- ISBN: 9780136085928 | 013608592X
- Cover: Paperback
- Copyright: 10/26/2011
Computer Vision: A Modern Approach, 2e,is appropriate for upper-division undergraduate- and graduate-level courses in computer vision found in departments of Computer Science, Computer Engineering and Electrical Engineering. This textbook provides the most complete treatment of modern computer vision methods by two of the leading authorities in the field. This accessible presentation gives both a general view of the entire computer vision enterprise and also offers sufficient detail for students to be able to build useful applications. Students will learn techniques that have proven to be useful by first-hand experience and a wide range of mathematical methods
David A. Forsyth received the D.Phil. degree in computer science from Oxford University. He is currently a Professor in the Computer Science Division at the University of California at Berkeley. He has co-authored over eighty technical papers on computer vision, computer graphics and machine learning and has co-edited two books.
Jean Ponce received the Ph.D. degree in Computer Science from the University of Paris Orsay. He is currently a Professor in the Department of Computer Science and the Beckman Institute at the University of Illinois at Urbana Champaign. Professor Ponce has written over a hundred conference and journal papers and co-edited two books on a range of subjects including computer vision and robotics.
I IMAGE FORMATION 1
1 Geometric Camera Models 3
1.1 Image Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Pinhole Perspective . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Weak Perspective . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Cameras with Lenses . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.4 The Human Eye . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Intrinsic and Extrinsic Parameters . . . . . . . . . . . . . . . . . . . 14
1.2.1 Rigid Transformations and Homogeneous Coordinates . . . . 14
1.2.2 Intrinsic Parameters . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.3 Extrinsic Parameters . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.4 Perspective Projection Matrices . . . . . . . . . . . . . . . . . 19
1.2.5 Weak-Perspective Projection Matrices . . . . . . . . . . . . . 20
1.3 Geometric Camera Calibration . . . . . . . . . . . . . . . . . . . . . 22
1.3.1 ALinear Approach to Camera Calibration . . . . . . . . . . . 23
1.3.2 ANonlinear Approach to Camera Calibration . . . . . . . . . 27
1.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Light and Shading 32
2.1 Modelling Pixel Brightness . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.1 Reflection at Surfaces . . . . . . . . . . . . . . . . . . . . . . 33
2.1.2 Sources and Their Effects . . . . . . . . . . . . . . . . . . . . 34
2.1.3 The Lambertian+Specular Model . . . . . . . . . . . . . . . . 36
2.1.4 Area Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Inference from Shading . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.1 Radiometric Calibration and High Dynamic Range Images . . 38
2.2.2 The Shape of Specularities . . . . . . . . . . . . . . . . . . . 40
2.2.3 Inferring Lightness and Illumination . . . . . . . . . . . . . . 43
2.2.4 Photometric Stereo: Shape from Multiple Shaded Images . . 46
2.3 Modelling Interreflection . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.1 The Illumination at a Patch Due to an Area Source . . . . . 52
2.3.2 Radiosity and Exitance . . . . . . . . . . . . . . . . . . . . . 54
2.3.3 An Interreflection Model . . . . . . . . . . . . . . . . . . . . . 55
2.3.4 Qualitative Properties of Interreflections . . . . . . . . . . . . 56
2.4 Shape from One Shaded Image . . . . . . . . . . . . . . . . . . . . . 59
2.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 Color 68
3.1 Human Color Perception . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.1.1 Color Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.1.2 Color Receptors . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 The Physics of Color . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2.1 The Color of Light Sources . . . . . . . . . . . . . . . . . . . 73
3.2.2 The Color of Surfaces . . . . . . . . . . . . . . . . . . . . . . 76
3.3 Representing Color . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3.1 Linear Color Spaces . . . . . . . . . . . . . . . . . . . . . . . 77
3.3.2 Non-linear Color Spaces . . . . . . . . . . . . . . . . . . . . . 83
3.4 AModel of Image Color . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4.1 The Diffuse Term . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.4.2 The Specular Term . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5 Inference from Color . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5.1 Finding Specularities Using Color . . . . . . . . . . . . . . . 90
3.5.2 Shadow Removal Using Color . . . . . . . . . . . . . . . . . . 92
3.5.3 Color Constancy: Surface Color from Image Color . . . . . . 95
3.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
II EARLY VISION: JUST ONE IMAGE 105
4 Linear Filters 107
4.1 Linear Filters and Convolution . . . . . . . . . . . . . . . . . . . . . 107
4.1.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2 Shift Invariant Linear Systems . . . . . . . . . . . . . . . . . . . . . 112
4.2.1 Discrete Convolution . . . . . . . . . . . . . . . . . . . . . . . 113
4.2.2 Continuous Convolution . . . . . . . . . . . . . . . . . . . . . 115
4.2.3 Edge Effects in Discrete Convolutions . . . . . . . . . . . . . 118
4.3 Spatial Frequency and Fourier Transforms . . . . . . . . . . . . . . . 118
4.3.1 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . 119
4.4 Sampling and Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.4.1 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.4.2 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.4.3 Smoothing and Resampling . . . . . . . . . . . . . . . . . . . 126
4.5 Filters as Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.5.1 Convolution as a Dot Product . . . . . . . . . . . . . . . . . 131
4.5.2 Changing Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.6 Technique: Normalized Correlation and Finding Patterns . . . . . . 132
4.6.1 Controlling the Television by Finding Hands by Normalized
Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.7 Technique: Scale and Image Pyramids . . . . . . . . . . . . . . . . . 134
4.7.1 The Gaussian Pyramid . . . . . . . . . . . . . . . . . . . . . 135
4.7.2 Applications of Scaled Representations . . . . . . . . . . . . . 136
4.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5 Local Image Features 141
5.1 Computing the Image Gradient . . . . . . . . . . . . . . . . . . . . . 141
5.1.1 Derivative of Gaussian Filters . . . . . . . . . . . . . . . . . . 142
5.2 Representing the Image Gradient . . . . . . . . . . . . . . . . . . . . 144
5.2.1 Gradient-Based Edge Detectors . . . . . . . . . . . . . . . . . 145
5.2.2 Orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.3 Finding Corners and Building Neighborhoods . . . . . . . . . . . . . 148
5.3.1 Finding Corners . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.3.2 Using Scale and Orientation to Build a Neighborhood . . . . 151
5.4 Describing Neighborhoods with SIFT and HOG Features . . . . . . 155
