 | Sutton M.A. Image correlation for shape, motion and deformation measurements: basic concepts, theory and applications / M.A.Sutton, J.-J.Orteu, H.Schreier. - New York: Springer, 2009. - xx, 321 p.: ill. - Ref.: p.299-316. - Ind.: p.317-321. - ISBN 978-0-387-78746-6
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1 Introduction ................................................. 1
1.1 Literature Survey ....................................... 1
1.1.1 Early History .................................... 1
1.1.2 Photogrammetry, 1850-Present ..................... 1
1.1.3 Digital Image Correlation - Background and
Related Activities ............................... 2
1.1.4 Digital Image Correlation ........................ 3
1.2 Discussion ............................................. 10
2 Elements of Geometrical Optics .............................. 13
2.1 Optics of a Camera ..................................... 13
2.1.1 Thin Lens ....................................... 13
2.1.2 Thick Lens ...................................... 22
2.1.3 Interpretation .................................. 24
2.1.4 Front Image Plane or Symmetry Interpretation .... 25
3 Single Camera Models and Calibration Procedures in
Computer Vision ............................................. 27
3.1 Camera Models .......................................... 27
3.1.1 Pinhole Camera .................................. 27
3.1.2 Distortion ...................................... 33
3.2 Calibration - Full Camera Model Parameter Estimation ... 44
3.2.1 Camera Parameter Estimation without
Distortion ...................................... 44
3.2.2 Bundle Adjustment for Distortion and Camera
Parameter Estimation ............................ 48
3.2.3 Equations and Procedures for Optimization
Structure of Matrices ........................... 51
3.2.4 Solution Details ................................ 55
3.2.5 Discussion ...................................... 60
4 Two-Dimensional and Three-Dimensional Computer Vision ....... 65
4.1 Two-Dimensional Computer Vision ........................ 65
4.1.1 Model ........................................... 65
4.1.2 Bundle Adjustment for Plane-to-Plane Imaging .... 67
4.1.3 Solution Details ................................ 67
4.1.4 Discussion ...................................... 69
4.2 Three-Dimensional Computer Vision ...................... 70
4.2.1 Geometry of a Stereovision System ............... 71
4.2.2 Epipolar Constraint ............................. 72
4.2.3 Calibration in Stereovision ..................... 73
4.2.4 Discussion ...................................... 79
5 Digital Image Correlation (DIC) ............................. 81
5.1 Introduction to Image Matching ......................... 81
5.1.1 The Aperture Problem ............................ 81
5.1.2 The Correspondence Problem ...................... 82
5.1.3 Speckle Pattern ................................. 83
5.2 Image Matching Methods ................................. 84
5.2.1 Differential Methods ............................ 84
5.2.2 Template Matching ............................... 87
5.3 Subset Shape Functions ................................. 88
5.3.1 Polynomial Shape Functions ...................... 90
5.3.2 Shape Functions for Stereo Matching ............. 92
5.3.3 Summary ......................................... 95
5.4 Optimization Criteria for Pattern Matching ............. 95
5.4.1 Offset in Lighting .............................. 96
5.4.2 Scale in Lighting ............................... 97
5.4.3 Offset and Scale in Lighting .................... 97
5.4.4 Concluding Remarks .............................. 98
5.5 Efficient Solution Methods ............................. 99
5.5.1 Efficient Update Rules for Planar Motion ........ 99
5.5.2 Extension to General Shape Functions ........... 101
5.6 Matching Bias ......................................... 103
5.6.1 Interpolation Bias ............................. 103
5.6.2 Bias Due to Noise .............................. 110
5.7 Statistical Error Analysis ............................ 113
5.7.1 Derivation for the One-Dimensional Case ........ 113
5.7.2 Confidence Margins from the Covariance
Matrix ......................................... 116
6 In-Plane Measurements ...................................... 119
6.1 Constraints and Applicability ......................... 119
6.1.1 Object Planarity Constraints ................... 119
6.1.2 Object Deformation Constraints ................. 119
6.2 Uniaxial Tension of Planar Specimen ................... 120
6.2.1 Experimental Considerations .................... 120
6.2.2 Imaging Considerations ......................... 120
6.2.3 Experimental Results ........................... 122
6.2.4 Discussion ..................................... 124
6.3 Out-of-Plane Motion ................................... 127
6.3.1 Standard Lens Systems for Single Camera
Measurements ................................... 127
6.3.2 Telecentric Lens System for Single Camera
