Preface to the Second Edition .................................. xi
Preface to the First Edition ................................. xiii
1 Images, Arrays, and Matrices ................................. 1
1.1 Multispectral Satellite Images .......................... 2
1.2 Algebra of Vectors and Matrices ......................... 5
1.2.1 Elementary Properties ............................ 6
1.2.2 Square Matrices .................................. 8
1.2.3 Singular Matrices ............................... 10
1.2.4 Symmetric, Positive Definite Matrices ........... 11
1.2.5 Linear Dependence and Vector Spaces ............. 12
1.3 Eigenvalues and Eigenvectors ........................... 13
1.4 Singular Value Decomposition ........................... 16
1.5 Vector Derivatives ..................................... 18
1.6 Finding Minima and Maxima .............................. 19
1.7 Exercises .............................................. 25
2. Image Statistics ........................................... 27
2.1 Random Variables ....................................... 27
2.1.1 Discrete Random Variables ....................... 28
2.1.2 Continuous Random Variables ..................... 29
2.1.3 Normal Distribution ............................. 32
2.2 Random Vectors ......................................... 34
2.3 Parameter Estimation ................................... 39
2.3.1 Sampling a Distribution ......................... 39
2.3.2 Interval Estimation ............................. 42
2.3.3 Provisional Means ............................... 43
2.4 Hypothesis Testing and Sample Distribution Functions ... 44
2.4.1 Chi-Square Distribution ......................... 48
2.4.2 Student-t Distribution .......................... 49
2.4.3 F-Distribution .................................. 50
2.5 Conditional Probabilities, Bayes' Theorem, and
Classification ......................................... 51
2.6 Ordinary Linear Regression ............................. 55
2.6.1 One Independent Variable ........................ 55
2.6.2 More Than One Independent Variable .............. 57
2.6.3 Regularization, Duality, and the Gram Matrix .... 60
2.7 Entropy and Information ................................ 62
2.7.1 Kullback-Leibler Divergence ..................... 64
2.7.2 Mutual Information .............................. 64
2.8 Exercises .............................................. 65
3 Transformations ............................................. 69
3.1 Discrete Fourier Transform ............................. 69
3.2 Discrete Wavelet Transform ............................. 73
3.2.1 Haar Wavelets ................................... 75
3.2.2 Image Compression ............................... 79
3.2.3 Multiresolution Analysis ........................ 82
3.2.3.1 Dilation Equation and Refinement
Coefficients ........................... 83
3.2.3.2 Cascade Algorithm ...................... 84
3.2.3.3 Mother Wavelet ......................... 85
3.2.3.4 Daubechies D4 Scaling Function ......... 87
3.3 Principal Components ................................... 89
3.3.1 Primal Solution ................................. 91
3.3.2 Dual Solution ................................... 91
3.4 Minimum Noise Fraction ................................. 93
3.4.1 Additive Noise .................................. 93
3.4.2 Minimum Noise Fraction Transformation in
ENVI ............................................ 96
3.5 Spatial Correlation .................................... 98
3.5.1 Maximum Autocorrelation Factor .................. 98
3.5.2 Noise Estimation ............................... 101
3.6 Exercises ............................................. 103
4 Filters, Kernels, and Fields ............................... 107
4.1 Convolution Theorem ................................... 107
4.2 Linear Filters ........................................ 111
4.3 Wavelets and Filter Banks ............................. 113
4.3.1 One-Dimensional Arrays ......................... 115
4.3.2 Two-Dimensional Arrays ......................... 120
4.4 Kernel Methods ........................................ 122
4.4.1 Valid Kernels .................................. 124
4.4.2 Kernel PCA ..................................... 127
4.5 Gibbs-Markov Random Fields ............................ 130
5 Image Enhancement and Correction ........................... 139
5.1 Lookup Tables and Histogram Functions ................. 139
5.2 Filtering and Feature Extraction ...................... 141
5.2.1 Edge Detection ................................. 141
5.2.2 Invariant Moments .............................. 145
5.3 Panchromatic Sharpening ............................... 150
5.3.1 HSV Fusion ..................................... 151