5.4.1 SIFT Features . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.4.2 HOG Features . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.5 Computing Local Features in Practice . . . . . . . . . . . . . . . . . 160
5.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6 Texture 164
6.1 Local Texture Representations Using Filters . . . . . . . . . . . . . . 166
6.1.1 Spots and Bars . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.1.2 From Filter Outputs to Texture Representation . . . . . . . . 168
6.1.3 Local Texture Representations in Practice . . . . . . . . . . . 170
6.2 Pooled Texture Representations by Discovering Textons . . . . . . . 171
6.2.1 Vector Quantization and Textons . . . . . . . . . . . . . . . . 172
6.2.2 K-means Clustering for Vector Quantization . . . . . . . . . . 172
6.3 Synthesizing Textures and Filling Holes in Images . . . . . . . . . . 176
6.3.1 Synthesis by Sampling Local Models . . . . . . . . . . . . . . 176
6.3.2 Filling in Holes in Images . . . . . . . . . . . . . . . . . . . . 179
6.4 Image Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.4.1 Non-local Means . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.4.2 Block Matching 3D (BM3D) . . . . . . . . . . . . . . . . . . 183
6.4.3 Learned Sparse Coding . . . . . . . . . . . . . . . . . . . . . 184
6.4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.5 Shape from Texture . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.5.1 Shape from Texture for Planes . . . . . . . . . . . . . . . . . 187
6.5.2 Shape from Texture for Curved Surfaces . . . . . . . . . . . . 190
6.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
III EARLY VISION: MULTIPLE IMAGES 195
7 Stereopsis 197
7.1 Binocular Camera Geometry and the Epipolar Constraint . . . . . . 198
7.1.1 Epipolar Geometry . . . . . . . . . . . . . . . . . . . . . . . . 198
7.1.2 The Essential Matrix . . . . . . . . . . . . . . . . . . . . . . . 200
7.1.3 The Fundamental Matrix . . . . . . . . . . . . . . . . . . . . 201
7.2 Binocular Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 201
7.2.1 Image Rectification . . . . . . . . . . . . . . . . . . . . . . . . 202
7.3 Human Stereopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.4 Local Methods for Binocular Fusion . . . . . . . . . . . . . . . . . . 205
7.4.1 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.4.2 Multi-Scale Edge Matching . . . . . . . . . . . . . . . . . . . 207
7.5 Global Methods for Binocular Fusion . . . . . . . . . . . . . . . . . . 210
7.5.1 Ordering Constraints and Dynamic Programming . . . . . . . 210
7.5.2 Smoothness and Graphs . . . . . . . . . . . . . . . . . . . . . 211
7.6 Using More Cameras . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
7.7 Application: Robot Navigation . . . . . . . . . . . . . . . . . . . . . 215
7.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8 Structure from Motion 221
8.1 Internally Calibrated Perspective Cameras . . . . . . . . . . . . . . . 221
8.1.1 Natural Ambiguity of the Problem . . . . . . . . . . . . . . . 223
8.1.2 Euclidean Structure and Motion from Two Images . . . . . . 224
8.1.3 Euclidean Structure and Motion from Multiple Images . . . . 228
8.2 Uncalibrated Weak-Perspective Cameras . . . . . . . . . . . . . . . . 230
8.2.1 Natural Ambiguity of the Problem . . . . . . . . . . . . . . . 231
8.2.2 Affine Structure and Motion from Two Images . . . . . . . . 233
8.2.3 Affine Structure and Motion from Multiple Images . . . . . . 237
8.2.4 From Affine to Euclidean Shape . . . . . . . . . . . . . . . . 238
8.3 Uncalibrated Perspective Cameras . . . . . . . . . . . . . . . . . . . 240
8.3.1 Natural Ambiguity of the Problem . . . . . . . . . . . . . . . 241
8.3.2 Projective Structure and Motion from Two Images . . . . . . 242
8.3.3 Projective Structure and Motion from Multiple Images . . . . 244
8.3.4 From Projective to Euclidean Shape . . . . . . . . . . . . . . 246
8.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
IV MID-LEVEL VISION 253
9 Segmentation by Clustering 255
9.1 Human Vision: Grouping and Gestalt . . . . . . . . . . . . . . . . . 256
9.2 Important Applications . . . . . . . . . . . . . . . . . . . . . . . . . 261
9.2.1 Background Subtraction . . . . . . . . . . . . . . . . . . . . . 261
9.2.2 Shot Boundary Detection . . . . . . . . . . . . . . . . . . . . 264
9.2.3 Interactive Segmentation . . . . . . . . . . . . . . . . . . . . 265
9.2.4 Forming Image Regions . . . . . . . . . . . . . . . . . . . . . 266
9.3 Image Segmentation by Clustering Pixels . . . . . . . . . . . . . . . 268
9.3.1 Basic Clustering Methods . . . . . . . . . . . . . . . . . . . . 269
9.3.2 The Watershed Algorithm . . . . . . . . . . . . . . . . . . . . 271
9.3.3 Segmentation Using K-means . . . . . . . . . . . . . . . . . . 272
9.3.4 Mean Shift: Finding Local Modes in Data . . . . . . . . . . . 273
9.3.5 Clustering and Segmentation with Mean Shift . . . . . . . . . 275
9.4 Segmentation, Clustering, and Graphs . . . . . . . . . . . . . . . . . 277
9.4.1 Terminology and Facts for Graphs . . . . . . . . . . . . . . . 277
9.4.2 Agglomerative Clustering with a Graph . . . . . . . . . . . . 279
9.4.3 Divisive Clustering with a Graph . . . . . . . . . . . . . . . . 281
9.4.4 Normalized Cuts . . . . . . . . . . . . . . . . . . . . . . . . . 284
9.5 Image Segmentation in Practice . . . . . . . . . . . . . . . . . . . . . 285
9.5.1 Evaluating Segmenters . . . . . . . . . . . . . . . . . . . . . . 286
9.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
10 Grouping and Model Fitting 290
10.1 The Hough Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 290
10.1.1 Fitting Lines with the Hough Transform . . . . . . . . . . . . 290
10.1.2 Using the Hough Transform . . . . . . . . . . . . . . . . . . . 292
10.2 Fitting Lines and Planes . . . . . . . . . . . . . . . . . . . . . . . . . 293
10.2.1 Fitting a Single Line . . . . . . . . . . . . . . . . . . . . . . . 294
10.2.2 Fitting Planes . . . . . . . . . . . . . . . . . . . . . . . . . . 295
10.2.3 Fitting Multiple Lines . . . . . . . . . . . . . . . . . . . . . . 296
10.3 Fitting Curved Structures . . . . . . . . . . . . . . . . . . . . . . . . 297
10.4 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
10.4.1 M-Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
10.4.2 RANSAC: Searching for Good Points . . . . . . . . . . . . . 302
10.5 Fitting Using Probabilistic Models . . . . . . . . . . . . . . . . . . . 306
10.5.1 Missing Data Problems . . . . . . . . . . . . . . . . . . . . . 307
10.5.2 Mixture Models and Hidden Variables . . . . . . . . . . . . . 309
10.5.3 The EM Algorithm for Mixture Models . . . . . . . . . . . . 310
10.5.4 Difficulties with the EM Algorithm . . . . . . . . . . . . . . . 312
10.6 Motion Segmentation by Parameter Estimation . . . . . . . . . . . . 313
10.6.1 Optical Flow and Motion . . . . . . . . . . . . . . . . . . . . 315
10.6.2 Flow Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
10.6.3 Motion Segmentation with Layers . . . . . . . . . . . . . . . 317
10.7 Model Selection: Which Model Is the Best Fit? . . . . . . . . . . . . 319
10.7.1 Model Selection Using Cross-Validation . . . . . . . . . . . . 322
10.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
11 Tracking 326
11.1 Simple Tracking Strategies . . . . . . . . . . . . . . . . . . . . . . . . 327
11.1.1 Tracking by Detection . . . . . . . . . . . . . . . . . . . . . . 327
11.1.2 Tracking Translations by Matching . . . . . . . . . . . . . . . 330
