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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 
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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/LeastSquares 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 OutOfPlane 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 OneDimensional 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 OutOfPlane 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 OutOfPlane 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 twodimensional and threedimensional 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 NewtonRaphson based iterative algorithm for computation of displacements and displacement gradients with fewer calculations than are required by the traditional coarsefine search. The NewtonRaphson 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 inplane displacements and strains with reduced computation. The Perturbation/LeastSquares (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 centerpoint displacements in regions sufficiently far apart that the strains can be computed accurately. Average normal strains are computed in three nonparallel directions and then transformed into normal strains and a shear strain using straintransformation 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 8bit 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 inplane 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 inplane 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 nonparallel 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 inplane 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 inplane and outofplane 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 noncontacting, 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 solidstate 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 inplane 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 singlebeam technique for the measurement of inplane 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 whitelight 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 Elasticplastic 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 NewtonRaphson algorithm that reduces the amount of computation required to implement correlation methods. Chen and Chiang [1992] applied Fourier Transform analysis to the study of Displacementonly 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 whitelight 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 whitelight 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 preload 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/LeastSquares (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 DT2953 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/LeastSquares (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 outofplane 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 onedimensional 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 realvalued 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 nonzero. 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 nonzero. 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 coiocI(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 x1: 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 R1, 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 ocI 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 = yoAoc1 Box = yoioiocx 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 = YI oAoc Box = yoBoxoc 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 = yoBoxocI and the linearity ofy imply A A Box = yoBoxoc I = a(yoAoc1) 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( mint(R/2) pixels forall m,n {O,..,R1} n nint(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,.,Rl}} 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: L91 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/LeastSquares (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/LeastSquares 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.nB 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 ndimensional 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 NewtonRaphson 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 realvalued 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(BoxC),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 (;ABA2BmnJE 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 highlevel 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 outofplane displacement 2. errors in inplane 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. Onedimensional 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 Displacementonly 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 centerpoint displacement when 12bit 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 subpixel 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 OutOfPlane Displacement It is clear that changes in apparent object size due to outofplane displacement should cause errors in normal strains measured using vision systems, unless the outofplane displacement is measured and considered as part of the analysis. The effects of outofplane 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 outofplane motions has been developed by Choa et al. [1989]. This section presents an order of magnitude analysis of the effects of outofplane displacement on measured normal strains when the outofplane 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 outofplane 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 outofplane 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 onedimensional 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 2hAg (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 onedimensional 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. OneDimensional Images And Motions Consider a onedimensional body which occupies positions described by the sets SA =[Xoh,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 zeromean 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)(BxB'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 8bit 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 onedimensional body which occupies positions described by the sets SA =[Xoh,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[XXo] 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 :Xoh we have h a(X)b(X)dX < Ja(X)b(X)(XXo)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.BlxO((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 straindisplacement 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 straindisplacement 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 straindisplacement 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 straindisplacement uncertainty relation can be written as lAul 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 straindisplacement 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 outofplane 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 DT2953 frame grabber for the IBM Microchannel Architecture 5. one Sony XC77CE CCD television camera with PS12SU power supply 6. one Sony PVM1342Q video monitor 7. one IBM 4019 Laser Printer and a Microsoft serial mouse. The video system connections are shown in detail in Figure 12. XC77CE Camera PS12SU Power Supply PVM1342Q 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 16111/4.] * 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 multifiber 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 selfloading. 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 61"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 rightdown 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 downright coordinate system as opposed to a rightup coordinate system. SPECKLE uses the downright 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 7.50E02 Point 2 3.19E+00 3.22E02 Point 3 3.15E+00 1.09E01 PLS Point 1 3.19E+00 7.42E02 Point 2 3.21 E+00 2.20E02 Point 3 3.15E+00 1.09E01 Figure 24. RIGIDC displacement results. RIGIDD Horizontal Vertical PC Point 1 3.09 9.47E02 Point 2 3.05 7.60E02 Point 3 3.05 1.03E01 PLS Point 1 3.11 9.73E02 Point 2 3.08 5.72E02 Point 3 3.07 9.24E02 Figure 25. RIGIDD displacement results. RIGIDE Horizontal Vertical PC Point 1 3.18 1.17E01 Point 2 3.15 7.61E02 Point 3 3.14 1.33E01 PLS Point 1 3.16 1.15E01 Point 2 3.10 1.20E01 Point 3 3.11 1.43E01 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 preload impossible when using the DT2953 frame grabber. The DT2953 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 freeend 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 freeend 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 freeend 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 freeend 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 ' S500 S1000 1500 2000 P 0 A/PC S A/PLS 0o 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 downright 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 preload 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 OutOfPlane Displacement This section provides verification of the order of magnitude analysis of the effects of outofplane 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 5E00 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 XC77 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 outofplane displacement on normal strains are of the expected order of magnitude and that the effect of outofplane displacement on shear strain is small, as expected. The equivalent pinhole model may prove to be useful for the estimation of allowable outofplane displacements. Effects Of OutOfPlane Displacement 2500+ 2000 1500 ^' I 1000 0 500 0 S500 1000 S1500 2000 2500 o Edd  Err Edr  Theory 0.04 0.02 0 0.02 0.04 Displacement (Inches) Figure 43. Effect of outofplane 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 inplane 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/LeastSquares (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 NewtonRaphson 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 inplane 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 straindisplacement 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 inplane displacement and strain. The next logical step is to explore further how the PLS and PC methods compare with each other, with the NewtonRaphson approach to Digital Correlation developed by Bruck et al. [1989], and with a coarsefine 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 leastsquares 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 Displacementonly Correlation based on a coarsefine 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 straindisplacement 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 straindisplacement 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. OutOfPlane Displacement Detection Analytic and experimental results presented in this document indicate that the normal strains measured using SPECKLE are affected by outofplane displacement of the test specimen. The errors in normal strains are on the order of the outofplane displacement divided by the distance between the object and image planes. It should be possible to detect such outofplane 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 outofplane 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 downright 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 nonlinear 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 