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PATTERN MAPPING IN PLANE MOTION ANALYSIS BY R. WALLACE FAIL 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 1987 Copyright 1987 by R. Wallace Fail To the Glory of God Digitized by the Internet Archive in 2010 with funding from University of Florida, George A. Smathers Libraries with support from Lyrasis and the Sloan Foundation http://www.archive.org/details/patternmappingin00fail ACKNOWLEDGMENTS I would like to express my appreciation to the members of my supervisory committee, Dr. S. S. Ballard, Dr. P. Hajela, Dr. U. H. Kurzweg, Dr. E. K. Walsh, and Dr. C. E. Taylor, chairman. Support of the graduate program by former department head, Dr. K. Millsaps, and current department head, Dr. M. A. Eisenberg, is gratefully acknowledged. Special thanks go to Mr. Steve McNeil and Mr. Marc Paquette at the University of South Carolina for their help and extending the use of their laboratory facilities to digitize the images used in this study. I am indebted to my wife, Robyn, for her unwavering support and love. TABLE OF CONTENTS Page ACKNOWLEDGMENTS................................................. iv LIST OF TABLES.................................................. vii LIST OF FIGURES.................................................. .ix ABSTRACT........................................................... xi CHAPTERS I INTRODUCTION............................................. 1 I.1 Background........................................ 1 1.2 Purpose of Present Work.............................. 3 1.3 Scope of Present Work................................ 3 1.4 Survey of Previous Work.............................5 1.4.1 Solid Mechanics .............................5 1.4.2 Image Processing/Pattern Recognition.........7 II THEORETICAL FOUNDATIONS...................................8 [I.1 Solid Mechanics..................................... II.2 Image Processing and Pattern Recognition............14 11.2.1 Digital Image.............................. 14 11.2.2 Preprocessing..............................18 11.2.3 Segmentation ...............................18 11.2.4 Image Features.............................. 24 11.2.5 Feature Extraction.........................26 11.2.6 Known Patterns.............................39 11.2.7 Syntactic Pattern Recognition..............41 11.2.8 Approximation of Syntactic Pattern Mapping.................................... 42 III IMAGE GENERATION AND ANALYSIS ............................58 II1.1 Synthetic Images...................................58 III.2 Image Analysis.................................. ..60 IV EXPERIMENTS.............................................. 78 IV.1 Test Specimen......................................78 IV.2 Test Equipment..................................... 78 IV.3 Experimental Procedure..............................82 V ANALYSIS OF RESULTS........................................84 V.1 Numerical.......................................... 84 V.2 Experimental............................................. 89 VI CONCLUSIONS AND RECOMMENDATIONS...........................94 VI.1 Conclusions........................................94 VI.2 Recommendations...................................95 APPENDIX BORDER FOLLOWING ALGORITHM..............................98 REFERENCES.........................................................99 BIOGRAPHICAL SKETCH...............................................104 vi LIST OF TABLES Table Page 1 Unordered spot coordinates in the undeformed image..........45 2 Unordered spot coordinates in the deformed image.............46 3 Ordered spot coordinates in the undeformed image............51 4 Ordered spot coordinates in the deformed image..............52 5 Spot distance from origin in undeformed image...............53 6 Spot distance from origin in deformed image.................54 7 Spot distance from Icon in undeformed image.................55 8 Spot distance from Icon in deformed image...................56 9 Input file for synthetic image generator....................59 10 Image analysis options file.................................60 11 Displacement of each spot...................................61 12 Displacement gradients......................................70 13 Lagrangian strain from displacement gradients...............71 14 Strain from Taylor series...................................72 15 Deformation gradient...................................... 73 16 Green deformation tensor...................................74 17 Right stretch tensor........................................75 18 Rotation tensor......................................... .. 76 19 Lagrangian strain from deformation gradient.................77 20 Summary of strain analysis for synthetic images.............86 21 The effect of SNR on translation and strain calculations....87 22 The effect of gray level difference on translation and strain calculations............................. ........... 87 23 The effect of spot radius on translation and strain calculations............................................. 88 24 Images used for rigidbody motion experiments...............89 25 Results of rigidbody rotation experiments...................90 26 Results of rigidbody translation experiments...............92 viii LIST OF FIGURES Figure Page 1 The motion of body B through space.......................... 9 2 The motion of the neighborhood of particle p...............10 3 A typical PCbased image processing system.................15 4 A spot and its digital image................................16 5 A pixel and its eightneighbors............................17 6 The bimodal histogram of figure 4 .........................19 7 Black and white regions in a bimodal histogram............21 8 Frame of spot in figure 4.................................. 24 9 Image artifacts........................................... 25 10 Onedimension model.......................................30 11 The discretization of KT and B ........................ ...33 12 Icon rotation measured from the vertical...................35 13 Icon rotation is determined to within a constant by equation (40) ............................................. 37 14 Patterns that are easily recognized and analyzed............40 15 Patterns of Figure 14 with Icons............................40 16 A pattern and its contextsensitive language................42 17 Image addresses and offsets.................................43 18 A simple example of motion................................. 44 19 The deformed image and its reference position...............48 20 Beam fabricated from CR39.................................79 21 Spot pattern on beam.......................................79 22 Experimental setup of load frame.........................80 23 Setup for fourpoint bending..............................80 24 Setup for cantilever beam.................................81 25 Rigidbody motion setup..................................81 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 PATTERN MAPPING IN PLANE MOTION ANALYSIS BY R. WALLACE FAIL May 1987 Chairman: C. E. Taylor Major Department: Engineering Sciences A new, highly automated method for measuring plane motion, pattern mapping, has been developed for rigidbody motion and strain analysis. Pattern mapping employed image processing and syntactic pattern recognition principles to recognize a known pattern before and after motion. Using the Lagrangian definition of motion, points in the two images were mapped, and the map was used to determine rigidbody motion and strain. Wholeimage analysis was thoroughly demonstrated. Known patterns were, in this case, rectangular grids of photo dots. These patterns were applied to the specimen by contact print ing. An icon was included in the pattern, thereby accommodating rigidbody rotations up to one revolution. Motion functions were selected by the user and easily changed to suit the particular application. Function coefficients were deter mined by least squares. Accuracy depended primarily on the signaltonoise ratio and the gray level difference between "black" and "white." An increase in these variables improved accuracy. User and computer time requirements were quite modest. Images were processed with little or no user interaction, and output speci fications were automatically read from a disk file. Depending on output specifications, computer time (VAX 11/750) for an 11X11 pattern in a 384X374 image was 25 to 38 CPU seconds. CHAPTER I INTRODUCTION I.1 Background The primary purpose in experimental solid mechanics is to deter mine stress and motion (strain, rigidbody translation, and rotation) from measurements of physical quantities. For example, stress may be computed from measurements of load and crosssectional area, and strain is measured by changes in characteristic lengths. Many methods (mechanical, electrical, optical, etc.) are available to analyze stress and motion [14]. Optical methods play an important role in experimental mechan ics. They have a number of advantages which include [5,6] 1. Noncontacting measurements do not interfere with material response. 2. Fullfield analysis is usually possible. 3. Highly accurate results are usually possible. 4. Response times are as fast as light, thereby accommodating dynamic investigations. 5. Actual structures under live loads can frequently be ana lyzed. 6. Effects of actual boundary conditions can be studied in detail. Although optical methods offer many advantages, they do have several undesirable features. For example, some methods are sensi tive to small extraneous vibrations and require expensive, delicate equipment. Perhaps the most undesirable feature is the fact that most of these methods do not directly yield their wealth of informa tion about stress or motion in a readily usable form. The extraction of this information is timeconsuming and re quires special skills. An expert may spend hours reducing data and, all too often, the amount of data analyzed is limited simply because automated analysis is not available. Furthermore, results can be affected by the investigator's particular techniques. Digital computers are opening doors to automation, increased accuracy of existing methods, and previously intractable or unknown techniques. Some of the latest innovations employ digital image processing and pattern recognition (IP/PR) technologies which utilize digital computers and video imaging equipment for direct data acqui sition. The fields of IP/PR are rapidly expanding and are well established in many areas. In experimental mechanics, their poten tial for efficiently gathering vast quantities of data consistently and accurately is being quickly developed. These technologies are freeing experimentalists from hours of painstaking labor and provid ing them with a new avenue to existing numerical methods and analyti cal tools [714]. 