Electronic speckle metrology

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Electronic speckle metrology
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Table of Contents
    Title Page
        Page i
        Page ii
    Dedication
        Page iii
    Acknowledgement
        Page iv
    Table of Contents
        Page v
        Page vi
        Page vii
        Page viii
    Abstract
        Page ix
        Page x
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
    Basic theory and notation
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
    Displacement refinement
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
    Analytic results on errors
        Page 36
        Page 37
        Page 38
        Page 39
        Page 40
        Page 41
        Page 42
        Page 43
        Page 44
        Page 45
        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
        Page 53
        Page 54
    Preparation for experiments
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
        Page 60
        Page 61
        Page 62
        Page 63
        Page 64
        Page 65
        Page 66
    Experimental results
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
        Page 82
        Page 83
    Conclusions and recommendations
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
        Page 90
        Page 91
    Appendix. The speckle user interface
        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
        Page 118
    Reference list
        Page 119
        Page 120
        Page 121
    Biographical sketch
        Page 122
        Page 123
        Page 124
Full Text











ELECTRONIC SPECKLE METROLOGY


By

WILHELM KURT SCHWAB














A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1992



























Copyright 1992

by

Wilhelm Kurt Schwab



























To my teacher,

William F. Carpenter













ACKNOWLEDGEMENTS


I would like to thank the chairman of my advisory committee, Dr. Charles E.

Taylor, for his belief in my abilities and for allowing me the freedom to explore.

I also wish to thank Dr. V.M. Popov for patiently checking the mathematical

descriptions presented in this document. Dr. Popov suggested changes to the original

manuscript that improved the presentation of the basic theory.

Dr. Corin Segal, Dr. John Abbitt, and Dr. Bruce Carroll contributed greatly to this

project through their generosity and hospitality. The use of their laboratory and

equipment was of great help in locating the source of several problems during the course

of my research.

I wish to thank Dr. Harold Doddington for his friendship and expert technical

assistance throughout the course of this project. I am also greatly indebted to Ron Brown

for his assistance with the construction of mounting hardware, and to Linda A. LeGrand

for proofreading the manuscript.

Finally I wish to acknowledge the constant moral support provided by my friends

Lee Herbst, William Miller, and Teresa Parrish, and by my parents.














TABLE OF CONTENTS

ACKNOWLEDGEMENTS ............................................... iv
ABSTRACT ................................................................ ix
INTRODUCTION ................................. .......................... 1
The Concepts Of Stress And Strain ....................................... 1
Techniques Of Experimental Stress Analysis .............................. 2
Electrical Resistance Strain Gages .................................... 2
Optical M ethods ..................................................... 3
Goal Of The Current Study ............................................... 4
Previous W ork .......................................................... 4
Speckle Photography And Computer Speckle Interferometry .......... 4
Digital Correlation Methods ......................................... 5
Automated Grid Methods ........................................... 6
H ybrid M ethods ................................. ....... ...... ...... 7
Electronic Speckle Metrology ............................................ 7
Experimental Configuration ........................................... 7
The SPECKLE Program .............................................. 9
Dynamic Histogramming .......................................... 10
Overview Of This Document ........................................... 11
Basic Theory And Notation ......................................... I I
Displacement Refinement ........................................... 12
Analytic Results On Errors .......................................... 12
Preparation For Experiments ........................................ 12
Experimental Results ............................................... 12
The SPECKLE User Interface ...................................... 13
Sum m ary ............................................................... 13
BASIC THEORY AND NOTATION ...................................... 14
M athematical Notation ................................................. 14
Function Definitions .............................................. 14
Sets And Images Of Sets ......................................... 15


v









Operations On Functions ........................................... 15
Topological Descriptions ............................................... 16
Spatial Relationships ................................................ 16
Description of Intensities ......................................... 18
The Use Of Logical Image Plane Coordinates ....................... 20
Computational Descriptions ............................................ 22
The Point And Region Of Interest .................................. 22
Sequence Of Approximate Motions ................................. 23
Sampled Data ...................................................... 23
Digital Images ...................................................... 24
Sum m ary ............................................................... 25
DISPLACEMENT REFINEMENT ...................................... 26
Iterative Computation Of Displacement ............................... 26
The Perturbation/Least-Squares Method ................................ 28
Heuristic Development ............................................. 29
Precise Formulation Of The PLS Method ........................... 30
The Perturbation/Correlation Method ................................... 31
The Newton Raphson Method Applied To Digital Correlation ....... 31
Heuristic Development ............................................. 32
Precise Development Of The PC Method ........................... 34
Sum m ary ............................................................... 34
ANALYTIC RESULTS ON ERRORS .................................... 36
Published Quantitative Results .......................................... 36
Effects Of Out-Of-Plane Displacement .................................. 37
Measurement Of Strain Using A Pinhole Camera .................... 38
Pinhole Equivalents Of Optical Systems ............................. 39
Order Of Magnitude Analysis Of Displacement Accuracy ............... 40
Dependence Of Displacement On Noise Magnitude ..................... 41
One-Dimensional Images And Motions ............................. 42
Comments On Scales ............................................... 42
Correlation Function And Measured Displacement .................. 43
Determination Of The Displacement Estimate ....................... 43
Interpretation Of The Derivative .................................. 45
Uncertainty Relation For Correlation ................................... 48








One Dimensional Images And Motions .............................. 49
The Correlation Function And Inner Products ....................... 49
Differentiation Of The Correlation Function ......................... 50
Sensitivity Analysis ................................................ 51
Comments On The Indeterminacy At The Exact Solution ............ 53
Summ ary ............................................................... 53
PREPARATION FOR EXPERIMENTS ................................... 55
Experimental Configuration ............................................ 55
Computer And Television System Details ........................... 55
Experimental Configuration And Procedures ........................ 56
Aspect Ratio Correction Calibration .................................... 59
Aspect Ratio Test 1 ................................................ 59
Aspect Ratio Test 2 ................................................ 60
Aspect Ratio Test 3 ................................................ 60
Summary Of Aspect Ratio Calibration .............................. 61
A Lucite Beam ......................................................... 61
M achining The Beam ............................................... 61
Loading The Beam ................................................. 62
Translating The Beam .............................................. 63
A Tensile Specimen ................................. ................... 64
Mechanical Specifications .......................................... 64
Loading The Tensile Specimen .................................... 64
Shear Specim en ........................................................ 64
Sum m ary ............................................................... 65
EXPERIMENTAL RESULTS ........................................... 67
Rigid Body Translation ................................................. 67
Experimental Procedure ............................................ 68
Results ............................................................. 69
Axial Strains In A Cantilever Beam ..................................... 71
Experimental Arrangement And Procedure ......................... 71
Results ............................................................. 72
Comparison With Beam Theory Predictions ......................... 72
Tension Test ........................................................... 75
Experimental Arrangement And Procedure ......................... 75









R results ............................................................. 76
Comparison With Theoretical Predictions ........................... 77
Shear Strain Test ....................................................... 78
Experimental Arrangement And Procedure ......................... 79
Results ............................................................. 80
Comparison With Theoretical Values ............................... 80
Effects Of Out-Of-Plane Displacement ................................. 81
Sum m ary ............................................................... 83
CONCLUSIONS AND RECOMMENDATIONS .......................... 84
Conclusions ............................................................ 84
Experimental Results ............................................... 85
Analytic Studies ..................................... .............. 85
Recommendations For Future Work .................................. 86
Verification Of The PLS And PC Methods ......................... 86
Extensions Of The PLS And PC Methods ........................... 87
Rigid Rotation Tolerance ........................................... 87
Out-Of-Plane Displacement Detection .............................. 88
Modification Of SPECKLE .......................................... 89
Sum m ary ............................................................... 91
APPENDIX THE SPECKLE USER INTERFACE ........................ 92
REFERENCE LIST ...................................................... 119
BIOGRAPHICAL SKETCH .............................................. 122













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

ELECTRONIC SPECKLE METROLOGY

By

Wilhelm Kurt Schwab

December, 1992

Chairperson: Charles E. Taylor
Major Department: Aerospace Engineering, Mechanics & Engineering Science

Researchers at the University of South Carolina have developed Digital

Correlation as a technique for the measurement of both two-dimensional and

three-dimensional rigid and deformable body motion. Errors in displacement gradients as

computed directly by Digital Correlation are generally on the order of 1000 micro strain.

Penalty Method Finite Element smoothing has been used to generate displacement

surfaces which may be analytically differentiated to compute displacement gradients more

accurately. The primary disadvantage of the Digital Correlation method is that it is

computationally intensive. Bruck developed a Newton-Raphson based iterative algorithm

for computation of displacements and displacement gradients with fewer calculations than

are required by the traditional coarse-fine search. The Newton-Raphson method still

requires Finite Element smoothing to produce displacement gradients that are useable

when actual displacement gradient component magnitudes are smaller than one percent.









The primary goal of the current study is to develop an iterative technique for the

determination of in-plane displacements and strains with reduced computation. The

Perturbation/Least-Squares (PLS) and Perturbation/Correlation (PC) algorithms are

developed to allow the iterative computation of displacement components. Published

results, and analytic models of the correlation process, developed in the current study,

indicate that displacement gradients should generally not be computed by direct

correlation. The program SPECKLE, developed by the author for the current study,

computes strains from center-point displacements in regions sufficiently far apart that the

strains can be computed accurately. Average normal strains are computed in three

non-parallel directions and then transformed into normal strains and a shear strain using

strain-transformation equations. The resulting technique avoids much of the

computational overhead of Digital Correlation and does not require Finite Element

smoothing.

The program SPECKLE is a Microsoft Windows program that runs on personal

computers based on the Intel 80386 or higher processor. SPECKLE implements the PLS

and PC methods of displacement computation, and normal and shear strain calculation

based on strain transformation equations. Experiments performed in the current study

indicate that SPECKLE may be used with an 8-bit frame grabber to measure normal and

shear strains with maximum errors on the order of 100 micro strain.














INTRODUCTION


Experimental Stress Analysis is a broad discipline, a branch of which is concerned

with the measurement of both stress and strain in structures and machines. The current

study is concerned with the experimental determination of in-plane displacements and

strains on the free surfaces of structural elements. Data is acquired and processed using a

personal computer equipped with digital image capture hardware connected to a television

system.

The Concepts Of Stress And Strain

Detailed descriptions of the theories of stress and strain are beyond the scope of

this document; however, some general comments are appropriate. Stress is a measure of

internal force per unit area in a continuous medium [Eisenberg, 1980, p. 57]. Strain is a

purely geometric measure of deformation. Normal strains are related to the change of

length per unit length, and shear strains are related to the reduction of what were

originally right angles [Eisenberg, 1980, p. 78]. The in-plane strains on the free surface of

a structural component are generally described by choosing a set of orthogonal coordinate

axes in the plane and then indicating normal strains in each of the coordinate directions

and a shear strain. These strains can be measured experimentally by measuring normal

strains in three non-parallel directions in the plane. It is then possible to use

transformation equations to determine the normal strains and the shear strain in any









desired coordinate system [Eisenberg, 1980, p. 158]. If the material in question is

isotropic and its Poisson's ratio is known, then the normal strain in the direction

perpendicular to the surface may be determined. The principal strains and maximum shear

strain may then be computed [Dally and Riley, 1978, p. 320]. It must be mentioned that it

is not possible to measure strain at a point. One must measure average normal strain over

some length. As the length decreases the average strain approaches the value of the

normal strain at the point of interest. Unfortunately, as the gage length decreases, so does

the accuracy to which the average normal strain can be determined [Dally and

Riley, 1978, p. 129].

Techniques Of Experimental Stress Analysis

This section provides a brief overview of some of the most common techniques for

the measurement of in-plane strains for objects in states of plane stress.

Electrical Resistance Strain Gages

Electrical resistance strain gages are widely used to measure strains on free

surfaces of structural elements. Strain gages can generally be mounted directly on a

prototype component. It is also possible to incorporate strain gages into instruments such

as load cells [Dally and Riley, 1978, p. 262]. The primary disadvantage of the use of

strain gages is that one or more gages must be mounted at each point on the structural

element under test at which the strain is to be determined. It is also necessary to know in

advance the points at which strain must be measured.








Optical Methods

Many optical methods for the determination of both in-plane and out-of-plane

displacements have been developed. Optical methods generally have the following

advantages:


1. Optical methods generally involve the formation of an image of some type that

allows data to be collected over a large region in one test.

2. Optical methods are non-contacting, so they do not alter the behavior of the test

specimen.

Optical methods generally suffer the following disadvantages:


1. The information collected is in the form of interference fringes that might not be

directly related to the value of the quantity of interest.

