Crossratios Twoview projective SFM Multiview geometry More projective SFM Planches httpwwwdiensfr poncegeomvislect4pptx httpwwwdiensfrponcegeomvislect4pdf ID: 555063
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Slide1
The end of projective cameras
Cross-ratios Two-view projective SFM Multi-view geometry More projective SFM
Planches
:
http://www.di.ens.fr/~
ponce/geomvis/lect4.pptx
http://www.di.ens.fr/~ponce/geomvis/lect4.pdfSlide2
Projective Spaces: (Semi-Formal) DefinitionSlide3
A Model of
P( R )3Slide4
Projective Subspaces and Projective CoordinatesSlide5
Affine and Projective SpacesSlide6
Affine and
ProjectiveCoordinatesSlide7
Affine and
ProjectiveCoordinatesSlide8
Cross-Ratios
Collinear points
Pencil of coplanar lines
Pencil of planes
{A,B;C,D}=
sin(
+
)sin(
+
)
sin(
+
+
)sin
Slide9
Cross-Ratios and Projective Coordinates
Along a line equipped with the basisIn a plane equipped with the basis
In 3-space equipped with the basis
*
*
*Slide10
Projective Transformations
Bijective linear map:
Projective transformation:
( = homography )
Projective transformations map projective subspaces onto
projective subspaces and preserve projective coordinates.
Projective transformations map lines onto lines and
preserve cross-ratios.Slide11
Perspective Projections induce projective transformations
between planes.Slide12
Projective Shape
Two point sets S and S’ in some projective space X are projectively equivalent
when there exists a projective
transformation y:
X
X
such that
S’ = y ( S ).
Projective structure from motion = projective shape recovery.
= recovery of the corresponding motion equivalence classes.Slide13
Epipolar
Geometry
Epipolar Plane
Epipoles
Epipolar Lines
BaselineSlide14
Geometric Scene
ReconstructionIdea: use (A,B,C,D,F) as a projective basis and reconstruct O’ and O’’, assuming that the epipolesare known.
A
B
C
D
F
G
H
I
J
K
E
O’
O’’Slide15
Geometric Scene
Reconstruction IIIdea: use (A,O”,O’,B,C)as a projective basis, assuming again that the
epipoles are known.Slide16
Epipolar Geometry
Epipolar Plane
Epipoles
Epipolar Lines
BaselineSlide17
Epipolar Constraint
Potential matches for p have to lie on the corresponding epipolar line l’.
Potential matches for
p’
have to lie on the corresponding
epipolar line
l
.Slide18
Epipolar Constraint: Calibrated Case
Essential Matrix
(
Longuet
-Higgins, 1981)Slide19
Properties of the Essential Matrix
E p’ is the epipolar line associated with p’. E p is the epipolar line associated with p. E e’=0 and E e=0. E is singular.
E has two equal non-zero singular values
(Huang and Faugeras, 1989).
T
TSlide20
Epipolar Constraint: Small Motions
To First-Order:
Pure translation:
Focus of ExpansionSlide21
Epipolar Constraint: Uncalibrated Case
Fundamental Matrix
(Faugeras and Luong, 1992)Slide22
Properties of the Fundamental Matrix
F p’ is the epipolar line associated with p’. F p is the epipolar line associated with p. F e’=0 and F e=0. F is singular.
T
TSlide23
The Eight-Point Algorithm (Longuet-Higgins, 1981)
|
F
|
=1.
Minimize:
under the constraint
2Slide24
Non-Linear Least-Squares Approach (Luong et al., 1993)
Minimize
with respect to the coefficients of F , using an
appropriate rank-2 parameterization.Slide25
The Normalized Eight-Point Algorithm (Hartley, 1995)
Center the image data at the origin, and scale it so themean squared distance between the origin and the data points is 2 pixels: q = T p , q’ =
T’ p’
.
Use the eight-point algorithm to compute F from the
points
q
and q’ . Enforce the rank-2 constraint.
Output T F T’
.Ti
ii
i
i
iSlide26
Data courtesy of R. Mohr and B. Boufama.Slide27
Without normalization
With normalization
Mean errors:
10.0pixel
9.1pixel
Mean errors:
1.0pixel
0.9pixel