1.2 Purpose of Present Work The IP/PR technologies have their price. For example, IP/PR programs that use fast Fourier transforms or correlation (template matching) are frequently considered computationally expensive because they entail large numbers of (relatively slow) floatingpoint calcu lations. Interactive programs provide great flexibility, but they need the user's constant attention. Iterative and backtracking (systematically retracing previous steps) methods can require sub stantial amounts of computer memory for intermediate data storage [1518]. Significantly reducing these time and memory requirements is certainly a desirable goal. Perhaps this is possible by rethinking the problem and developing a highly automated method which capital izes on the strengths of the digital computer. Thus, the purpose of this work is to develop such a method, pattern mapping, to obtain data via digital imaging equipment and subsequently determine plane motion. 1.3 Scope of Present Work Although it was not apparent in the beginning, the concepts developed in this investigation are readily applied to the photodot (grid) method [19]. The scope includes this photodot adaptation. Actually, this investigation encompasses a variety of scientific disciplines. Perhaps the best approach is to consider the attributes of each and work toward a goal that is obtainable in a reasonable amount of time. Specific objectives included in the scope are to 1. guarantee unique mapping 2. provide subpixel registration of 0.1 pixel 3. provide a wide range of motion measurement 4. provide wholeimage analysis 5. reduce computer and user time as much as possible 6. provide highly automated analysis. (Pixels are discrete elements in a digital image, and wholeimage implies results are computed for every photodot.) Special emphasis is on accuracy and increased computation efficiency. Several laboratory experiments using a CR39 (plastic) prismatic beam were included in the study. The experiments were translation, rotation, general rigidbody motion, fourpoint bending, and cantilever beam. All experiments were static cases of motion in a plane normal to the optical axis. A rectangular (11X11) pattern of photodots was contact printed on the lateral side of the beam. For convenience, motion is defined only in terms of the Lagrangian formulation. Evaluation of motion functions are limited to a linear and a second order approximation. The least squares method of curve fitting is used exclusively, and normal equations are solved by matrix inversion using GaussJordan elimination. Synthetic images are used to debug analysis programs and inves tigate the influence of various factors. Motion in the images in clude affine transformations and rigidbody rotation about the image center. Interpolation of betweenpixel intensities is bilinear, and systeminduced distortions are assumed eliminated during preprocess ing. Noise has a zero mean and selectable Gaussian or uniform 5 distributions. Image size is 256X256 pixels which are unsigned, 16 bits or less. The camera position is assumed fixed. Segmentation and feature extraction (dividing the image into subregions and obtaining useful information) include neither prefil tering nor histogramming (gray levels) and are based on the random selection of a few pixels. Processing is automatic and direct because known patterns in the image are always completely visible and recognizable. Feature space includes gray levels, areas, centroids, and second moments. The pattern language consists of only one sen tence, and it is contextsensitive. 1.4 Survey of Previous Work 1.4.1 Solid Mechanics Early applications of image processing in experimental mechanics involved the analyses of fringes. The fringes were from photoelasti city, moire, or holographic interferometry. Burger [7] and Chen [8] describe several of these applications in detail. Another image processing technique applied to mechanics is cor relation. Researchers [9,2024] determine plane displacements (ui, i=1,2) and their gradients (3ui/aXj, i,j=1,2) by comparing digitally recorded images of the same random pattern. For example, a pattern of black speckles (e.g., minute specks of black paint) on a white field is recorded as integer gray levels at discrete points (pixels). Two images are used in the analysis: one before and one after motion. By comparing (spatially registering) small subimages from each, displacements and their gradients are determined at selected points in the (original) pattern. Sigler and Haworth [25] combine holography with correlation to determine motion. Thus, the correlation method has the advantages of using coherent or conventional light sources and easily applied patterns. The grid method is one of the oldest and simplest methods used in the analysis of strain. One simply applies reference marks (lines, spots, scratches, etc.) to the surface of a specimen and records the original distances between marks. Next, the specimen is deformed and the distances are again recorded. Then, changes in the distances are determined and strain is computed [1,2,2632]. Although simple, the grid method is tedious and traditionally limited to larger strains. Typical strain measurements are greater than 0.001 inchesperinch; the results depend on the care exercised and the instrumentation [19,26,28]. Even when optical comparators are used for the smaller strains, many manhours are frequently needed to complete the analysis of large grids. Examining a specimen through a comparator and determining the position of a grid mark is also a subjective process. Some investi gators define a mark's position by its left edge, some by its right edge, and some by a variety of definitions. All these definitions are subject to error caused by rotation, the nonlinear characteris tics of the recording medium (if photographically recorded), grid application techniques, lighting, operator fatigue, etc. [19]. Parks [32] suggested computer technology might improve the accuracy of the method and would certainly reduce the labor involved. Previous efforts to fully computerize the grid method have not been reported in the literature. Sevenhuijesen [33,34] reported some feasibility studies with a Reticon photodiode array camera. Using an eightbit digitizer and digitizing the signal from each diode, he obtained a resolution of a few hundred micro inchesperinch. Images were recorded photographi cally without the aid of an image processing system. 1.4.2 Image Processing/Pattern Recognition Several researchers active in IP/PR have been interested in two and threedimensional motion analysis. Most efforts have concentrat ed on the motion of rigid bodies, clouds, the human body, optical flow, etc. Motion parameters were obtained by a variety of iterative processes [3538]. Huang and Tsai [3943] proposed a direct method of mapping the threedimensional motion of a plane using two, timesequential images. Mapping parameters were determined uniquely by solving a set of linear equations. Unfortunately, motion parallel to the optical axis could not be determined, and "small" approximations were made for strain and rotation. Mathematical proofs guaranteeing unique mapping between images were included. CHAPTER II THEORETICAL FOUNDATIONS II.1 Solid Mechanics The description of motion used here is limited to the Lagrangian description and is consistent with the classical field theories of modern continuum mechanics [4448]. The Lagrangian description is sufficient to determine the directions of the principal axes of strain and the magnitudes of the principal stretches. It is, there fore, a fully general measure of strain and is capable of describing small or large strains. For convenience, all reference frames are Cartesian in threedimensional Euclidean space. In Figure 1, body $ consists of a set of particles which assume a continuous progression of configurations in time. The undeformed configuration is chosen as the reference state where particle P occupies the "material" position X at time t=to. Subsequent motion carries particle P to the "spatial" position x in the deformed or "after motion" state at time t. Although each state may be defined in terms of its own coordinate system, material and spatial coordi nates are usually measured with respect to the same coordinate axis. The motion of a point is described by the relation of the coordinates between the two states. Mathematically, motion is ex pressed by a onetoone mapping x symbolized as x = x(X,t) or x. = xi(X IX2,X3 t) 2. 1 2 3' Figure 1. The motion of body B through space. The deformation derivative or gradient gradient of the motion x is defined as the of x at X and is denoted by 3x F = or  aX 3x. F. . ii DXj where F is the fundamental quantity for the analysis of local proper ties of deformation, and the physical components are, of course, dimensionless as are all measures of relative configuration. 10 Alternatively, motion written in terms of the displacement vector u is x = X + u(X,t) or xi = X. + ui(XI,X 2X ,t) 2. i 1 2' 3 And the derivative of u, the displacement gradient, is au J = X or J. ij 1 SX. If two particles in the reference configuration are an infini tesimal distance dX apart, then these two particles are an infinites imal distance dx apart in the deformed configuration (see Figure 2). Tensor F maps the neighborhood of particle P in the Figure 2. The motion of the neighborhood of particle P. reference configuration to its neighborhood in the deformed configur ation. Thus, the linear transformation is dx = FdX or dx. = F..dX. S 1 13 j Unique mapping is guaranteed as long as the Jacobian J = IFI is greater than zero and finite. Since vector dX has length dS and vector dx has length ds, Lagrangian strain E may be defined in terms of these lengths squared as (ds) (dS)2 = 2dXE.dX The Green deformation tensor C, referred to the undeformed con figuration, gives the new squared length (ds)2 of the element into which the given element dX is deformed. Symbolically, the relation ship is (ds)2 = dX dX (ds) = dXCdX where C = FT.F I Conversely, the Cauchy deformation tensor B gives the initial squared length (dS) of an element dx in the deformed configuration. This relationship is written as (dS)2 = dx.B *dx where 1 (F T F B : (F ) F Lagrangian strain written in terms of the Green deformation tensor is E = [C1] 2 2 (11) In terms of the displacement gradient, E is I ui au 3u auk au Eij = 2[X + i + X] ij 2 jX. 2X. 3X. jX. 3 3 1. 