11.1.3 Using Affine Transformations to Confirm a Match . . . . . . 332
11.2 Tracking Using Matching . . . . . . . . . . . . . . . . . . . . . . . . 334
11.2.1 Matching Summary Representations . . . . . . . . . . . . . . 335
11.2.2 Tracking Using Flow . . . . . . . . . . . . . . . . . . . . . . . 337
11.3 Tracking Linear Dynamical Models with Kalman Filters . . . . . . . 339
11.3.1 Linear Measurements and Linear Dynamics . . . . . . . . . . 340
11.3.2 The Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . 344
11.3.3 Forward-backward Smoothing . . . . . . . . . . . . . . . . . . 345
11.4 Data Association . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
11.4.1 Linking Kalman Filters with Detection Methods . . . . . . . 349
11.4.2 Key Methods of Data Association . . . . . . . . . . . . . . . 350
11.5 Particle Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
11.5.1 Sampled Representations of Probability Distributions . . . . 351
11.5.2 The Simplest Particle Filter . . . . . . . . . . . . . . . . . . . 355
11.5.3 The Tracking Algorithm . . . . . . . . . . . . . . . . . . . . . 356
11.5.4 A Workable Particle Filter . . . . . . . . . . . . . . . . . . . . 358
11.5.5 Practical Issues in Particle Filters . . . . . . . . . . . . . . . 360
11.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
V HIGH-LEVEL VISION 365
12 Registration 367
12.1 Registering Rigid Objects . . . . . . . . . . . . . . . . . . . . . . . . 368
12.1.1 Iterated Closest Points . . . . . . . . . . . . . . . . . . . . . . 368
12.1.2 Searching for Transformations via Correspondences . . . . . . 369
12.1.3 Application: Building Image Mosaics . . . . . . . . . . . . . . 370
12.2 Model-based Vision: Registering Rigid Objects with Projection . . . 375
12.2.1 Verification: Comparing Transformed and Rendered Source
to Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
12.3 Registering Deformable Objects . . . . . . . . . . . . . . . . . . . . . 378
12.3.1 Deforming Texture with Active Appearance Models . . . . . 378
12.3.2 Active Appearance Models in Practice . . . . . . . . . . . . . 381
12.3.3 Application: Registration in Medical Imaging Systems . . . . 383
12.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
13 Smooth Surfaces and Their Outlines 391
13.1 Elements of Differential Geometry . . . . . . . . . . . . . . . . . . . 393
13.1.1 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
13.1.2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
13.2 Contour Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
13.2.1 The Occluding Contour and the Image Contour . . . . . . . . 402
13.2.2 The Cusps and Inflections of the Image Contour . . . . . . . 403
13.2.3 Koenderink’s Theorem . . . . . . . . . . . . . . . . . . . . . . 404
13.3 Visual Events: More Differential Geometry . . . . . . . . . . . . . . 407
13.3.1 The Geometry of the Gauss Map . . . . . . . . . . . . . . . . 407
13.3.2 Asymptotic Curves . . . . . . . . . . . . . . . . . . . . . . . . 409
13.3.3 The Asymptotic Spherical Map . . . . . . . . . . . . . . . . . 410
13.3.4 Local Visual Events . . . . . . . . . . . . . . . . . . . . . . . 412
13.3.5 The Bitangent Ray Manifold . . . . . . . . . . . . . . . . . . 413
13.3.6 Multilocal Visual Events . . . . . . . . . . . . . . . . . . . . . 414
13.3.7 The Aspect Graph . . . . . . . . . . . . . . . . . . . . . . . . 416
13.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
14 Range Data 422
14.1 Active Range Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . 422
14.2 Range Data Segmentation . . . . . . . . . . . . . . . . . . . . . . . . 424
14.2.1 Elements of Analytical Differential Geometry . . . . . . . . . 424
14.2.2 Finding Step and Roof Edges in Range Images . . . . . . . . 426
14.2.3 Segmenting Range Images into Planar Regions . . . . . . . . 431
14.3 Range Image Registration and Model Acquisition . . . . . . . . . . . 432
14.3.1 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
14.3.2 Registering Range Images . . . . . . . . . . . . . . . . . . . . 434
14.3.3 Fusing Multiple Range Images . . . . . . . . . . . . . . . . . 436
14.4 Object Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
14.4.1 Matching Using Interpretation Trees . . . . . . . . . . . . . . 438
14.4.2 Matching Free-Form Surfaces Using Spin Images . . . . . . . 441
14.5 Kinect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
14.5.1 Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
14.5.2 Technique: Decision Trees and Random Forests . . . . . . . . 448
14.5.3 Labeling Pixels . . . . . . . . . . . . . . . . . . . . . . . . . . 450
14.5.4 Computing Joint Positions . . . . . . . . . . . . . . . . . . . 453
14.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
15 Learning to Classify 457
15.1 Classification, Error, and Loss . . . . . . . . . . . . . . . . . . . . . . 457
15.1.1 Using Loss to Determine Decisions . . . . . . . . . . . . . . . 457
15.1.2 Training Error, Test Error, and Overfitting . . . . . . . . . . 459
15.1.3 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . 460
15.1.4 Error Rate and Cross-Validation . . . . . . . . . . . . . . . . 463
15.1.5 Receiver Operating Curves . . . . . . . . . . . . . . . . . . . 465
15.2 Major Classification Strategies . . . . . . . . . . . . . . . . . . . . . 467
15.2.1 Example: Mahalanobis Distance . . . . . . . . . . . . . . . . 467
15.2.2 Example: Class-Conditional Histograms and Naive Bayes . . 468
15.2.3 Example: Classification Using Nearest Neighbors . . . . . . . 469
15.2.4 Example: The Linear Support Vector Machine . . . . . . . . 470
15.2.5 Example: Kernel Machines . . . . . . . . . . . . . . . . . . . 473
15.2.6 Example: Boosting and Adaboost . . . . . . . . . . . . . . . 475
15.3 Practical Methods for Building Classifiers . . . . . . . . . . . . . . . 475
15.3.1 Manipulating Training Data to Improve Performance . . . . . 477
15.3.2 Building Multi-Class Classifiers Out of Binary Classifiers . . 479
15.3.3 Solving for SVMS and Kernel Machines . . . . . . . . . . . . 480
15.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
16 Classifying Images 482
16.1 Building Good Image Features . . . . . . . . . . . . . . . . . . . . . 482
16.1.1 Example Applications . . . . . . . . . . . . . . . . . . . . . . 482
16.1.2 Encoding Layout with GIST Features . . . . . . . . . . . . . 485
16.1.3 Summarizing Images with Visual Words . . . . . . . . . . . . 487
16.1.4 The Spatial Pyramid Kernel . . . . . . . . . . . . . . . . . . . 489
16.1.5 Dimension Reduction with Principal Components . . . . . . . 493
16.1.6 Dimension Reduction with Canonical Variates . . . . . . . . 494
16.1.7 Example Application: Identifying Explicit Images . . . . . . 498
16.1.8 Example Application: Classifying Materials . . . . . . . . . . 502
16.1.9 Example Application: Classifying Scenes . . . . . . . . . . . . 502
16.2 Classifying Images of Single Objects . . . . . . . . . . . . . . . . . . 504
16.2.1 Image Classification Strategies . . . . . . . . . . . . . . . . . 505
16.2.2 Evaluating Image Classification Systems . . . . . . . . . . . . 505
16.2.3 Fixed Sets of Classes . . . . . . . . . . . . . . . . . . . . . . . 508
16.2.4 Large Numbers of Classes . . . . . . . . . . . . . . . . . . . . 509
16.2.5 Flowers, Leaves, and Birds: Some Specialized Problems . . . 511
16.3 Image Classification in Practice . . . . . . . . . . . . . . . . . . . . . 512
16.3.1 Codes for Image Features . . . . . . . . . . . . . . . . . . . . 513
