
more susceptible to environmental noise, especially the vibrations of the building. The
DSPI setup was still located on the table. Real noise signals were recorded at various
points and bigger variances than cases A and B were observed for these noise signals.

Figure 3.27 shows one noise example.

 

Figure 3.27. Noise sample obtained under case C.

66

In this situation, an alternative calibration method was used, measuring in-plane
displacement of a plate. The recorded noise signals were added to each point over all the
pixels in the image during displacement measurement by the IMMS algorithm. The same
displacement calibration set up as shown in Figure 3.9 was used to introduce known in-
plane displacement. Table 3.3 gives some examples of these test results. They indicate

that the IMMS worked well even under these high noise conditions.

Table 3.3. Some examples of displacement measurements.

 

 

 

 

 

Variance of Measured displacement by Induced known

noise signals the IMMS (um) displacement(pm) 2:193:15:
30.62 1.85 1.86 -0.75
37.75 1.75 1.68 4.04
30.23 2.82 2.70 4.29

 

 

 

 

 

All the above simulations were repeated tens of times at different dates and times of
day. The IMMS method proved to have a good ability to reduce some specific noise
effects. Especially, the IMMS method has very reliable ability to remove high frequency
noise and small amplitude low frequency noise. This ability can make the DSPI technique

useful in more practical environments than before.

67

Chapter 4. A technique for calibrating and evaluating phase shifter

4.1 Introduction

As mentioned, a phase shifter is commonly used in DSPI setups to evaluate phase
angle. The working status of the phase shifter, such as the necessary driving voltage, non-
linearity, stability of the PZT, and tilt of the mirror, greatly affect the accuracy of the
ultimate measurement results of the DSPI technique. Therefore, calibration of the phase
shifier is a very important step in the DSPI technique. In the past decades, several
methods were developed to calibrate the phase shifter. Two methods are commonly used.
The first one is a fringe-tracking method and the second one is the Carré method as
described by Brug (1999). These methods have some obvious drawbacks and limitations.
The fringe-tracking method is suited only for accurate 21: phase steps and is difficult to
control. The Carré method uses several phase-stepped images to calculate the phase
shifting angle with an assumption that the phase shifter is linear. These techniques cannot

evaluate the linearity, the tilt, and the non-uniformity of the motion of phase shifters.

Based on the IMMS technique, an online method to calibrate the phase shifter is
developed here. This improved algorithm provides an easy and convenient way to inspect
the linearity and the non-uniformity of the phase shifter. It is suitable to be used to check

the working reliability of the phase shifting setup in real time.

68

4.2 Scanning method for PZT’s calibration

As described in Chapter 3, after the determination of the maximum and minimum
intensities Imax and 1mm, the phase value can be computed by equation 3.10. The same

computation can be applied at each point along the entire intensity curve. Therefore, the
plot of phase values versus driving voltages was obtained with 21: ambiguities as seen in
Figure 4.1. The real diagram of phases versus the driving voltage was obtained by
performing phase unwrapping with respect to the zero driving voltage point. Figure 4.2
presents an example of a phase vs. driving-voltage diagram afier phase unwrapping. The
phase steps induced by the driving voltages can be determined from this diagram (Figure
4.2). For instance, on this sample curve, 2.5 volts to the PZT will result in a 2-radian
phase shift. Furthermore, from this phase diagram, the driving voltage to the PZT to get a
specified phase step at this pixel point can be obtained by interpolation of the obtained
data. As well, the non-linearity of the phase shifting at this point is visually displayed on
the curve. To calibrate a specific phase step, i.e. 1t/2 phase step, the above calibration can
be performed at a number of pixel points over the entire image. An average driving

voltage to get that phase step can be computed.

69

 

Figure 4.2. Phase step V.S. driving voltage.