3 Strain, computed by a Taylor series [1], is E au 8u)2 av 2 ,w 2 S11 V 3X 9X aX X / Byv 2 au 2 aw 2' E = A1+2 +() +() +() I 22 1 Y a Y aY aY (12) (13) au yv au u u v Dv Dw aw + + + + 1 r Y x xs a 3X aY 3X aY arcsin[(1 )( 2 (1+E )(1+E22) 11 22 (10) 12 = 21 13 For many engineering applications, the socalled "small dis placement" theory is adequate and much easier to implement. By this theory, strain is au 11 ax 1 au av 12 2 Y ax (14) C21 = E12 av 22 3X If the Jacobian J is greater than zero and finite, the deforma tion gradient written in terms of a rotation tensor R and the right stretch tensor U, a polar decomposition, is F = R.U (15) Tensors R and U are unique and represent rigidbody rotation and stretch, respectively. Rotation tensor R is orthogonal and produces a rigidbody rotation between the principal axes of C at X and the 1 principal axes of B at x. The necessary and sufficient condition for no local rigidbody rotation (sometimes called "pure strain") during motion is R=1. Symmetric, positivedefinite U produces the changes in vector lengths during motion and produces rotation, in addition to R, of all vectors except those in the principal directions of U. Note that U=1 is necessary and sufficient to define locally "pure" rigidbody motion (neglecting the trivial case when no motion occurs), and translation does not change a vector with respect to the common reference axis. Tensors U and R, written in terms of F and C, are U = C1 R = F.U_1 R F.U (16) (17) As a final note, the deformation at any point may result from translation, a rigidbody rotation of the principal axes of strain, and stretches along these axes. These motions may occur in any order, but their tensorial measures may not. Mathematically, Lagrangian motion is assumed as a successive application of 1. stretch by U 2. rigidbody rotation by R 3. translation to x. 11.2 Image Processing and Pattern Recognition II.2.1 Digital Images A typical image processing system is shown in Figure 3. Conver sion of the intensity distribution of a camera image to a form suit able for computers, digitizing, is accomplished by the digitizer. After digitization, the image is transferred to the computer for further processing. How an image is processed depends, of course, on the hardware configuration [15,18,35,4951]. CCD Camera F ~ Object Microcomputer Digitizer with Internal Monitor Digitizer Figure 3. A typical PCbased image processing system. A digital image is an approximation of a real, continuous image. For example, the real image of the spot in Figure 4a is re presented in digital form as the matrix of integer numbers in Figure 4b. Each matrix element (or pixel which is dimensionless) corre sponds to a discrete point in the real image, and the value of the element (gray level) is the approximation of the average local illum ination intensity. Notice that black has lower gray levels than white and that the edge region (boundary) between the two has inter mediate values. The variation of gray levels in regions of constant illumination is the effect of noise. Gray levels range from zero to some maximum which depends on the hardware (e.g., camera, digitizer). 149 151 155 146 163 159 159 N 15 15 162 151 142 145 146 156 145 166 148 165 157 145 156 142 146 150 127 148 158 166 151 153 142 142 157 149 153 148 143 150 155 156 145 146 150 159 166 56152 54 153 155 132 148 144 151 169 164 144 165 156 138 15 146 148 129 1~ 98 115 127 150 15 137 149 152 151 153 15 151 147 156 148 138 83 47 36 51 56 60 3# 142 154 147 163 146 15 146 152 146 134 138 66 45 52 48 49 47 51 74 52 139 152 154 15 146 145 154 14 141 94 56 56 47 50 43 51 46 53 59 91 151 153 153 151 146 159 143 138 53 61 51 50 55 43 48 51 45 56 44 135 154 146 154 150 150 150 10 51 48 36 44 55 49 51 58 53 50 49 105 141 144 15 153 139 152 96 50 50 44 52 48 56 57 50 54 46 47 1t1 15 143 143 156 155 155 18 40 43 46 49 59 41 48 49 55 51 42 193 146 158 152 153 148 166 137 47 51 47 41 48 37 51 50 50 47 33 137 156 153 143 147 148 151 153 95 45 5 55 49 50 51 48 44 43 82 165 147 153 160 147 135 155 159 138 76 55 45 57 53 54 47 63 132 151 158 147 148 145 148 151 162 159 144 82 50 55 54 53 39 70 141 148 152 148 159 152 151 153 156 153 148 132 148 132 149 96 99 143 154 150 145 154 163 159 151 144 141 149 157 148 156 145 158 145 154 142 139 145 145 147 154 161 148 149 157 150 141 144 158 149 166 150 151 142 142 158 145 156 150 150 151 147 149 156 158 146 160 148 140 148 149 151 144 149 149 150 15 153 152 149 157 160 155 147 150 131 150 150 163 150 158 153 147 146 163 136 151 154 153 152 153 151 16 146 148 144 152 153 148 16Q 160 136 157 144 144 14 162 146 148 154 Real Image Digital Image a) b) Figure 4. A spot and its digital image. The values, I(x,y), in the digital image correspond to illumination intensity in the real image. Noise corrupts the image gray levels, and the sources may be classified into two groups: optical and electronic. Two examples of optical noise are particulate matter in the air and laser speckle. Electronic noise comes from the discretization process, thermal sources in the hardware, interference from extraneous electromagnetic sources, etc. Noise is usually assumed statistically random and normally distributed with a zero mean, i, and a standard devia tion, a Unfortunately, no known method completely eliminates the effects of noise. Each array element has neighbors which are commonly called the eightneighbors [52]. In Figure 5, the center element, pixel (x,y), is the reference element or element of interest. The element directly above it is arbitrarily labeled neighbor 0, and the other elements are named in a counterclockwise direction. Although ele ments along the image's outer array edge do not have all Figure 5. A pixel and its eightneighbors. 1 0 7 2 Pixel 6 (x.y) 3 4 5 8 neighbors, the neighbors they do have are also labeled according to Figure 5. Spatial and illumination sampling must be sufficient to repre sent adequately the real image. Too low a spatial sampling frequency averages intensity over too large an area (see Figure 2.7 in refer ence 15), while too high a sampling frequency results in the loss of information outside the field of view. In general, increasing inten sity corresponds to increased gray levels, but intensities beyond the capability of the equipment are limited to the maximum gray level. An increase in the range of gray levels increases image detail (see Figure 2.8 in reference 15). 11.2.2 Preprocessing After digitization, the digital image is corrected for geometric and radiometric distortion. Distortion, introduced by the processing system, is removed by using the appropriate calibration procedure [49,53]. This type of correction is frequently called preprocessing and images in this study are assumed corrected unless otherwise stated. II.2.3 Segmentation The next step in image processing, segmentation, consists of identifying regions and boundaries of interest in the image [18,54]. Of course, the difficulty in segmenting an image depends on its complexity. Since the images developed in this study are rela tively simple, they are easily segmented. Histogramming. Segmentation usually begins with histogram ming. When the frequency distribution of the gray levels in an image 19 is computed for the entire image, contrast and the presence of multi ple modes are readily observed. Figure 6, the bimodal histogram of Figure 4, has an average black of 50 and an average white of 150. Note, the histogram yields no direct information about the location of the gray levels in the image [55]. 50  Contrast Range 40 Pixel 30 Count 20 10 0 qLA A. .I 0 20 40 60 80 100 120 140 160 180 200 Gray Levels Figure 6. The bimodal histogram of Figure 4. Usually, computing the histogram for the entire image is not considered computationally expensive. This is not true in the present example because the percent of the total computer time devoted to histogramming the entire image is significant. Since the images used here are designed to have histograms similar to Figure 6, the needed information is easily estimated without actually computing a histogram. Thresholding. Thresholding follows histograimming and includes the task of determining the gray level which, in this case, reason ably separates regions of black from white. This is frequently accomplished interactively or by use of a priori information, and the choice of the optimum threshold is a subject of considerable interest [5662]. The alternative approach in this study requires neither inter active thresholding nor an optimum threshold. Instead, a good esti mate of gray levels that are "definitely black" and an estimate of the average white level are used. The estimates are derived from histogram statistics and based on the following assumption: the histogram is strongly bimodal. In Figure 6, gray level 53 is assumed black and the average white gray level is about 151. The task of estimating these levels is a fourstep process. First, pixels are randomly selected and their average gray level IA is computed along with the standard deviation oA. A sample size of 30 or more is statistically sufficient to estimate the mean and standard deviation of the entire histogram [63,64]. Obviously, darker pixels are to the left of the average and lighter pixels are to the right (see Figure 7). In the second step, the standard deviation is subtracted from the average, resulting in IB. Algebraically, this is I = IA A (18) 21 50 40 30 I Pixel 1 B A Count 20 0 20 40 60 80 100 120 140 160 180 200 Gray Levels Figure 7. Black and white regions in a bimodal histogram (IA=132, I=98, I1=53, 12=151, It=102). Another set of 30 or more pixels whose values are below IB are ran domly selected and used to compute the average I1. Thus, the average black level is approximated by I1. Chebychev's theorem [63] states that, if a probability distribu tion has a mean p and a standard deviation a, the probability of obtaining a value x which deviates from the mean by at least K stand ard deviations is at most Mathematically, this is expressed as K2 KK P(xU > Ko) <  (19) K Since only one standard deviation is subtracted from the average, finding pixels darker than Ig is highly probable. A skew of the histogram to the left further improves the odds of finding pixels less than Ig. The third step is the initial estimation of the average white gray level, I2. A sufficient random sample whose values are greater than IA is used to compute the initial estimate of 12. Finally, the threshold value It is computed using the estimates for the black and white levels. Symbolically, 1 + 12 I = 2 (20) t 2 Images designed for this study are assumed to have two regions: black and white. In a sense, they are binary (region) images. Thus, segmentation is easily accomplished by locating and isolating the black regions of interest from the white background. Raster scanning and border following. Beginning at the origin, the upper left corner, the image is raster scanned (i.e., along a row or column) until a black pixel is found. If at least one of the neighbors is black, a possible black region is assumedspurious noise is assumed if the pixel has no black neighbors. If a black neighbor is found, the border between black and white is followed around the black region thereby isolating it from the neighboring white. Rosenfeld [18] has developed a generalpurpose borderfollowing