2. Optical methods often require the processing of a photographic film or plate before

data can be extracted from the collected images.

The strengths and weaknesses of optical methods vary widely between methods

and often must be assessed in terms of specific applications. An excellent example is

provided by the fact that photoelasticity [Dally and Riley, 1978, p. 406] generally suffers

the disadvantage that a model of the object under test must be constructed. This

statement does not apply when the prototype material is a birefringent material, as is true

for some transparent plastics.








Goal Of The Current Study

The introduction of computer hardware capable of capturing digital images from

solid-state television cameras provides an opportunity for the development of new

experimental techniques for the measurement of displacements and strains in solids.

Interferometric optical methods of displacement measurement are sensitive to

displacements on the order of the wavelength of light and often require vibration isolation.

Many applications, however, do not require interferometric sensitivity. The goal of the

current study is to develop a method for the determination of in-plane displacements and

strains that does not require vibration isolation and exploits recent advances in personal

computer technology. A secondary goal of the current study is to investigate the

sensitivity of displacement gradients calculated via Digital Correlation methods [Chu et

al., 1985] to changes in the computed rigid body displacement.

Previous Work

Electronic Speckle Metrology was first conceived as an electronic analog to

Speckle Photography. As the work progressed it began to resemble the Digital

Correlation and Automated Grid methods.

Speckle Photography And Computer Speckle Interferometry

Speckle Photography is a single-beam technique for the measurement of in-plane

displacements. A single beam of coherent light is used to illuminate the object under test.

The object is photographed in its undeformed configuration, loaded and photographed

again on the same photographic plate. The resulting double exposure specklegram locally

has the appearance of the sum of a speckle pattern with a translated copy of itself.









Interference fringes result when such a double exposure pattern is Fourier filtered [Burch

and Tokarski, 1968]. Speckle Photography has been highly developed as an optical

method for the measurement of displacements. Most applications of Speckle Photography

do not require vibration isolation, which can be a distinct advantage over methods such as

Holographic Interferometry [Jones and Wykes, 1989, p. 64]. Chiang et al. [1982] provide

a detailed explanation of the application of Speckle Photography. Khetan and Chiang

[1976] describe single aperture recording methods, and Chiang and Khetan [1979]

describe multiple aperture recording methods for speckle photography. Strains are

obtained by differentiation of the measured displacement fields [Khetan and

Chiang, 1976]. Chen and Chiang [1990] developed an electronic analog of Speckle

Photography, called Computer Speckle Interferometry. The book by Jones and Wykes

[1989] provides an excellent description of optical methods for experimental stress

analysis.

Digital Correlation Methods

Digital Correlation methods using white-light speckle patterns have been applied

to both rigid body and deformnable body mechanics problems, most prominently by

researchers at the University of South Carolina. Peters and Ranson [1982] described the

advantages of digital correlation over methods that require the formation and analysis of

fringe patterns. Peters et al. [1982] used Digital Correlation to measure displacements,

which were subsequently used as input to a boundary integral equation method to

determine surface traction on a contour. Stresses within the boundary defined by the

contour were computed from the surface tractions. Sirkis and Taylor [1990] used a









similar technique for Elastic-plastic stress analysis, and noted that bounds on the accuracy

of displacement measurements made with digital computers should be expressed in terms

of pixels for comparison. Peters et al. [1983] used Digital Correlation to solve problems

in rigid body mechanics. Chu et al. [1985] applied Digital Correlation to deformable body

mechanics, and noted that the computed displacement gradients were very sensitive to the

computed values of the displacement components. Sutton et al. [1991] developed a

method for smoothing the displacement fields obtained by correlation, allowing

displacement gradients to be computed. The primary disadvantage of Digital Correlation

methods is that they are computationally intensive. Bruck et al. [1989] developed a

Newton-Raphson algorithm that reduces the amount of computation required to

implement correlation methods. Chen and Chiang [1992] applied Fourier Transform

analysis to the study of Displacement-only Digital Correlation with laser speckle patterns,

arriving at theoretical limits for the accuracy of such correlation methods and bounds for

allowable displacement gradient magnitudes.

Automated Grid Methods

Electronic Speckle Metrology is closely related to Automated Grid methods,

which have been studied extensively. Sirkis [1990] provides a summary of previous work

in grid methods. Sirkis [1990] also provides a detailed analysis of a particular Automated

Grid method, and concludes that the fill factor, signal to noise ratio and grid spot diameter

control the accuracy of the method considered. Sirkis and Taylor [1990] and Fail and

Taylor [1990] developed experimental techniques, based on the use of CCD cameras,









which can be classified as grid methods. Sirkis and Lim [1991] studied the accuracy of

automated grid methods.

Hybrid Methods

Optical methods often provide information in the form of fringe patterns, which

must be analyzed to provide useful information. Many hybrid techniques that use image

processing to analyze fringe patterns have been developed. Chen [1985] and Gillies

[1988] applied image processing to the analysis of photoelastic fringes. Erbeck [1985]

and Ineichen et al. [1980] applied image processing techniques to Speckle Photography

fringes. Electronic Speckle Pattern Interferometry (ESPI) is a technique that combines an

optical interferometer with a television system to eliminate the photographic processing

step required by Speckle Photography. ESPI was first demonstrated by Butters and

Leendertz, and is described in detail by Jones and Wykes [1989, p. 165].

Electronic Speckle Metrology

This section provides an overview of Electronic Speckle Metrology as

implemented for the current study. The program SPECKLE, developed by the author to

support the current study, implements the computational methods described in this

document. The appendix contains a detailed description of SPECKLE's user interface.

Experimental Configuration

Electronic Speckle Metrology is based on the assumption that the test specimen

has some intrinsic texture which moves with the test specimen during deformation. The

intrinsic texture may be added as part of the specimen preparation, resulting in a









white-light speckle pattern [Asundi and Chiang, 1982], or it may be produced by laser

speckle. The use of laser speckle proved to be impossible with the optics available to the

author, so white-light speckle patterns were used in all experiments described in this

document. It should be noted that laser speckle can be resolved by CCD cameras given

appropriate optics [Chen and Chiang, 1992]. The experimental configuration includes the

following hardware:


1. a light source

2. the test specimen and any required loading hardware

3. a television camera, frame grabber, and computer.

The camera should be oriented normal to the plane in which the test specimen is to be

deformed. A schematic of the experimental configuration is shown in Figure 1.

SLight Source

H Camera and Lens




Object




Computer and Frame Grabber Monitor

Figure 1. Experimental configuration.


The experimental procedure is essentially as follows:









1. Prepare the specimen for the test, and assemble the specimen loading apparatus.

2. Place any required pre-load on the test specimen.

3. Capture a digital image of the test specimen.

4. Load the test specimen as required to produce the desired deformation.

5. Capture a digital image of the test specimen in its deformed configuration.

6. Analyze the undeformed and deformed image pair using SPECKLE.

The SPECKLE Program

The program SPECKLE is a Microsoft Windows application, developed by the

author for the current study, that provides a basic implementation of Electronic Speckle

Metrology. Figure 2 shows the SPECKLE main window with its undeformed and

deformed image viewers, which must always be open, and optional scan line and

histogram windows. The user may size the windows freely, and use the scroll bars on the

image viewers to view any desired portion of the image. SPECKLE uses the positions of

the scroll bar thumbs to interpret mouse clicks by the user, so it is not necessary to scroll

the two windows to the same location. The user may load undeformed and deformed

images of an object and determine displacements and average strains in regions as desired.

The user must indicate the locations of three features in both the undeformed and

deformed images. The locations do not need to be precise since SPECKLE uses either the

Perturbation/Least-Squares (PLS) or Perturbation/Correlation (PC) method, both of

which are developed in this document, to improve the estimates of displacement resulting









from the user's input. The resulting displacements are used to determine the average

strains in the region in which points were selected by the user


Figure 2. A view of the SPECKLE user interface.

Dynamic Histogramming

The images used in the current study were captured with the program DT2953,

which was written by the author to support the Data Translation DT-2953 frame grabber

for the IBM Micro Channel architecture. The program DT2953 allows the user to

activate a real time histogram of a portion of the frame grabber. The histogram display is

updated at approximately two hertz, allowing the user to set the light source intensity and

camera aperture. The lower right window shown in Figure 2 contains a typical histogram

for black dots on a white surface when the light level is chosen properly. Note that most

of the range of the frame grabber is being used. Assuming that the test specimen is white

with black dots, the light level is controlled by first aligning and focusing the camera and

then adjusting the light source and lens aperture to admit as much light as possible without


fie Wlndow Modes Launch
_-I ^~m.., ..,- ,-,.. __image -I_1
0255



JL+ I_ a* 59
" Undeformed Image I- -* ."- Undeformed Image "I I ^
IW + 0 13782



LA I I o 255









clipping, which occurs when light intensity levels exceed the allowable input range. The

adjustment is simple since the histogram responds to changes almost immediately. A

typical dynamic histogram is shown in Figure 3. No peaks corresponding to dark features

are visible because the scanned region does not include any such features. The most

important regions to scan are the bright regions, so that clipping may be avoided.


03 1259




0 255


Figure 3. A dynamic histogram window.

Overview Of This Document

This section briefly describes the purpose of each of the chapters in the body of

this document and of the appendix.

Basic Theory And Notation

The chapter "Basic Theory And Notation" describes a continuum model of the

operation of the frame grabber and defines notation which is used in subsequent chapters.

The fundamental assumption of the methods described in this document is that the image

of the object deforms with the object. It is shown that this property is preserved when the

frame grabber distorts the image.








Displacement Refinement

The chapter "Displacement Refinement" develops an algorithm for the iterative

computation of displacements, and also develops the Perturbation/Least-Squares (PLS)

and Perturbation/Correlation (PC) methods of displacement refinement.

Analytic Results On Errors

The chapter "Analytic Results On Errors" presents analytic studies of errors due to

out-of-plane displacement and noise. It also explores the numerical conditioning of the

use of correlation for the determination of displacement and displacement gradients. The

models considered are all one-dimensional and idealized. The emphasis of the chapter is

on obtaining qualitative results of the greatest possible generality; therefore, the numerical

bounds obtained are often not precise.

Preparation For Experiments

The chapter "Preparation For Experiments" describes the calibration of the

experimental system and the construction of test specimens and their loading or translating

mechanisms.

Experimental Results

The chapter "Experimental Results" presents the results of experiments performed

to verify the correctness of the PLS and PC methods, and the implementation of these

methods in SPECKLE.








The SPECKLE User Interface

The Appendix provides a detailed description of SPECKLE from a user's

perspective. It also describes the theory of SPECKLE's calibration method.

Summary


This chapter provides an overview of Experimental Stress Analysis, and the place

of Electronic Speckle Metrology among other experimental methods. The current study

has two distinct goals: the development of a new experimental method, and the study of

sensitivities in existing methods. The review of literature presented in this chapter is

extended throughout this document as appropriate.













BASIC THEORY AND NOTATION


The purpose of this chapter is to provide a foundation for the development of

iterative displacement correction algorithms in subsequent chapters. Abstractions which

facilitate the use of the experimental data are defined. In particular the concept of the

motion function of Continuum Mechanics is extended to coordinates that are natural for

use with the experimental data. The resulting continuum model is used as a guide for the

interpretation of discrete experimental data throughout this document. The fundamental

assumption of Electronic Speckle Metrology is that the image of the object deforms with

the object. It is shown that this property is preserved when the idealized frame grabber

distorts the images.

Mathematical Notation

This section provides a brief discussion of notation commonly used by

mathematicians. The conventions described are used throughout this document to allow

complicated expressions to be written in compact forms. The benefits of writing

expressions using this notation is apparent in the chapters "Displacement Refinement" and

"Analytic Results On Errors."

Function Definitions

We will consider a function to be a mapping from one set into another set

[Bruck, 1978, p. 21]. Suppose that U and V are sets, and fis a function from U into V.









Then we will write f: U -+ V. The element of V corresponding to a given element of U is

sometimes indicated by the notation
f:U->V
u:- f(u)
where f(u) is replaced by the value given by fwhen applied to the element u. The set U is

called the domain of f. The symbol 91 is used to denote the set (-co,oo).

Sets And Images Of Sets

A common notation for sets is { x : P(x) }, which is read as "the set of all x such

that the property P(x) is true" [Herstein, 1975, p. 2]. This notation allows very compact

and readable statements. Another concept which is used in the developments to follow is

that of the image of a set under a function. Let f: U-+V be a given function, and let

WcU. Then we define the image of W under fby
f(W) = {f(u) V: u W}
which is the set of all values that result when f is applied to each element of U.

