Foreword by Raanan Fattal .................................... xiii
Preface ........................................................ xv
Cover Image .............................................. xvi
Acknowledgments .......................................... xvi
I Background ................................................... 1
1 Introduction ................................................. 3
1.1 Statistics as Priors ....................................... 6
1.2 Statistics as Image Descriptors ............................ 7
1.3 Statistical Pipeline ...................................... 10
1.4 Natural Images ............................................ 11
1.5 Discussion ................................................ 11
2 The Human Visual System ................................... 15
2.1 Radiometric and Photometric Terms ......................... 15
2.2 Human Vision .............................................. 17
2.3 The Eyes .................................................. 18
2.3.1 Optical Media and the Retina ....................... 19
2.3.2 Photoreceptors ..................................... 20
2.3.3 Horizontal and Bipolar Cells ....................... 25
2.3.4 Amacrine and Ganglion Cells ........................ 26
2.4 The Lateral Geniculate Nucleus and Cortical Processing .... 27
2.5 Implications of Human Visual Processing ................... 29
2.5.1 Visual Acuity ...................................... 30
2.5.2 Temporal Resolution ................................ 31
2.5.3 Contrast ........................................... 31
2.5.4 Color Processing ................................... 32
2.5.5 Visual Illusions ................................... 34
3 Image Collection and Calibration .......................... 41
3.1 Image Capture ............................................. 42
3.1.1 Photographer and Camera Bias ....................... 43
3.1.2 High Dynamic Range Imaging ......................... 43
3.1.3 Field of View ...................................... 46
3.1.4 Multispectral and Hyperspectral Imaging ............ 49
3.1.5 Depth and Range Capture ............................ 51
3.2 Post-processing and Calibration ........................... 54
3.2.1 Radiometric Calibration ............................ 54
3.2.2 Lens Aberrations ................................... 55
3.2.3 Noise .............................................. 59
3.3 Image Databases ........................................... 60
3.3.1 Van Hateren's Natural Image Database ............... 60
3.3.2 University of Texas at Austin Databases ............ 61
3.3.3 UPenn Natural Image Database ....................... 61
3.3.4 The Barcelona Calibrated Images Database ........... 62
3.3.5 HDR Photographic Survey ............................ 62
3.3.6 IPL Calibrated Color Image Database ................ 63
3.3.7 McGill Calibrated Color Image Database ............. 64
3.3.8 Hyperspectral Images of Natural Scenes ............. 64
3.3.9 Real-World Hyperspectral Images Database ........... 65
3.3.10 Bristol Hyperspectral Images Database .............. 66
3.3.11 Amsterdam Library of Object Images (ALOI) .......... 66
3.3.12 Caltech-256 Object Category Dataset ................ 67
3.3.13 Brown Range Image Database ......................... 67
II Image Statistics .......................................... 69
4 First-Order Statistics .................................... 71
4.1 Histograms and Moments .................................... 72
4.1.1 Image Moments and Moment Invariants ................ 74
4.1.2 Histogram Adjustments .............................. 75
4.2 Moment Statistics and Average Distributions ............... 77
4.3 Material Properties ....................................... 81
4.4 Nonlinear Compression in Art .............................. 83
4.5 Dark-Is-Deep Paradigm ..................................... 85
4.6 Summary ................................................... 86
5 Gradients, Edges, and Contrast ............................ 89
5.1 Real-World Considerations ................................. 89
5.1.1 Perceptual Consequences ............................ 90
Image Space Consequences .................................. 90
5.2 Gradients ................................................. 90
5.2.1 The Forward Difference Method ...................... 91
5.2.2 The Backward Difference Method ..................... 92
5.2.3 The Central Difference Method ...................... 93
5.2.4 The Sцbel Operator ................................. 94
5.2.5 Second Derivative Methods .......................... 95
5.2.6 Gradient Magnitude ................................. 96
5.2.7 Gradient Statistics ................................ 97
5.2.8 Single-Image Gradient Statistics ................... 97
5.3 Edges ..................................................... 99
5.3.1 Definition of an Edge ............................. 100
5.3.2 Edge Detection Processes .......................... 100
5.3.3 Edge Statistics ................................... 103
5.4 Linear Scale Space ....................................... 104
5.4.1 The N-Jet and Feature Detectors ................... 108
5.4.2 Implications for Human Perception ................. 109
5.4.3 Scale-Space Statistics ............................ 110
5.5 Contrast in Images ....................................... 111
5.6 Image Deblurring ......................................... 114
5.7 Superresolution .......................................... 116
5.8 Inpainting ............................................... 116
6 Fourier Analysis ......................................... 119
6.1 Autocorrelation .......................................... 121
6.2 The Fourier Transform .................................... 123
6.3 The Wiener-Khintchine Theorem ............................ 128
6.4 Power Spectra ............................................ 128
6.4.1 Slope Computation ................................. 129
6.4.2 Spectral Slope Analysis ........................... 132
6.4.3 Dynamic Range ..................................... 137
6.4.4 Dependence on Image Representation ................ 139
6.4.5 Angular Dependence ................................ 140
6.4.6 Temporal Dependence ............................... 140
6.4.7 1/ƒ Failures ...................................... 141
