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Tables of Contents for The Geometry of Multiple Images
Chapter/Section Title
Page #
Page Count
Preface
xiii
 
Notation
xix
 
A tour into multiple image geometry
1
62
Multiple image geometry and three-dimensional vision
2
2
Projective geometry
4
7
2-D and 3-D
11
3
Calibrated and uncalibrated capabilities
14
2
The plane-to-image homography as a projective transformation
16
2
Affine description of the projection
18
1
Structure and motion
19
2
The homography between two images of a plane
21
1
Stationary cameras
22
1
The epipolar constraint between corresponding points
23
1
The Fundamental matrix
24
2
Computing the Fundamental matrix
26
2
Planar homographies and the Fundamental matrix
28
3
A stratified approach to reconstruction
31
1
Projective reconstruction
31
4
Reconstruction is not always necessary
35
1
Affine reconstruction
36
3
Euclidean reconstruction
39
5
The geometry of three images
44
1
The Trifocal tensor
45
4
Computing the Trifocal tensor
49
2
Reconstruction from N images
51
3
Self-calibration of a moving camera using the absolute conic
54
2
From affine to Euclidean
56
1
From projective to Euclidean
57
2
References and further reading
59
4
Projective, affine and Euclidean geometries
63
64
Motivations for the approach and overview
65
4
Projective spaces: basic definitions
66
1
Projective geometry
67
1
Affine geometry
68
1
Euclidean geometry
69
1
Affine spaces and affine geometry
69
4
Definition of an affine space and an affine basis
69
1
Affine morphisms, affine group
70
2
Change of affine basis
72
1
Affine subspaces, parallelism
73
1
Euclidean spaces and Euclidean geometry
73
5
Euclidean spaces, rigid displacements, similarities
74
1
The isotropic cone
75
3
Projective spaces and projective geometry
78
15
Basic definitions
78
1
Projective bases, projective morphisms, homographies
79
9
Projective subspaces
88
5
Affine and projective geometry
93
8
Projective completion of an affine space
93
2
Affine and projective bases
95
3
Affine subspace Xn of a projective space Pn
98
1
Relation between PLG(X) and AG(X)
99
2
More projective geometry
101
15
Cross-ratios
101
5
Duality
106
6
Conics, quadrics and their duals
112
4
Projective, affine and Euclidean geometry
116
6
Relation between PLG(X) and S(X)
117
2
Angles as cross-ratios
119
3
Summary
122
2
References and further reading
124
3
Exterior and double or Grassmann-Cayley algebras
127
46
Definition of the exterior algebras of the join
129
8
First definitions: The join operator
129
4
Properties of the join operator
133
4
Plucker relations
137
3
Derivation of the Plucker relations
137
2
The example of 3D lines: II
139
1
The example of 3D planes: II
140
1
The meet operator: The Grassmann-Cayley algebra
140
6
Definition of the meet
140
4
Some planar examples
144
1
Some 3D examples
145
1
Duality and the Hodge operator
146
22
Duality
147
6
The example of 3D lines: III
153
2
The Hodge operator
155
3
The example of 2D lines: II
158
4
The example of 3D planes: III
162
2
The example of 3D lines: IV
164
4
Summary and conclusion
168
1
References and further reading
169
4
One camera
173
74
The Projective model
175
21
The pinhole camera
176
4
The projection matrix
180
5
The inverse projection matrix
185
6
Viewing a plane in space: The single view homography
191
3
Projection of a line
194
2
The affine model: The case of perspective projection
196
11
The projection matrix
197
3
The inverse perspective projection matrix
200
2
Vanishing points and lines
202
5
The Euclidean model: The case of perspective projection
207
9
Intrinsic and Extrinsic parameters
207
4
The absolute conic and the intrinsic parameters
211
5
The affine and Euclidean models: The case of parallel projection
216
16
Orthographic, weak perspective, para-perspective projections
216
6
The general model: The affine projection matrix
222
5
Euclidean interpretation of the parallel projection
227
5
Departures from the pinhole model: Nonlinear distortion
232
4
Nonlinear distortion of the pinhole model
232
2
Distortion correction within a projective model
234
2
Calibration techniques
236
4
Coordinates-based methods
237
2
Using single view homographies
239
1
Summary and discussion
240
2
References and further reading
242
5
Two views: The Fundamental matrix
247
68
Configurations with no parallax
250
8
The correspondence between the two images of a plane
251
4
Identical optical centers: Application to mosaicing
255
3
The Fundamental matrix
258
20
Geometry: The epipolar constraint
259
5
Algebra: The bilinear constraint
264
2
The epipolar homography
266
4
Relations between the Fundamental matrix and planar homographies
270
5
The S-matrix and the intrinsic planes
275
3
Perspective projection
278
14
The affine case
278
2
The Euclidean case: Epipolar geometry
280
2
The Essential matrix
282
4
Structure and motion parameters for a plane
286
1
Some particular cases
287
5
Parallel projection
292
8
Affine epipolar geometry
292
2
Cyclopean and affine viewing
294
3
The Euclidean case
297
3
Ambiguity and the critical surface
300
9
The critical surfaces
301
2
The quadratic transformation between two ambiguous images
303
4
The planar case
307
2
Summary
309
1
References and further reading
310
5
Estimating the Fundamental matrix
315
44
Linear methods
317
4
An important normalization procedure
317
1
The basic algorithm
318
2
Enforcing the rank constraint by approximation
320
1
Enforcing the rank constraint by parameterization