Operations On Functions

It is often convenient to consider functions which arise from combinations of

existing functions. Let functions f: F -+ 91 and g: G -> 9? be given. Define the sum of f

and g by
f+g: F rG-- 9R
x :-> f(x) + g(x)
The difference, product and quotient of real-valued functions are defined similarly.

Another very common way that new functions arise is by composition. Given the

functions f and g defined above, define the composite function
fog:H-+9?
x :-> f(g(x))









where the set H is the set of all points x in the domain of G such that g(x) is in the domain

of f The symbol fo g may be read as "f compose g" or "f composed with g." Section

1.6 of the book by Leithold [1981] contains a description of the operations described in

this section. Sum, difference, and composite functions and function inner products [Boyce

and DiPrima, 1977, p. 463] are exploited throughout this document to simplify

expressions involving integrals.

Topological Descriptions

In this section we formulate a continuum model of the operation of the frame

grabber. This model will be applied, in subsequent chapters, to the measurement of the

plane motion of a test specimen from images of the specimen in its undeformed and

deformed configurations. We begin by considering the light intensity patterns to which the

light sensitive elements of the camera are exposed and the mapping of these intensity

patterns to gray levels in a logical image space on the frame grabber. We then describe

conditions which allow us to compute motions using digital image coordinates. By

analogy to the theory of continuum mechanics, we assume that all functions entering the

development are continuous [Malvemrn, 1969, p. 1].

Spatial Relationships

The physical image plane is the plane containing the light sensitive surface of the

camera. The logical image plane is the imaginary plane through the sampled data

produced by the frame grabber. This section defines sets and functions which allow us to

describe the spatial mapping of images from the physical image plane to the logical image

plane. We begin with the following definitions:









1. The sets P and L are the physical and logical image planes, respectively.

2. The camera spatial function c: P -- L, assumed to be invertible and continuously

differentiable, maps each point in the physical image plane to the corresponding

point in the logical image plane.


3. SA is the set of all points in the physical image plane such that the light intensity

due to the test specimen in its undeformed configuration is non-zero.

A
4. SB is the set of all points in the physical image plane such that the light intensity

due to the test specimen in its deformed configuration is non-zero.

5. SA = c(SA)is the set of all points in the logical image of the test specimen in its

undeformed configuration.

6. SB = cSBj) is the set of all points in the logical image of the test specimen in its

deformed configuration.

7. We assume that there exists a unique invertible and continuously differentiable

function x : SA -- SB, called the physical image motion, that maps each point in

the physical undeformed image to the corresponding point in the physical

deformed image.

8. The logical image motion
X: SA -- SB
X coioc-I(X)

maps each point in the logical undeformed image to the corresponding point in the

logical deformed image.









We make the additional assumptions that c' and 97' are continuously differentiable.

Note that since c, c' and are invertible we have
x-1: SB -+ SA
X:-*cox- oc-(


Since c, c1, k, and *7' are continuous and since compositions of continuous functions are

continuous it follows that x and x' are continuous. The continuous differentiability ofx

and x"' follows from the continuous differentiability of c, c"', *, and R-1, and the chain rule.

See Rudin [1964] for discussions of continuity, continuous differentiability and the chain

rule. The sets and functions described above are shown in Figure 4.



---SO, \ x\-- ^ ~. --





C
P L


Figure 4. Spatial mapping from the physical to logical planes.

Description of Intensities

We now add descriptions of the luminous intensity patterns impinging on the light

sensitive surface of the camera, and of the corresponding gray level patterns. The

functions which must be added to the existing topological description are as follows:









1. The camera intensity function y: [0, oo) -+ [0, oo), assumed to be invertible and

continuously differentiable, maps luminous intensity values in the physical image

plane to gray level values in the logical image plane.

2. The physical undeformed intensity A: SA -+> [0, oo), assumed to be continuously

differentiable, describes the light intensity pattern impinging on the sensitive

surface of the camera when the test specimen is in its undeformed configuration.

A A1
3. The physical deformed intensity B: SB -+ [0, oo), assumed to be continuously

differentiable, describes the light intensity pattern impinging on the sensitive

surface of the camera when the test specimen is in its deformed configuration.

4. The logical undeformed intensity
A: SA -+ [0, Oo)
X :-+yoAoc-(X)

describes the gray level pattern that corresponds to the physical undeformed

intensity.

5. The logical deformed intensity
B: SB -+ [0, oo)
X :- yoBo oc-I o x-'(X)

describes the gray level pattern that corresponds to the physical deformed

intensity.

Note that by the assumptions and results above and the chain rule, the functions A and B

are continuously differentiable. The complete topological description is shown in

Figure 5, which may be more easily understood by comparison with Figure 4.























Luminous Gray
Intensity Level


Figure 5. Topological description of intensities.


Note that the physical and logical images must satisfy the equations

A = yoAoc-1
Box = yoioioc-x

and

A = y- loAoc
Bofi = yoBoxoc

The Use Of Logical Image Plane Coordinates

Experimental techniques which use photographs of a test specimen to measure its

motion are based on the assumption that the image of the object deforms in a predictable

way as the test specimen deforms. For the purposes of the current study we expect one of

two scenarios:









1. If the light source intensity remains constant then we expect the physical deformed

intensity composed with the physical image motion function to be equal to the

physical undeformed intensity.

2. If the light source intensity changes between exposures then we expect the physical

deformed intensity composed with the physical image motion function to be a

scalar multiple to the physical undeformed intensity.

Both of the above statements assume that the object is plane, the object deforms in its

plane, and that the physical image plane is parallel to the object plane. We now state and

prove a Lemma specifying conditions which are sufficient to allow us to compute

deformations in logical image coordinates.


Lemma. In terms of the notation described above

1. Box = A ifandonlyifBoi = A.

2. If the camera intensity function y is linear then

i. If there exists a real number, a, such that B o x = aA then B o x = aA.
*A A
ii. Ifthere exists real number, a, suchthat Box =aA then B o x = aA.


Proof The proof of 1 is as follows:.

Suppose B o x = A. Then the relations
A = yoAoc-~
Box = yoBoXoc-~
imply B o x = A. Suppose B o x =A. The the relations
A = Y-I oAoc
Box = y-oBoxoc
imply B ox = A.









To prove 2.i, suppose there exists a real number, a, such that B o x = aA. Note that the
relations

A = yoAoc-'
A
Box = yoBoxoc-I

and the linearity ofy imply
A A
Box = yoBoxoc I


= a(yoAoc-1)
aA

The proof of 2.ii is similar. QED.


The above lemma states conditions sufficient to guarantee that the logical image

deforms with the logical image motion if and only if the physical image deforms with the

physical image motion. These results allow us to compute the motion of the object in the

logical image plane without considering distortions caused by the camera and frame

grabber. It is necessary to consider these distortions when interpreting the results in terms

of the object.

Computational Descriptions

In this section we develop notation to simplify the use of input data in the form of

digital images. Unless otherwise stated, all formulations are in terms of logical image

coordinates.

The Point And Region Of Interest

The displacements and strains computed by the techniques described in this

document are all computed as averages over a region. The region, called the Region of









Interest, is a square array of points and is specified by a center point, called the Point of

Interest, and a linear dimension. We denote the Point of Interest by XV) and the points in

the Region of Interest are given by

I) =x(- m-int(R/2) pixels forall m,n {O,-..,R-1}
n n-int(R/2) )

where R is the linear size of the Region of Interest, and int is the greatest integer function

[Leithold, 1981, p. 44]. Note that the Point of Interest is truly the center of the region

only if the region size is odd. For convenience we define the index set

A={(m,n): m,nE {0,---.,R-l}}

to index the Region of Interest. Note that X)#X').

Sequence Of Approximate Motions

The techniques for the determination of displacement to be considered in this

document are iterative. To facilitate the discussion of such schemes, we define a sequence

of approximate motions of the form
x(k) : SA SB
X:-. X+u(k)

for all positive integers k, where the constant displacement vectors u(),u(),... are to be

determined.

Sampled Data

The governing equations for computation developed in the following chapter

involve values of or integrals involving the functions A, and B o x(k00, D I B o x(k), and

D2B o x(k) for all positive integers, k. To facilitate the development of the governing









equations, define samples of the undeformed and deformed intensities and of the partial

derivatives of the deformed intensity
A = x = A(X("n))

DB D = B o x (X(mC))
D IB(Mn = D B o x0)(X(mn))

D2B" = D2B ox(Ck)(X())

for all (m, n) E A and for all k { 1,2, ... It is necessary to determine approximations

to these values based on the experimental data. The remainder of this section defines

quantities which may be used to approximate the samples of the logical intensities.

Digital Images

Consider the problem of approximating the logical intensities by data collected in

the form of digital images. Let S be the set of all points at which sampled data is collected

using the frame grabber. Let functions
a, 3: S -+ [0, 0)

describe the digital image intensities of the test specimen in its undeformed and deformed

configurations, respectively. Let the functions
A,B: L-91

be interpolated extensions of the functions a and (3, respectively, where L is the logical

image plane. The method of interpolation is arbitrary. SPECKLE currently supports only

bilinear interpolation, but nonlinear interpolation methods are often preferable

[Sutton, 1988]. Define samples of the interpolated intensities by
-= oXA(X(mn))
fQ(k = -. X ^)(Xm








for all (in, n) e A and for all positive integers, k. For each positive integer, k, the

approximations, AiBm and A2Bm for all (rm,n) e A, to the partial derivatives of the

deformed intensity are obtained either through finite difference approximations based on

the values of B9 for all (em,n) e A or by differentiation of the interpolated functions.

Summary

This chapter provides the foundation for the iterative displacement computation

algorithms to be developed in the next chapter. The collection of data using a television

camera is described in terms of point set topology. The light intensity patterns on the light

sensitive portion of the camera and their transformation into gray level patterns on a

logical image plane coincident with the frame grabber are described using a continuum

model. The chapter concludes with a development of notation to facilitate the use of

digital images for computation.














DISPLACEMENT REFINEMENT


This chapter describes the Perturbation/Least-Squares (PLS) and

Perturbation/Correlation (PC) methods for the determination of displacement in two

dimensions. Both methods require a pair of images of the test specimen, one image each

before and after deformation. The implementations of the two methods share many

common features, and these common features have been used as a guide in the

development of SPECKLE.

Iterative Computation Of Displacement

This section provides an overview of the iterative computation of displacement

using digital images of a test specimen in its undeformed and deformed configurations.

SPECKLE relies on the user to indicate a base displacement as an integral number of

pixels. The user may use a mouse to click on a given feature in both the undeformed and

deformed images or enter the displacement components in a dialog box. The details of the

techniques for indicating a base displacement are described in detail in the Appendix. The

estimated displacement provided by the user is refined through a sequence of iterations as

follows:


1. Select the interpolation and differentiation algorithms, and the displacement

refinement algorithm. SPECKLE currently supports only bilinear interpolation and

a second order central difference differentiation algorithm. A displacement








refinement algorithm is selected by choosing the corresponding Displacement

Engine. See the Appendix for details.

2. Initialize the approximate displacement u) using the estimate provided by the user.

Use the undeformed and deformed intensity samples to initialize the values of

An and Bn for all (m, n) e A. Use the selected numerical differentiation

algorithm to compute the values of A1B and A2B() for all (m,n) e A, based

on the values of B for all (m, n) e A. Set the iteration number, k, equal to one.

3. Compute u(1" using the selected displacement refinement algorithm. The

displacement refinement algorithms available to users of SPECKLE are developed

in this chapter.

4. If Ju')-uck) is sufficiently small or if the maximum allowed number of iterations

has been exceeded then stop.

5. Use the interpolation algorithm to compute 1) for all (m, n) e A, based on the

values of BR for all (mn) e Aand the value ofu ). Use the selected numerical
differentiation algorithm to determine Al,&B, and A2B'(k) for all (mn) A,
based on the values of B=Ik1m for all (m,n) e A.

6. Increment k and go to Step 2.

The PLS and PC methods are described in terms of a single step that may be used

in Step 2 of the displacement computation algorithm, a flow chart of which is shown in

Figure 6.


























Iteration Failure


Figure 6. Flow chart of displacement iteration algorithm.

The Perturbation/Least-Squares Method

In this section we develop the PLS method governing equations, which are based

on the assumption that the deformed intensity composed with the motion should equal the

undeformed intensity. We use tangent plane approximations to the deformed intensity

resulting in a perturbation equation, which relates the displacement components to the

difference between the undeformed and deformed intensities. We then determine the

required changes in the displacement components to minimize the squared error between

the expected gray level differences and the observed gray level differences. The









development of the method begins with a heuristic argument and then proceeds to a

precise development of a numerical algorithm. The notation developed in the chapter

"Basic Theory and Notation" is used throughout the development.