6.5 Phase Spectra ............................................ 142
6.6 Human Perception ......................................... 144
6.7 Fractal Forgeries ........................................ 145
6.8 Image Processing and Categorization ...................... 145
6.9 Texture Descriptors ...................................... 146
6.10 Terrain Synthesis ........................................ 147
6.11 Art Statistics ........................................... 148
7 Dimensionality Reduction ................................. 153
7.1 Principal Component Analysis ............................. 156
7.1.1 Whitening ......................................... 159
7.1.2 PCA on Pixels ..................................... 159
7.1.3 PCA on Patches .................................... 159
7.1.4 PCA on Images ..................................... 163
7.1.5 Eigenfaces ........................................ 164
7.2 Independent Components Analysis .......................... 166
7.3 ICA on Natural Images .................................... 169
7.4 Gaussian Mixture Models .................................. 172
8 Wavelet Analysis ......................................... 175
8.1 Wavelet Transform ........................................ 176
8.2 Multiresolution Analysis ................................. 178
8.3 Signal Processing ........................................ 181
8.4 Other Bases .............................................. 185
8.5 2D Wavelets .............................................. 187
8.6 Contourlets, Curvelets, and Ridgelets .................... 190
8.7 Coefficient Histograms ................................... 190
8.8 Scale Invariance ......................................... 193
8.9 Correlations between Coefficients ........................ 194
8.10 Complex Wavelets ......................................... 197
8.11 Correlations between Scales .............................. 200
8.12 Application: Image Denoising ............................. 202
8.13 Application: Progressive Reconstruction .................. 203
8.14 Application: Texture Synthesis ........................... 203
9 Markov Random Fields ..................................... 207
9.1 Image Interpretation ..................................... 208
9.2 Graphs ................................................... 210
9.2.1 Neighborhood Systems .............................. 210
9.2.2 Cliques ........................................... 212
9.3 Probabilities and Markov Random Fields ................... 212
9.3.1 Gibbs Distributions ............................... 213
9.3.2 Auto-Models ....................................... 214
9.4 MAP-MRF .................................................. 215
9.5 Applications ............................................. 216
9.5.1 Image Restoration ................................. 216
9.5.2 Object Segmentation ............................... 217
9.6 Complex Models and Patch-Based Regularities .............. 218
9.6.1 Products of Experts ............................... 218
9.6.2 Fields of Experts ................................. 219
9.7 Statistical analysis of MRFs ............................. 219
III Beyond Two Dimensions .................................... 225
10 Color .................................................... 227
10.1 Trichromacy and Metamerism ............................... 228
10.2 Color as a 3D Space ...................................... 229
10.3 Opponent Processing ...................................... 230
10.4 Color Transfer ........................................... 235
10.4.1 Color Transfer through Simple Moments ............. 236
10.4.2 Color Transfer through Higher-Order
Manipulation ...................................... 238
10.4.3 Histogram Features at Different Scales ............ 239
10.4.4 Color Transfer as a 3D Problem .................... 241
10.5 Color Space Statistics ................................... 242
10.5.1 Color Space Normalization ......................... 244
10.5.2 Correlation Analysis .............................. 245
10.6 Color Constancy and White Balancing ...................... 248
10.6.1 Computational Color Constancy as the Minkowski
Norm .............................................. 251
10.6.2 White-Balance Algorithm Selection ................. 254
10.7 Summary .................................................. 254
11 Depth Statistics ......................................... 257
11.1 The "Dead Leaves" Model .................................. 259
11.2 Perception of Scene Geometry ............................. 260
11.2.1 Length Perception ................................. 261
11.2.2 Orientation and Angle Perception .................. 262
11.3 Correlations between 2D and Range Statistics ............. 264
11.4 Depth Reconstruction ..................................... 267
12 Time and Motion .......................................... 269
12.1 The Statistics of Time ................................... 269
12.2 Motion ................................................... 274
12.2.1 Correlation-Based Motion Detection ................ 274
12.2.2 Gradient-Based Motion Detection ................... 275
12.3 Applications Using Statistical Motion Regularities ....... 275
12.3.1 Specular Highlights ............................... 276
12.3.2 Markov Random Fields .............................. 276
12.3.3 People Detection and Biological Motion ............ 276
12.3.4 Motion Blur ....................................... 277
12.3.5 Shape from Dynamic Occlusion ...................... 277
12.4 Optical Flow ............................................. 278
12.4.1 Probabilistic Optical Flow ........................ 281
12.4.2 Statistics of Real-World Motion and Retinal Flow .. 282
12.4.3 Statistics of Optical Flow ........................ 283
A Basic Definitions ......................................... 285
A.l Probabilities and Bayes' Rule ............................. 285
A.2 Gaussian Distribution ..................................... 286
A.3 Kullback-Leibler Divergence ............................... 286
Bibliography .................................................. 287
Index ......................................................... 341
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