70

Tilt and non-uniformity of the phase shifter movement are two other important
concerns that significantly affect the accuracy of phase shifiing. In the illumination area,
two lines are drawn. One is along the x-axis direction, and the other one is along the y-
axis direction as seen in Figure 4.3. At a specific given driving voltage, such as the
calibrated 2.5 volts to obtain a 2-radian phase step, phase shifting results corresponding
to this driving voltage at all points along these two perpendicular edges of the image are
plotted in a 3-dimensional graph to visually illustrate the tilt and non-uniformity of the

movement of the phase shifier.

 

Figure 4.3. Two lines are drawn on the image.

71

To implement the idea, the IMMS signal-processing algorithm deve10ped in
Chapter 3 was employed to obtain the phase shifting curve at each point along these two
edges. The phase shifting values corresponding to the calibrated n/2 driving voltage were
determined from the fitting curve. The computed phase shifting values were plotted in a
3-D graph, on which the tilt and non-uniformity of the phase shifter can be inspected
visually and directly. Figure 4.4 is an example of such a 3-D graph. After rotation of
Figure 4.4, a 2-D side view of the Figure 4.4 was obtained as in Figure 4.5. On this 2-D
graph, the trend line and the distribution of the phase shifting values indicate the tilt and
uniformity of the phase shifting motion. For instance, if there is an angle between the
trend line and horizontal line in the graph, it can be inferred that tilt occurs along this
edge, the bigger the angle, the more serious the tilt. If the phase shifting value points are
distributed far away from a straight trend line, we can say the uniformity of the phase
shifting motion is bad. If all points concentrate on a straight trend line closely, the
uniformity of the phase shifting motion is good. Therefore, Figure 4.5 can show the tilt
and non-uniformity of the movement of the phase shifter along the x-axis direction
clearly. Figure 4.6 is a 2-D side view along the vertical direction to show the tilt and

non-uniformity of the phase shifter motion in the Y direction.

72

Angle

Phase

2.5

2.0

1.5

1.0

0.5

0.0

 

 

 

 

 

Figure 4.4. A 3-D plot of the obtained phase shifting results.

73

Phase Angle

 

 

 

 

 

 

 

 

 

Figure 4.5. Tilt and non- uniformity in x-direction.

74

 

DJ-

 

!"-"
U1

M
Phase Angle

 

...-._.1..
U1

[3
Ln

 

 

 

 

Figure 4.6. Tilt and non- uniformity in y-direction.

75

4.3 Calibrations of two phase shifters

The method described above has been used to calibrate and inspect the phase shifter

used in a DSPI setup, which is a small mirror mounted on a PZT.

Actually, the figures from 4.1 to 4.5 are from a calibration test of a real piezo
electronic transducer phase shifter. This phase shifter is a small mirror mounted on a
PZT. Figure 4.7 represents a recorded intensity wave. After filtering and curve fitting, a

smooth waveform was obtained as shown in Figure 4.8.

The calibration result has been shown in Figure 4.2. For this specific experimental
setup, the driving voltage to get 1r/2 phase step is 2.09V. Figure 4.2 shows that the
linearity of this phase shifter is good as well. Figures 4.3, 4.4 and 4.5 indicate that the tilt

and non-uniformity of this specific PZT phase shifter is very small.

From the obtained results of hundreds of tests, the average driving voltage to get
1t/2 phase step is determined at 2.13V for this apparatus. The standard deviation for these

tests is 0.104 V, which implies very good stability for this PZT phase shifter setup.

76

 

Figure 4.7. Recorded intensity waveform.

77

 

Figure 4.8. Fitting curve obtained by IMMS.

78

A modified phase shifter is presented in Figure 4.9. A large mirror is attached to the
PZT by some adhesive putty. The behavior of this PZT mirror was checked by
performing the newly developed calibration method. A recorded intensity wave is
presented in Figure 4.10. After signal processing, a smooth curve is obtained as seen in
Figure 4.11. The proposed calibration method is performed to obtain the phase-driving
voltage diagram as shown in Figure 4.12. The linearity is seen to be good in this graph.
Figure 4.13 presents the 3-D graph of phase shifting along the two edges of the image. In
Figures 4.14 and 4.15, the tilt and non-uniformity of the phase shifting motion can be
seen more clearly. This phase shifiing setup is not so uniform as the previous one, and tilt
is obvious in both X and Y directions. The calibration test was repeated 30 times for this

setup. Serious tilt of the mirror is always observed. Linearity is not consistent either.