Heuristic Development

Suppose that k steps of the displacement iteration have been completed, so that the

displacement vectors
um),..., U(k)

are known. Assume that the speckle pattern moves with the test specimen during

deformation and that the light source output remains constant. Then we expect the

undeformed and deformed intensities to satisfy
Box=A

To improve the current estimate of displacement we seek the value of a constant

displacement vector u such that the approximate equation
B(x(k)(X) +u) wA(X)

is as true as possible throughout the Region of Interest. Substitution of a tangent plane

approximation for the value of the deformed intensity results in the approximate equation
A(X) B o x(k)(X) + D1B o x(k)(X)ul +D2B o x(X)u2

for all X in the Region of Interest. The above motivates us to seek u such that
A.n-B m DiB^uj +D2Bmu2
in the Least Squares sense for all (m, n) E A. To complete the iteration step, let
U(k+l) = U(k) +u

At this point control is returned to the displacement iteration algorithm.








Precise Formulation Of The PLS Method

Assume that k iterations of the displacement computation have been completed, so

that the displacement vectors
U ) ...,u )

are known. Using the preceding heuristic development as a guide, define matrices M, MNf

and N2) by
M~~ -A B
M(I) -- AIR 0)
mnm Bmn
Mn =,&2B=
for all (m, n) e A. Define a squared error function,
E: 92-,
v :-+ llviMC) + v2M2) M112

Let u be a displacement vector which satisfies
E(u) = if{E(v) :v e= 2}

It can be shown that the space of n-dimensional matrices with real coefficients is an inner

product space with inner product, :, defined by

a:b=EZ;,j (aijbj)]

for all matrices a and b [Malvern, 1969, p 35] [Herstein, 1975, p. 191]. Hence it suffices

to solve the linear algebraic system [Davis, 1975, p. 176]
C M1): M(I) M(1): M(2) ui') ( M<1): M
M(2): M() M(2): M(2) u2 M(2): M

To complete the iteration step let
u(k+l) = u(k) +u
At this point control is returned to the displacement computation algorithm.








The Perturbation/Correlation Method

In this section we develop the Perturbation/Correlation (PC) method governing

equations, which are based on the assumption that the deformed intensity composed with

the motion is proportional to the undeformed intensity. An important advantage of Digital

Correlation techniques is that they can be made tolerant of changes in light intensity

between the capture of the undeformed and deformed images by normalization of the

correlation function. The development of the method begins with a heuristic argument

and then proceeds to a precise development of a numerical algorithm. The notation

developed in the chapter "Basic Theory and Notation" is used throughout the

development.

The Newton Raphson Method Applied To Digital Correlation

The primary disadvantage of Digital Correlation methods has been that they are

computationally intensive. Bruck et al. [1989] developed a Newton-Raphson based

algorithm for determining displacements and displacement gradients with reduced

computation by digital correlation, but it requires the evaluation of first and second order

derivatives of the correlation function. Displacement gradients must be on the order of

0.01 to be accurately determined by Bruck's method, unless a penalty method finite

element smoothing scheme is used to smooth the computed displacements [Sutton, 1991].

The PC method also provides a differential correction algorithm for Digital Correlation,

but requires estimates of only the first order partial derivatives of the sampled images. It

should be noted that the current form of the PC method does not determine values of the

displacement gradients.







Heuristic Development

We begin by applying the Schwarz Inequality to the undeformed intensity and a

tangent plane approximation to the deformed intensity. The tangent plane approximation

allows direct solution for a displacement correction which minimizes the difference

between the two sides of the resulting inequality. For integrals of real-valued functions, a

and b, the Schwarz Inequality has the form
(f a(X)b(X)do) 2 < (f a2(X)doC) (f b2(X)do)
with equality holding if and only if a and b are scalar multiples of one another. Let the

functions A and B describe the undeformed and deformed intensities respectively, let u)

be the approximate displacement vectors, and let the functions x() be the approximate
motions. See the chapter "Basic Theory And Notation" for details. To develop the

iterative algorithm, assume that k iterations have already been completed. Then the

displacement vectors
U(1),.., U(k)

are known. Let u be a constant displacement vector and note that
B(x>(k)X)+u) Bo x)(X) +DiBo x(k)(X)u, +D2Bo x, (X)U2

The Schwarz Inequality gives

[i A(X)(B o x(X) + D B o x(k)(X)ui + D2B o x()(X)u2) do]

< [IfJA2(X)d][J(B o x(k,)(X) + DB x(k)(X)Ul +D2B o x
which, expressing the integrals as inner products, becomes
[(A, B o x) + (A, DiB o x(k))ui +(A,D2B o x(k))U2]2
< IIAII2IIBox +uD Box(kC) +u2D2Box(k)112
We seek the displacement, u, that minimizes the difference between the two sides of the








above inequality. Define an error function,
E": 912 -+ [0, 00)
v :-+ IIAII2IIBox) +vDB o x (k) + vIDIB v2D2B x(k)112
[(A, Box(k))+(A, DiBox(k))vi +(A,D2Box(k))v2 ]2

so that E(u) represents the quantity to be minimized. The displacement vector u should be

chosen such that
E(u) = inf{E(v): v e 912}

Hence we choose u such that
DiE(u) = 0
D2E(u) = 0

It can be shown that under these conditions u must satisfy the following equations:
IIAII2(B x(k), DB o x))- (A B x(k))(A, DIB x(k))
= [(ADIB ox(k))2 IIAI2lIDiBoxxk)Il2] u
+ [(AD2Box(k))(A,DlB o x(k)) IIA112(D2B o x(k), D I B o x(k)]u2

(la)
IIAI12(B-oxC),D2Box(k))- (ABox(k))(AD2Box(k))
= [(A, DiBox(k))(A, D2Box(k)) lIAl12(DB ox(k), D2Bo x(k))]u
+ [(A, D2B o x(k))2 IIAIl2ID2B ox(k)112]u2

(lb)

The above equations provide two simultaneous linear algebraic equations in the

components ofu. To complete the iteration step, determine the components ofu which

satisfy the above equations and let
u(k+l) = u) +u
At this point control is returned to the displacement iteration algorithm.







Precise Development Of The PC Method

Assume that k iterations of the displacement computation have been completed, so
the displacement vectors
u(),. .,u(k)
are known. Approximating the integrals in equations la and lb by Riemann sums, let u
be the displacement vector which satisfies the matrix equation Mu = C where






MC2 = (X1)2 [ ( A ) ] ( ) ( n )]
M [Ii=|E .AuAB J -k) 2][Yu _(k))2]

B=I=

Ml = [E (A B) T]2_[ (;A) 2][Y (Ak) 21
M22 = [I (;ABA2BmnJ-E Am (B=)

and all sums are over all (mn, n) r A. To complete the iteration step, let
u(k+l) = u(k) +u
At this point control is returned to the displacement computation algorithm.

Summary

This chapter describes the foundation of the displacement computation algorithm
used by SPECKLE, beginning with a description of a high-level iterative algorithm for the
computation of the average displacement in a specified region. The displacement
computation algorithm is written in terms of an unspecified displacement refinement
algorithm. The remainder of the chapter develops the PLS and PC methods for
displacement refinement. Both methods use tangent plane approximations to allow direct









solution for an estimate of the required displacement correction. It is important to note

that no conditions for convergence of the PLS and PC methods have been determined;

however, the methods have performed well in practice, as results summarized in the

chapter "Experimental Results" demonstrate. It is also important to note that both

methods produce zero corrections if the exact motion is determined, as can be seen by

substitution into the governing equations:

1. The PLS method produces a zero correction if Box0o = A.

2. The PC method produces a zero correction if there exists a constant, a, such that
Box0 = aA.














ANALYTIC RESULTS ON ERRORS


This chapter represents an attempt to explain the errors which are inherent in the

use of digital imaging systems to determine the motion of a test specimen from changes in

observed gray levels. The chapter begins with a brief summary of published numerical and

experimental work directed toward understanding errors in the use of vision systems in

experimental solid mechanics. Qualitative analyses of the following are presented:


1. errors in normal strain due to out-of-plane displacement

2. errors in in-plane displacement measurements due to quantization errors and noise

3. sensitivity of the displacement gradients computed by the Digital Correlation

method to changes in the assumed rigid body displacement.

One-dimensional continuum models are used throughout for simplicity. It should be

emphasized that the purpose of all analyses presented in this chapter is to obtain general

qualitative results. Combined numerical and experimental studies have been performed by

Fail and Taylor [1990], Sirkis [1991], and Sutton et al. [1988]. A combined analytic and

experimental study has been published by Chen and Chiang [1992].

Published Quantitative Results

The work published by Sirkis of current interest has been in the area of Automated

Grid methods. Sirldkis found that the fill factor, signal to noise ratio, and the diameter and









intensity profile of grid spots significantly affect the accuracy of Automated Grid methods

[1990]. Additional numerical and experimental studies were performed by Fail and Taylor

[1990]. Chen and Chiang [1992] used the Shannon Sampling Theorem to determine

theoretical limits for the accuracy of displacements measured by Displacement-only Digital

Correlation using laser speckle. Sutton et al. [1988] performed numerical and

experimental analyses of errors in displacements and displacement gradients measured

using the Digital Correlation method. Their conclusions included the following:


1. Up to a magnitude of 0.05, strains do not affect the accuracy of the computed

center-point displacement when 12-bit intensity quantization is used.

2. To achieve optimal accuracy in both displacement and strain calculations, it is

necessary to sample the signals at a frequency that is high relative to the frequency

of the signal. It is also necessary to use at least a 12 bit A/D converter and to use

nonlinear interpolation for sub-pixel intensity reconstruction.

The purpose of this chapter is to obtain qualitative results which are as

independent of the exact form of the undeformed and deformed intensities as possible.

Numerical estimates, when obtained, are not precise.

Effects Of Out-Of-Plane Displacement

It is clear that changes in apparent object size due to out-of-plane displacement

should cause errors in normal strains measured using vision systems, unless the

out-of-plane displacement is measured and considered as part of the analysis. The effects

of out-of-plane displacement on the measurement of normal strains using vision systems









have been described by Sirkis and Lim [1991] and by Peters et al. [1989]. A stereoscopic

vision system capable of measuring out-of-plane motions has been developed by Choa et

al. [1989]. This section presents an order of magnitude analysis of the effects of

out-of-plane displacement on measured normal strains when the out-of-plane displacement

is not considered in the computation. The chapter "Experimental Results" describes an

experiment which verifies the analysis of this section.

Measurement Of Strain Using A Pinhole Camera

Consider a pinhole camera imaging an object as shown in Figure 7.





Image
Plane .>
^^ H
f ^ ^^ ---- >
hi h2 Object
_LL





Figure 7. Definition sketch for out-of-plane displacement effects.


The object is moved a distance 8 away from the pinhole between exposures. The heights




H h
L 1








H _h+Ah
L+5 1

The measured strain, e, is then given by
Ah
h
_-5
L+8
5
L

(2)

which provides an estimate of the error to be expected in normal strain components, since

the body did not deform.

Pinhole Equivalents Of Optical Systems

We now consider a method for approximating optical systems by pinholes to

simplify estimates of the errors, due to out-of-plane displacements, in normal strains as

measured using vision systems. Consider a lens being used to form an image of an object

as shown in Figure 8. Suppose that the field of view has width W and the image has width

w. Note that D=L+l and that by similar triangles we have
W w
L 1

Hence the distance, L, from the object to the pinhole which produces an image of the

same size as the original lens is given by

L= D
1+-w
w(3)
(3)


where D is the distance from the object plane to the image plane.






40

A
Image Object
Plane Plane

-I
w/2W








Figure 8. Definition sketch for the pinhole equivalent to a lens.

Order Of Magnitude Analysis Of Displacement Accuracy

The purpose of this section is to provide an order of magnitude analysis of the

errors in displacement measurement due to noise. We begin by considering the effect of

noise in intensity data on the accuracy with which we may determine displacements.

Consider a one-dimensional image with a linear gray level distribution as shown in

Figure 9.


g
!- /
/ g=(G/h)X


h X


Figure 9. Change In Gray Level And Apparent Change In Position.








Note that a change in gray level Ag corresponds to a change in horizontal position AX,
and
AX Ag
h G

by similar triangles. We expect the error in displacement to be bounded as follows:
IAul| 21AXl

2h|Ag|

(4)
Using typical scales from current research, we choose Ag = 1 gl h = 10 pixels and

G=100 gl. We then have
lAu 2 10 pixels (
l100gl (Igi)

< 0.2 pixels

The next section of this chapter helps to explain how we are able to obtain displacements

more accurately than this conservative estimate might lead us to expect.