Measurements ................................... 128
6.3.3 Out-of-Plane Translation Experiments ........... 128
6.3.4 Stereo-vision Calibration ...................... 130
6.3.5 Experimental Results ........................... 133
6.3.6 Discussion ..................................... 133
6.3.7 Remarks ........................................ 137
6.4 Development and Application of Far-Field Microscope
for Microscale Displacement and Strain Measurements ... 138
6.4.1 Problem Description: Measurement of Crack
Closure Load During Fatigue Crack Growth ....... 138
6.4.2 Fatigue Specimen Geometry, Material and
Surface Preparation ............................ 139
6.4.3 Validation Specimen and Preparation ............ 140
6.4.4 Pattern Application ............................ 140
6.4.5 Optical Setup for Imaging ...................... 142
6.4.6 Validation Experiment .......................... 143
6.4.7 Fatigue Experiment ............................. 144
6.4.8 Post-processing to Determine COD ............... 145
6.4.9 Experimental Results ........................... 146
6.4.10 Discussion ..................................... 148
6.5 Inverse Methods: Material Property Measurements
Using Full-Field Deformations and the Virtual Fields
Method ................................................ 150
6.5.1 Virtual Fields Method .......................... 150
6.5.2 Derivation of VFM Equations for Cantilever
Beam Undergoing Planar Deformations ............ 151
6.5.3 Virtual Fields for Cantilever Beam Specimen .... 152
6.5.4 Experimental Studies ........................... 154
6.5.5 Smoothing by 2D Finite Element Methods ......... 156
6.5.6 Material Property Results ...................... 157
6.5.7 Discussion ..................................... 157
6.6 Accurate 2D Deformation Measurements in a Scanning
Electron Microscope: Basic Concepts and Application ... 159
6.6.1 Imaging in an SEM .............................. 159
6.6.2 Pattern Development and Application ............ 160
6.6.3 Digital Image Correlation to Quantify SEM
Imaging System Errors .......................... 162
6.6.4 Digital Image Correlation for Elastic
Deformation Measurements ....................... 167
6.6.5 Tensile Experiment ............................. 169
6.6.6 Experimental Results ........................... 170
6.6.7 Discussion ..................................... 171
6.6.8 Remarks ........................................ 172
7 Stereo-vision System Applications .......................... 175
7.1 Stereovision System Design Considerations ............. 175
7.2 Quasi-Static Experiment: Four-Camera Stereo-Vision
System for Large Deformation Measurements in
Specimen Subjected to Out-of-Plane Bending ............ 176
7.2.1 Problem Description: Compression and
Out-of-Plane Bending of Flawed Polymer
Sheets ......................................... 176
7.2.2 Geometry of Specimen ........................... 177
7.2.3 Experimental Considerations and Arrangement
of Two Stereovision Systems .................... 177
7.2.4 Calibration of the Camera Systems .............. 179
7.2.5 Post-processing and Special Considerations ..... 181
7.2.6 Experiments .................................... 186
7.2.7 Experimental Results ........................... 186
7.2.8 Discussion and Practical Considerations ........ 188
7.3 Dynamic Experiment: High Speed Stereovision System
for Large Deformation Measurements of Specimen
Subjected to Combined Tension-Torsion Impact
Loading ............................................... 191
7.3.1 Problem Description: Impact Loading of Single
Edge Cracked Specimen Subjected to Mixed Mode
I/III Impact Loading ........................... 192
7.3.2 Specimen Geometry and Preparation .............. 192
7.3.3 Experimental Considerations and Arrangement
of Two Stereovision Systems .................... 192
7.3.4 Calibration of the Stereovision System ......... 194
7.3.5 Post-processing ................................ 195
7.3.6 Experiments .................................... 195
7.3.7 Experimental Results ........................... 195
7.3.8 Discussion and Practical Considerations ........ 199
7.4 Development and Application of 3D Digital Image
Correlation Principles in Stereomicroscopy for
Microscale Shape and Deformation Measurements ......... 199
7.4.1 Problem Description: Shape and Deformation
Measurements for Small, Soft Specimens
Subjected to Mechanical Loading ................ 200
7.4.2 Specimen Geometry and Preparation .............. 200
7.4.3 Optical Components in Stereo-Microscope and
Experimental Considerations .................... 202
7.4.4 Distortion Correction and Stereo Calibration ... 202
7.4.5 Post Processing ................................ 204
7.4.6 Validation Experiment .......................... 204
7.4.7 Mouse Carotid Experiments ...................... 205
7.4.8 Results ........................................ 206
7.4.9 Discussion ..................................... 206
8 Volumetric Digital Image Correlation (VDIC) ................ 209
