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A tour into multiple image geometry
1.1 Multiple image geometry and three-dimensional vision
1.2 Projective geometry
1.3 2-D and 3-D
1.4 Calibrated and uncalibrated capabilities
1.5 The plane-to-image homography as a projective transformation
1.6 Affine description of the projection
1.7 Structure and motion
1.8 The homography between two images of a plane
1.9 Stationary cameras
1.10 The epipolar constraint between corresponding points
1.11 The Fundamental matrix
1.12 Computing the Fundamental matrix
1.13 Planar homographies and the Fundamental metrix
1.14 A stratified approach to reconstruction
1.15 Projective reconstruction
1.16 Reconstruction is not always necessary
1.17 Affine reconstruction
1.18 Euclidean reconstruction
1.19 The geometry of three images
1.20 The Trifocal tensor
1.21 Computing the Trifocal tensor
1.22 Reconstruction from N images
1.23 Self-calibration of a moving camera using the absolute conic
1.24 From affine to Euclidean
1.25 From projective to Euclidean
1.26 References and further reading
Projective, afflne and Euclidean geometries
2.1 Motivations for the approach and overview
2.1.1 Projective spaces: basic definitions
2.1.2 Projective geometry
2.1.3 Affine geometry
2.1.4 Euclidean geometry
2.2 Affine spaces and affine geometry
2.2.1 Definition of an affine space and an affine basis
2.2.2 Affine morphisms, affine group
2.2.3 Change of affine basis
2.2.4 Affine subspaces, parallelism
2.3 Euclidean spaces and Euclidean geometry
2.3.1 Euclidean spaces, rigid displacements, similarities
2.3.2 The isotropic cone
2.4 Projective spaces and projective geometry
2.4.1 Basic definitions
2.4.2 Projective bases, projective morphisms, homographies
2.4.3 Projective subspaces
2.5 Affine and projective geometry
2.5.1 Projective completion of an affine space
2.5.2 Affine and projective bases
2.5.3 Affine subspace Xn of a projective space Pn
2.5.4 Relation between PLG(X) and A4(X)
2.6 More projective geometry
2.6.1 Cross-ratios
2.6.2 Duality
2.6.3 Conics, quadrics and their duals
2.7 Projective, affine and Euclidean geometry
2.7.1 Relation between PCG(X) and S(X)
2.7.2 Angles as cross-ratios
2.8 Summary
2.9 References and further reading
Exterior and double or Grassmann-Cayley algebras
3.1 Definition of the exterior algebra of the join
3.1.1 First definitions: The join operator
3.1.2 Properties of the join operator
3.2 Pliicker relations
3.2.1 Derivation of the Pluicker relations
3.2.2 The example of 3D lines: II
3.2.3 The example of 3D planes: II
3.3 The meet operator: The Grassmann-Cayley algebra
3.3.1 Definition of the meet
3.3.2 Some planar examples
3.3.3 Some 3D examples
3.4 Duality and the Hodge operator
3.4.1 Duality
3.4.2 The example of 3D lines: III
3.4.3 The Hodge operator
3.4.4 The example of 2D lines: II
3.4.5 The example of 3D planes: III
3.4.6 The example of 3D lines: IV
3.5 Summary and conclusion
3.6 References and further reading
One camera
4.1 The Projective model
4.1.1 The pinhole camera
4.1.2 The projection matrix
4.1.3 The inverse projection matrix
4.1.4 Viewing a plane in space: The single view homography
4.1.5 Projection of a line
4.2 The affine model: The case of perspective projection
4.2.1 The projection matrix
4.2.2 The inverse perspective projection matrix
4.2.3 Vanishing points and lines
4.3 The Euclidean model: The case of perspective projection
4.3.1 Intrinsic and Extrinsic parameters
4.3.2 The absolute conic and the intrinsic parameters
4.4 The affine and Euclidean models: The case of parallel projection
4.4.1 Orthographic, weak perspective, para-perspective projections
4.4.2 The general model: The affine projection matrix
4.4.3 Euclidean interpretation of the parallel projection
4.5 Departures from the pinhole model: Nonlinear distortion
4.5.1 Nonlinear distortion of the pinhole model
4.5.2 Distortion correction within a projective model
4.6 Calibration techniques
4.6.1 Coordinates-based methods
4.6.2 Using single view homographies
4.7 Summary and discussion
4.8 References and further reading
Two views: The Fundamental matrix
5.1 Configurations with no parallax
5.1.1 The correspondence between the two images of a plane
5.1.2 Identical optical centers: Application to mosaicing
5.2 The Fundamental matrix .
5.2.1 Geometry: The epipolar constraint
5.2.2 Algebra: The bilinear constraint
5.2.3 The epipolar homography .
5.2.4 Relations between the Fundamental matrix and planar homo-
graphies.