Dependence Of Displacement On Noise Magnitude

In this section we consider an idealized one-dimensional example involving a rigid

body displacement with noise superimposed on the image intensities. We will investigate

the dependence of the displacement determined by correlation on the amplitude of the

noise. The measured displacement is assumed to depend analytically on the noise

amplitude, and all functions are assumed to be continuously differentiable. We will use

implicit differentiation to compute the derivative of the measured displacement with

respect to a measure of the noise amplitude, and then perform an asymptotic analysis of

the resulting derivative for the case of sinusoidal noise functions of high frequency.








One-Dimensional Images And Motions

Consider a one-dimensional body which occupies positions described by the sets

SA =[Xo-h,X0+h] and SB in its undeformed and deformed configurations, respectively. Let

the function B: SB -> 91 describe the deformed intensity. For simplicity, we will consider

a rigid motion. Fix a displacement, 0, and let the motion be
x: SA -> SB
X:_+X+A

Let the undeformed intensity be
A: SA -> 91
X :--+ B o x(X) + an(X)
where ac, called the noise parameter, is a measure of the noise amplitude and n: SA -* 91

is a zero-mean noise shape function. The undeformed intensity is thus equal to the

deformed intensity composed with the motion when the noise amplitude is zero. For each

displacement, u, define the approximate motion
Xu : SA 91

X:->X+u

Comments On Scales

The functions defined above describe the undeformed and deformed image

intensities and a noise superimposed on the undeformed image. The actual noise

superimposed on the undeformed intensity is given by an. Later in this section we will

express the displacement determined by correlation as a power series in the parameter ca,

and neglect terms of second and higher order in a. We will assume that the values of A,

B, and In[ vary from zero to two hundred. A noise of two gray levels is thus obtained by

choosing a on the order of 0.01, which should be sufficiently small to justify neglecting








the higher order terms. The size scales have been chosen to match typical values from the

experiments performed as part of the current study.

Correlation Function And Measured Displacement

Define an inner product and a norm by
(a, b) = Js^ a(X)b(X)dX


for all integrable a, b: SA -+ 91. For each a e 91 define a correlation function


:-+ (A,BoxU)2
IIAII2 IIBoxuII2


The measured displacement is found by choosing the displacement which maximizes the

above correlation function. Assume there exists a unique function U: 91 -+ 91 such that

Ca(U(a)) =maxC.

for all ae9l?. Given a noise parameter, a, U(a) is the measured displacement. Note that

in particular XU(o) = x, which states that the measured displacement is equal to the exact

displacement when the noise amplitude is zero.

Determination Of The Displacement Estimate

The goal of this section is to determine how the measured displacement varies for

small values of the noise parameter, a. The measured displacement may be written as
U(a) = UI+U(O)a + O(a2)

(5)
Recall that the scales have been chosen so that a two gray level noise results when

a=--0.01. The equation which implicitly defines the estimated displacement is








C' ((a))= 0

To facilitate the computation of the derivative of the correlation function, define for each
a 6 91 functions
f : 9l1>9?
u :- (B o x+ an,B oXu)2
g.: 9-- 91
u :-4 IIBox+anll2I1BoxuI2

fc
sothatCa=-. It can be shown that

fi(u) = 2(B x + an, B o xu)(B o x + an, B/ o xu)
g9(u) 211Bo x + anl12(B o xu., B' o x)

The quotient rule for differentiation gives
C/ (u) -"(u)g(u) fa(u)ga(u)
^aW 2 U
ga(u)

Since CChas an extreme value at U(a), we have C,(U(a)) = 0, which implies

f/(U(a)) ga(u(a)) = f.(U(a)) g/(U(a))

Substitution of the expressions for f, and g'a gives
(B o x + an, B' o xu(a))IIB o xu()|I| = (B3o x + an,B xu(a))(B xu(a), B/ o xu(a))

which implicitly defines the measured displacement in terms of the noise parameter.

Differentiation with respect to a and subsequent evaluation when a=0 results in
[(n, B' o x) + (B o x, B1 o x)U'(0)]JIB o x112
+ (Box,B' ox)[2(B ox, B'ox)U(0)]
= [(n, B o x) + (B o x, B' o x)U'(0)](B o x, B' o x)
+ IIBoxll2[IIB'oxI2 + (B ox, B1ox)]U'(0)








which yields

u(o) = IIB xll 2(nB'ox)-(B-xB'ox)(nBo x)
lIBoxll2,B' oxll2 -(0BoxB'ox)2

(10)

Note that the Schwarz Inequality gives
lIB oxIll2IB' o xll2 (Box,B'ox)2
with equality holding if and only if B o x and B' o x are proportional. Hence IU'(0) is

defined provided B and B' are not proportional. IfB and B' are proportional then 1J(0) is

of the form 0/0, but should still be definable as a limit.

Interpretation Of The Derivative

We now consider the effects of sinusoidal noise functions of large frequency,

through an asymptotic analysis of U(0). The resulting expression is then used to provide

an order of magnitude estimate of the displacement error observed in the use of vision

systems for the measurement of plane motion. It will be seen that the effects of noise on

the measured displacement decrease as the frequency of the noise increases.

Asymptotic Analysis. Let Bo=200 represent a scale for the image intensities. The

noise function is then assumed to be of the form
n: SA 91
X :-+ B0 sin(coX)
where co is a constant spatial frequency. By successive integration by parts, it can be

shown that for all sufficiently smooth functions f

(n, f) = O(1)

as o->oo [Bleistein and Handelsman, 1986, p. 80]. Hence we have
U'(0) = o( ) as co -+ 0








This result is consistent with the findings, in a particular example, of Sutton et. al. [1988]

that rapid sampling is essential for accurate results.


Leading Terms. Inspection of the asymptotic expansion of the integral of the

product of a sinusoid of large frequency and a smooth function [Bleistein and

Handelsman, 1986, p. 80] leads to the order of magnitude estimates

|(n,Box) -B B21
OD
I(n,B'ox)I -BoBo-
for large cD, where B' is a characteristic value for the derivative of B, which is assumed to

satisfy Bo ~ B/h. The magnitudes of the remaining integrals appearing in Equation 10

may be estimated as follows:
JIB oX112 B2 h0
llB'ox112-[ ~ h

I(Box,B'ox)I -BoBBh
The above estimates indicate that

BohcD
which gives

Iu'(0)l1 -
The frequency of the noise may be more conveniently expressed in terms of the number of

cycles in the interval of interest, which we denote by N. The frequency then becomes

c= .- and hence, for large values of N,
h
IU'(0)- h
icN








Numerical estimate of error. The author's experience indicates that the difference

in gray level between two successive images of an object in the same configuration is

typically two gray levels. The corresponding value of the noise parameter is a=O.0 1.

Since the effects of noise decrease with increasing frequency of noise, any estimates of the

effect of noise should be based on a low frequency noise. Assume that the noise oscillates

only twice in the correlation interval, so N=2, and assume the half gage length is given by

h=10 pixels. The estimate of the error in displacement, |Aul is then given by
Aul =aIU'(o0)I
0.01--
RtN
~0.02 pixels
The use of the large frequency expansion for |U'(0)I requires justification.

Figure 11 shows a graph, created by Mathcad, which compares the exact value of |U'(0)

and the asymptotic estimate, for large N, given by I U(0) I ~ -- versus the number of

noise oscillations, N. The example was computed assuming a parabolic image and a

sinusoidal noise function. The comparison is favorable even at one oscillation over the

interval.


The results of this section indicate that displacement errors on the order of 0.02

pixels are expected given the region size typically used for Digital Correlation, and the

noise characteristics of the CCD camera and 8-bit frame grabber used to collect data for

the current study.







48


.53.0516 60



40



20-
a


.O=0590177, o / OI
0 1 2 3 4 5 6
.0.01. 11N.5



Figure 11. A test of the asymptotic estimate of Equation 6.

Uncertainty Relation For Correlation


This section represents an attempt to explain the strong dependence of strain

components determined by Digital Correlation on the estimate of displacement, as

observed by Chu et al. [1985] and by Sutton et al. [1991]. We will fix undeformed and

deformed images in one dimension, and define a class of approximate motions functions,

parameterized in terms of the approximate displacement and strain. We will then define a

correlation function and study the relationship between changes in the approximate strain

and displacement near maxima of the correlation function. The result will be an inequality

relating the uncertainties in the approximate displacement and strain, and the size of the

region over which the correlation is performed. Note that in the following analysis we will

assume that all functions considered are continuously differentiable.








One Dimensional Images And Motions

Choose a position X0 and a length h. Consider a one-dimensional body which

occupies positions described by the sets SA =[Xo-h,Xo+h] and SB in its undeformed and

deformed configurations, respectively. For all displacements, u, and for all strains, e,

define an approximate motion
xUS :SA --9
X:-+ X+u+s[X-Xo]

Let the functions
A:SA- 91R
B: SB --9
be the undeformed and deformed intensities, respectively.

The Correlation Function And Inner Products

Define a correlation function
C: 9 x9?-+9%

(fSAA(X)B(xu(X))dx)2
(1 )- fAA2 (X)dX) f AB2 x 1;(X))Jdx)

which must be maximized to determine the displacement and strain. The analysis to

follow is facilitated by the use of inner product notation. Let a and b be functions defined

on the set SA. Define inner products
(a, b) = s^a(X)b(X)dX
= fs^ a(X)b(X)(X Xo)dX

and a norm
lal| =(a ?a
for all integrable a, b : SA -+ 91. Using the above notation the correlation function may be








written as
u s-) = (AB ox)2 for all (u,e) r 9x9
[[AJJ'IJBox ]]21


We now state and prove an important property of the inner products we have defined.

Lemma. Let a, b : SA -+ 91 be integrable. Then
|(a,b)> h|(ab)I

Proof. Note that since
SA= {Xe 91 :Xo-h ={X E9: IX-Xo| Ih}
we have
-h a(X)b(X)dX < Ja(X)b(X)(X-Xo)dX < h a(X)b(X)dX

where all integrals are over the set SA. Then
-h(a, b):5 (a, b)5 h(a, b)
which is equivalent to the result to be proven.

Differentiation Of The Correlation Function

The analysis to follow requires the first partial derivatives of the correlation

function with respect to both strain and displacement. To facilitate the calculation of these

derivatives, define functions
f: 91 x 9R-- 91
(u,e):-) (A,Box )2
g:9?x 9-,91
(us): IAIIJIBoxl2
so that C--=fg. It can be shown that
D f(u, e) = 2(A, B o xu)(A, B' o xu)
D2f(u, ) = 2(A, B o xXA, BI oxu)
Dig(u,8) = 2IiAJI2(B o xU,B/ oxU)







D2g(u,s) = 211AII2(Boxu,B/ oxU)

Sensitivity Analysis

In the following analysis we assume that the solution for the maximum of the

correlation is not exact. The expressions below are not defined if the solution is exact.

The implications of the indeterminacy at the exact solution of the estimates obtained are

considered below. Suppose that numerical maximization of the correlation function value

has resulted in the approximate displacement u and the approximate strain e. To estimate

the sensitivity of the strain computed by the correlation algorithm to the assumed rigid

body displacement, u, change the displacement by an amount Au and compute the

resulting change, As, in the computed strain assuming that the value of the correlation

function remains constant at first order. The requirement that the value of the correlation

function be constant at first order implies
0 = DiC(u, e)Au + D2C(u, s)AS

which implies that
A DiC(u,s)
Aul D2C(Ue)
SD1 f(u, e)g(u, e) f(u, s)D 1 g(u, 8)
D2f(u, e)g(u, e) f(u, s)D2g(u, e)
Substitution of the expressions for the partial derivatives gives
Aer (x~loxHA.,B -xu)XB xu, B/ )
( A, BI o xu)JJB o xull 0 A, ox0 o xU ox
I A& 1 .11 B ox )
Au (A,B'ox)IllBoxII (A,Box)(Boxu, Bn oxu)
which may be written as
(JI E (l~xlA.Bl-xO-((ABoxu)BoxuBoxru,)
I Au2
l (JIBoxIA ( A, B ox ) oxg)B o xu) ,B/oxu)
(iB o Xul2 A- (A, B xu)B o xuBo xurx)
(lIB o xll-(A,B- x)Bo xs, BIo xc)









Thus by the above Lemma we have the strain-displacement uncertainty relation
AS> IAul
h

(7)

The implications of this result are discussed below.

Implications for the computation of strain. Typical values from current research

indicate that the uncertainty in displacement should be approximately 0.01 pixels and the

gage length should be approximately twenty pixels. Under these conditions the

strain-displacement uncertainty relation predicts an uncertainty in strain on the order of

1000 micro strain. This agrees well with the standard deviation of 0.007 found by Bruck

et al. [1989]. It should be noted that the strain-displacement uncertainty relation provides

a lower bound for the uncertainty in strain. It does not indicate that it is impossible to

determine strain accurately, but it does indicate that one's confidence in the strain

estimates obtained should be low.