 

Figure 4.9.The phase shifier.

79

Waveform Chart PM 0 ‘1!

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it

I"

 

Figure 4.10. Intensity curve at one pixel point obtained by phase shifting.

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80

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Figure 4.12. Phase step versus driving voltages.

81

    

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82

 

 

 

 

Phase Angle

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.14. Tilt and non-uniformity in x-direction.

83

 

 

 

 

 

 

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p...-
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[III

in...

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84

 

4.4 Advantages of the new technique

A new technique to calibrate the phase shifting has been derived from the IMMS
method, and its applications to PZT phase shifters were successfiilly demonstrated in
repeated experiments. This algorithm shows several practical advantages. First, it
eliminates the drawbacks caused by the common assumption of the linearity and
uniformity of the phase shifter, thus efficiently minimizing the movement error of the
PZT. Secondly, it can be used to inspect several working parameters of the phase shifter

simultaneously, such as non-linearity, non-uniformity and tilt of the phase shifter.

Overall, this new method is a convenient and ideal way to calibrate the phase shifter,
to inspect the quality of the phase shifter, and to evaluate the reliability of the
experimental setup. Whenever the experimental setup is adjusted, such as the
illumination angle or the path length of beam, this algorithm can implement calibration

and inspection online very conveniently.

85

Chapter 5. Simplified DSPI setups

Through use of the IMMS method developed in Chapter 3, the in-plane DSPI setup
can be simplified and perform quantitative measurements. As well, a simple out-of-plane
setup patented by Cloud (2000) can measure out-of-plane displacement quantitatively by

the application of the IMMS method.
5.1 A simple in-plane DSPI set up

5.1.1 Introduction

Since the first demonstration of Electronic Speckle Pattern Interferometry (ESPI)
almost simultaneously by Macovski, Ramsey, and Schaefer (1971) in the United States
and by Butters and Leendertz (1971) in England, the majority of research work on ESPI
techniques has focused on the development of the phase shifting techniques, phase
unwrapping algorithms and digital filter applications (e. g. Creath 1985, Fomaro, et al.
1996, Hibino 1999, Kaufmann, 1998, Sesselmann and Goncalves 1998, Surrel 2000). The
goals of these investigations have largely been to obtain more accurate numerical results
or to simplify the digital signal processing. Relatively little research has been published
about modifying the system of apparatus of the ESPI. Siebert, Wegner and Ettemeyer
(2001) introduced a compact ESPI sensor that can be attached to the test object, but the
construction of this sensor is still a trade secret. Two basic well-known practical in-
plane sensitive ESPI setups were introduced by Jones and Wykes (1983) as seen in

Figure 5.1. Most research work to date has been done using the illumination arrangement

86

of Figure 5.1 (a) or variations thereof. The setup of Figure 5.1 (b) is similar to that used

by Post, Han, and Ifiu (1994) to generate virtual gratings for moire' interferometry.

As far as the author can discern, no quantitative DSPI measurement results based on
the second arrangement shown in Figure 5.1. (b) have been reported. The most plausible
reason is that it is very difficult to employ phase shifting techniques to obtain quantitative
results because: (1) it is hard to make a big PZT mirror that is stiff, stable, and of
reasonable mass, (2) driving a large reflection mirror to do phase shifting is much more
difficult than driving a very small mirror such as is commonly used in the first setup. The
tilt, twist, and deformation of the large mirror are difficult to handle with conventional
phase shifting and data reduction processes. However, the advantages of this setup are
very attractive for practical applications of DSPI. This arrangement forms a nearly-
common-path interferometer, which implies excellent noise immunity and fewer
components used than are needed in the first arrangement. The Improved Max-Min
Scanning phase shifting algorithm has a good immunity to tilt, non-linearity and non-
uniformity of phase shifting and vibration noise as reported in Chapter 3. Consequently,
this IMMS method can be a good choice to make the Figure 5.1 (b) setup work
quantitatively. The next section demonstrates that, with the IMMS method, the in-plane
ESPI setup can be simplified very much and the noise tolerance of DSPI can be improved

significantly.