8.1 Imaging and Discrete Pattern Recording ................ 209
8.2 System Calibration for VDIC ........................... 211
8.3 Volumetric Shape and Deformation Measurements ......... 213
8.4 Volumetric Shape Functions ............................ 214
8.5 Volumetric Image Reconstruction ....................... 218
8.6 Case Study: Quantifying Micro-damage in Polymeric
Foam Undergoing Uniaxial Compression .................. 218
8.6.1 Background ..................................... 218
8.6.2 Specimen and Experimental Considerations ....... 219
8.6.3 CT Imaging and VDIC Parameters ................. 222
8.6.4 Experimental Results ........................... 222
8.6.5 CT-Imaging Discussion .......................... 224
9 Error Estimation in Stereo-vision .......................... 225
9.1 Sub-optimal Position Estimator ........................ 225
9.2 Optimal Position Estimator ............................ 226
9.3 Variance in 3D Position ............................... 228
9.4 Discussion ............................................ 228
10 Practical Considerations for Accurate Measurements with
DIC ........................................................ 229
10.1 General Imaging Considerations ........................ 229
10.1.1 Depth of Field and Field of View ............... 229
10.1.2 Image Artifacts ................................ 231
10.1.3 Subset Patterning .............................. 233
10.1.4 Exposure Time .................................. 243
10.2 Practical Considerations for 2D-DIC Measurements ...... 244
10.2.1 Out-of-Plane Motion ............................ 244
10.2.2 Depth of Field at High Magnification ........... 245
10.2.3 Measurement Accuracy and Intensity Noise ....... 245
10.2.4 Thermal Effects at Moderate Temperatures ....... 246
10.2.5 Other Factors .................................. 246
10.3 Practical Considerations for 3D-DIC Measurements ...... 247
10.3.1 Camera and Lens Selection ...................... 247
10.3.2 Stereo-vision System Configuration ............. 248
10.3.3 Individual Camera Calibration .................. 250
10.3.4 Single Lens Stereo-vision System ............... 250
10.4 Stereo-vision Measurement Process ..................... 252
10.4.1 Subset Selection and Matching .................. 252
A Continuum Mechanics Formulation for Deformations ........... 255
A.1 Strain Tensors ........................................ 257
A.2 Strain Rate Tensors ................................... 258
A.3 Lagrangian ............................................ 260
A.4 Eulerian .............................................. 260
В Elements of Linear Algebra ................................. 261
B.1 Orthonormal Bases for Column Space, СA ................ 263
B.2 Orthonormal Bases for Row Space, RA ................... 263
B.3 Matrices Constructed from Rotation and Warping
Components ............................................ 264
B.4 Singular Value Decomposition of a Matrix .............. 266
С Method for Local Surface Strain Estimation ................. 269
D Methods of Non-linear Optimization and Parameter
Estimation ................................................. 273
D.l Levenberg-Marquardt and Non-linear Optimization ....... 273
D.1.1 Mathematical Background ........................ 273
D.l.2 Hessian Matrix Approximation and Least
Squares ........................................ 274
D.1.3 Gradient Search Process ........................ 275
D.1.4 Local Quadratic Functional Form ................ 275
D.1.5 Combined Steepest Descent and Newton-Raphson ... 276
D.1.6 Least Squares with 2D Image Positions .......... 277
D.2 Least Squares for Optimal Parameter Estimation ........ 279
E Terminology in Statistics and Probability .................. 281
E.1 Expectation ........................................... 281
E.2 Mean Value ............................................ 281
E.3 Variance .............................................. 282
E.4 Approximate Expectation and Variance Expressions ...... 283
F Basics of Projective Geometry .............................. 285
F.1 Homogeneous Coordinates ............................... 285
F.2 Why are Homogeneous Coordinates Interesting in
Computer Vision: An Example ........................... 285
F.3 Lines in  2 ........................................... 286
G Rotation Tensor Formulations ............................... 287
H Spline Functions ........................................... 289
H.1 Two-Dimensional Case: Spline Curve .................... 289
H.1.1 Spline Definition .............................. 289
H.1.2 B-Spline Definition ............................ 290
H.2 Three-Dimensional Case: Spline Surface ................ 292
H.3 Spline Derivatives .................................... 293
I Triangulation - Location of 3D Position with Skew Rays ..... 295
References .................................................... 299
Index ......................................................... 317
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