16.3.2 Image Classification Datasets . . . . . . . . . . . . . . . . . . 513
16.3.3 Dataset Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
16.3.4 Crowdsourcing Dataset Collection . . . . . . . . . . . . . . . 515
16.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
17 Detecting Objects in Images 519
17.1 The Sliding Window Method . . . . . . . . . . . . . . . . . . . . . . 519
17.1.1 Face Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 520
17.1.2 Detecting Humans . . . . . . . . . . . . . . . . . . . . . . . . 525
17.1.3 Detecting Boundaries . . . . . . . . . . . . . . . . . . . . . . 527
17.2 Detecting Deformable Objects . . . . . . . . . . . . . . . . . . . . . . 530
17.3 The State of the Art of Object Detection . . . . . . . . . . . . . . . 535
17.3.1 Datasets and Resources . . . . . . . . . . . . . . . . . . . . . 538
17.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
18 Topics in Object Recognition 540
18.1 What Should Object Recognition Do? . . . . . . . . . . . . . . . . . 540
18.1.1 What Should an Object Recognition System Do? . . . . . . . 540
18.1.2 Current Strategies for Object Recognition . . . . . . . . . . . 542
18.1.3 What Is Categorization? . . . . . . . . . . . . . . . . . . . . . 542
18.1.4 Selection: What Should Be Described? . . . . . . . . . . . . . 544
18.2 Feature Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544
18.2.1 Improving Current Image Features . . . . . . . . . . . . . . . 544
18.2.2 Other Kinds of Image Feature . . . . . . . . . . . . . . . . . . 546
18.3 Geometric Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
18.4 Semantic Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
18.4.1 Attributes and the Unfamiliar . . . . . . . . . . . . . . . . . . 550
18.4.2 Parts, Poselets and Consistency . . . . . . . . . . . . . . . . . 551
18.4.3 Chunks of Meaning . . . . . . . . . . . . . . . . . . . . . . . . 554
VI APPLICATIONS AND TOPICS 557
19 Image-Based Modeling and Rendering 559
19.1 Visual Hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
19.1.1 Main Elements of the Visual Hull Model . . . . . . . . . . . . 561
19.1.2 Tracing Intersection Curves . . . . . . . . . . . . . . . . . . . 563
19.1.3 Clipping Intersection Curves . . . . . . . . . . . . . . . . . . 566
19.1.4 Triangulating Cone Strips . . . . . . . . . . . . . . . . . . . . 567
19.1.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
19.1.6 Going Further: Carved Visual Hulls . . . . . . . . . . . . . . 572
19.2 Patch-Based Multi-View Stereopsis . . . . . . . . . . . . . . . . . . . 573
19.2.1 Main Elements of the PMVS Model . . . . . . . . . . . . . . 575
19.2.2 Initial Feature Matching . . . . . . . . . . . . . . . . . . . . . 578
19.2.3 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
19.2.4 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
19.2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
19.3 The Light Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584
19.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
20 Looking at People 590
20.1 HMM’s, Dynamic Programming, and Tree-Structured Models . . . . 590
20.1.1 Hidden Markov Models . . . . . . . . . . . . . . . . . . . . . 590
20.1.2 Inference for an HMM . . . . . . . . . . . . . . . . . . . . . . 592
20.1.3 Fitting an HMM with EM . . . . . . . . . . . . . . . . . . . . 597
20.1.4 Tree-Structured Energy Models . . . . . . . . . . . . . . . . . 600
20.2 Parsing People in Images . . . . . . . . . . . . . . . . . . . . . . . . 602
20.2.1 Parsing with Pictorial Structure Models . . . . . . . . . . . . 602
20.2.2 Estimating the Appearance of Clothing . . . . . . . . . . . . 604
20.3 Tracking People . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
20.3.1 Why Human Tracking Is Hard . . . . . . . . . . . . . . . . . 606
20.3.2 Kinematic Tracking by Appearance . . . . . . . . . . . . . . . 608
20.3.3 Kinematic Human Tracking Using Templates . . . . . . . . . 609
20.4 3D from 2D: Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
20.4.1 Reconstruction in an Orthographic View . . . . . . . . . . . . 611
20.4.2 Exploiting Appearance for Unambiguous Reconstructions . . 613
20.4.3 Exploiting Motion for Unambiguous Reconstructions . . . . . 615
20.5 Activity Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
20.5.1 Background: Human Motion Data . . . . . . . . . . . . . . . 617
20.5.2 Body Configuration and Activity Recognition . . . . . . . . . 621
20.5.3 Recognizing Human Activities with Appearance Features . . 622
20.5.4 Recognizing Human Activities with Compositional Models . . 624
20.6 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
20.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626
21 Image Search and Retrieval 627
21.1 The Application Context . . . . . . . . . . . . . . . . . . . . . . . . . 627
21.1.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 628
21.1.2 User Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
21.1.3 Types of Image Query . . . . . . . . . . . . . . . . . . . . . . 630
21.1.4 What Users Do with Image Collections . . . . . . . . . . . . 631
21.2 Basic Technologies from Information Retrieval . . . . . . . . . . . . . 632
21.2.1 Word Counts . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
21.2.2 Smoothing Word Counts . . . . . . . . . . . . . . . . . . . . . 633
21.2.3 Approximate Nearest Neighbors and Hashing . . . . . . . . . 634
21.2.4 Ranking Documents . . . . . . . . . . . . . . . . . . . . . . . 638
21.3 Images as Documents . . . . . . . . . . . . . . . . . . . . . . . . . . 639
21.3.1 Matching Without Quantization . . . . . . . . . . . . . . . . 640
21.3.2 Ranking Image Search Results . . . . . . . . . . . . . . . . . 641
21.3.3 Browsing and Layout . . . . . . . . . . . . . . . . . . . . . . 643
21.3.4 Laying Out Images for Browsing . . . . . . . . . . . . . . . . 644
21.4 Predicting Annotations for Pictures . . . . . . . . . . . . . . . . . . 645
21.4.1 Annotations from Nearby Words . . . . . . . . . . . . . . . . 646
21.4.2 Annotations from the Whole Image . . . . . . . . . . . . . . 646
21.4.3 Predicting Correlated Words with Classifiers . . . . . . . . . 648
21.4.4 Names and Faces . . . . . . . . . . . . . . . . . . . . . . . . 649
21.4.5 Generating Tags with Segments . . . . . . . . . . . . . . . . . 651
21.5 The State of the Art of Word Prediction . . . . . . . . . . . . . . . . 654
21.5.1 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655
21.5.2 Comparing Methods . . . . . . . . . . . . . . . . . . . . . . . 655
21.5.3 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 656
21.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659
VII BACKGROUND MATERIAL 661
22 Optimization Techniques 663
22.1 Linear Least-Squares Methods . . . . . . . . . . . . . . . . . . . . . . 663
22.1.1 Normal Equations and the Pseudoinverse . . . . . . . . . . . 664
22.1.2 Homogeneous Systems and Eigenvalue Problems . . . . . . . 665
22.1.3 Generalized Eigenvalues Problems . . . . . . . . . . . . . . . 666
22.1.4 An Example: Fitting a Line to Points in a Plane . . . . . . . 666
22.1.5 Singular Value Decomposition . . . . . . . . . . . . . . . . . . 667
22.2 Nonlinear Least-Squares Methods . . . . . . . . . . . . . . . . . . . . 669
22.2.1 Newton’s Method: Square Systems of Nonlinear Equations. . 670
22.2.2 Newton’s Method for Overconstrained Systems . . . . . . . . 670
22.2.3 The Gauss—Newton and Levenberg—Marquardt Algorithms . 671
22.3 Sparse Coding and Dictionary Learning . . . . . . . . . . . . . . . . 672