87

11

 

   

 

-0- }- -.-.-.-—--.-O-u—O-C-c-c-0-

 

(a) (b)

Figure 5.1. Two basic practical illumination apparatus of in-plane speckle interferometry.

5.1.2 Setup and Tests

Figure 5.2 is a diagram of the in-plane ESPI setup developed in this work. A plane

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.2. The arrangement used in this work.

88

mirror is placed in front of the object and is perpendicular to the object surface. This
mirror reflects half of the illumination beam to provide a second in-plane illuminating
beam while the other half illuminates directly in a symmetrical direction. The nearly
common path of the two illuminating beams gives this setup a better environmental
tolerance. Furthermore, the reflection mirror could be attached to the object. This should
further improve the noise immunity of the setup. Figure 5.3 is a photo of the entire simple
setup. The reflection mirror is attached to a small PZT by adhesive putty (right part of the

Figure 5.3).

 

Figure 5.3. The experimental setup and the PZT mirror.

89

To check the behavior of the PZT-reflection mirror, the phase shifter calibration
technique proposed in Chapter 4 was used to calibrate the behavior of the PZT-drive
mirror. In Figure 5.4 and 5.5, the tilt and variance of the spots indicate that the tilt and
non-uniformity of this PZT-mirror are very significant over the entire image area. It is
impossible to obtain reliable phase measurement by using the common phase shifting

techniques such as four-step phase shifting.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Figure 5.4. The tilt and non-linearity of the PZT mirror in the horizontal direction.

90

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.5. The tilt and non-linearity of the PZT mirror in the vertical direction.

To compare the immunity to noise of the setup, both in-plane DSPI arrangements
shown in Figure 5.1 were used to construct correlation fringe maps. In this test, the
specimen was placed on a normal table as seen in Figure 5.3 instead of the vibration—
insulated table. Consequently, the vibration of floor will severely affect the measurement
results. Figure 5.6 presents the comparison of the correlation fringe patterns obtained by
both arrangements simultaneously. In the fringe graph, the left part of the vertical curve
in this picture was obtained by the Figure 5.1 (b) arrangement. The right part of it was
obtained by Figure 5.1 (a) arrangement. The comparison in this figure indicates that it is
impossible for the Figure 5.1 (a) setup to provide a stable correlation fringe map, which

indicates it cannot perform measurements under such a noisy condition. The Figure 5.1

91

(b) arrangement produces more stable fringe maps and presents an excellent immunity to

noisy environments.

 

Figure 5.6. The comparison between two arrangements of DSPI.

To make the simple in-plane setup work quantitatively, the phase shifiing method
must be robust to tilt, non-uniformity and non-linearity. The previous work about the
IMMS phase shifting method demonstrated that the IMMS method meets the above
demands. Therefore, the IMMS phase shifting method was employed to process the
acquired signals. In-plane displacement measurements were performed to test this
apparatus. The displacement calibration plate was placed on a normal table, which is

very vulnerable to vibrations.

92

Known displacement induced by a micrometer was measured by the setup in Figure
5.3 with IMMS technique. Table 5.1 shows the comparison between the DSPI
measurements and induced displacements. Even under such a noisy condition, good

agreement is observed.