321
4
Parameterizing by the epipolar homography
322
2
Computing the Jacobian of the parameterization
324
1
Choosing the best map
325
1
The distance minimization approach
325
4
The distance to epipolar lines
326
1
The Gradient criterion and an interpretation as a distance
327
1
The ``optimal'' method
328
1
Robust Methods
329
6
M-Estimators
330
2
Monte-Carlo methods
332
3
An example of Fundamental matrix estimation with comparison
335
6
Computing the uncertainty of the Fundamental matrix
341
5
The case of an explicit function
341
1
The case of an implicit function
342
1
The error function is a sum of squares
343
2
The hyper-ellipsoid of uncertainty
345
1
The case of the Fundamental matrix
345
1
Some applications of the computation of ΛF
346
7
Uncertainty of the epipoles
346
3
Epipolar Band
349
4
References and further reading
353
6
Stratification of binocular stereo and applications
359
50
Canonical representations of two views
361
1
Projective stratum
362
18
The projection matrices
362
3
Projective reconstruction
365
2
Dealing with real correspondences
367
1
Planar parallax
368
4
Image rectification
372
4
Application to obstacle detection
376
2
Application to image based rendering from two views
378
2
Affine stratum
380
13
The projection matrices
381
4
Affine reconstruction
385
1
Affine parallax
386
1
Estimating H∞
387
4
Application to affine measurements
391
2
Euclidean stratum
393
10
The projection matrices
393
2
Euclidean reconstruction
395
1
Euclidean parallax
396
1
Recovery of the intrinsic parameters
397
4
Using knowledge about the world: Point coordinates
401
2
Summary
403
2
References and further reading
405
4
Three views: The trifocal geometry
409
60
The geometry of three views from the viewpoint of two
411
8
Transfer
412
4
Trifocal geometry
416
3
Optical centers aligned
419
1
The Trifocal tensors
419
20
Geometric derivation of the Trifocal tensors
419
6
The six intrinsic planar morphisms
425
3
Changing the reference view
428
1
Properties of the Trifocal matrices Gni
429
6
Relation with planar homographies
435
4
Prediction revisited
439
2
Prediction in the Trifocal plane
439
1
Optical centers aligned
440
1
Constraints satisfied by the tensors
441
5
Rank and epipolar constraints
442
1
The 27 axes constraints
442
3
The extended rank constraints
445
1
Constraints that characterize the Trifocal tensor
446
8
The Affine case
454
1
The Euclidean case
454
5
Computing the directions of the translation vectors and the rotation matrices
454
4
Computing the ratio of the norms of the translation vectors
458
1
Affine cameras
459
2
Projective setting
459
1
Euclidean setting
460
1
Summary and Conclusion
461
3
Perspective projection matrices, Fundamental matrices and Trifocal tensors
462
1
Transfer
463
1
References and further reading
464
5
Determining the Trifocal tensor
469
32
The linear algorithm
471
12
Normalization again!
471
1
The basic algorithm
472
4
Discussion
476
1
Some results
477
6
Parameterizing the Trifocal tensor
483
8
The parameterization by projection matrices
485
1
The six-point parameterization
486
2
The tensorial parameterization
488
1
The minimal one-to-one parameterization
489
2
Imposing the constraints
491
4
Projecting by parameterizing
492
1
Projecting using the algebraic constraints
492
1
Some results
493
2
A note about the ``change of view'' operation
495
1
Nonlinear methods
495
5
The nonlinear scheme
496
1
A note about the geometric criterion
497
2
Results
499
1
References and further reading
500
1
Stratification of n ≥ 3 views and applications
501
38
Canonical representations of n views
503
1
Projective stratum
503
22
Beyond the Fundamental matrix and the Trifocal tensor
504
2
The projection matrices: Three views
506
6
The projection matrices: An arbitrary number of views
512
13
Affine and Euclidean strata
525
4
Stereo rigs
529
7
Affine calibration
529
4
Euclidean calibration
533
3
References and further reading
536
3
Self-calibration of a moving camera: From affine or projective calibration to full Euclidean calibration
539
54
From affine to Euclidean
542
7
Theoretical analysis
542
4
Practical computation
546
1
A numerical example
546
2
Application to panoramic mosaicing
548
1
From projective to Euclidean
549
11
The rigidity constraints: Algebraic formulations using the Essential matrix
550
3
The Kruppa equations: A geometric interpretation of the rigidity constraint
553
4
Using two rigid displacements of a camera: A method for self-calibration
557
3
Computing the intrinsic parameters using the Kruppa equations
560
5
Recovering the focal lengths for two views
560
2
Solving the Kruppa equations for three views
562
1
Nonlinear optimization to accumulate the Kruppa equations for n > 3 views: The ``Kruppa'' method
563
2
Computing the Euclidean canonical form
565
7
The affine camera case
565
2
The general formulation in the perspective case
567
5
Computing all the Euclidean parameters
572
10
Simultaneous computation of motion and intrinsic parameters: The ``Epipolar/Motion'' method
573
4
Global optimization on structure, motion, and calibration parameters
577
1
More applications
578
4
Degeneracies in self-calibration
582
6
The spurious absolute conics lie in the real plane at infinity
583
4
Degeneracies of the Kruppa equations
587
1
Discussion
588
1
References and further reading
589
4
A Appendix
593
4
A.1 Solution of minx||Ax||2 subject to ||x||2 = 1
593
1
A.2 A note about rank-2 matrices
594
3
References
597
38
Index
635