22.3.1 Sparse Coding . . . . . . . . . . . . . . . . . . . . . . . . . . 672
22.3.2 Dictionary Learning . . . . . . . . . . . . . . . . . . . . . . . 673
22.3.3 Supervised Dictionary Learning . . . . . . . . . . . . . . . . . 675
22.4 Min-Cut/Max-Flow Problems and Combinatorial Optimization . . . 675
22.4.1 Min-Cut Problems . . . . . . . . . . . . . . . . . . . . . . . . 676
22.4.2 Quadratic Pseudo-Boolean Functions . . . . . . . . . . . . . . 677
22.4.3 Generalization to Integer Variables . . . . . . . . . . . . . . . 679
22.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682
Bibliography 684
Index 737
List of Algorithms 760
1 Geometric Camera Models 3
1.1 Image Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Pinhole Perspective . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Weak Perspective . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Cameras with Lenses . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.4 The Human Eye . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Intrinsic and Extrinsic Parameters . . . . . . . . . . . . . . . . . . . 14
1.2.1 Rigid Transformations and Homogeneous Coordinates . . . . 14
1.2.2 Intrinsic Parameters . . . . . . . . . . . . . . . . . . . . . . . 16
1.2.3 Extrinsic Parameters . . . . . . . . . . . . . . . . . . . . . . . 18
1.2.4 Perspective Projection Matrices . . . . . . . . . . . . . . . . . 19
1.2.5 Weak-Perspective Projection Matrices . . . . . . . . . . . . . 20
1.3 Geometric Camera Calibration . . . . . . . . . . . . . . . . . . . . . 22
1.3.1 ALinear Approach to Camera Calibration . . . . . . . . . . . 23
1.3.2 ANonlinear Approach to Camera Calibration . . . . . . . . . 27
1.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Light and Shading 32
2.1 Modelling Pixel Brightness . . . . . . . . . . . . . . . . . . . . . . . 32
2.1.1 Reflection at Surfaces . . . . . . . . . . . . . . . . . . . . . . 33
2.1.2 Sources and Their Effects . . . . . . . . . . . . . . . . . . . . 34
2.1.3 The Lambertian+Specular Model . . . . . . . . . . . . . . . . 36
2.1.4 Area Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2 Inference from Shading . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.1 Radiometric Calibration and High Dynamic Range Images . . 38
2.2.2 The Shape of Specularities . . . . . . . . . . . . . . . . . . . 40
2.2.3 Inferring Lightness and Illumination . . . . . . . . . . . . . . 43
2.2.4 Photometric Stereo: Shape from Multiple Shaded Images . . 46
2.3 Modelling Interreflection . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.3.1 The Illumination at a Patch Due to an Area Source . . . . . 52
2.3.2 Radiosity and Exitance . . . . . . . . . . . . . . . . . . . . . 54
2.3.3 An Interreflection Model . . . . . . . . . . . . . . . . . . . . . 55
2.3.4 Qualitative Properties of Interreflections . . . . . . . . . . . . 56
2.4 Shape from One Shaded Image . . . . . . . . . . . . . . . . . . . . . 59
2.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3 Color 68
3.1 Human Color Perception . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.1.1 Color Matching . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.1.2 Color Receptors . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 The Physics of Color . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2.1 The Color of Light Sources . . . . . . . . . . . . . . . . . . . 73
3.2.2 The Color of Surfaces . . . . . . . . . . . . . . . . . . . . . . 76
3.3 Representing Color . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3.1 Linear Color Spaces . . . . . . . . . . . . . . . . . . . . . . . 77
3.3.2 Non-linear Color Spaces . . . . . . . . . . . . . . . . . . . . . 83
3.4 AModel of Image Color . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4.1 The Diffuse Term . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.4.2 The Specular Term . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5 Inference from Color . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5.1 Finding Specularities Using Color . . . . . . . . . . . . . . . 90
3.5.2 Shadow Removal Using Color . . . . . . . . . . . . . . . . . . 92
3.5.3 Color Constancy: Surface Color from Image Color . . . . . . 95
3.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
II EARLY VISION: JUST ONE IMAGE 105
4 Linear Filters 107
4.1 Linear Filters and Convolution . . . . . . . . . . . . . . . . . . . . . 107
4.1.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2 Shift Invariant Linear Systems . . . . . . . . . . . . . . . . . . . . . 112
4.2.1 Discrete Convolution . . . . . . . . . . . . . . . . . . . . . . . 113
4.2.2 Continuous Convolution . . . . . . . . . . . . . . . . . . . . . 115
4.2.3 Edge Effects in Discrete Convolutions . . . . . . . . . . . . . 118
4.3 Spatial Frequency and Fourier Transforms . . . . . . . . . . . . . . . 118
4.3.1 Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . 119
4.4 Sampling and Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.4.1 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.4.2 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.4.3 Smoothing and Resampling . . . . . . . . . . . . . . . . . . . 126
4.5 Filters as Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.5.1 Convolution as a Dot Product . . . . . . . . . . . . . . . . . 131
4.5.2 Changing Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.6 Technique: Normalized Correlation and Finding Patterns . . . . . . 132
4.6.1 Controlling the Television by Finding Hands by Normalized
Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
4.7 Technique: Scale and Image Pyramids . . . . . . . . . . . . . . . . . 134
4.7.1 The Gaussian Pyramid . . . . . . . . . . . . . . . . . . . . . 135
4.7.2 Applications of Scaled Representations . . . . . . . . . . . . . 136
4.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5 Local Image Features 141
5.1 Computing the Image Gradient . . . . . . . . . . . . . . . . . . . . . 141
5.1.1 Derivative of Gaussian Filters . . . . . . . . . . . . . . . . . . 142
5.2 Representing the Image Gradient . . . . . . . . . . . . . . . . . . . . 144
5.2.1 Gradient-Based Edge Detectors . . . . . . . . . . . . . . . . . 145
5.2.2 Orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.3 Finding Corners and Building Neighborhoods . . . . . . . . . . . . . 148
5.3.1 Finding Corners . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.3.2 Using Scale and Orientation to Build a Neighborhood . . . . 151
5.4 Describing Neighborhoods with SIFT and HOG Features . . . . . . 155
5.4.1 SIFT Features . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.4.2 HOG Features . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.5 Computing Local Features in Practice . . . . . . . . . . . . . . . . . 160
5.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6 Texture 164
6.1 Local Texture Representations Using Filters . . . . . . . . . . . . . . 166
6.1.1 Spots and Bars . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.1.2 From Filter Outputs to Texture Representation . . . . . . . . 168
6.1.3 Local Texture Representations in Practice . . . . . . . . . . . 170
6.2 Pooled Texture Representations by Discovering Textons . . . . . . . 171
6.2.1 Vector Quantization and Textons . . . . . . . . . . . . . . . . 172
6.2.2 K-means Clustering for Vector Quantization . . . . . . . . . . 172
6.3 Synthesizing Textures and Filling Holes in Images . . . . . . . . . . 176