Table 5.1. The comparison between induced and measured results.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. Induced Measured Difference
Rotation angle (Rad) displacement (um) displacement (um) (%)

2.15254e-4 0.98552 1.01 -2.48397
3.44407e-4 1.62560 1.70 -4.57677
4.30508e-4 2.07264 2.19 -5.66234
6.45763e-4 3.07848 3.12 -1.34872
8.61017e-4 3.57632 3.47 2.972888
8.61017e-4 4.14528 4.08 1.574803
6.45763e-4 5.18160 5.18 0.030878
8.61017e-4 2.43840 2.60 -6.6273
4.30508e-4 2.56032 2.74 -7.01787

 

93

5.2 A simple out-of-plane DSPI setup

Several variants of a simple out-of-plane sensitive DSPI setup were reported and
patented by Cloud (2000, 2003). Figure 5.7 is a schematic of one of these arrangements.
In this setup, a glass scattering plate is placed in front of the specimen. When the laser
beam passes through the scattering plate to illuminate the specimen, part of the
illuminating beam is reflected directly into the imaging system to provide the reference
beam. The camera has a very narrow viewing angle with the illumination beam, allowing
this arrangement to be sensitive to out-of-plane displacement. The nearly common path
character of this arrangement gives this setup an excellent noise tolerance, which was
observed while obtaining correlation fringe maps without an optics table. As mentioned
by Cloud (2000), to make this simple setup able to measure out-of-plane displacement
quantitatively, an appropriate method for phase map determination is desired. A driver to
move or tilt the ground-glass plate seems the simplest way to perform phase shifting
technique in this apparatus. However, the mass and limited stiffness of the plate will
result in severe unwanted errors for phase evaluation, such as vibration and uneven
movement throughout the entire plate, when conventional processing is used.
Considering the characteristics of the new IMMS method developed in this work, it can

be a good alternative algorithm to perform phase evaluations for this set up.

94

 

 

 

 

 

Ground Glass .
Imaging System

 

 

 

 

 

 

 

 

 

 

 

IBM Compatible

Figure 5.7. Schematic of the simple out-of-plane sensitive DSPI system.

Figure 5.8 shows such an out-of-plane DSPI device. In this apparatus, a PZT is
attached to the plate by adhesive putty to drive the plate to perform phase shifting. Figure
5.9 is the picture of the PZT plate. The plate is flexibly suspended at three points, two at
the bottom edge, and one close to the middle point of the upper edge. The driving point is
nearly the center of the plate. It is obvious that the driving motion involves both

deformation and tilt of the glass plate, so it is definitely not uniform throughout the area

of interest.

95

 

 

Figure 5.9. The phase shifting plate.

96

The calibration method developed in Chapter 4 is performed to check the properties
of the phase shifting motion. Figure 5.10 is the initial speckle image. The PZT transducer
shows in the right part in this image. Figure 5.11 shows that the linearity is not bad at this
pixel point of the image. Figure 5.12 is a 3-D graph of the phase shifting along two edges
of the region of interest. Figure 5.13 indicates that the tilt is severe along the X direction.
Figure 5.14 indicates that the tilt along Y direction is less than that in the X direction, but
still severe. Serious non-uniformity of the phase shifting can be seen in both graphs. It is
known that the IMMS method can minimize the effect of the tilt and non-uniformity of

the phase shifting movement.

97

 

Figure 5.10. A speckle picture with the PZT driving setup on the right.

98

 

Figure 5.11. The linearity of the phase shifting.

99

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100

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101

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.14. The tilt and non-uniformity of the phase shifting along the y-
direction.

102

The IMMS method is applied with this out-of-plane setup to perform signal
processing to measure some induced out-of-plane displacement on a calibration plate.
Figure 5.15 is a phase change map obtained by IMMS method from an out-of-plane rigid
body rotation. Figure 5.16 is the fringe map after filtering of Figure 5.15. Figure 5.17 is
the measured distribution of the displacement field. A 3-D graph of the out-of-plane
displacement field is presented in Figure 5.17. The measurement tests were repeated,
Table 5.2 gives the comparison of the DSPI results and induced displacements. The good
agreement between them indicates that this simple out-of-plane DSPI setup works well to

measure out-of-plane displacement quantitatively.

Phase Difltence .” ap

 

 

Figure 5.15. A raw phase difference map.

103

Phase Dihence Map diet Smoothing

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104

 

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Figure 5.17. The measured displacement field in gray scale.

105

 

 

 

 

 

 

Figure 5.18. The 3-D map of out-of-plane rigid body rotation.