Implications for the computation of displacement. The strain-displacement

uncertainty relation can be written as
lAul which provides an upper bound for the uncertainty in displacement given an uncertainty in

strain. Note that a strain uncertainty IAsel=0.001 results in an estimate for the uncertainty

in displacement of the form
IAuI < 0.01 pixels
given a gage length of twenty pixels, which corresponds to h=10 pixels. Hence, when

computing displacements, it may be advisable to simply ignore the effects of strain at or









below the 1000 micro strain level, since the errors induced should be smaller than those

resulting from noise and computational errors. Once the displacements have been

determined to a known accuracy, it is possible to select sufficiently long gage lengths to

determine average strains to a prescribed tolerance based on center point displacements.

Comments On The Indeterminacy At The Exact Solution

The strain-displacement uncertainty relation is not defined if the motion

determined by the correlation algorithm maximizes the correlation function. The ratio of

strain error to displacement error is of the form 0/0 when the correlation function is

maximized. This is expected, since we have asked how we must change the strain in

response to changes in displacement in order to maintain a constant value of the

correlation function. When the solution for the displacement and strain which maximize

the value of the correlation function is exact, it should not be possible to alter the strain in

response to a change in displacement in such a way that the correlation function value is

held constant.

Summary

This chapter contains qualitative analyses of the effects of out-of-plane

displacement on normal strains and the effects of noise on the accuracy with which

displacements may be determined using correlation. An analysis of the simultaneous

determination of displacement and strain by correlation is also presented. The results

obtained indicate the following:









1. The error of measured displacements due to noise, neglecting cancellation, can

easily approach 0.1 pixels when eight bit quantization is used.

2. The effect of noise on the displacement computed by correlation decreases as the

noise frequency increases.

3. Displacement errors on the order of 0.02 pixels are expected given typical

conditions for current applications of Digital Correlation.

4. Given the fact that Digital Correlation has a displacement uncertainty of 0.01

pixels, the expected strain uncertainty is on the order of 1000 micro strain when

correlation is performed over a region with a radius of approximately ten pixels.

It is important to note that the analyses presented in this chapter do not consider the

effects of interpolation. The lack of consideration of interpolation is important for the

following reasons:


1. An important source of error has not been included in the models.

2. The error estimates presented in this chapter represent "best case" results, and it

should not be possible to reduce errors below the predicted levels by improved

interpolation algorithms.













PREPARATION FOR EXPERIMENTS


This chapter describes the preparation for experiments performed to provide

verification of the PLS and PC methods of displacement refinement. The discussion

begins with a description of the procedure used to calibrate SPECKLE's aspect ratio

correction mechanism. The construction of the test specimens and their translation or

loading mechanisms is then described.

Experimental Configuration

This section describes the basic experimental configuration used for all

experiments described in this document. The television system is described in detail and

then the basic experimental system is described.

Computer And Television System Details

The computer system consisted of an IBM PS/2 Model 70 B21 (Intel 80486 CPU)

personal computer with 4MB RAM and the following components:


1. one 120MB hard drive

2. two 3.5 inch internal high density floppy disk drives and one external 5.25 inch

high density floppy disk drive

3. one IBM 8514/A video adapter and one IBM 8514 monitor









4. one Data Translation DT-2953 frame grabber for the IBM Microchannel

Architecture

5. one Sony XC-77CE CCD television camera with PS-12SU power supply

6. one Sony PVM-1342Q video monitor

7. one IBM 4019 Laser Printer and a Microsoft serial mouse.

The video system connections are shown in detail in Figure 12.


XC-77CE Camera


PS-12SU Power Supply PVM-1342Q Monlitor

Figure 12. Video system connections.

Experimental Configuration And Procedures

The experimental technique described in this document is similar to both the

Digital Correlation [Sutton, 1991] and Automated Grid methods [Sirkis, 1991]. The

differences are in the method of computation and the preparation of the test specimen

surface.








Experimental configuration. The object under test is assumed to have a planar

surface and to deform in the plane of that surface. The television camera is mounted with

its optical axis perpendicular to the surface of the test specimen. The camera must be

connected to a computer containing a frame grabber, and the computer must be running

image capture software. The experimental configuration is shown in Figure 13.

Light Source

H Camera and Lens



Object



Computer and Frame Grabber Monitor

Figure 13. Experimental configuration.

Optics. Except where noted, all images were captured using a Fujinon Television

zoom lens H6X12.5R, which is an f/1.2 lens with focal lengths ranging from 12.5 mm to

75 mm. The lens was mounted to the XC 77CE camera with a 10 mm extender ring, and

was typically set at a focal length of 75 mm and focused at approximately 3 m. The actual

object plane distances were smaller than 3 m due to the presence of the extender ring.

This configuration placed the image plane approximately 48 cm from the object plane.

Experimental Procedure. The system shown in Figure 13 was assembled on a

moveable base. Given a test specimen and a loading fixture, the assembly of the system









for a given test consists of aligning and focusing the camera. See the Appendix "The

SPECKLE User Interface" for descriptions of the programs DT2953 and SPECKLE. The

procedure for all experiments (except the aspect ratio calibration) described in this

document is as follows:


1. If the test specimen has a dark or metallic surface then paint it with a light color

spray paint.

2. Assemble the test specimen and its loading mechanism.

3. Position and focus the camera to obtain the desired image, aligning the optical axis

of the camera lens with the normal to the surface of the test specimen.

4. Draw a dot pattern on the object using a felt tip pen.

5. Set the desired options for DT2953, as described in the Appendix, "The

SPECKLE User Interface."

6. Use the dynamic histogram feature of DT2953 to determine the correct light

source intensity and position, and the correct lens aperture setting.

7. For the first test with a given alignment of the camera, capture two images of the

test specimen in its undeformed configuration and analyze the image pair using

SPECKLE to determine the base line displacement and strain errors. If the results

are not acceptable, then realign the camera and repeat this step until the results are

acceptable.









8. Capture and save an image of the object in its undeformed configuration, load the

object, and capture and save and image of the object in its deformed configuration.

9. Analyze the undeformed and deformed image pair to determine the desired

displacements and strains.

10. To perform multiple repetitions of the current test, go to step 8.

Aspect Ratio Correction Calibration

SPECKLE's aspect ratio compensation mechanism was calibrated using three

images of objects of known dimensions, which were imaged using different optical

systems. The calibration results suggest the aspect ratio is dependent primarily on the

combination of the CCD camera and the frame grabber [Sirkis, 1991]. The tests described

below were all performed using the following procedure:


1. Image the test specimen with the lens under test.

2. Capture the image and save an image file using DT2953.

3. Load the image into SPECKLE and use the Scan Line Dialog Box to measure the

horizontal and vertical dimensions, in pixels, of a feature of the object with known

physical dimensions.

4. Use the resulting dimensions to compute an Aspect Ratio as described in the

section "Aspect Ratio Calibration" of the Appendix.

Aspect Ratio Test 1

The first aspect ratio test was performed by imaging a two inch by two inch object

using a television lens with a focal length of 75 mm. The resulting image was saved in the










file ASPECT1.SUB. The data from ASPECT1.SUB are shown in Figure 14. Note that

the physical dimensions are in inches and the logical dimensions are in pixels.


File ASPECT1.SUB

Trial Horizontal Vertical
Pixels Physical Pixels Physical
1 283 2 321 2
2 284 2 319 2
3 284 2 319 2
Average 283.6667 2 319.6667 2

Ratio 1.12691

Figure 14. Results of the first aspect ratio test.


Aspect Ratio Test 2


The second aspect ratio test was performed using a fixed focus lens with a focal

length of 12.5 mm imaging a two inch by two inch object. The data from the second

aspect ratio test are shown in Figure 15, where the physical dimensions are in inches and

the logical dimensions are in pixels.

File ASPECT2.SUB

Trial Horizontal Vertical
Pixels Physical Pixels Physical
1 207 2 235 2
2 206 2 235 2
3 206 2 234 2
Average 206.3333 2 234.6667 2

Ratio 1.137318


Figure 15. Results of the second aspect ratio test.

Aspect Ratio Test 3

The third aspect ratio test was performed using a zoom lens set at a 75 mm focal

length with a 20 mm extender ring. The target object was 0.5 in by 0.5 in. The data are









shown in Figure 16, where the physical dimensions are in inches and the logical

dimensions are in pixels.

File ASPECT3.SUB

Trial Horizontal Vertical
Pixels Physical Pixels Physical
1 275 0.5 315 0.5
2 276 0.5 315 0.5
3 276 0.5 315 0.5
Average 275.6667 0.5 315 0.5

Ratio 1.142684

Average Ratio 1.135637

Figure 16. Results of the third aspect ratio test.

Summary Of Aspect Ratio Calibration

The average aspect ratio from the three aspect ratio tests is 1.135637. This value

was entered into SPECKLE.INI as 1.13 to calibrate the Aspect Ratio Compensation

Mechanism.

A Lucite Beam

The Lucite beam described in this section was used in both the rigid body

translation and cantilever beam tests. This section describes the machining of the beam

and the mechanisms used to translate it and to load it as a cantilever beam.

Machining The Beam

The Lucite beam was machined from a 0.36 inch thick sheet of Lucite. The sheet

was rough cut on a band saw and then milled according to the specifications indicated in

Figure 17. Note that the specified depth of the beam was 17/16 inch, but the final physical

depth was 1.08 inch.








62





d
aJ
&


M1/4 C-
lo hol OVI7
4 C-







417/9 16-111/4.]
*---- 9 1/0 7-- -i 114 !






Figure 17. Mechanical specifications for the Lucite beam.



Pulley
Lucite
Beon







'- Weight

V se




Figure 18. Loading mechanism for the cantilever beam.


Loading The Beam


The loading mechanism used to load the Lucite beam as a cantilever beam is


shown schematically in Figure 18. A steel vise and a loading frame were bolted to an


optical table and the Lucite beam was clamped using the vise. The loading frame was


used to suspend a pulley with its top level with the loading hole of the beam. A 1/4 inch


diameter bolt was placed through the hole in the beam and securely fastened to the beam









with washers and nuts. A multi-fiber steel cable was attached to hooks and installed as a

loading cable with one hook attached to each side of the bolt, as close to the beam as

possible. The loading cable was looped through a load holder, which was suspended from

the pulley.

Translating The Beam

The Lucite beam was used as the test specimen in the rigid body translation test.

The beam was clamped to a loading frame as shown schematically in Figure 19. The beam

is clamped to the moving portion of the frame. The beam may be translated by turning the

wheel on the left side of the schematic. The wheel is attached to a rod threaded at 18

threads per inch, which moves relative to the frame as it is turned and which is attached to

the moving portion of the frame with a thrust bearing.


Support Support

Figure 19. Rigid translation mechanism.









A Tensile Specimen

An aluminum tensile specimen was obtained and prepared for use in strain tests

with SPECKLE. This section describes the mechanical specifications of the tensile

specimen and the arrangement for loading it.

Mechanical Specifications

The tensile specimen is 0.040 inch thick and has physical dimensions as shown in

Figure 20.

I ---- r 0475 -- |
1.0000

5,0000
--- 9.0000

ALuminum TensiLe Specimen.
ALL dimensions In inches,


Figure 20. Physical dimensions of the tensile specimen.

Loading The Tensile Specimen

The loading mechanism for the tensile specimen is shown schematically in

Figure 21. The tensile specimen was placed in the wedge grips, and the loading weights

were suspended from the right end of the lever arm. The mechanical advantage of the

system was ten to one.

Shear Specimen

A shear strain demonstration device was used to evaluate the ability of SPECKLE

to identify a shear strain field. The shear strain demonstrator was manufactured by










Ann Arbor Instrument Works and is self-loading. It has dimensions as shown in

Figure 22. The shear strain demonstration device shears a 0.5 inch thick natural rubber

sheet when the nut at the top of the frame is turned. The nut and the threaded shaft,

which is attached to the member fixed to the center of the rubber sheet, are threaded at 28

threads per inch. Mechanical support for the shear strain demonstration device was

provided by the loading frame used for the rigid body translation test.


Loading Frw V th
Tens&& Spl.in NountaOg
Hardwe
6-1"-1?
AU daOWnIonM In hckd.
SLe Exploded View


Figure 21. Loading the tensile specimen.

Summary


This chapter describes the preparation required for the experiments described in

the next chapter. The aspect ratio calibration procedure need only be performed once for

each camera and frame grabber combination. The remainder of this chapter describes the

specimens and translation and loading systems used to produce known deformations for











tests designed to verify the accuracy of the PLS and PC displacement refinement methods,

and SPECKLE's implementation of these methods.