6.3.1 Synthesis by Sampling Local Models . . . . . . . . . . . . . . 176
6.3.2 Filling in Holes in Images . . . . . . . . . . . . . . . . . . . . 179
6.4 Image Denoising . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.4.1 Non-local Means . . . . . . . . . . . . . . . . . . . . . . . . . 183
6.4.2 Block Matching 3D (BM3D) . . . . . . . . . . . . . . . . . . 183
6.4.3 Learned Sparse Coding . . . . . . . . . . . . . . . . . . . . . 184
6.4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.5 Shape from Texture . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.5.1 Shape from Texture for Planes . . . . . . . . . . . . . . . . . 187
6.5.2 Shape from Texture for Curved Surfaces . . . . . . . . . . . . 190
6.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
III EARLY VISION: MULTIPLE IMAGES 195
7 Stereopsis 197
7.1 Binocular Camera Geometry and the Epipolar Constraint . . . . . . 198
7.1.1 Epipolar Geometry . . . . . . . . . . . . . . . . . . . . . . . . 198
7.1.2 The Essential Matrix . . . . . . . . . . . . . . . . . . . . . . . 200
7.1.3 The Fundamental Matrix . . . . . . . . . . . . . . . . . . . . 201
7.2 Binocular Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 201
7.2.1 Image Rectification . . . . . . . . . . . . . . . . . . . . . . . . 202
7.3 Human Stereopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.4 Local Methods for Binocular Fusion . . . . . . . . . . . . . . . . . . 205
7.4.1 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.4.2 Multi-Scale Edge Matching . . . . . . . . . . . . . . . . . . . 207
7.5 Global Methods for Binocular Fusion . . . . . . . . . . . . . . . . . . 210
7.5.1 Ordering Constraints and Dynamic Programming . . . . . . . 210
7.5.2 Smoothness and Graphs . . . . . . . . . . . . . . . . . . . . . 211
7.6 Using More Cameras . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
7.7 Application: Robot Navigation . . . . . . . . . . . . . . . . . . . . . 215
7.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8 Structure from Motion 221
8.1 Internally Calibrated Perspective Cameras . . . . . . . . . . . . . . . 221
8.1.1 Natural Ambiguity of the Problem . . . . . . . . . . . . . . . 223
8.1.2 Euclidean Structure and Motion from Two Images . . . . . . 224
8.1.3 Euclidean Structure and Motion from Multiple Images . . . . 228
8.2 Uncalibrated Weak-Perspective Cameras . . . . . . . . . . . . . . . . 230
8.2.1 Natural Ambiguity of the Problem . . . . . . . . . . . . . . . 231
8.2.2 Affine Structure and Motion from Two Images . . . . . . . . 233
8.2.3 Affine Structure and Motion from Multiple Images . . . . . . 237
8.2.4 From Affine to Euclidean Shape . . . . . . . . . . . . . . . . 238
8.3 Uncalibrated Perspective Cameras . . . . . . . . . . . . . . . . . . . 240
8.3.1 Natural Ambiguity of the Problem . . . . . . . . . . . . . . . 241
8.3.2 Projective Structure and Motion from Two Images . . . . . . 242
8.3.3 Projective Structure and Motion from Multiple Images . . . . 244
8.3.4 From Projective to Euclidean Shape . . . . . . . . . . . . . . 246
8.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
IV MID-LEVEL VISION 253
9 Segmentation by Clustering 255
9.1 Human Vision: Grouping and Gestalt . . . . . . . . . . . . . . . . . 256
9.2 Important Applications . . . . . . . . . . . . . . . . . . . . . . . . . 261
9.2.1 Background Subtraction . . . . . . . . . . . . . . . . . . . . . 261
9.2.2 Shot Boundary Detection . . . . . . . . . . . . . . . . . . . . 264
9.2.3 Interactive Segmentation . . . . . . . . . . . . . . . . . . . . 265
9.2.4 Forming Image Regions . . . . . . . . . . . . . . . . . . . . . 266
9.3 Image Segmentation by Clustering Pixels . . . . . . . . . . . . . . . 268
9.3.1 Basic Clustering Methods . . . . . . . . . . . . . . . . . . . . 269
9.3.2 The Watershed Algorithm . . . . . . . . . . . . . . . . . . . . 271
9.3.3 Segmentation Using K-means . . . . . . . . . . . . . . . . . . 272
9.3.4 Mean Shift: Finding Local Modes in Data . . . . . . . . . . . 273
9.3.5 Clustering and Segmentation with Mean Shift . . . . . . . . . 275
9.4 Segmentation, Clustering, and Graphs . . . . . . . . . . . . . . . . . 277
9.4.1 Terminology and Facts for Graphs . . . . . . . . . . . . . . . 277
9.4.2 Agglomerative Clustering with a Graph . . . . . . . . . . . . 279
9.4.3 Divisive Clustering with a Graph . . . . . . . . . . . . . . . . 281
9.4.4 Normalized Cuts . . . . . . . . . . . . . . . . . . . . . . . . . 284
9.5 Image Segmentation in Practice . . . . . . . . . . . . . . . . . . . . . 285
9.5.1 Evaluating Segmenters . . . . . . . . . . . . . . . . . . . . . . 286
9.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
10 Grouping and Model Fitting 290
10.1 The Hough Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 290
10.1.1 Fitting Lines with the Hough Transform . . . . . . . . . . . . 290
10.1.2 Using the Hough Transform . . . . . . . . . . . . . . . . . . . 292
10.2 Fitting Lines and Planes . . . . . . . . . . . . . . . . . . . . . . . . . 293
10.2.1 Fitting a Single Line . . . . . . . . . . . . . . . . . . . . . . . 294
10.2.2 Fitting Planes . . . . . . . . . . . . . . . . . . . . . . . . . . 295
10.2.3 Fitting Multiple Lines . . . . . . . . . . . . . . . . . . . . . . 296
10.3 Fitting Curved Structures . . . . . . . . . . . . . . . . . . . . . . . . 297
10.4 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
10.4.1 M-Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
10.4.2 RANSAC: Searching for Good Points . . . . . . . . . . . . . 302
10.5 Fitting Using Probabilistic Models . . . . . . . . . . . . . . . . . . . 306
10.5.1 Missing Data Problems . . . . . . . . . . . . . . . . . . . . . 307
10.5.2 Mixture Models and Hidden Variables . . . . . . . . . . . . . 309
10.5.3 The EM Algorithm for Mixture Models . . . . . . . . . . . . 310
10.5.4 Difficulties with the EM Algorithm . . . . . . . . . . . . . . . 312
10.6 Motion Segmentation by Parameter Estimation . . . . . . . . . . . . 313
10.6.1 Optical Flow and Motion . . . . . . . . . . . . . . . . . . . . 315
10.6.2 Flow Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
10.6.3 Motion Segmentation with Layers . . . . . . . . . . . . . . . 317
10.7 Model Selection: Which Model Is the Best Fit? . . . . . . . . . . . . 319
10.7.1 Model Selection Using Cross-Validation . . . . . . . . . . . . 322
10.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
11 Tracking 326
11.1 Simple Tracking Strategies . . . . . . . . . . . . . . . . . . . . . . . . 327
11.1.1 Tracking by Detection . . . . . . . . . . . . . . . . . . . . . . 327
11.1.2 Tracking Translations by Matching . . . . . . . . . . . . . . . 330
11.1.3 Using Affine Transformations to Confirm a Match . . . . . . 332
11.2 Tracking Using Matching . . . . . . . . . . . . . . . . . . . . . . . . 334
11.2.1 Matching Summary Representations . . . . . . . . . . . . . . 335
11.2.2 Tracking Using Flow . . . . . . . . . . . . . . . . . . . . . . . 337
11.3 Tracking Linear Dynamical Models with Kalman Filters . . . . . . . 339
11.3.1 Linear Measurements and Linear Dynamics . . . . . . . . . . 340
11.3.2 The Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . 344
11.3.3 Forward-backward Smoothing . . . . . . . . . . . . . . . . . . 345
11.4 Data Association . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
11.4.1 Linking Kalman Filters with Detection Methods . . . . . . . 349
11.4.2 Key Methods of Data Association . . . . . . . . . . . . . . . 350