algorithm. First, gray levels are reduced by thresholding to O's for black or 1's for white. (This is quickly done if one has the appropriate hardware.) By comparing neighbors and locating the next edge pixel, the algorithm follows the border counterclockwise for an outer border and clockwise for an inner one. Since borders may overlap, provisions are made for backtracking. In this study, the generality of Rosenfeld's algorithm is un necessary, and the algorithm is modified for faster execution. The modified algorithm (see appendix) is designed to follow outer edges (no inner edges exist) in the clockwise direction, and reducing the image is eliminated by combining thresholding with border following. The only information saved during border following is the y coordinate of the spot's leftmost pixel, L; the y coordinate of the rightmost pixel, R; the x coordinate of the highest pixel, T; and the x coordinate of the lowest pixel, B. Since the black region extends a little beyond the approximate border, 2 pixels are sub tracted from L and T, while 2 pixels are added to the R and B (see Figure 8). Thus, the local region of interest (black and inter mediate gray) is "framed" for further processing. Raster scanning resumes at the bottom of the frame and continues until another black region is found and it is framed. Scanning and framing continue until the entire image is covered. Thus, the image is completely segmented and the information needed for feature ex traction is available. rY 149 151 155 14 1 15 1 1 149 151 155 146 1A3 1SB ISB 1S1 157 150 144 150 146 145 146 150 153 Edge 15& 147 147 145 151 Edge T 150 ~ A 1441141 149 157 148 156 145 158 145 154 1571150 141 144 158 149 166 150 151 142 156 158 146 160 148 140 148 149 1511144 155 147 150 131 150 150 163 150 158153 151 160 146 148 144 152 153 148 160 160 1 1 149 147 136 Frame 142 145 146 156 145 160 148 165 153 142 148 157 149 153 148 143 153 155 132 148 144 151 169 164 127 150 154 137 149 152 151 153 60 90 142 154 147 163 146 150' 51 70 52 139 152 150 150 146 46 53 59 91 151 153 153 151 51 45 56 44 135 154 146 154 58 53 50 49 105 141 144 150 50 54 46 47 101 150 143 143 49 55 51 42 103 146 158 1 2 50 50 47 33 137 156 153 13 Edge 48 44 43 82 165 147 153 160 47 50 63 132 151 158 147 148 39 70 141 148 152 148 159 152 143 154 150 145 154 163 159 151 139 145 145 147 154 161 148 149 150 145 156 150 150 151 147 149 149 150 151 153 152 149 157 160 146 163 136 151 154 153 152 153 157 144 144 141 162 146 148 154 Figure 8. Frame of spot in Figure 4. II.2.4 Image Features Images contain various types of information known as features, and, when properly selected, they adequately represent the images. Here, it is assumed that natural features are intrinsic to the real scene and artifacts are added as desired to aid subsequent proces sing. Features, both natural and artifact, are generally grouped into three classes: physical, structural, and mathematical [65]. 145 156 142 146 150 127 155 156 145 146 150 159 165 156 138 155 146 148 151 147 156 148 138 83 152 146 134 138 66 45 154 140 141 84 56 56 159 143 138 53 61 51 150 150 100 51 48 36 139 15i 96 50 50 44 155 155 108 40 43 46 148 166 137 47 51 47 148 151 153 85 45 50 135 155 159 138 76 55 148 151 162 159 144 82 153 156 153 148 132 148 . . . j . Physical and structural features are, in this case, the arti facts shown in Figure 9 plus the pattern they form in the image. For simplicity, the (single) icon is considered a "special" spot, and all other spots are assumed uniform in size. Spot Icon Figure 9. Image artifacts. The icon is a "special" spot. Mathematical features are frame areas, spot centroids, and those concerning icon identification and rotation. These features allow extensive integer programming which speeds the majority of the cal culations and improves accuracy. 11.2.5 Feature Extraction Here, feature extraction consists of isolating the artifacts and computing their desired mathematical features. Since the artifacts are extracted (by framing) during segmentation, all that remains is the computation. Frame area. Frame (integer) area is easily computed using the information obtained during the segmentation process. The computa tion is A = (RL)*(BT) (21) Since the icon's frame area is much larger than the other spots', classifying each artifact as "spot" or "icon" is a simple, binary task. Spot centroids. The centroids of all spots are used as defini tions of spot position. If the gray levels in a spot's frame are assumed to be "weights," the x and y coordinates of the centroid are calculated [18] by B R E E xI(x,y) x=T y=L B R E E I(x,y) x=T y=L (22) R B E Z yI(x,y) y=L x=T R B E E I(x,y) y=L x=T Henceforth, these summation limits are implied unless otherwise noted. How do gray levels, noise, and spot size affect spatial resolu tion of the centroid? Here, it is evident (see Figure 8) that merely summing with gray levels I(x,y) does not yield the proper result because the white pixels shift the spot's computed centroid away from the correct value. A transformation of all gray levels in the frame by the equation I'(x,y) = I I(x,y), (23) w where I, is the local average white level and I'(x,y) is the noisy gray level difference (between black and white), gives the approxi mate centroid coordinates. And since pixels on the frame edge are white, their average gives a good, local estimate for I,. The sub stitution of I'(x,y) into equations (22) leads to E E x I'(x,y) S=xy E E I'(x,y) xy (24) Z E y I'(x,y) yx E E I'(x,y) yx To account for noise, assume each gray level in the image con sists of the actual (i.e., uncorrupted by noise) value, r(x,y), plus zeromean Gaussian noise, n(x,y). Thus, I'(x,y) = I r(x,y) + n(x,y) W I'(x,y) = H(x,y) + n(x,y) (25) where H(x,y) = I r(x,y) is the actual gray level difference. Substituting equation (25) into equations (24) leads to EExH + ZZxn xy xy ZEH + ZEn xy xy (26) EZyH + EZyn  yx yx E H + EE yx yx If the expected value [63] for the noise is assumed EjnI(x,y)}  EEn(x,y) xy  ECn(xy) yx En(x,y) x  En(x,y) y (27) and substituted into equations (26), then the expected centroid is ZZxH ZxEn E{X = xy x y EEH + EZn xy xy ZExH + Ejn}Zx xy xy (28) EZyH + E{n}Ey E{Y1} yx y EZH + Enr} yx Since E{n}=O, equations (28) become EExH E{} =xy ECH xy (29) ElyH EY} yx EZH yx In general, the summations in equations (27) are not equal to zero, which introduces random error. But if Zn = 0, then X = E{X}, y and if Zn = 0, then Y = E{Y}. Thus, the summation process used to x compute the centroids greatly reduces, although in general it does not eliminate, the effect of noise. To understand better how noise, spot diameter, and number of gray levels interact to affect spatial resolution, consider the one dimension model in Figure 10. This model is essentially a column in the twodimensional image (see Figure 8) with frame edges at x=T and x=B. For simplicity, edge pixels are always located at x=T+2 and x=B2. (The extension to the general twodimension case is straight forward.) Similar to equation (26), the centroid calculation is ExH(x) + Zxn(x) Sx x SH(x) + En(x) (30) x x _T_ T T+2 T+3 \ line length I v x B3 B2  F. Frame Figure 10. Onedimension model. Assume a Ii KT H x=T+2 KHc x=B2 H(x) = H(x) H T+3 0 otherwise 1 O 0<(K< <1 o Bl and T<6 where Hc, a constant over the line length, is the gray level differ ence and Hm is the maximum possible gray level difference (contrast) between black and white. The influence of noise and the edge pixels can be described in terms of resultants. The noise resultant, 6n is the product of random variables 6 and no, i.e., Exn(x)=6n and En(x)=nc. Dimen x x sionless parameter KT is the fraction of the x=T+2 edge pixel covered by the line; and KTH is the resultant. Assume the same for and KBH at edge x=B2. During digitization, KT and tized (see Figure 11) to K' and <', respectively. For a given line segment at a specified location, KT and KB are constants. The sub stitution of these values into equation (30) yields B3 Z xH +H [T (T+2)+K (B2)]+6n Sx=T+3 B3 Z H(x)+H ( + If x=T+3 B3 E x = (T+3)+(T+4)+(T+5)+ ... +(B5)+(B4)+(B3) x=T+3 is added to 33 Z x = (B3)+(B4)+(B5)+ ... +(T+5)+(T+4)+(T+3) x=T+3 then B3 2 Z x = (T+B)+(T+B)+(T+B)+ ... +(T+B)+(T+B)+(T+B) x=T+3 B3 E x = (T+B) x=T+3 where n is the number of terms in the sum. n = (B3) (T+3) + 1 Since B3>T+3 and n>1, n = BT5. Therefore, B3 B3 (BT5)(T+B) E x = x2T x=T+3 B3 E H(x) = (BT5)H x=T+3 Subsequently, equation (30) becomes (BT5)(T+B)Hc+2{6nc+Hc[ c c T B After dividing the top and bottom by Hc, equation (31) becomes (BT5)(T+B)+2[6c+KT(T+2)+ 2[(BT5)+E+KT + n where E = H Note, the E ratio is C Pratt's [66] signaltonoise ratio, similar to the inverse of H 2 SNR = (c) o (33) 1 I 1 I C Fraction of Edge Pixel covered Figure 11. The discretization of T and K An increase in He im proves spatial resolution of the edges. (31) (32) After studying Figure 10 and equation (32), it is observed that 6=T or 6=B gives the most noise error while (BT5)(T+B)+2[CE(T+2)+K (B2)] 6 = T[BB_5+TB (314) 2[(BT5)+K +BK gives none. If a line length of d=BT5 is assumed, equation (32) becomes (d)(d+2T+5)+2[6+K (T+2)+BKg(d+T+3)] X = (35) 2[d+S+KT+ If &=0, (d)(d+2T+5)+2[KT(T+2)+K (d+T+3)] xJ= (36) 2[d+K +K B which is the "exact" centroid. If T=0 is assumed and K' and K' are T B substituted into equation (35), the resolution error is (d)(d+5)+2[2KT+K (d+3)] (d)(d+5)+2[6+2K+K'(d+3)] R = (37) e 2[d+KT+KB] 2[d+S+<+ The general relationship between E, d, and Re is clearer if equation (37) is simplified. To assume the line position is a random variable implies the following expected values: EIKT} = E{ After some algebra, equation (37) becomes Re (d+526) e 2(d++1) (38) For E much smaller than d, equation (38) suggests an almost linear relation between & and R A spatial dependency (perhaps difficult to verify experimentally) is also indicated. to verify experimentally) is also indicated. Icon Centroid Figure 12. Icon rotation measured from the vertical. Icon rotation. Next, the icon's angle of rotation is computed (see Figure 12) using the second moment of inertia. The analysis is as follows: I = EE I'(x,y) d' xx y xy I = EE I'(x,y) d2 (39) yy x xy P = LE I'(x,y) d d xy xy xy where Ixx = moment of inertia about icon's xaxis I = moment of inertia about icon's yaxis Pxy = product of inertia about icon's centroid dx = x distance from pixel to icon centroid d = y distance from pixel to icon centroid. Icon rotation is 2P S= ~Tan 1 x ) (40) 2 I I yy xx The above equation gives the angle to within a constant. For example, the analysis gives the same angle for both icon orientations in Figure 13. The difference is resolved by constructing line "AA" through the icon centroid and computing the centroids of the upper and lower segments. Line "AA" has the form A = mxA+b where xA and yA are points on "AA" with m = TAN(e) and b = conmxcon Icon Icon SY x +6 K "A" ,' / Icon "A"/ +6 Upper Segment Upper Segment Icon Lower Segment Figure 13. Icon rotation is determined to within a constant by equa tion (40). The centroid for the upper segment is XA YA E E xI'(x,y) S= x=T y=L XA YA E E I'(x,y) x=T y=L XA YA E E yI'(x,y) x=T y=L XA YA E Z I'(x,y) x=T y=L (41) and for the lower segment, B R E E xI'(x,y) X=XA Y=YA XL B R E E I'(x,y) X=XA Y=YA (42) B R E E yI'(x,y) XX A Y=YA L B R E E I'(x,y) X=XA YYA The distances from the icon centroid to the segment centroids are 2  2' d = U /(X X ) +(Y Y con U U Icon U Icon (43) ( m on2 2 d Icon ) I+(Y co L Icon L Icon If dU>dL, 9 computed by equation (40) is correct; otherwise 0 must be shifted by r9ir 8>0 e  (44) fe+ r 8<0 Thus, a maximum rotation of one revolution is measurable. Information saved during feature extraction is, of course, frame area, the centroid coordinates of each spot, the number of spots, which spot is the icon, and icon rotation. The areas and the cen troids are stored as linear arrays in the order the spots were found. Lastly, the problem of sorting this raw information and using it to match spots in the before and aftermotion images is solved using pattern recognition principles. 