5.2.5 The S-matrix and the intrinsic planes
5.3 Perspective projection
5.3.1 The affine case
5.3.2 The Euclidean case: Epipolar geometry
5.3.3 The Essential matrix
5.3.4 Structure and motion parameters for a plane
5.3.5 Some particular cases
5.4 Parallel projection
5.4.1 Affine epipolar geometry
5.4.2 Cyclopean and affine viewing
5.4.3 The Euclidean case
5.5 Ambiguity and the critical surface
5.5.1 The critical surfaces
5.5.2 The quadratic transformation between two ambiguous images
5.5.3 The planar case
5.6 Summary .
5.7 References and further reading
Estimating the Fundamental matrix
6.1 Linear methods
6.1.1 An important normalization procedure
6.1.2 The basic algorithm
6.1.3 Enforcing the rank constraint by approximation
6.2 Enforcing the rank constraint by parameterization
6.2.1 Parameterizing by the epipolar homography
6.2.2 Computing the Jacobian of the parameterization
6.2.3 Choosing the best map
6.3 The distance minimization approach
6.3.1 The distance to epipolar lines
6.3.2 The Gradient criterion and an interpretation as a distance
6.3.3 The "optimal" method
6.4 Robust Methods
6.4.1 M-Estimators
6.4.2 Monte-Carlo methods
6.5 An example of Fundamental matrix estimation with comparison
6.6 Computing the uncertainty of the Fundamental matrix
6.6.1 The case of an explicit function
6.6.2 The case of an implicit function
6.6.3 The error function is a sum of squares
6.6.4 The hyper-ellipsoid of uncertainty
6.6.5 The case of the Fundamental matrix
6.7 Some applications of the computation of AF
6.7.1 Uncertainty of the epipoles
6.7.2 Epipolar Band
6.8 References and further reading
Stratification of binocular stereo and applications
7.1 Canonical representations of two views
7.2 Projective stratum
7.2.1 The projection matrices
7.2.2 Projective reconstruction
7.2.3 Dealing with real correspondences
7.2.4 Planar parallax
7.2.5 Image rectification
7.2.6 Application to obstacle detection
7.2.7 Application to image based rendering from two views
7.3 Affine stratum
7.3.1 The projection matrices
7.3.2 Affine reconstruction
7.3.3 Affine parallax
7.3.4 Estimating Ho.
7.3.5 Application to affine measurements
7.4 Euclidean stratum
7.4.1 The projection matrices
7.4.2 Euclidean reconstruction
7.4.3 Euclidean parallax
7.4.4 Recovery of the intrinsic parameters
7.4.5 Using knowledge about the world: Point coordinates
7.5 Summary .
7.6 References and further reading
8 Three views: The trifocal geometry
8.1 The geometry of three views from the viewpoint of two
8.1.1 Transfer
8.1.2 Trifocal geometry
8.1.3 Optical centers aligned
8.2 The Trifocal tensors
8.2.1 Geometric derivation of the Trifocal tensors
8.2.2 The six intrinsic planar morphisms
8.2.3 Changing the reference view
8.2.4 Properties of the Trifocal matrices G
8.2.5 Relation with planar homographies
8.3 Prediction revisited
8.3.1 Prediction in the Trifocal plane
8.3.2 Optical centers aligned
8.4 Constraints satisfied by the tensors
8.4.1 Rank and epipolar constraints
8.4.2 The 27 axes constraints
8.4.3 The extended rank constraints
8.5 Constraints that characterize the Trifocal tensor
8.6 The Affine case
8.7 The Euclidean case
8.7.1 Computing the directions of the translation vectors and the
rotation matrices
8.7.2 Computing the ratio of the norms of the translation vectors
8.8 Affine cameras . .
8.8.1 Projective setting
8.8.2 Euclidean setting .
8.9 Summary and Conclusion .
8.9.1 Perspective projection matrices, Fundamental matrices and
Trifocal tensors
8.9.2 Transfer
8.10 References and further reading
Determining the Trifocal tensor
9.1 The linear algorithm
9.1.1 Normalization again!
9.1.2 The basic algorithm
9.1.3 Discussion
9.1.4 Some results
9.2 Parameterizing the Trifocal tensor
9.2.1 The parameterization by projection matrices
9.2.2 The six-point parameterization
9.2.3 The tensorial parameterization
9.2.4 The minimal one-to-one parameterization .
9.3 Imposing the constraints .
9.3.1 Projecting by parameterizing .
9.3.2 Projecting using the algebraic constraints .
9.3.3 Some results . .
9.4 A note about the "change of view" operation
9.5 Nonlinear methods . . .
9.5.1 The nonlinear scheme .
9.5.2 A note about the geometric criterion
9.5.3 Results
9.6 References and further reading
Stratification of n > 3 views and applications
10.1 Canonical representations of n views
10.2 Projective stratum
10.2.1 Beyond the Fundamental matrix and the Trifocal tensor
10.2.2 The projection matrices: Three views .
10.2.3 The projection matrices: An arbitrary number of views
10.3 Affine and Euclidean strata
10.4 Stereo rigs
10.4.1 Affine calibration
10.4.2 Euclidean calibration
10.5 References and further reading
Self-calibration of a moving camera: From affine or projective
calibration to full Euclidean calibration
11.1 From affine to Euclidean .
11.1.1 Theoretical analysis
11.1.2 Practical computation .
11.1.3 A numerical example .
11.1.4 Application to panoramic mosaicing
11.2 From projective to Euclidean
11.2.1 The rigidity constraints: Algebraic formulations using the
Essential matrix .
11.2.2 The Kruppa equations: A geometric interpretation of the
rigidity constraint
11.2.3 Using two rigid displacements of a camera: A method for
self-calibration .
11.3 Computing the intrinsic parameters using the Kruppa equations
11.3.1 Recovering the focal lengths for two views
11.3.2 Solving the Kruppa equations for three views .
11.3.3 Nonlinear optimization to accumulate the Kruppa equations
for n > 3 views: The "Kruppa" method .
11.4 Computing the Euclidean canonical form
11.4.1 The affine camera case .
11.4.2 The general formulation in the perspective case
11.5 Computing all the Euclidean parameters
11.5.1 Simultaneous computation of motion and intrinsic parame-
ters: The "Epipolar/Motion" method .
11.5.2 Global optimization on structure, motion, and calibration
parameters
11.5.3 More applications .
11.6 Degeneracies in self-calibration .
11.6.1 The spurious absolute conics lie in the real plane at infinity
11.6.2 Degeneracies of the Kruppa equations .
11.7 Discussion .
11.8 References and further reading
Appendix
A.1 Solution of minx JlAxjj2 subject to ||1x12 = 1
A.2 A note about rank-2 matrices