./Nut



/ J- Threaded
Rod
Frame Rod




1.5 1 12 in


Figure 22. The shear demonstration device.













EXPERIMENTAL RESULTS


This chapter describes tests involving rigid and deformable body mechanics that

were performed to provide verification of the PLS and PC methods of displacement

refinement. Note that displacements reported by SPECKLE are given as ordered pairs of

the form (right,down) with the components measured in pixels. The displacements are

reported without correction for the aspect ratio of the video system. The right-down

coordinate system is natural for use with computer graphics, but is not acceptable for

reporting strain tensor components, which should be reported in a right hand coordinate

system. In a personal communication, Dr. M.A. Sutton suggested the use of a down-right

coordinate system as opposed to a right-up coordinate system. SPECKLE uses the

down-right coordinate system for reporting strain tensor components. The horizontal

components of both image coordinates and displacements are corrected for the aspect

ratio of the video system before the strains are computed.

Rigid Body Translation

The first test of SPECKLE to be presented is a test involving a rigid body

translation. The Lucite beam was painted with multiple coats of Krylon Flat White

Interior and Exterior Enamel with sanding between coats, and dots were drawn on its








surface. Two of the three rows of dots are shown in Figure 23.



9*




Figure 23. Dots on the Lucite beam for the translation test.


The test specimen was mounted to the moving portion of a loading frame that was in turn

mounted to an optical table. The arrangement is shown in Figure 19. The loading frame

is actuated by turning a ring mounted to a rod threaded at 18 threads per inch. The

loading frame was mounted to the optical table so that the displacement produced by the

frame was approximately horizontal. A dial indicator with a scale calibrated in 0.001 inch

increments was mounted to measure the displacement of the object.

Experimental Procedure

The rigid body displacement tests were performed using the following procedure:


1. Move the test specimen to its reference position at which the dial indicator scale

has been zeroed and capture the undeformed image.

2. Move the test specimen until the dial indicator reads 0.010 inches and capture the

"deformed" image.

It is important when returning the test specimen to its reference position, to move it past

the zero reading and then bring it back to the zero reading, since otherwise the slack in the









threads of the loading frame will cause inconsistent results when the test specimen is

displaced to its "deformed" position.


Results


Three repetitions were performed, and the images were saved in files with root

names RIGIDC, RIGIDD, and RIGIDE. The results from the three tests are shown in

Figure 24 through Figure 26, respectively.

RIGIDC Horizontal Vertical
PC Point 1 3.16E+00 -7.50E-02
Point 2 3.19E+00 -3.22E-02
Point 3 3.15E+00 -1.09E-01
PLS Point 1 3.19E+00 -7.42E-02
Point 2 3.21 E+00 -2.20E-02
Point 3 3.15E+00 -1.09E-01


Figure 24. RIGIDC displacement results.


RIGIDD Horizontal Vertical
PC Point 1 3.09 -9.47E-02
Point 2 3.05 -7.60E-02
Point 3 3.05 -1.03E-01
PLS Point 1 3.11 -9.73E-02
Point 2 3.08 -5.72E-02
Point 3 3.07 -9.24E-02

Figure 25. RIGIDD displacement results.


RIGIDE Horizontal Vertical
PC Point 1 3.18 -1.17E-01
Point 2 3.15 -7.61E-02
Point 3 3.14 -1.33E-01
PLS Point 1 3.16 -1.15E-01
Point 2 3.10 -1.20E-01
Point 3 3.11 -1.43E-01

Figure 26. RIGIDE displacement results.


Note that the displacements in the above tables are expressed in terms of pixels. The

average of all horizontal components of displacements is 3.13 pixels. The width of the






70

beam is 1.08 inches, and the width of the image of the beam is 332 pixels. The physical

displacement corresponding to the average horizontal displacement is then 0.01 inches.

The error relative to the 0.010 inch displacement measured with a dial indicator is 1.81%.

The results are graphed in Figure 27. The legend of Figure 27 shows the icons and line

styles used to identify each line. The lines, except the average value line, are coded by the

last letter of the test name and the method of computation of displacement. The average

value line corresponds to the average of all indicated values. Note that a different line

style is used for each test, and that most of the scatter appears to be between tests, rather

than between the individual values for each given test.


Rigid Body Displacement Results
Horizontal Component

_ 3.25E+00
I 3.20E+00oL -
f 3.15E+00 --- ---- ---
| 3.1OE+00 <............. -............ ------------
E ........................ ......... X ------------------------- ..
0 3.05E+00- -- -----------------------------
'- 3.00E+00
o 2.95E+0
0m 2.95E+00


Point Number


--UC/PC

-- C--- C/PLS
........ ........ D/PC
........ X ....... D/PLS

--+--- E/PC

------ E/PLS

Avg


Figure 27. Rigid body displacement results.








Axial Strains In A Cantilever Beam

The second test of SPECKLE to be discussed is a measurement of the average

strain due to bending in a cantilever beam. This test demonstrates the use of SPECKLE in

situations involving a strain gradient in a known direction.

Experimental Arrangement And Procedure

The dot pattern shown in Figure 28 was applied to the beam with an overhead

projector marker.



.8' .








Figure 28. Dots for the cantilever beam test.


The beam was prepared for loading as shown in Figure 18. The camera was positioned

normal to the face of the beam, and the camera lens was focused on the dot pattern. The

undeformed images were captured with no load on the beam, and the deformed images

were captured as soon as possible after the application of a ten pound load. The beam

creeps under load, making the use of a pre-load impossible when using the DT-2953 frame

grabber. The DT-2953 is not capable of holding multiple frames in its frame store

memory, and the beam would creep significantly in the time it would take to save the









undeformed image to disk or machine memory in preparation for the capture of the

deformed image.

Results

Three repetitions were performed. In each case an image was captured with the

beam in its undeformed configuration, the beam was loaded and an image of the beam in

its deformed configuration was captured. The load was removed from the beam as

quickly as possible to minimize the effect of creep. The images corresponding to the three

repetitions were saved in files with root names GRADA, GRADB, and GRADC,

respectively. SPECKLE was used to compute the average strain along lines parallel to the

center line of the beam on the tension side, along the neutral axis, and on the compression

side. Both the PLS and PC methods were used to compute displacements that were

subsequently used to compute the average strains. The resulting normal strains are shown

in micro strain in Figure 29.
Test Method T N C
GRADA PC 1332 -94 -1343
PLS 1316 -120 -1376
GRADB PC 1460 -109 -1481
PLS 1482 -89 -1459
GRADC PC 1379 -70 -1636
PLS 1472 -65 -1515
Average PC 1390 -91 -1487
PLS 1423 -91 -1450
Combined 1407 -91 -1468

Figure 29. Cantilever beam average normal strains.

Comparison With Beam Theory Predictions

A dial indicator was used to measure the elastic free-end displacement, d, of the

beam in response to a 10 pound load, which is approximately 0.15 inch. Consider the






73

definition sketch shown in Figure 30.



| p "
_____4_ -T



-- x2 -
-- L-


Figure 30. Definition sketch for the cantilever beam.

The two circles in the sketch represent the end points of a line along which the average

strain is to be computed. It can be shown that the displacement of the free-end of a

cantilever beam made of a homogenous elastic material with constant cross section is

given by
d= PL3
3EI

where P is the applied load, L is the length of the beam, E is the Young's Modulus of the

material, and I is the centroidal moment of inertia of the area of the cross section of the

beam. The above may be used to determine the elastic modulus in terms of the free-end

displacement. It can further be shown that the axial strain due to bending is given by
P L
e(x, y) = -jy(x -L)

The average strain along the line between the points (x,,h) and (x2,h) in the reference

configuration is equal to the strain at the midpoint, which can be written as

E(xI, X2,h) = -ih[ X2 +XI LI









The parameters for the line selected along the tension side of the beam are summarized in

Figure 31. See Figure 30 for a definition sketch of the beam. The dot pattern used in the

cantilever beam tests is shown in Figure 28. The values ofx, and x2 (see Figure 30) are

the distances from the fixed end of the beam to the dots in the bottom and top,

respectively, of the beam.

Beam Parameters
a 0.36 in
b 1.08 in
xl 2.25 in
x2 3.25 in
L 9.25 in
h 0.40625 in
P 10 Ib
d 0.15 in


Figure 31. Beam parameters.


Given the measured free-end displacement, the elastic modulus is 465 ksi and the

corresponding theoretical value for the average strain on the tension side is then 1501

micro strain, which agrees well with the experimental results. A zero average strain is

expected along the neutral axis, and along the compression side the average strain is

expected to be the negative of the value computed along the tension side. The results are

tabulated in Figure 29 and plotted in Figure 32. At each position on the beam, the

columns from left to right in Figure 32 correspond to the legend entries from top to

bottom. The legend entries are coded by the last letter of the test name and the method of

displacement computation.











Cantelever Beam Test Results


2000
% 1500 x
S1000
I) 500
0
I '-
S-500
S-1000
-1500
-2000


P


-0--- A/PC

S- --A/PLS
0--o- B/PC
-- B/PLS

0- C/PC

-o0- C/PLS
N C
-)(- Theory
position Tension, Neutral, Compression -ho

Figure 32. Axial strains for the cantilever beam.


Tension Test

This section describes a test in which the aluminum tensile specimen was loaded in

pure tension, with the horizontal and vertical axes of the camera aligned with the principal

axes of the test specimen. Recall that SPECKLE reports the strain tensor components in a

down-right coordinate system. The dot pattern imaged for the principal axes tension test

is shown in Figure 33. The pattern was imaged using the 75 mm zoom lens with a 20 numm

extender ring. The zoom lens was set for a focal length of 75 mm and its focus was set at

approximately one meter. The distance between the object and image planes was

approximately nine inches.

Experimental Arrangement And Procedure

The tensile specimen was mounted as shown in Figure 21. The surfaces of the

tensile specimen were painted with multiple coats of Krylon Flat White Interior and








Exterior Enamel with sanding between coats. The dot pattern is as shown in Figure 33.












Figure 33. Dots for the tensile test.


The dots were drawn with an Overhead Projector Pen with an Extra Fine point. Each

image pair was digitized as follows:


1. An 8 pound pre-load was placed on the loading arm to remove the slack from the

loading frame, and the "undeformed" image was captured.

2. Two ten pound weights were placed on the loading arm and the deformed image

was captured.

The mechanical advantage of the loading frame is 10:1, so the loads on the specimen were

80 lb and 280 lb, respectively. The calculations below are based on the principle of

superposition and assume that the loads were zero and 200 pounds, respectively.

Results

Three repetitions of the procedure described above were performed. The images

captured were saved in files with root names VERTE, VERTF, and VERTG. The data

generated by SPECKLE are shown in Figure 34. The next section compares the measured









values with the theoretical values.
Test Label PC PLS
VERTE Edd 1198 1167
Err -434 -377
Edr -17 -55
VERTF Edd 1145 1138
Err -300 -307
Edr -134 -101
VERTG Edd 1156 1138
Err -371 -335
Edr -13 -11
Average Edd 1166 1148
Err -368 -340
Edr -55 -56


Figure 34. Tensile specimen strain results.

Comparison With Theoretical Predictions


The physical dimensions of the aluminum tensile specimen are shown in Figure 20.

The tension test parameters are shown in Figure 35, where E is Young's Modulus, P is the

difference in the applied load between the final and reference configurations, t is the

thickness of the specimen, and w is the width of the specimen.
Tensile Specimen Parameters
E 1.00E+07 psi
P 200 Ib
t 0.04 in
w 0.475 in


Figure 35. Tensile specimen parameters.


The nominal value for the incremental axial strain is given by
P
dd -twE

and Hooke's Law gives


Sir = -VSdd


where v is Poisson's Ratio, which is 0.33 for Aluminum. The theoretical value for the









incremental normal strains are shown in Figure 36.
Theoretical Normal Strains
Edd 1053 micro strain
Err -351 micro strain


Figure 36. Theoretical strains.


The theoretical value for the shear strain is zero. The results are tabulated in Figure 34

and plotted in Figure 37. For each strain component in Figure 37, the columns from left

to right correspond to the legend entries from top to bottom. The legend entries are

coded by the last letter of the test name and the method of displacement computation.


Tensile Specimen Results

1200 -II E/PC
1000
'E" 800 El E/PLS
6 S00 M F/PC
2400
4200 D F/PLS
.,200-
0.j I O B |G/PC
3 -200- G/PLS
-400
-60o0 Avg
Edd Err Edr D] Theory
Strain Tensor Component Edd, Err, Edr


Figure 37. Tensile specimen strain results.