11.2.6 Known Patterns In this study, patterns are classified as either "known" or "unknown." Known patterns contain structural features (natural, artifact, or both) that are recognizable before and after motion. Unknown patterns are patterns which never have recognizable structure or those in which features are obscured beyond recognition. Known patterns containing only artifacts are considered here. Artifacts and their arrangement are selected to improve segmen tation, feature extraction, mapping, and subsequent numerical analysis. Examples of simple, easily analyzed patterns are shown in Figure 14. These patterns are fine for small rotations and are easily recognized because the relative order of the spots does not change, e.g., the upper left spot remains in the same position relative to the other spotsduring motion. If rotation is 45 degrees or more for the square patterns and 90 degrees or more for the others, the relative order is lost. The patterns in Figure 15 allow spot identification after rotations up to one revolution. Once the patterns in the undeformed and deformed images have been 66 56555 ewes .me * sue... * @505mg * wsswg S 5 66 m ewe.... 6 6 ) 6 5 Figure 14. Patterns that are easily recognized and analyzed. Frames are for reference and not part of the pattern. *ee ee * U U U 0C0 eeO S sew. wee sees Figure 15. Patterns of Figure 14 with icons. The icons are located for convenience. 000000 0 * U U U 00000 *0 * 3 6 S @6 *5e66 ewe.... 66 @6 Se * *..L@ e U sew.... .me.... ewe.... * C 9 * w * S S * 6 5 recognized, subsequent analysis is simplified (especially for the Lagrangian description) because the undeformed spacing is approxi mately constant. 11.2.7 Syntactic Pattern Recognition Pattern recognition is usually divided into two classes: deci siontheoretic and syntactic. In the first class, a pattern is represented by feature vectors and recognized by a stochastic parti tioning of feature space [6769]. Syntactic recognition uses pattern structure and linguistic rules for analysis [70,71]. Syntactic recognition assumes a pattern is described by a language which contains valid sentences composed from a finite alpha bet or set of symbols. The language is governed by "pattern grammar" or set of grammar rules. In contextfree languages, pattern grammar is unrestricted. Contextsensitive languages restrict pattern grammar on the number of sentences, sentence construction, or both. Here, the pattern language is contextsensitive and has only one sentence which is composed of a fixed number of ordered symbols. The alphabet symbols are the spots' relative positions whose order is determined in the undeformed image. A spot's context is determined by its 0,2,4,6 (or alternatively by 0 through 7) neighbors. For example, (2,2) has neighbors (1,2), (2,1), (3,2), and (2,3). An attempt to map (2,2) in the undeformed image to, say (1,1), in the deformed is "out of context" with the neighbors (see Figure 16). Thus, correct pattern mapping is assured if every spot maps in con text with its neighbors. * (1,1) (1,2) (1.3) (1,4) (1,5) (2. 1) (2,2) (2.3) (2.4) (2.5) S ** (3.1) (3.2) (3.3) (3.4) (3.5) 0 0 4 (4,1) (4,2) (4.3) (4,4) (4.5) (5.) (5 ) (52) (5.3) (5,4) (5.5) Pattern Onesentence Language Figure 16. A pattern and its contextsensitive language. Note the alphabet (symbols) and context in the language. Higher dimensional pattern grammars [7072] provide the capabil ity to formally describe, recognize, and map patterns similar to those in Figure 15. Since a record of each spot and its neighbors must be maintained and manipulated, generating and parsing formal grammar for large patterns becomes an intense process. If possible, trading some of the formal elegance for an economic approximation is highly desirable. 11.2.8 Approximation of Syntactic Pattern Mapping Fortunately for many motions, spot relative positions change very little, which allows context to be determined from spot 43 "addresses." If both images are carefully divided into a number of addresses subregionss), then a spot has the same address before and after motionor nearly so. The determination of a spot's address is a simple calculation involving centroid coordinates, and recognition is reduced to merely arranging the spot addresses in proper order. Thus, context is recognized. "Offsets" are the spot addresses relative to the icon address. All offsets are defined according to an icon address of 0 (see Figure 17). Image Border  K Spot Addresses * 18 10 2 6 14 * 17 9 1 7 15 01 * 16 8 0 8 16 * 15 7 1 9 17 *0 1 14 6 2 10 18 "Offee se Figure 17. Image addresses and offsets. image edges. Some addresses overlap Pattern geometry affects the upper limits of measurable strain because large strain displacements may change the order of the off sets in the deformed image. Spreading spots further apart (in the undeformed image) increases the size of a spot's subregion. A larger subregion allows the spot more strain displacement before any change in offset order occurs. The details are best illustrated by example (see Figure 18). It is assumed that the undeformed image is analyzed first and, with one exception, both images receive the same preliminary analysis. * * * emiS* * Undeformed U U 0 + o U+ 0 U eo Deformed Figure 18. A simple example of motion (au./3X.=0.05, u=v=10 pixels, e=1200, E..=0.0525). The origin is in the upper left corner of 1ach image. Successful mapping is initially assumed and later verified by four final checks. First, the image is segmented and the features are extracted. Scanning is by column and the spot centroids are stored in the order they were found. (See columns marked "Raw" in Tables 1 and 2.) The icon is identified, and it's rotation is also computed. Table 1. Unordered spot coordinates in the undeformed image. Reference From Icon X Y X Y X Y 58.0101 93.0079 163.0518 197.9896 127.9832 58.0481 93.0216 197.9616 127.9396 163.0107 128.7025 57.9599 93.0284 198.0660 63.0010 92.9964 127.9581 162.9653 197.9768 57.9856 58.0635 92.9969 128.0099 163.0027 197.9936 57.9660 58.0056 57.9950 58.0041 57.9801 93.0576 92.9791 93.0132 92.9724 92.9915 127.9873 128.0263 128.0002 128.0575 127.9894 163.0297 162.9798 163.0147 162.9531 162.9875 198.0315 198.0169 198.0273 198.0136 197.9982 57.4541 92.4517 162.4955 197.4332 127.4271 57.4187 92.3923 197.3321 127.3103 162.3813 128.0000 57.2575 92.3260 197.3632 162.2984 92.2207 127.1824 162.1896 197.2011 57.2101 57.2148 92.1481 127.1610 162.1538 197.1446 57.8310 57.9438 58.0797 58.1617 57.9914 92.9227 92.9173 93.1757 92.9836 93.0760 128.0000 127.8911 127.9383 128.2151 128.0738 162.9676 162.9909 163.0989 163.1105 162.8523 197.8964 197.9548 198.0384 198.0978 198.1556 70.5459 35.5483 34.4955 69.4332 0.5729 70.5813 35.6077 69.3321 0.6897 34.3813 0.0000 70.7425 35.6740 69.3632 34.2984 35.7793 0.8176 34.1896 69.2011 70.7899 70.7852 35.8519 0.8390 34.1538 69.1446 70.1690 70.0562 69.9203 69.8383 70.0086 35.0773 35.0827 34.8243 35.0164 34.9240 0.0000 0.1089 0.0617 0.2151 0.0738 34.9676 34.9909 35.0989 34.1105 34.8523 69.8964 69.9548 70.0384 70.0978 70.1556 NOTE: Units are pixels. "Raw" coordinates are the spot centroids in the image. "Reference" and "From Icon" values are computed by equations (46) and (47), respectively. See Figure 18. Table 2. Unordered spot coordinates in the deformed image. Reference From Icon X Y X Y X Y 98.5336 131.2724 78.6761 163.9343 111.3736 196.6288 58.7113 144.0612 229.4016 91.4833 176.7262 123.7393 38.9059 209.4617 71.5500 156.8817 18.9873 104.3150 189.5733 51.6854 136.9841 84.3163 169.6761 117.1079 149.7807 33.4158 50.2682 64.3727 67.1092 81.1960 84.0388 95.2677 98.0816 100.8559 112.1065 114.9435 129.6776 126.2522 131.8162 143.1225 145.8710 157.2419 160.0107 162.8025 174.0585 176.8253 190.9335 193.7303 207.7552 224.6135 51.9634 53.0816 88.4532 54.2228 89.5630 55.4298 124.9335 90.7333 56.5016 126.0252 91.8919 128.0000 161.4258 93.0300 162.6010 128.3497 197.9716 163.7403 129.5597 199.0750 164.8545 200.2602 166.0404 201 .3280 202.4800 192.1880 155.3833 196.7857 118.6530 160.0309 81 .8551 201.5070 123.2581 45.0350 164.6782 86.5161 128.0000 206.0458 49.7056 169.3190 91.1147 210.6846 132.4753 54.3186 173.9321 95.7498 137.2151 58.9647 100.3761 63.6284 76.0366 74.9184 39.5468 73.7772 38.4370 72.5702 3.0665 37.2667 71.4984 1.9748 36.1081 0.0000 33.4258 34.9700 34.6010 0.3497 69.9716 35.7403 1.5597 71 .0750 36.8545 72.2602 38.0404 73.3280 74.4800 64.1880 27.3833 68.7857 9.3470 32.0309 46.1449 73.5070 4.7419 82.9650 36.6782 41.4839 0.0000 78.0458 78.2944 41.3190 36.8853 82.6846 4.4753 73.6814 45.9321 32.2502 9.2151 69.0353 27.6239 64.3716 NOTE: Units are pixels. "Raw" coordinates are the spot centroids in the image. "Reference" and "From Icon" values are computed by equations (46) and (47), respectively. See Figure 18. All spots are moved into a "reference position" by translation and rotation (see Figure 19). Translation is accomplished by adding X to the X and YT to the Y of each spot; X and YT are X = X X T cp Icon (45) Y = Y Y T cp Icon I where X and Y are the coordinates of the icon in reference pos cp cp tion (typically the image center). Rotation for each spot computed from the x and y distance between it and the icon. T computation is X = XX p Icon Y = YY p Icon p = X + Y P P 1 p _ = Tan () z X P p XRef = X+p[cos( z0)cos z ]+XT YRe = Y+p[sin(w 6)sin w ]+Y Ref z z T where XRef and YRef are the new coordinates of each reference position. The results of translation and shown in Tables 1 and 2 under the "Reference" column. Position relative to the icon, i.e., distance from spot in rotation the icon, XRel XRef XIcon Y =Y YIcon Rel Ref Icon (46) the are is (47) 48 0  ) e * S* Deformed Image Reference Position Figure 19. The deformed image and its reference position. The