5.3.2 Brovey Fusion .................................. 152
5.3.3 PCA Fusion ..................................... 153
5.3.4 DWTFusion ...................................... 154
5.3.5 Á Trous Fusion ................................. 155
5.3.6 Quality Index .................................. 157
5.4 Topographic Correction ................................ 159
5.4.1 Rotation, Scaling, and Translation ............. 159
5.4.2 Imaging Transformations ........................ 160
5.4.3 Camera Models and RFM Approximations ........... 161
5.4.4 Stereo Imaging and Digital Elevation Models .... 163
5.4.5 Slope and Aspect ............................... 167
5.4.6 Illumination Correction ........................ 170
5.5 Image-Image Registration .............................. 175
5.5.1 Frequency-Domain Registration .................. 176
5.5.2 Feature Matching ............................... 177
5.5.2.1 High-Pass Filtering ................... 178
5.5.2.2 Closed Contours ....................... 179
5.5.2.3 Chain Codes and Moments ............... 179
5.5.2.4 Contour Matching ...................... 180
5.5.2.5 Consistency Check ..................... 180
5.5.2.6 Implementation in IDL ................. 181
5.5.3 Resampling and Warping ......................... 182
5.6 Exercises ............................................. 183
6. Supervised Classification: Part 1 .......................... 187
6.1 Maximum a Posteriori Probability ...................... 188
6.2 Training Data and Separability ........................ 189
6.3 Maximum Likelihood Classification ..................... 193
6.3.1 ENVI's Maximum Likelihood Classifier ........... 195
6.3.2 Modified Maximum Likelihood Classifier ......... 196
6.4 Gaussian Kernel Classification ........................ 198
6.5 Neural Networks ....................................... 202
6.5.1 Neural Network Classifier ...................... 207
6.5.2 Cost Functions ................................. 209
6.5.3 Backpropagation ................................ 212
6.5.4 Overfitting and Generalization ................. 216
6.6 Support Vector Machines ............................... 219
6.6.1 Linearly Separable Classes ..................... 220
6.6.1.1 Primal Formulation .................... 221
6.6.1.2 Dual Formulation ...................... 222
6.6.1.3 Quadratic Programming and Support
Vectors ............................... 224
6.6.2 Overlapping Classes ............................ 225
6.6.3 Solution with Sequential Minimal
Optimization ................................... 227
6.6.4 Multiclass SVMs ................................ 228
6.6.5 Kernel Substitution ............................ 230
6.6.6 Modified SVM Classifier ........................ 231
6.7 Exercises ............................................. 232
7 Supervised Classification: Part 2 .......................... 237
7.1 Postprocessing ........................................ 237
7.1.1 Majority Filtering ............................. 238
7.1.2 Probabilistic Label Relaxation ................. 238
7.2 Evaluation and Comparison of Classification
Accuracy .............................................. 240
7.2.1 Accuracy Assessment ............................ 241
7.2.2 Model Comparison ............................... 246
7.3 Adaptive Boosting ..................................... 250
7.4 Hyperspectral Analysis ................................ 257
7.4.1 Spectral Mixture Modeling ...................... 259
7.4.2 Unconstrained Linear Unmixing .................. 261
7.4.3 Intrinsic End-Members and Pixel Purity ......... 261
7.5 Exercises ............................................. 263
8 Unsupervised Classification ................................ 267
8.1 Simple Cost Functions ................................. 268
8.2 Algorithms That Minimize the Simple Cost Functions .... 270
8.2.1 K-Means Clustering ............................. 271
8.2.2 Kernel K-Means Clustering ...................... 271
8.2.3 Extended K-Means Clustering .................... 273
8.2.4 Agglomerative Hierarchical Clustering .......... 278
8.2.5 Fuzzy K-Means Clustering ....................... 280
8.3 Gaussian Mixture Clustering ........................... 282
8.3.1 Expectation Maximization ....................... 283
8.3.2 Simulated Annealing ............................ 286
8.3.3 Partition Density .............................. 286
8.3.4 Implementation Notes ........................... 287
8.4 Including Spatial Information ......................... 289
8.4.1 Multiresolution Clustering ..................... 289
8.4.2 Spatial Clustering ............................. 289
8.5 Benchmark ............................................. 292