106

 

Table 5.2. The comparison between induced and measured results

 

 

 

 

 

 

Rotation angle (Rad) Induced gmlacement Measurectsliiplacement Difference (%)
4.84322e-5 1.0129 1.06 -4.648
7.5339e-5 1.5372 1.54 -0. 18076
7.5339e-5 0.7686 0.73 5.0234
8.61017e-5 1.9764 1.91 3.36098
8.61017e-5 2.0203 1.96 2.9870

 

 

 

 

 

 

5.3 Conclusions

In this chapter, both the in-plane DSPI and out-of-plane DSPI were simplified very
much compared to the commonly used setups. This makes the DSPI setup more cost
effective. These simplified setups were shown to work quantitatively with the IMMS
phase shifting method. Very good noise immunity of the presented setups was seen. This
broadens the application of the DSPI technique to some noisy service environments

outside the laboratory.

107

Chapter 6. Summary and discussions

In this study, an IMMS method was developed for phase evaluation. The IMMS
method demonstrated a good immunity to some particular noise. As well, a new
technique based on the IMMS was developed to calibrate the behavior of PZT phase
shifters in real-time. By applying the IMMS technique, both in-plane and out—of-plane

DSPI setups were simplified.

An improved Max-Min scanning method for phase determination has been
presented in Chapter 3, and its application to the determination of phase maps was
successfully demonstrated in some DSPI experiments to measure crack length on a CT

specimen and surface displacement fields on a calibration plate.

The new IMMS algorithm has several practical advantages over the other traditional
methods. The least-square curve-fitting method and the digital filters applied in this
technique allow the maximum and minimum intensity values to be estimated more
accurately than with other techniques. This improvement is achieved because the
intensity waveform can be fitted well from recorded signals through more cycles of the
intensity waveform. Under ideal test conditions, such as on a vibration-insulated table in
a laboratory and an excellent linear phase shifter, this new algorithm does not have
advantages over other phase shifting techniques. However, in some noisy situations, since
signal-processing techniques in the time domain can be employed to the recorded
intensity signals, the improved algorithm is proved to have a better environment

tolerance. For instance, the low pass filter used in this work could remove vibration

108

noise efficiently. Furthermore, the sign of the phase angle can be determined by the
gradient of the fitting curve directly, so the requirement for additional images to
determine signs becomes unnecessary, meaning the elimination of the difficult steps to
manipulate the phase shifter carefully to obtain some tiny phase shifting steps as

mentioned by Vikhagen (1990).

With the IMMS method, the requirement to calibrate the phase shifter is removed
completely, because the phase steps are arbitrary and need not even be known. Since
only the intensity changes are used to determine maximum and minimum intensity and
the phase shifting step length is less important, the non-linearity of the phase shifter does
not significantly affect the accuracy of results. Based on point-by-point analysis of the
intensity wave, non-uniform phase shifting over the whole image does not significantly
affect the result. Compared to the common assumption for most traditional methods that
the phase shifting has a good linearity, these features of the IMMS minimize the
requirements for the phase shifter very much and enhance the noise tolerance of the phase

evaluations.

Even more significantly, with slow deformation, such as deformation under thermal
loading or digitally controlled loading, ideal intensity signals can be recorded during the
object deformation, so that the phase shifter could be entirely removed from the
experimental setup. The same is true if there are path length changes caused by, say,
thermal drift in optic elements. So, the method uses to advantage what have been sources

of error or problems in conventional approaches.

It has to be clearly pointed out that the intensity waveform at each pixel plays a key

role in this algorithm. The curve fitting technique is an ideal method to obtain a good

109

 

intensity waveform. A sinusoid least square fitting method was published many years
ago by Ransom, et al. (1986) and Macy, et al. (1983). In the sinusoid fitting method, the
fitting model is fixed to I(v)=a(v)+b(v)Cos((p(v)). Afier solving a series of equations,
parameters, a(v), b(v) and (p(v) are determined. In the improved method presented in this
work, the fitting parameters are not the goal of the performance of the fitting techniques.
As mentioned in chapter 3, only the intensity-changing curve is desired from fitting
techniques. Therefore, the fitting model is very flexible. Furthermore, there are more
options of commercial curve fitting software that are appropriate to the task. More
significantly, some options have better tolerance of non-linearity of the PZT compared to

the sinusoid fitting method.