distances from the icon are shown in Tables 1 and 2 under the columns marked "From Icon." Although all spots have been located relative to the icon, recognizing the pattern they form and arranging them in proper order (by address) is not always easyespecially if rotation is more than a few degrees. Address computation is based on "learning" the undeformed image. The spots nearest and farthest from the icon row and column are located in the undeformed image and used to calculate the address "offset," Aoffset, by the following: X = MAX(X) max m = MIN(X) Y = MAX(Y) max Y min = MIN(Y) X N = max N = NINT( ) x X mln Y max N = NINT( ) Smin X N (X ) SX + max x min X = X + space min N +1 X Y N (Y ) Ymax y min Y = Y + space min N +1 N N = NINT( ) + 1 shift X space Xel YRel Offset = NINT( ) + NINT( )(N ) offset X space sift space space where Xmax Xmin Ymax Ymin N = = maximum = minimum = maximum = minimum number of X centroid X centroid Y centroid Y centroid rows above or below icon (48) (49) (50) Ny = number of columns left or right of icon Nrows = number of rows in the image Xspac = approximate row spacing Yspace = approximate column spacing Nshift = number of possible columns in image. Function NINT rounds numbers to the nearest integer value. Values Space' space' and Nshift are saved from the undeformed image and used in the deformed. Fortunately, arranging the spots in proper sequence is a simple matter of bubble sorting [73] the offsets in ascending order. By equation (50), offsets (see column "Offset" in Tables 3 and 4) increase down columns and left to right. Column "Key" is the order in which the spots were found. Note, the twodimension pattern context is now represented by a onedimension array, XRel, which corresponds to the ordered spot addresses. Recognition of the twodimension context is accomplished by determining the rowcolumn order of the spots. The leftmost column is assumed to begin with the lowest offset and continue with ascend ing offset until the X coordinate decreases. Column 2 begins with the next highest offset and continues until the X coordinate again decreases. The process continues until the offset array is exhaust ed. By incrementing row and column counters, the relative spot order is determined which establishes context (see Tables 58). Until now, both undeformed and deformed images were processed the same (except for learning the undeformed image) and assumed to have the same context. This assumption is verified by the following Table 3. Ordered spot coordinates in the undeformed image. Reference From Icon Y 57.8310 57.9438 57.9914 58.0797 58.1617 92.9227 92.9173 92.9836 93.0760 93.1757 127.8911 127.9383 128.0000 128.0738 128.2151 162.8523 162.9676 162.9909 163.0989 163.1105 197.8964 197.9548 198.0384 198.0978 198.1556 X Y Offset 18 17 16 15 14 10 9 8 7 6 2 1 0 1 2 6 7 8 9 10 14 15 16 17 18 Key 1 2 5 3 4 6 7 9 10 8 12 13 11 15 14 20 16 17 18 19 21 22 23 24 25 NOTE: Units are pixels. Offsets are computed by equation (50). 70.5459 35.5483 0.5729 34.4955 69.4332 70.5813 35.6077 0.6897 34.3813 69.3321 70.7425 35.6740 0.0000 34.2984 69.3632 70.7899 35.7793 0.8176 34.1896 69.2011 70.7852 35.8519 0.8390 34.1538 69.1446 70.1690 70.0562 70.0086 69.9203 69.8383 35.0773 35.0827 35.0164 34.9240 34.8243 0.1089 0.0617 0.0000 0.0738 0.2151 34.8523 34.9676 34.9909 35.0989 35.1105 69.8964 69.9548 70.0384 70.0978 70.1556 X 57.4541 92.4517 127.4271 162.4955 197.4332 57.4187 92.3923 127.3103 162.3813 197.3321 57.2575 92.3260 128.0000 162.2984 197.3632 57.2101 92.2207 127.1824 162.1896 197.2011 57.2148 92.1481 127.1610 162.1538 197.1446 Keys are the order in which the spots were found. Undeformed spacing is 35 pixels in x and y. See Figure 18. Table 4. Ordered spot coordinates in the deformed image. Reference From Icon Offset Key X Y X Y 18 9 56.5016 45.0350 71.4984 82.9650 17 14 93.0300 49.7056 34.9700 78.2944 16 19 129.5597 54.3186 1.5597 73.6814 15 23 166.0404 58.9647 38.0404 69.0353 14 25 202.4800 63.6284 74.4800 64.3716 10 6 55.4298 81.8551 72.5702 46.1449 9 11 91.8919 86.5161 36.1081 41.4839 8 16 128.3497 91.1147 0.3497 36.8853 7 21 164.8545 95.7498 36.8545 32.2502 6 24 201.3280 100.3761 73.3280 27.6239 2 4 54.2228 118.6530 73.7772 9.3470 1 8 90.7333 123.2581 37.2667 4.7419 0 12 128.0000 128.0000 0.0000 0.0000 1 18 163.7403 132.4753 35.7403 4.4753 2 22 200.2602 137.2151 72.2602 9.2151 6 2 53.0816 155.3833 74.9184 27.3833 7 5 89.5630 160.0309 38.4370 32.0309 8 10 126.0252 164.6782 1.9748 36.6782 9 15 162.6010 169.3190 34.6010 41.3190 10 20 199.0750 173.9321 71.0750 45.9321 14 1 51.9634 192.1880 76.0366 64.1880 15 3 88.4532 196.7857 39.5468 68.7857 16 7 124.9335 201.5070 3.0665 73.5070 17 13 161.4258 206.0458 33.4258 78.0458 18 17 197.9716 210.6846 69.9716 82.6846 NOTE: Units are pixels. Offsets are computed by equation (50). Keys are the order in which the spots were found. See Figure 18. Table 5. Spot distance from origin in undeformed image. COL 1 2 3 4 5 ROW 1 X: 58.0101 58.0481 57.9599 57.9856 58.0635 Y: 57.9660 93.0576 128.0263 162.9875 198.0315 COL 1 2 3 4 5 ROW 2 X: 93.0079 93.0216 93.0284 92.9964 92.9969 Y: 58.0056 92.9791 128.0002 163.0297 198.0169 COL 1 2 3 4 5 ROW 3 X: 127.9832 127.9396 128.7025 127.9581 128.0099 Y: 57.9801 92.9724 127.9873 162.9798 198.0273 COL 1 2 3 4 5 ROW 4 X: 163.0518 163.0107 163.0010 162.9653 163.0027 Y: 57.9950 92.9915 127.9894 163.0147 198.0136 COL 1 2 3 4 5 ROW 5 X: 197.9896 197.9616 198.0660 197.9763 197.9936 Y: 58.0041 93.0182 128.0575 162.9531 197.9982 NOTE: Units are pixels. The origin is the upper left corner of the image. See Figure 18. Table 5. Spot distance from origin in deformed image. COL 1 2 3 4 5 ROW 1 X: 229.4016 196.6288 163.9343 131.2724 98.5336 Y: 100.8559 84.0388 67.1092 50.2682 33.4158 COL 1 2 3 4 5 ROW 2 X: 209.4617 176.7262 144.0612 111.3736 78.6761 Y: 131.8162 114.9435 98.0816 81.1960 64.3727 COL 1 2 3 4 5 ROW 3 X: 189.5733 156.8817 123.7393 91.4833 58.7113 Y: 162.8025 145.8710 129.6776 112.1065 95.2677 COL 1 2 3 4 5 ROW 4 X: 169.6761 136.9841 104.3150 71.5500 38.9059 Y: 193.7303 176.8253 160.0107 143.1225 126.2522 COL 1 2 3 4 5 ROW 5 X: 149.7807 117.1079 84.3163 51.6854 18.9873 Y: 224.6135 207.7552 190.9335 174.0585 157.2419 NOTE: Units are pixels. The origin is the upper left corner of the image. See Figure 18. Table 7. Spot distance from icon in undeformed image. COL 1 2 3 4 5 ROW 1 X: 70.5459 70.5813 70.7425 70.7899 70.7852 Y: 70.1690 35.0773 0.1089 34.8523 69.8964 COL 1 2 3 4 5 ROW 2 X: 35.5483 35.6077 35.6740 35.7793 35.8519 Y: 70.0562 35.0827 0.0617 34.9676 69.9548 COL 1 2 3 4 5 ROW 3 X: 0.5729 0.6897 0.0000 0.8176 0.8390 Y: 70.0086 35.0164 0.0000 34.9909 70.0384 COL 1 2 3 4 5 ROW 4 X: 34.4955 34.3813 34.2984 34.1896 34.1538 Y: 69.9203 34.9240 0.0738 35.0989 70.0978 COL 1 2 3 4 5 ROW 5 X: 69.4332 69.3321 69.3632 59.2011 69.1446 Y: 69.8383 34.8243 0.2151 35.1105 70.1556 NOTE: Units are pixels. Spacing is 35 pixels in x and y. See Figure 18. Table 8. Spot distance from icon in deformed image. COL 1 2 3 4 5 ROW 1 X: 71.4984 72.5702 73.7772 74.9184 76.0366 Y: 82.9650 46.1449 9.3470 27.3833 64.1880 COL 1 2 3 4 5 ROW 2 X: 34.9700 36.1081 37.2667 38.4370 39.5468 Y: 78.2944 41.4839 4.7419 32.0309 68.7857 COL 1 2 3 4 5 ROW 3 X: 1.5597 0.3497 0.0000 1.9748 3.0665 Y: 73.6814 36.8853 0.0000 36.6782 73.5070 COL 1 2 3 4 5 ROW 4 X: 38.0404 36.8545 35.7403 34.6010 33.4258 Y: 69.0353 32.2502 4.4753 41.3190 78.0458 COL 1 2 3 4 5 ROW 5 X: 74.4800 73.3280 72.2602 71.0750 69.9716 Y: 64.3716 27.6239 9.2151 45.9321 82.6845 NOTE: Units are pixels. See Figure 18. checks: 1. same number of rows in each image 2. same number of columns in each image 3. same number of spots in each image 4. rows times columns equal number of spots. Of course, if any check fails, the images are "out of con text." If strains are large enough, context similarities between images cannot be recognized and an error message is printed. Assum ing large, local rotation does not occur in the material, recognizing similar context between the two patterns that does not exist is very remote. Thus if no check fails, the approximation algorithm estab lishes a onetoone relationship, or "context mapping," between spots in the two images. CHAPTER III IMAGE GENERATION AND ANALYSIS III.1 Synthetic Images Computer generated images, or "synthetic" images, were used as the primary development tool and to study the influence of various parameters. The options data file shown in Table 9 was used to gen erate these 256X256 pixel images. Motion is limited to affine transformations plus rigidbody rotation about the image center. The transformations are x = c X + c2Y + 3 (51) y = c4X + c5Y + 6 There are three image output options available. The "BINARY" option is the machine representation of the image. "DECIMAL" is the machine version converted to a readable form (see, for example, Figure 4b). "PLOT" displays a thresholded image, suitable for hard copy, on a graphics terminal (see, for example, Figure 15). "U123.DAT," "NIMG.DAT," "D123.DAT," and "DIMG.DAT" are userdefined file names. The image generator automatically reads the input file and generates the specified output thereby allowing the user a choice of interactive or batch processing. Table 9. Input file for synthetic image generator. IMAGE GENEREATOR INPUT DATA SPOT GEOMETRY RADIUS XSPACING 4.00 35.00 PATTERN SIZE XCOLS YROWS 5 5 YSPACING 35.00 GRAY LEVELS BLACK WHITE 00000 00200 NOISE DATA DISTRIBUTION TYPE: N NORMAL STANDARD DEVIATION: 007.000 UNIFORM IMAGE MOTION MOTION TYPE: 2 1POLYNOMIAL ONLY 2POLYNOMIAL AND ROTATION 3ROTATION ONLY POLYNOMIAL X Y TRANSLATION x: +1.05000 +0.05000 y: +0.05000 +1.05000 ROTATION (DEGREES): +120.000 010.0000 010.0000 IMAGE OUTPUT FORMAT BINARY: DECIMAL: PLOT: UNDEFORMED U123.DAT NIMG.DAT *SCREEN* DEFORMED D123.DAT DIMG.DAT *SCREEN* Y/N Y N N III.2 Image Analysis Similar to image generation, image analysis begins with an op tions file which is shown in Table 10. If "USE THESE FILE NAMES?" is answered no, the user is automatically prompted for the file name. Since no other information is required by the program, the user is free to choose interactive or batch processing. Table 10. Image analysis options file. ******** IMAGE ANALYSIS INPUT/OUTPUT OPTIONS ******** INPUT SPECIFICATIONS IMAGE FILE NAME: USE THESE FILE NAMES?: IMAGE OUTPUT UNORDERED SPOT DATA: DISTANCE FROM ORIGIN: DISTANCE FROM ICON: DISPLACEMENT ANALYSIS TOTAL DISPLACEMENT: DISPLACEMENT GRADIENTS: LAGRANGIAN STRAIN: TAYLOR SERIES STRAIN: MOTION ANALYSIS DEFORMATION GRADIENT: GREEN DEFORM. TENSOR: RIGHT STRETCH TENSOR: ROTATION TENSOR: LAGRANGIAN STRAIN: UNDEFORMED + U123.DAT N UNDEFORMED + Y Y Y DEFORMED + D123.DAT N DEFORMED  Y Y Y OUTPUT Y Y Y Y OUTPUT Y Y Y Y Y After segmentation, feature extraction, and context recognition, automatic processing continues with motion analysis. The motion of each spot is resolved and the displacement is computed by equation (3) (see