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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