8.6 Kohonen Self-Organizing Map ........................... 295
8.7 Image Segmentation .................................... 297
8.7.1 Segmenting a Classified Image .................. 299
8.7.2 Object-Based Classification .................... 300
8.7.3 Mean Shift ..................................... 303
8.8 Exercises ............................................. 304
9 Change Detection ........................................... 311
9.1 Algebraic Methods ..................................... 311
9.2 Postclassification Comparison ......................... 313
9.3 Principal Components Analysis ......................... 313
9.3.1 Iterated PCA ................................... 313
9.3.2 Kernel PCA ..................................... 314
9.4 Multivariate Alteration Detection ..................... 319
9.4.1 Canonical Correlation Analysis ................. 320
9.4.2 Orthogonality Properties ....................... 322
9.4.3 Scale Invariance ............................... 324
9.4.4 Iteratively Reweighted MAD ..................... 325
9.4.5 Correlation with the Original Observations ..... 327
9.4.6 Regularization ................................. 328
9.4.7 Postprocessing ................................. 330
9.5 Decision Thresholds and Unsupervised Classification
of Changes ............................................ 331
9.6 Radiometric Normalization ............................. 336
9.7 Exercises ............................................. 338
Appendix A: Mathematical Tools ................................ 343
A.l Cholesky Decomposition ................................ 343
A.2 Vector and Inner Product Spaces ....................... 345
A.3 Least Squares Procedures .............................. 347
A.3.1 Recursive Linear Regression .................... 347
A.3.2 Orthogonal Linear Regression ................... 350
Appendix B: Efficient Neural Network Training Algorithms ...... 355
B.l Hessian Matrix ........................................ 355
B.l.l R-Operator ...................................... 356
B.l.1.1 Determination of Rυ{n} ................. 358
B.l.1.2 Determination of Rυ{δ0} ................ 359
B.l.1.3 Determination of Rυ{δh} ................ 359
B.1.2 Calculating the Hessian ......................... 360
B.2 Scaled Conjugate Gradient Training .................... 360
B.2.1 Conjugate Directions ........................... 362
B.2.2 Minimizing a Quadratic Function ................ 363
B.2.3 Algorithm ...................................... 366
B.3 Kalman Filter Training ................................. 368
B.3.1 Linearization .................................. 371
B.3.2 Algorithm ...................................... 372
B.4 A Neural Network Classifier with Hybrid Training ...... 379
Appendix C: ENVI Extensions in IDL ............................ 381
C.l Installation .......................................... 381
C.2 Extensions ............................................ 382
C.2.1 Kernel Principal Components Analysis ........... 384
C.2.2 Discrete Wavelet Transform Fusion .............. 386
C.2.3 Á Trous Wavelet Transform Fusion ............... 388
C.2.4 Quality Index .................................. 389
C.2.5 Calculating Heights of Man-Made Structures in
High-Resolution Imagery ........................ 390
C.2.6 Illumination Correction ........................ 392
C.2.7 Image Registration ............................. 393
C.2.8 Maximum Likelihood Classification .............. 394
C.2.9 Gaussian Kernel Classification ................. 396
C.2.10 Neural Network Classification .................. 397
C.2.11 Support Vector Machine Classification .......... 399
C.2.12 Probabilistic Label Relaxation ................. 399
C.2.13 Classifier Evaluation and Comparison ........... 401
C.2.14 Adaptive Boosting a Neural Network
Classifier ..................................... 402
C.2.15 Kernel K-Means Clustering ...................... 404
C.2.16 Agglomerative Hierarchical Clustering .......... 405
C.2.17 Fuzzy K-Means Clustering ....................... 406
C.2.18 Gaussian Mixture Clustering .................... 407
C.2.19 Kohonen Self-Organizing Map .................... 409
C.2.20 Classified Image Segmentation .................. 410
C.2.21 Mean Shift Segmentation ........................ 411
C.2.22 Multivariate Alteration Detection .............. 412
C.2.23 Viewing Changes ................................ 415
C.2.24 Radiometric Normalization ...................... 416
Appendix D: Mathematical Notation ............................. 419
References .................................................... 421
Index ......................................................... 429
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