Table 11). Next, the deformation and displacement gradients at each spot are computed by differentiating the functions used to describe motion and displacement. Table 11. Displacement of each spot. COL 1 2 3 4 5 ROW 1 X: 171.3916 138.5808 105.9744 73.2868 40.4701 Y: 42.8899 9.0188 60.9171 112.7193 164.6158 COL 1 2 3 4 5 ROW 2 X: 116.4538 83.7046 51.0328 18.3772 14.3208 Y: 73.8106 21.9643 29.9186 81.8337 133.6442 COL 1 2 3 4 5 ROW 3 X: 61.5900 28.9421 4.9632 35.4747 69.2986 Y: 104.8223 52.8985 1.6903 50.8733 102.7596 COL 1 2 3 4 5 ROW 4 X: 6.6243 26.0267 58.6861 91.4154 124.0968 Y: 135.7353 83.8338 32.0212 19.8922 71 .7614 COL 1 2 3 4 5 ROW 5 X: 48.2088 80.8538 113.7497 146.2914 179.0063 Y: 166.6094 114.7370 62.8761 11.1054 40.7563 NOTE: Units are pixels. Compare images in Figure 18. Displacements are computed using the data in Tables 5 and 6 in equation (3). The choice of a mapping model depends, of course, on the application. Intuition, a priori knowledge, elasticity theory, numerical efficiency, etc. are used to select the approximating func tions. In this study, simple polynomials, linear or second order, adequately approximate the motion and displacement in the synthetic and real images. The models are linear x = 11X + c21 + 31Y y = c12X + c22 32Y (52) u = c13X + c23 c33Y v 0 C14X + c24 + C34Y and second order 2 2 x= 11 + X c+ 31 + c41 + cY + + c XY 11 21 31 41 51 01 2 2 y = C12 + c22X + 32 + cY2 + 52Y + c6XY u = 13X + c23X + c33 + cY cY Y + c XY 13 23 33 143 53 63 (53) 2 2 v = c1X2 c2X + c3 + 44Y + c54Y + c64XY These models are fitted to a small neighborhood of spots and differentiated to obtain the displacement and deformation 63 gradients. Typically, the neighborhoods are 3X3, 5X5, or 7X7 and the gradients are computed for the center spot. Exceptions are the spots near the edge of the pattern and their gradients are computed like a forwardbackward difference. For both models, the coefficients are determined by least squares evaluation. Based on a modified form of the Conte and Deboor [74] notation, the normal equations are T c = f (54) where X X2 ." XN = 1 ... Linear Y1 Y2 N (55) 2 2 2 X X2 ... XN 1 2 N X1 X2 XN 1 1 1 2 2 2 Second Y Y ... Y Order Y Yx ... YN X1Y1 X2Y2 ... XNYN Linear c 01 c14 c 31 c 34_ L_ _3 j (56) C = i. 61 I: x 1 f = xN "N Second order 1 u061 Y1 Ul V1 and N is the number of spots in the neighborhood. T S= A and Equations (54) become Ac = Z which are solvable by GaussJordan elimination. (57) (58) f = z Hornbeck [75] points out GaussJordan elimination may be used to solve equations (58), but unfortunately the set is extremely illconditioned. Taking his advice, double precision arithmetic is used to cope with the large variation in the magnitudes of the co efficients in any given row. In addition to his suggestions, 0 and f are scaled before solution and the c's are reconverted during gradient computation. For a variety of programming reasons, solution of equations (58) is not performed directly by GaussJordan elimination. Instead, the inverse of A is computed using Hornbeck's [75] GaussJordan matrix inversion subroutine, with the addition of appropriate error traps, and c is solved by 1 c = A Z (59) How accurate are the values of c and how well do equations (52) or (53) fit the data? Debugging experience indicates the number of reliable digits (numerically speaking, not experimentally) runs in the teens. In this study the solutions are known, but if they were unknown, goodnessoffit and the contribution of each term could be easily tested by the methods suggested by Miller and Freund [63]. The example of Figure 18 is continued with the calculation of c. Assume the linear model of equation (52) and a 3X3 neighborhood centered at spot (3,3). A substitution of data from Table 5 into equation (55) produces T I = 93.0216 93.0284 92.9964 127.9396 128.7025 127.9581 163.0107 163.0010 162.9653 1 .0000 1.0000 1.0000 1.0000 1 .0000 1.0000 1 .0000 1.0000 1.0000 92.9791 128.0002 163.0297 92.9724 127.9873 162.9798 92.9915 127.9894 163.0147 (60) Likewise, data from Tables 6 and 11 in equation (57) lead to 176.7262 144.0612 111 .3736 156.8817 123.7393 91.4833 136.9841 104.3150 L71 .5500 114.9435 98.0816 81.1960 145.8710 129.6776 112.1065 176.8253 160.0107 143.1225 83.7046 51.0328 18.3772 28.9421 4.9632 36.4747 26.0267 58.6861 91 .4154 21.9643 29.9186 81.8337 52.8985 1 .6903 50.8733 83.8338 32.0212 19.8922 (61) The solution for c is 0.56849 c = 316.52130 _0.93425 0.88427 77.50840 0.48175 1.56849 316.52130 0.93425 0.88427 77.50840 1.48165 (62) The motion functions (i.e., the "mathematical" maps) in equa tions (52) become x = 0.56849X + 316.52130 0.93425Y (63) y = 0.88427X + 77.50840 0.48175Y I and the displacement maps in equations (52) become u = 1.56849X + 316.52130 0.93425Y (64) v = 0.88427X + 77.5084 1.48175Y The deformation and displacement gradients, computed by equa tions (2) and (4), respectively, become 0.56849 0.88427 1 .56849 0.88427 0.93425 0.48175 0.93425 1.48175] (65) (66) Strain, computed by equations (11) or (12), is .05255 0[.05256 0.05256 0.05246] (67) By the Taylor series method of equation (13), strain is 0.05124 0.04763 0.0476 3 0.05115 (68) 68 Strain computed according to small displacement theory is, in this case, quite inaccurate. By equations (14), 1 .56849 e = 0.02499 Rigidbody rotation is computed the Green deformation tensor is [ .10509 0.10511 0.02499 1.48175_ (69) as follows. By equation (8), 0.10511 1.10491 (70) The eigenvalues of C are 1 .21009 S= .00000 ~p 0.00000 0.00000 0.99990 (71) 1/2 and C 2 is p 1/2 ~p 1 .10004 0.00000 0.00000 0.99995 (72) 1/2 Since U = C then p p 1.05000 0.05005 0.05005 1.05000 (73) 69 Rotation tensor R, computed by equation (17), is F0.50016 0.86596 0.86596 0.50011 The remainder of the output for the example in Figure 18 is shown in Tables 12 through 19. Table 12. Displacement gradients. COL 1 2 3 4 5 ROW 1 au/ax: 1.56849 1.56849 1.56849 1.56849 1.56849 3v/ax: 0.88427 0.88427 0.88427 0.88427 0.88427 au/3Y: 0.93425 0.93425 0.93425 0.93425 0.93425 av/aY: 1 .48175 1.48175 1.48175 1 .48175 1.48175 COL 1 2 3 4 5 ROW 2 3u/aX: 1.56849 1.56849 1.56849 1.56849 1.56849 av/ax: 0.88427 0.88427 0.88427 0.88427 0.88427 3u/3Y: 0.93425 0.93425 0.93425 0.93425 0.93425 3v/3Y: 1.48175 1.48175 1.48175 1.48175 1.48175 COL 1 2 3 4 5 ROW 3 3u/aX: 1.56849 1.56849 1.56849 1.56849 1.56849 av/3X: 0.88427 0.88427 0.88427 0.88427 0.88427 3u/3Y: 0.93425 0.93425 0.93425 0.93425 0.93425 av/aY: 1.48175 1.48175 1.48175 1.48175 1.48175 COL 1 2 3 4 5 ROW 4 3u/3X: 1.56849 1.56849 1.56849 1.56849 1.56849 av/ax: 0.88427 0.88427 0.88427 0.88427 0.88427 au/3Y: 0.93425 0.93425 0.93425 0.93425 0.93425 av/3Y: 1.48175 1.48175 1.48175 1.48175 1.48175 COL 1 2 3 4 5 ROW 5 3u/3X: 1.56849 1.56849 1.56849 1.56849 1.56849 av/ax: 0.88427 0.88427 0.88427 0.88427 0.88427 au/3Y: 0.93425 0.93425 0.93425 0.93425 0.93425 3v/3Y: 1.48175 1.48175 1.48175 1.48175 1.48175 NOTE: Correct values are 3u/3X = 1.56830, 3v/3X = 0.88433, 3u/3Y = 0.93433, 3v/3Y = 1 .48170. Table 13. Lagrangian strain from displacement gradients. COL 1 2 3 4 5 ROW 1 E11: 52554. 52554. 52554. 52554. 52554. E12: 52557. 52557. 52557. 52557. 52557. E22: 52459. 52459. 52459. 52459. 52459. COL 1 2 3 4 5 ROW 2 E11: 52554. 52554. 52554. 52554. 52554. E12: 52557. 52557. 52557. 52557. 52557. E22: 52459. 52459. 52459. 52459. 52459. COL 1 2 3 4 5 ROW 3 E11: 52554. 52554. 52554. 52554. 52554. E12: 52557. 52557. 52557. 52557. 52557. E22: 52459. 52459. 52459. 52459. 52459. COL 1 2 3 4 5 ROW 4 E11: 52554. 52554. 52554. 52554. 52554. E12: 52557. 52557. 52557. 52557. 52557. E22: 52459. 52459. 52459. 52459. 52459. COL 1 2 3 4 5 ROW 5 E11: 52554. 52554. 52554. 52554. 52554. E12: 52557. 52557. 52557. 52557. 52557. E22: 52459. 52459. 52459. 52459. 52459. NOTE: Units are micro inchesperinch. Correct values are E  52,500. Table 14. Strain from Taylor series. COL 1 2 3 4 5 ROW 1 E11: 51241. 51241. 51241. 51241. 51241. E12: 47634. 47634. 47634. 47634. 47634. E22: 51151. 51151. 51151. 51151. 51151. COL 1 2 3 4 5 ROW 2 E11: 51241. 51241. 51241. 51241. 51241. E12: 47634. 47634. 47634. 47634. 47634. E22: 51151. 51151. 51151. 51151. 51151. COL 1 2 3 4 5 ROW 3 E11: 51241. 51241. 51241. 51241. 51241. E12: 47634. 47634. 47634. 47634. 47634. E22: 51151. 51151. 51151. 51151. 51151. COL 1 2 3 4 5 ROW 4 E11: 51241. 51241. 51241. 51241. 51241. E12: 47634. 47634. 47634. 47634. 47634. E22: 51151. 51151. 51151. 51151. 51.151. COL 1 2 3 4 5 ROW 5 E11: 51241. 51241. 51241. 51241. 51241. E12: 47634. 47634. 47634. 47634. 47634. E22: 51151. 51151. 51151. 51151. 51151. NOTE: Units are micro inchesperinch. Correct values are E1 = E22 = 51,190, E12 = E21 = 47,582. Table 15. Deformation gradient. COL 1 2 3 4 5 ROW 1 F11: 0.56849 0.56849 0.56849 0.56849 0.56849 F12: 0.93425 0.93425 0.93425 0.93425 0.93425 F21: 0.88427 0.88427 0.88427 0.88427 0.88427 F22: 0.48175 0.48175 0.48175 0.48175 0.48175 COL 1 2 3 4 5 ROW 2 F11: 0.56849 0.56849 0.56849 0.56849 0.56849 F12: 0.93425 0.93425 0.93425 0.93425 0.93425 F21: 0.88427 0.88427 0.88427 0.88427 0.88427 F22: 0.48175 0.48175 0.48175 0.48175 0.48175 COL 1 2 3 4 5 ROW 3 F11: 0.56849 0.56849 0.56849 0.56849 0.56849 F12: 0.93425 0.93425 0.93425 0.93425 0.93425 F21: 0.88427 0.88427 0.88427 0.88427 0.88427 F22: 0.48175 0.48175 0.48175 0.48175 0.48175 COL 1 2 3 4 5 ROW 4 F11: 0.56849 0.56849 0.56849 0.56849 0.56849 F12: 0.93425 0.93425 0.93425 0.93425 0.93425 F21: 0.88427 0.88427 0.88427 0.88427 0.88427 F22: 0.48175 0.48175 0.48175 0.48175 0.48175 COL 1 2 3 4 5 ROW 5 F11: 0.56849 0.56849 0.56849 0.56849 0.56849 F12: 0.93425 0.93425 0.93425 0.93425 0.93425 F21: 0.88427 0.88427 0.88427 0.88427 0.88427 F22: 0.48175 0.48175 0.48175 0.48175 0.48175 NOTE: Correct 0.88433, F22 = values are 0.48170. F11 =  0.56830, Fl2 =  0.93433, ?21 " Table 16. Green deformation tensor COL 1 2 3 4 5 ROW 1 C11: 1.10511 1.10511 1.10511 1.10511 1.10511 C12: 0.10511 0.10511 0.10511 0.10511 0.10511 C22: 1.10492 1.10492 1.10492 1.10492 1.10492 COL 1 2 3 4 5 ROW 2 C11: 1.10511 1.10511 1.10511 1.10511 1.10511 C12: 0.10511 0.10511 0.10511 0.10511 0.10511 C22: 1.10492 1.10492 1.10492 1.10492 1.10492 COL 1 2 3 4 5 ROW 3 C11: 1.10511 1.10511 1.10511 1.10511 1.10511 C12: 0.10511 0.10511 0.10511 0.10511 0.10511 C22: 1.10492 1.10492 1.10492 1.10492 1.10492 COL 1 2 3 4 5 ROW 4 C11: 1.10511 1.10511 1.10511 1.10511 1.10511 C12: 0.10511 0.10511 0.10511 0.10511 0.10511 C22: 1.10492 1.10492 1.10492 1.10492 1.10492 COL 1 2 3 45 ROW 5 C11: 1.10511 1.10511 1.10511 1.10511 1.10511 C12: 0.10511 0.10511 0.10511 0.10511 0.10511 C22: 1.10492 1.10492 1.10492 1.10492 1.10492 NOTE: Correct values are C11 = C22 = 1.10500, C12 = 0.10500. 