11.5 Particle Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
11.5.1 Sampled Representations of Probability Distributions . . . . 351
11.5.2 The Simplest Particle Filter . . . . . . . . . . . . . . . . . . . 355
11.5.3 The Tracking Algorithm . . . . . . . . . . . . . . . . . . . . . 356
11.5.4 A Workable Particle Filter . . . . . . . . . . . . . . . . . . . . 358
11.5.5 Practical Issues in Particle Filters . . . . . . . . . . . . . . . 360
11.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
V HIGH-LEVEL VISION 365
12 Registration 367
12.1 Registering Rigid Objects . . . . . . . . . . . . . . . . . . . . . . . . 368
12.1.1 Iterated Closest Points . . . . . . . . . . . . . . . . . . . . . . 368
12.1.2 Searching for Transformations via Correspondences . . . . . . 369
12.1.3 Application: Building Image Mosaics . . . . . . . . . . . . . . 370
12.2 Model-based Vision: Registering Rigid Objects with Projection . . . 375
12.2.1 Verification: Comparing Transformed and Rendered Source
to Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
12.3 Registering Deformable Objects . . . . . . . . . . . . . . . . . . . . . 378
12.3.1 Deforming Texture with Active Appearance Models . . . . . 378
12.3.2 Active Appearance Models in Practice . . . . . . . . . . . . . 381
12.3.3 Application: Registration in Medical Imaging Systems . . . . 383
12.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388
13 Smooth Surfaces and Their Outlines 391
13.1 Elements of Differential Geometry . . . . . . . . . . . . . . . . . . . 393
13.1.1 Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
13.1.2 Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
13.2 Contour Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
13.2.1 The Occluding Contour and the Image Contour . . . . . . . . 402
13.2.2 The Cusps and Inflections of the Image Contour . . . . . . . 403
13.2.3 Koenderink’s Theorem . . . . . . . . . . . . . . . . . . . . . . 404
13.3 Visual Events: More Differential Geometry . . . . . . . . . . . . . . 407
13.3.1 The Geometry of the Gauss Map . . . . . . . . . . . . . . . . 407
13.3.2 Asymptotic Curves . . . . . . . . . . . . . . . . . . . . . . . . 409
13.3.3 The Asymptotic Spherical Map . . . . . . . . . . . . . . . . . 410
13.3.4 Local Visual Events . . . . . . . . . . . . . . . . . . . . . . . 412
13.3.5 The Bitangent Ray Manifold . . . . . . . . . . . . . . . . . . 413
13.3.6 Multilocal Visual Events . . . . . . . . . . . . . . . . . . . . . 414
13.3.7 The Aspect Graph . . . . . . . . . . . . . . . . . . . . . . . . 416
13.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
14 Range Data 422
14.1 Active Range Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . 422
14.2 Range Data Segmentation . . . . . . . . . . . . . . . . . . . . . . . . 424
14.2.1 Elements of Analytical Differential Geometry . . . . . . . . . 424
14.2.2 Finding Step and Roof Edges in Range Images . . . . . . . . 426
14.2.3 Segmenting Range Images into Planar Regions . . . . . . . . 431
14.3 Range Image Registration and Model Acquisition . . . . . . . . . . . 432
14.3.1 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
14.3.2 Registering Range Images . . . . . . . . . . . . . . . . . . . . 434
14.3.3 Fusing Multiple Range Images . . . . . . . . . . . . . . . . . 436
14.4 Object Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
14.4.1 Matching Using Interpretation Trees . . . . . . . . . . . . . . 438
14.4.2 Matching Free-Form Surfaces Using Spin Images . . . . . . . 441
14.5 Kinect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
14.5.1 Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
14.5.2 Technique: Decision Trees and Random Forests . . . . . . . . 448
14.5.3 Labeling Pixels . . . . . . . . . . . . . . . . . . . . . . . . . . 450
14.5.4 Computing Joint Positions . . . . . . . . . . . . . . . . . . . 453
14.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
15 Learning to Classify 457
15.1 Classification, Error, and Loss . . . . . . . . . . . . . . . . . . . . . . 457
15.1.1 Using Loss to Determine Decisions . . . . . . . . . . . . . . . 457
15.1.2 Training Error, Test Error, and Overfitting . . . . . . . . . . 459
15.1.3 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . 460
15.1.4 Error Rate and Cross-Validation . . . . . . . . . . . . . . . . 463
15.1.5 Receiver Operating Curves . . . . . . . . . . . . . . . . . . . 465
15.2 Major Classification Strategies . . . . . . . . . . . . . . . . . . . . . 467
15.2.1 Example: Mahalanobis Distance . . . . . . . . . . . . . . . . 467
15.2.2 Example: Class-Conditional Histograms and Naive Bayes . . 468
15.2.3 Example: Classification Using Nearest Neighbors . . . . . . . 469
15.2.4 Example: The Linear Support Vector Machine . . . . . . . . 470
15.2.5 Example: Kernel Machines . . . . . . . . . . . . . . . . . . . 473
15.2.6 Example: Boosting and Adaboost . . . . . . . . . . . . . . . 475
15.3 Practical Methods for Building Classifiers . . . . . . . . . . . . . . . 475
15.3.1 Manipulating Training Data to Improve Performance . . . . . 477
15.3.2 Building Multi-Class Classifiers Out of Binary Classifiers . . 479
15.3.3 Solving for SVMS and Kernel Machines . . . . . . . . . . . . 480
15.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
16 Classifying Images 482
16.1 Building Good Image Features . . . . . . . . . . . . . . . . . . . . . 482
16.1.1 Example Applications . . . . . . . . . . . . . . . . . . . . . . 482
16.1.2 Encoding Layout with GIST Features . . . . . . . . . . . . . 485
16.1.3 Summarizing Images with Visual Words . . . . . . . . . . . . 487
16.1.4 The Spatial Pyramid Kernel . . . . . . . . . . . . . . . . . . . 489
16.1.5 Dimension Reduction with Principal Components . . . . . . . 493
16.1.6 Dimension Reduction with Canonical Variates . . . . . . . . 494
16.1.7 Example Application: Identifying Explicit Images . . . . . . 498
16.1.8 Example Application: Classifying Materials . . . . . . . . . . 502
16.1.9 Example Application: Classifying Scenes . . . . . . . . . . . . 502
16.2 Classifying Images of Single Objects . . . . . . . . . . . . . . . . . . 504
16.2.1 Image Classification Strategies . . . . . . . . . . . . . . . . . 505
16.2.2 Evaluating Image Classification Systems . . . . . . . . . . . . 505
16.2.3 Fixed Sets of Classes . . . . . . . . . . . . . . . . . . . . . . . 508
16.2.4 Large Numbers of Classes . . . . . . . . . . . . . . . . . . . . 509
16.2.5 Flowers, Leaves, and Birds: Some Specialized Problems . . . 511
16.3 Image Classification in Practice . . . . . . . . . . . . . . . . . . . . . 512
16.3.1 Codes for Image Features . . . . . . . . . . . . . . . . . . . . 513
16.3.2 Image Classification Datasets . . . . . . . . . . . . . . . . . . 513
16.3.3 Dataset Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . 515
16.3.4 Crowdsourcing Dataset Collection . . . . . . . . . . . . . . . 515
16.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517
17 Detecting Objects in Images 519
17.1 The Sliding Window Method . . . . . . . . . . . . . . . . . . . . . . 519
17.1.1 Face Detection . . . . . . . . . . . . . . . . . . . . . . . . . . 520
17.1.2 Detecting Humans . . . . . . . . . . . . . . . . . . . . . . . . 525
17.1.3 Detecting Boundaries . . . . . . . . . . . . . . . . . . . . . . 527
17.2 Detecting Deformable Objects . . . . . . . . . . . . . . . . . . . . . . 530
17.3 The State of the Art of Object Detection . . . . . . . . . . . . . . . 535
