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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