In Chapter 3, the noise tolerance of the developed IMMS method is discussed.
Noise signals obtained from different situations were added to some sample recorded
signals and the developed IMMS method was employed to process these noisy data. It is
shown that the IMMS method has a good tolerance to some specific noise signals,
particularly to high frequency noise and to small amplitude low frequency noise. This

brings an advance to DSPI technology’s application in factory environments.

A disadvantage of this method is that, to construct the intensity waveform, many
image frames need to be recorded, which means that large numbers of image signals have
to be dealt with. Current high-speed computers make the improved algorithm practicable

to determine phase values in a short time.

Based on the developed IMMS method, a new technique to calibrate the phase
shifting behavior has been derived in Chapter 4, and its applications to check the

behavior of a PZT phase shifter was successfully demonstrated in some experiments.

110

 

This technique uses the IMMS method to compute the phase values along the entire
intensity waveform to obtain the diagram of phase shifiing versus the driving voltages to
the PZT at some pixels so that the linearity of the phase shifting can be clearly displayed
on the diagram. By considering more pixel points along various directions over the
image, the tilt and non-uniformity of the phase shifting can be shown in a 3-D graph.

This calibration algorithm shows several practical advantages. First, it eliminates the
drawbacks caused by the common assumption of the linearity and uniformity of the phase
shifter, thus efficiently minimizing the movement error of the PZT. Secondly, it can be
used to inspect a few properties of the phase shifier simultaneously, such as non-linearity,

non-uniformity and tilt of the phase shifter.

Overall, this new method is a very convenient way to calibrate the phase shifter, to
inspect the quality of the phase shifter, and to assess the reliability of the experimental
setup. Whenever the experimental setup is adjusted, such as the illumination angle or the
path length of beam, this algorithm can implement calibration and inspection in real-time

very conveniently.

Since the IMMS method shows good environment tolerance, particularly
minimizing the effect of errors of phase shifting, both the in-plane DSPI and out-of-plane
DSPI were simplified very much compared to the commonly used setups. In Chapter 5,
these simplified setups were shown to work quantitatively with the IMMS phase shifting
method. Very good noise immunity of the presented setup was seen. This simplification
makes the DSPI setup more cost effective and broadens the application of the ESPI

technique to noisy service environment outside the laboratory.

111

Chapter 7. Conclusions and recommendations

0 A new algorithm for phase determination that uses digital filters and curve fitting
techniques, called the IMMS method, was developed and verified.

o The IMMS technique was successfully used to measure crack length on compact
tension (CT) specimen.

0 The IMMS technique shows good noise tolerance.

o The IMMS technique doesn’t require phase shifting calibration as needed for other
techniques.

0 The IMMS technique can be used to perform on-line calibration of the behavior of
PZT phase shifter including driving voltage calibration, effect of tilt, non-uniformity, and
no-linearity.

- In some circumstances, for instance, under conditions of thermal loading or digitally
controlled loading, the IMMS technique allows elimination of phase shifters.

o By employing the IMMS method, simple DSPI setups were developed and applied to
perform out-of-plane and in—plane displacement/deformation measurements.

0 The simple DSPI setups show good noise tolerance.

112

Recommendations for future work:

1. Based upon the simplified DSPI apparatus, it is possible to develop some compact
DSPI sensors to measure displacement and deformation. The sensors will be very
portable and even can be attached to objects.

2. A processing program can be developed to use a series of continuously recorded
images to measure large displacements, which can eliminate the limitation of

decorrelation and memory loss.

3. Apply the simplified DSPI setup to measure slow deformation, such as the deformation
under thermal loading or digitally controlled loading, so that the phase shifter can be

removed from the DSPI setup completely.

113

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