17.3.1 Datasets and Resources . . . . . . . . . . . . . . . . . . . . . 538
17.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
18 Topics in Object Recognition 540
18.1 What Should Object Recognition Do? . . . . . . . . . . . . . . . . . 540
18.1.1 What Should an Object Recognition System Do? . . . . . . . 540
18.1.2 Current Strategies for Object Recognition . . . . . . . . . . . 542
18.1.3 What Is Categorization? . . . . . . . . . . . . . . . . . . . . . 542
18.1.4 Selection: What Should Be Described? . . . . . . . . . . . . . 544
18.2 Feature Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544
18.2.1 Improving Current Image Features . . . . . . . . . . . . . . . 544
18.2.2 Other Kinds of Image Feature . . . . . . . . . . . . . . . . . . 546
18.3 Geometric Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
18.4 Semantic Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549
18.4.1 Attributes and the Unfamiliar . . . . . . . . . . . . . . . . . . 550
18.4.2 Parts, Poselets and Consistency . . . . . . . . . . . . . . . . . 551
18.4.3 Chunks of Meaning . . . . . . . . . . . . . . . . . . . . . . . . 554
VI APPLICATIONS AND TOPICS 557
19 Image-Based Modeling and Rendering 559
19.1 Visual Hulls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559
19.1.1 Main Elements of the Visual Hull Model . . . . . . . . . . . . 561
19.1.2 Tracing Intersection Curves . . . . . . . . . . . . . . . . . . . 563
19.1.3 Clipping Intersection Curves . . . . . . . . . . . . . . . . . . 566
19.1.4 Triangulating Cone Strips . . . . . . . . . . . . . . . . . . . . 567
19.1.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 568
19.1.6 Going Further: Carved Visual Hulls . . . . . . . . . . . . . . 572
19.2 Patch-Based Multi-View Stereopsis . . . . . . . . . . . . . . . . . . . 573
19.2.1 Main Elements of the PMVS Model . . . . . . . . . . . . . . 575
19.2.2 Initial Feature Matching . . . . . . . . . . . . . . . . . . . . . 578
19.2.3 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
19.2.4 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
19.2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 581
19.3 The Light Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584
19.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587
20 Looking at People 590
20.1 HMM’s, Dynamic Programming, and Tree-Structured Models . . . . 590
20.1.1 Hidden Markov Models . . . . . . . . . . . . . . . . . . . . . 590
20.1.2 Inference for an HMM . . . . . . . . . . . . . . . . . . . . . . 592
20.1.3 Fitting an HMM with EM . . . . . . . . . . . . . . . . . . . . 597
20.1.4 Tree-Structured Energy Models . . . . . . . . . . . . . . . . . 600
20.2 Parsing People in Images . . . . . . . . . . . . . . . . . . . . . . . . 602
20.2.1 Parsing with Pictorial Structure Models . . . . . . . . . . . . 602
20.2.2 Estimating the Appearance of Clothing . . . . . . . . . . . . 604
20.3 Tracking People . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
20.3.1 Why Human Tracking Is Hard . . . . . . . . . . . . . . . . . 606
20.3.2 Kinematic Tracking by Appearance . . . . . . . . . . . . . . . 608
20.3.3 Kinematic Human Tracking Using Templates . . . . . . . . . 609
20.4 3D from 2D: Lifting . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
20.4.1 Reconstruction in an Orthographic View . . . . . . . . . . . . 611
20.4.2 Exploiting Appearance for Unambiguous Reconstructions . . 613
20.4.3 Exploiting Motion for Unambiguous Reconstructions . . . . . 615
20.5 Activity Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
20.5.1 Background: Human Motion Data . . . . . . . . . . . . . . . 617
20.5.2 Body Configuration and Activity Recognition . . . . . . . . . 621
20.5.3 Recognizing Human Activities with Appearance Features . . 622
20.5.4 Recognizing Human Activities with Compositional Models . . 624
20.6 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
20.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626
21 Image Search and Retrieval 627
21.1 The Application Context . . . . . . . . . . . . . . . . . . . . . . . . . 627
21.1.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 628
21.1.2 User Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629
21.1.3 Types of Image Query . . . . . . . . . . . . . . . . . . . . . . 630
21.1.4 What Users Do with Image Collections . . . . . . . . . . . . 631
21.2 Basic Technologies from Information Retrieval . . . . . . . . . . . . . 632
21.2.1 Word Counts . . . . . . . . . . . . . . . . . . . . . . . . . . . 632
21.2.2 Smoothing Word Counts . . . . . . . . . . . . . . . . . . . . . 633
21.2.3 Approximate Nearest Neighbors and Hashing . . . . . . . . . 634
21.2.4 Ranking Documents . . . . . . . . . . . . . . . . . . . . . . . 638
21.3 Images as Documents . . . . . . . . . . . . . . . . . . . . . . . . . . 639
21.3.1 Matching Without Quantization . . . . . . . . . . . . . . . . 640
21.3.2 Ranking Image Search Results . . . . . . . . . . . . . . . . . 641
21.3.3 Browsing and Layout . . . . . . . . . . . . . . . . . . . . . . 643
21.3.4 Laying Out Images for Browsing . . . . . . . . . . . . . . . . 644
21.4 Predicting Annotations for Pictures . . . . . . . . . . . . . . . . . . 645
21.4.1 Annotations from Nearby Words . . . . . . . . . . . . . . . . 646
21.4.2 Annotations from the Whole Image . . . . . . . . . . . . . . 646
21.4.3 Predicting Correlated Words with Classifiers . . . . . . . . . 648
21.4.4 Names and Faces . . . . . . . . . . . . . . . . . . . . . . . . 649
21.4.5 Generating Tags with Segments . . . . . . . . . . . . . . . . . 651
21.5 The State of the Art of Word Prediction . . . . . . . . . . . . . . . . 654
21.5.1 Resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655
21.5.2 Comparing Methods . . . . . . . . . . . . . . . . . . . . . . . 655
21.5.3 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 656
21.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659
VII BACKGROUND MATERIAL 661
22 Optimization Techniques 663
22.1 Linear Least-Squares Methods . . . . . . . . . . . . . . . . . . . . . . 663
22.1.1 Normal Equations and the Pseudoinverse . . . . . . . . . . . 664
22.1.2 Homogeneous Systems and Eigenvalue Problems . . . . . . . 665
22.1.3 Generalized Eigenvalues Problems . . . . . . . . . . . . . . . 666
22.1.4 An Example: Fitting a Line to Points in a Plane . . . . . . . 666
22.1.5 Singular Value Decomposition . . . . . . . . . . . . . . . . . . 667
22.2 Nonlinear Least-Squares Methods . . . . . . . . . . . . . . . . . . . . 669
22.2.1 Newton’s Method: Square Systems of Nonlinear Equations. . 670
22.2.2 Newton’s Method for Overconstrained Systems . . . . . . . . 670
22.2.3 The Gauss—Newton and Levenberg—Marquardt Algorithms . 671
22.3 Sparse Coding and Dictionary Learning . . . . . . . . . . . . . . . . 672
22.3.1 Sparse Coding . . . . . . . . . . . . . . . . . . . . . . . . . . 672
22.3.2 Dictionary Learning . . . . . . . . . . . . . . . . . . . . . . . 673
22.3.3 Supervised Dictionary Learning . . . . . . . . . . . . . . . . . 675
22.4 Min-Cut/Max-Flow Problems and Combinatorial Optimization . . . 675
22.4.1 Min-Cut Problems . . . . . . . . . . . . . . . . . . . . . . . . 676
22.4.2 Quadratic Pseudo-Boolean Functions . . . . . . . . . . . . . . 677
22.4.3 Generalization to Integer Variables . . . . . . . . . . . . . . . 679
22.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682
Bibliography 684
Index 737
List of Algorithms 760
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