Shear Strain Test

This section describes a test performed using the shear strain demonstration device

shown in Figure 22.








Experimental Arrangement And Procedure

The shear strain demonstrator was clamped to the loading frame used in the rigid

body displacement test to provide mechanical support, and the dot pattern shown in

Figure 38 was drawn on its surface. See Figure 38 for the location of the dot pattern on

the shear strain demonstrator.

%%4


4 .





Figure 38. Dots for the shear strain test.

The camera was placed normal to the surface of the demonstrator and focused on the dot

pattern. Each undeformed and deformed image pair was captured using the following

procedure:


1. The wing nut on the shear device was loosened to completely unload the shear

specimen. The wing nut was then tightened to remove slack and the undeformed

image was captured.

2. The wing nut was tightened an additional one half turn and the deformed image

was captured.









Results

Three repetitions of the procedure described above were performed. The resulting

images were saved in files with root names DEMOB, DEMOC, and DEMOD. The strain

tensor components as determined by SPECKLE are shown in Figure 39.


Figure 39. Shear strain test results.

Comparison With Theoretical Values

Consider the definition sketch shown in Figure 40, and the parameters shown in

Figure 41. The shear strain may be estimated by considering the reduction in the right

angle between material lines originally along the d and r axes. The tensor shear strain

satisfies the equation
2e& = Lu

where u is the shearing displacement and L is the width of the shear specimen. Given the

parameters above, the expected tensor shear strain is 2232 micro strain, which agrees well

with the experimental results. For each strain component, the results shown in Figure 42

correspond to the legend entries from top to bottom. The legend entries are coded by the


Test Label PC PLS
DEMOB Edd 170 165
Err 248 261
Edr 2419 2417
DEMOC Edd 374 240
Err 132 150
_____Edr 2246 2263
DEMOD Edd 238 190
Err 474 504
____ Edr 2127 2116
Average Edd 260 198
Err 285 305
_____Edr 2264 2266










last letter of the test name and the method of displacement computation.




E

(5
L
D Deformed 4
WRegion Of
Interest

L
0
I/ --J |
| ___ I/. 0









Vd


Figure 40. Definition sketch for the shear strain test.

Parameter Value
Thread Pitch 28 threadinch
Turns 0.5 turns
Displacement 0.018 inch
Width 4 inch


Figure 41. Shear specimen parameters.

Effects Of Out-Of-Plane Displacement


This section provides verification of the order of magnitude analysis of the effects

of out-of-plane displacement on normal strains measured using vision systems when plane

motions are assumed. The Lucite beam was mounted on a translation stage so that it

could be moved parallel to the axis of the camera, which was fitted with the Fujinon zoom

lens with a 10 mm extender ring and set for a 75 mm focal length, with the focus adjusted








at approximately 10m.


Shear Strain Test Results E BPC
II B/PC

2500 - B/PLS
2000~ DC/PC
o 1500 El C/PLS
t3
1000 D D/PC

5E-00 E D/PLS

Edd Err Edrvg
Strain Tensor Component El Theory


Figure 42. Shear strain test results.


The object to image plane distance was approximately 50 cm. The horizontal size of the

field of view in the object plane was approximately 5.5 cm and the horizontal size of the

sensitive area of the Sony XC-77 CE is 0.88 cm. The distance from the object to the

pinhole equivalent to the lens, as described in the chapter "Analytic Results On Errors," is

then 16.97 in. Two images of the object were captured with the object in its reference

position. The object was displaced toward and away from the camera by 0.020 in, 0.030

in, and 0.040 in, and an image was captured in each position. The images were analyzed

in pairs using the first reference image as the undeformed image in all cases. The resulting

normal and shear strains are shown in Figure 43 that both normal strain components

should agree with the theoretical normal strain, and the shear strain should be zero. The

results confirm that the effects of out-of-plane displacement on normal strains are of the

expected order of magnitude and that the effect of out-of-plane displacement on shear









strain is small, as expected. The equivalent pinhole model may prove to be useful for the

estimation of allowable out-of-plane displacements.




Effects Of Out-Of-Plane Displacement


2500+
2000
1500 ^'
I 1000
0 500
0
S-500
-1000
S-1500
-2000
-2500


-o- Edd

-- Err
--Edr
--- Theory


-0.04 -0.02 0 0.02 0.04
Displacement (Inches)

Figure 43. Effect of out-of-plane displacement on measured strains.


Summary

This chapter describes experiments which provide verification of the PLS and PC

methods of displacement refinement, and of the use of these methods to estimate strains

from displacements. Tests involving rigid body displacement, bending of a beam, and

tensile and shear fields are described. Experience with using SPECKLE for the

measurement of displacements and strains has indicated that it is imperative to analyze a

zero motion test called a rating test to determine base line errors in displacements

before proceeding to the desired measurements. It is possible to measure strains with a

base line error of approximately one hundred micro strain, and rating tests provide a good

means to estimate the base line strain errors.













CONCLUSIONS AND RECOMMENDATIONS


The primary objective of the current study was to develop an iterative technique

for the determination of in-plane displacements and strains using digital images of a test

specimen before and after deformation. This objective has been accomplished with the

development of the Perturbation/Least-Squares (PLS) and Perturbation/Correlation (PC)

methods for the iterative refinement of displacement, and the development of the program

SPECKLE, which is a Microsoft Windows program that uses the PLS and PC methods to

compute displacements and to estimate strains. This chapter lists conclusions that can be

drawn from the results of the current study and recommendations for future work.

Conclusions

The scope of the current study includes the following:


1. The PLS and PC methods for the refinement of rigid body displacement estimates

were developed as alternatives to the Newton-Raphson approach developed by

Bruck.

2. Tests were performed to verify the PLS and PC method governing equations and

the iterative displacement computation algorithm.

3. Qualitative analytic studies of the errors inherent in the use of vision systems for

the computation of displacement and strain were conducted.








The remainder of this section presents some conclusions which can be drawn from

the results of the current study.

Experimental Results

The experiments described in this document indicate that the PLS and PC methods

can be used to determine in-plane displacements using digital images of a test specimen

before and after deformation. The experiments also indicate that the resulting

displacements can be used for the subsequent determination of normal and shear strains in

a constant strain field, and average normal strains in a varying strain field. Typical errors

in strain are on the order of 100 micro strain.

Analytic Studies

A secondary goal of the current study was to study analytic models of the effects

of image noise on the use of correlation to measure displacements and strains using images

of the test specimen before and after deformation. The results are summarized in the

chapter "Analytic Results On Errors." The analytic studies of errors are consistent with

published experimental and numerical studies, which conducted by other investigators.

Correlation methods for the computation of displacements and displacement gradients

function by locating extreme values of a correlation function [Sutton et al., 1988]. The

analytic studies indicate that effects of noise on the displacements computed by correlation

may be minimized by sampling the images of the test specimen at spatial frequencies that

are high relative to the spatial spectrum of the images. The strain-displacement

uncertainty relation derived for correlation predicts uncertainties in strain of the order of

magnitude observed in studies in which correlation was used to determine strains and









displacements simultaneously. When the magnitude of the displacement gradient

components are on the order of 0.001 and smaller, the results favor the computation of

displacements neglecting the localized effects of displacement gradients and the

subsequent determination of displacement gradients from the displacements computed in

widely separated regions. The required distance between the centers of the individual

correlation regions can be determined given the displacement error bounds and allowable

strain errors. The difficulty with this approach is that it severely limits the allowable rigid

body rotation between the undeformed and deformed configurations. The current solution

to the problems of displacement gradient measurement as implemented by Sutton et al.

[1991] involves the use of a finite element analysis to smooth the computed displacement

fields. It would be preferable to be able to determine the displacement gradients from the

acquired images without the computational overhead of finite element analysis and the

required assumptions about the constitutive behavior of the test specimen material.

Recommendations For Future Work

This section contains recommendations based on the author's experience and

results published by other researchers.

Verification Of The PLS And PC Methods

The results of the experiments described in this document indicate that the PLS

and PC methods of displacement refinement can be used to measure in-plane displacement

and strain. The next logical step is to explore further how the PLS and PC methods

compare with each other, with the Newton-Raphson approach to Digital Correlation

developed by Bruck et al. [1989], and with a coarse-fine approach to Digital Correlation








as developed by Peters et al. [1983] and Chu et al. [1985]. The comparisons should

consider accuracy, computational efficiency, experimental convenience and hardware

requirements.

Extensions Of The PLS And PC Methods

The PLS and PC methods for displacement refinement should be extended to

include rigid body rotation. One way to support rigid body rotation is to include

displacement gradients in the approximate motion used to derive the governing equations

for the methods. The results published by Sutton et al. [1991], the results of the current

study, and the author's experience indicate that the extension to computing displacement

gradients will be more difficult than simply including displacement gradients in the

equations. The PLS method was originally developed with strains and rigid rotation

included in the least-squares fit. Numerical tests of the original PLS method governing

equations demonstrated that the measured strains and rotation were unacceptably sensitive

to the computed rigid body displacement. The current versions of the PLS and PC

methods are limited to cases involving rotations of at most a few degrees.. The largest

rigid rotation that can be tolerated by the current versions is not known and should be

determined. This problem has been considered for Displacement-only Correlation based

on a coarse-fine search algorithm [Chen and Chiang, 1992].

Rigid Rotation Tolerance

The PLS and PC methods of displacement refinement have been implemented

neglecting the local effects of displacement gradients over the regions in which center

point displacements are to be computed. Strains are computed from the center point








displacements in regions sufficiently far apart that the expected error in displacement

divided by the distance between the region center points is small compared to the strain to

be measured. The following factors contributed to the decision to neglect the localized

effects of displacement gradients:


1. The author's negative experience with an early version of the PLS method which

included displacement gradients in the least squares fit.

2. The conclusion by Sutton et al. [1991] that displacement smoothing is required for

accurate determination of displacement gradient components.

3. Estimates based on the strain-displacement uncertainty relation derived in this

document that indicate a total loss of precision on displacement gradient

calculations when scales typical in the current study are used.

4. Estimates based on the strain-displacement uncertainty relation that indicate the

errors in displacement due to the neglecting of displacement gradients should be on

the order of 0.01 pixels.

The most severe limitation imposed by neglecting displacement gradients is that

rigid rotations between exposures must be kept very small.

Out-Of-Plane Displacement Detection

Analytic and experimental results presented in this document indicate that the

normal strains measured using SPECKLE are affected by out-of-plane displacement of the

test specimen. The errors in normal strains are on the order of the out-of-plane

displacement divided by the distance between the object and image planes. It should be









possible to detect such out-of-plane displacements by attaching a long thin rigid object to

the test specimen. The rigid object should have a dot pattern drawn on it, and be attached

to the test specimen at only one point. The rigid object will move with the test specimen,

but not deform with it. The average strain along the axis of the rigid object should be

computed. If it is significant, then out-of-plane motion should be suspected.

Modification Of SPECKLE

SPECKLE is a Microsoft Windows 3.x application that implements the PLS and

PC methods of displacement refinement. This section lists some enhancements to the

program which should be implemented before more data is processed.


Efficiency and usability issues. The current version of SPECKLE requires that the

user mark exactly three points, after which the following computations are performed:


1. An average displacement is computed in a region centered about each marked

point.

2. Normal strains are computed along three lines defined by the selected points.

3. Strain transformation equations are used to determine normal and shear strains in a

down-right coordinate system.

This structure can be inefficient. The cantilever beam test described in the chapter

"Experimental Results" did not require the computation of the general state of strain, but

these calculations were always performed. The current implementation should be

extended as follows:









1. The user should be able to specify the number of points to be marked.

2. The user should be able to associate pairs of points marked for displacement

calculations for calculation of normal strains.

3. The user should be able to specify names for the individual normal strains.

4. The user should be able to associate triads of normal strains for the computation of

strain fields.

5. The user should be able to control the format of the log files SPECKLE writes to

facilitate compilation of experimental results.

SPECKLE is an interactive program that allows the user to control the points at

which displacements are computed and the parameters used to control the iteration. The

current version of SPECKLE sometimes wastes time by computing quantities that should

have been saved for reuse. The primary advantage of building collections of computed

results is that only the results affected by changes made by the user would need to be

computed when the data base is updated to reflect changes made by the user. Such

modifications would allow the collection and processing of much more data on the

performance of the PLS and PC methods than has been included in this study.


Extended support for research. SPECKLE currently supports only bilinear

interpolation and should be extended to support non-linear interpolation methods.

SPECKLE should also be extended to provide options for the method used to estimate

derivatives of the deformed image. The use of arbitrary collections of displacement values

described above would allow SPECKLE to support the calculation of strains and small