0 Table 17. Right stretch tensor. COL 1 2 3 4 5 ROW 1 U11: 1.05005 1.05005 1.05005 1.05005 1.05005 U12: 0.05005 0.05005 0.05005 0.05005 0.05005 U22: 1.04996 1.04996 1.04996 1.04996 1.04996 COL 1 2 3 4 5 ROW 2 U11: 1.05005 1.05005 1.05005 1.05005 1.05005 U12: 0.05005 0.05005 0.05005 0.05005 0.05005 U22: 1.04996 1.04996 1.04996 1.04996 1.04996 COL 1 2 3 4 5 ROW 3 U11: 1.05005 1.05005 1.05005 1.05005 1.05005 U12: 0.05005 0.05005 0.05005 0.05005 0.05005 U22: 1.04996 1.04996 1.04996 1.04996 1.04996 COL 1 2 3 4 5 ROW 4 U11: 1.05005 1.05005 1.05005 1.05005 1.05005 U12: 0.05005 0.05005 0.05005 0.05005 0.05005 U22: 1.04996 1.04996 1.04996 1.04996 1.04996 COL 1 2 3 4 5 ROW 5 U11: 1.05005 1.05005 1.05005 1.05005 1.05005 U12: 0.05005 0.05005 0.05005 0.05005 0.05005 U22: 1.04996 1.04996 1.04996 1.04996 1.04996 NOTE: Correct values are U11 = U22 = 1.05000, U12 = 0.05000. Table 18. Rotation tensor. COL 1 2 3 4 5 ROW 1 R11: 0.50011 0.50011 0.50011 0.50011 0.50011 R12: 0.86596 0.86596 0.86596 0.86596 0.86596 R21: 0.86596 0.86596 0.86596 0.86596 0.86596 R22: 0.50011 0.50011 0.50011 0.50011 0.50011 COL 1 2 3 4 5 ROW 2 R11: 0.50011 0.50011 0.50011 0.50011 0.50011 R12: 0.86596 0.86596 0.86596 0.86596 0.86596 R21 0.86596 0.86596 0.86596 0.86596 0.86596 R22: 0.50011 0.50011 0.50011 0.50011 0.50011 COL 1 2 3 4 5 ROW 3 R11: 0.50011 0.50011 0.50011 0.50011 0.50011 R12: 0.86596 0.86596 0.86596 0.86596 0.86596 R21: 0.86596 0.86596 0.86596 0.86596 0.86596 R22: 0.50011 0.50011 0.50011 0.50011 0.50011 COL 1 2 3 4 5 ROW 4 R11: 0.50011 0.50011 0.50011 0.50011 0.50011 R12: 0.86596 0.86596 0.86596 0.86596 0.86596 R21: 0.86596 0.86596 0.86596 0.86596 0.86596 R22: 0.50011 0.50011 0.50011 0.50011 0.50011 COL 1 2 3 4 5 ROW 5 R11: 0.50011 0.50011 0.50011 0.50011 0.50011 R12: 0.86596 0.86596 0.86596 0.86596 0.86596 R21: 0.86596 0.86596 0.86596 0.86596 0.86596 R22: 0.50011 0.50011 0.50011 0.50011 0.50011 NOTE: Correct 0.86603, R22 = values are R11 = 0.50000, R12 = 0.50000. 0.86603, R21 = m Table 19. Lagrangian strain from deformation gradient. COL 1 2 3 4 5 ROW 1 E11: 52554. 52554. 52554. 52554. 52554. E12: 52557. 52557. 52557. 52557. 52557. E22: 52459. 52459. 52459. 52459. 52459. COL 1 2 3 4 5 ROW 2 E11: 52554. 52554. 52554. 52554. 52554. E12: 52557. 52557. 52557. 52557. 52557. E22: 52459. 52459. 52459. 52459. 52459. COL 1 2 3 4 5 ROW 3 E11: 52554. 52554. 52554. 52554. 52554. E12: 52557. 52557. 52557. 52557. 52557. E22: 52459. 52459. 52459. 52459. 52459. COL 1 2 3 4 5 ROW 4 E11: 52554. 52554. 52554. 52554. 52554. E12: 52557. 52557. 52557. 52557. 52557. E22: 52459. 52459. 52459. 52459. 52459. COL 1 2 3 4 5 ROW 5 E11: 52554. 52554. 52554. 52554. 52554. E2: 52557. 52557. 52557. 52557. 52557. E22: 52459. 52459. 52459. 52459. 52459. NOTE: Units are micro inchesperinch. Correct values are Eii = 52,500.  CHAPTER IV EXPERIMENTS IV.1 Test Specimen The prismatic beam shown in Figure 20 was used for all experi ments. A clear plastic, CR39, was chosen for its relatively low modulus of elasticity and capacity for large, elastic strain. The 11X11 test pattern on the beam was generated by the synthe tic image generator and applied by contact printing. First, a hard copy of the pattern was photographed using Kodalith film. After coating the beam with a buffer of polyurethane and naphtha, a Liquid Light emulsion was applied, contact printed, and chemically fixed (see Figure 21). IV.2 Test Equipment Three experimental setups were configured; two were for strain tests and the third for rigidbody motion. The strain setup was a typical photoelastic load frame with special fixtures (see Figure 22). The fixtures were for fourpoint bending (see Figure 23) and cantilever beam (see Figure 24) experi ments. The equipment shown in Figure 25 was used for the rigidbody motion tests. A photocopy of a protractor was taped to the back of the lens holder and rotation was measured relative to the tip of the S2 Drix '", 11xl Spot Pattern ~I~______ ____ I~~  ~c i S' 1 2"  2"  1"J  , __________________ 2 3 7"Figure 20. Beam fabricated from CR3. Figure 20. Beam fabricated from CR39. I *** i** ** J * 0, 0 i^ ***** **.** ***** ***** O O O O 0 0 O O oLe S.' 0* ( 00 0@( 0O( 00 00 1 S.( 00 t i ] ' I t i r * * l ** .. .. .. Figure 21. Spot pattern on beam. 4 #28 Drill  L _~~__~ ~ 80 Figure 22. Experimental setup of load frame. Figure 23. Setup for fourpoint bending. Figure 24. Setup for cantilever beam. Rigidbody motion setup. Figure 25. clamped pointer. Measurements were accurate to about 1 degree. A dial indicator measured translation in the horizontal direction with an accuracy of 0.0005 inch. Since image processing equipment was unavailable at the University of Florida, the equipment for the rigidbody experiments was transported to USC where the digital images were recorded. IV.3 Experimental Procedure Unfortunately, the strain tests were not done because the load frame was too large to transport to USC. The second order model in equation (53) was for these tests. The equipment shown in Figure 25 and an image processing system similar to the one shown in Figure 3 (based on the IBM PC) was used for the rigidbody experiments. The CCD camera was a Sony model XCM38 with a zoom lens, and the digitizer board was an eightbit Datacube "Frame Grabber" with 385 rows and 374 columns of pixels. The setup included several routine checks. First, alignment between the rail, table, and camera were checked to minimize experi mental error. Then, the zoom lens was adjusted to enlarge the pattern image as much as possible. Resolution in the horizontal direction was measured at 358 pixels per inch. Lighting was adjusted to provide the usual illumination for tests in the USC lab, and the transparent specimen caused some lighting problems. Contrast was impaired by the light transmitted through the beam. 83 After setup, images were digitized and stored on floppy disk. All images were transferred to magnetic tape via USC's VAX11/780 computer for transport to the University of Florida. CHAPTER V ANALYSIS OF RESULTS The analysis of results has been divided into two sections: numerical and experimental. The numerical section summarizes synthe tic image work and the experimental section summarizes the analysis of the images digitized at USC. V.1 Numerical The analysis of synthetic images included rigidbody motion, homogeneous strain, combined motion, and the influence of noise, gray levels, and spot radius. To ensure a good random sample, the results from a variety of spot spacings were averaged. The pattern was 5X5 with the icon being spot (3,3). Except as noted, the gray level difference was 200, and the zeromean noise was Gaussian with a standard deviation of 7.0 gray levels. The absolute values of noisy pixels with values less than zero were used instead of the negative gray levels. In the following tables, the average absolute error is defined as follows: average absolute error is the average of the absolute values of the errors. For example, if E11=E12=E22=0 and the computed values are E11=100p, E12=150p, and E22=50p, the average absolute error is 100p. Standard deviation, also in the tables, is the standard devia tion of the computed values. For the strain example above, the standard deviation is 132y. For translations of various amounts in the x and y directions, the average absolute error was 0.008 pixels with a standard deviation of 0.011 pixels. Translations near whole pixels had slightly less error. (Equation (38) suggested a spatial sensitivity.) Rigidbody rotation was about the image center which was approx imately twothirds of a pixel below the pattern center. Rotations ranged from 0 to 180 degrees, and the results, computed using the elements of the rotation tensor, were very consistent with an average absolute error of 0.0039 degrees and a standard deviation of 0.0008 degrees. Several strain cases were investigated and summarized in Table 20. Normal strains were E =E22 with E12=E21=0; shear strains were E =E 22=0 with E 12=E21; and combined strains were E 11=E22=E=E21' See equation (12) for strain definitions. Noise had a significant effect on results. As suggested by equation (38), increasing SNR decreased translation and strain error (see Table 21). Framing failures began occurring below SNR=100. Gray level was another significant variable because of the SNR and the digitization of illumination intensity. For constant noise, increasing gray level difference increased the SNR and thereby in creased accuracy. With a noise standard deviation of 7.00 gray levels, spots were frequently unidentifiable below a gray level Summary of strain analysis for synthetic images. Strain Displacement Strain Gradient Ave. Abs. Standard Case aui/axj Correct Error Deviation Normal 0.000010 10 17 10 0.000100 100 17 15 0.001000 1001 22 27 0.010000 10050 40 32 0.100000 105000 40 49 Shear 0.000010 10 24 0 0.000100 100 29 0 0.001000 1000 25 0 0.010000 10000 33 0 0.090000a 90000 34 0 Combined 0.000010 10 3 14 0.000100 100 7 28 0.001000 1000 23 26 a Limited by pattern geometry. NOTE: Rigidbody motion was perinch. SNR=816. zero. Strain units are micro inches Table 20. The effect of SNR on translation and strain calculations. Translation Strain Ave. Abs. Standard Ave. Abs. Standard SNR Error Deviation Error Deviation 100 0.0049 0.0362 39 58 200 0.0029 0.0253 19 42 400 0.0021 0.0176 20 35 600 0.0016 0.0143 14 32 800 0.0012 0.0123 10 26 1000 0.0012 0.0109 8 21 2500 0.0008 0.0068 5 8 5000 0.0006 0.0043 3 6 10000 0.0004 0.0034 2 5 S0.0001 0.0001 0 0 NOTE: Translations were 10.5 pixels in x and y. No rotation. All strains were Eij=100P. Table 22. The effect of gray level difference on translation and strain calculations. Translation Strain Ave. Abs. Standard Ave. Abs. Standard H Error Deviation Error Deviation 32 0.0257 0.0270 ** ** 64 0.0043 0.0349 66 62 128 0.0026 0.0172 20 45 256 0.0009 0.0089 3 16 512 0.0007 0.0043 5 8 1024 0.0003 0.0022 4 5 2048 0.0002 0.0011 2 2 4096 0.0001 0.0006 1 2 NOTE: Translations were 10.5 pixels strains were Eij=100I. in x and y. No rotation. All Table 21. difference of 32. Below a difference of 64, a strain of Eij=100U was undetectable (see Table 22). An increase in gray level difference reduced the error caused by digitization. Without noise, a difference of 16 gray levels produced an average absolute strain error of 12p (for Eij=100p) with a stand ard deviation 0.031 and, above a difference of 64, error was nil. However, SNR effects greatly overshadowed digitization error. The effect of spot radius decreased rapidly with increased radius (see Table 23). Equation (38) suggested this for small 5, and, in fact, almost no change occurred for radii above 10 pixels. Table 23. The effect of spot radius on translation and strain calculations. Translation Strain Spot Ave. Abs. Standard Ave. Abs. Standard Radius Error Deviation Error Deviation 2 0.0030 0.0233 29 20 4 0.0018 0.0272 10 32 6 0.0013 0.0085 7 15 8 0.0016 0.0069 8 13 10 0.0020 0.0052 8 19 12 0.0021 0.0059 5 11 NOTE: Translations were 10.5 pixels in x and y. No rotation. All strains were Eij=100p. SNR=816. 13 How much VAX 11/750 computer time did the analysis of a 256X256 pixel image require? For a 3X3 spot pattern, CPU time was about 4.9 