projective 2d geometry (cont’) course 3 multiple view geometry comp 290-089 marc pollefeys
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Projective 2D geometry (cont’)course 3
Multiple View GeometryComp 290-089Marc Pollefeys
Content
• Background: Projective geometry (2D, 3D), Parameter estimation, Algorithm evaluation.
• Single View: Camera model, Calibration, Single View Geometry.
• Two Views: Epipolar Geometry, 3D reconstruction, Computing F, Computing structure, Plane and homographies.
• Three Views: Trifocal Tensor, Computing T.• More Views: N-Linearities, Multiple view
reconstruction, Bundle adjustment, auto-calibration, Dynamic SfM, Cheirality, Duality
Multiple View Geometry course schedule(subject to change)
Jan. 7, 9 Intro & motivation Projective 2D Geometry
Jan. 14, 16
(no course) Projective 2D Geometry
Jan. 21, 23
Projective 3D Geometry Parameter Estimation
Jan. 28, 30
Parameter Estimation Algorithm Evaluation
Feb. 4, 6 Camera Models Camera Calibration
Feb. 11, 13
Single View Geometry Epipolar Geometry
Feb. 18, 20
3D reconstruction Fund. Matrix Comp.
Feb. 25, 27
Structure Comp. Planes & Homographies
Mar. 4, 6 Trifocal Tensor Three View Reconstruction
Mar. 18, 20
Multiple View Geometry MultipleView Reconstruction
Mar. 25, 27
Bundle adjustment Papers
Apr. 1, 3 Auto-Calibration Papers
Apr. 8, 10
Dynamic SfM Papers
Apr. 15, 17
Cheirality Papers
Apr. 22, 24
Duality Project Demos
Last week …
l'lx 0xl T x'xl T1,0,0l
Points and lines
0xx CT xl C0ll * CT 1* CC
Conics and dual conics
ll' -TH
-1-TCHHC ' THHCC **'
xx' HProjective transformations
Last week …
1002221
1211
y
x
taa
taa
1002221
1211
y
x
tsrsr
tsrsr
333231
232221
131211
hhh
hhh
hhh
1002221
1211
y
x
trr
trr
Projective8dof
Affine6dof
Similarity4dof
Euclidean3dof
Concurrency, collinearity, order of contact (intersection, tangency, inflection, etc.), cross ratio
Parallellism, ratio of areas, ratio of lengths on parallel lines (e.g midpoints), linear combinations of vectors (centroids). The line at infinity l∞
Ratios of lengths, angles.The circular points I,J
lengths, areas.
Projective geometry of 1D
x'x 22H
The cross ratio
Invariant under projective transformations
T21, xx
3DOF (2x2-1)
02 x
4231
43214321 x,xx,x
x,xx,xx,x,x,x Cross
22
11detx,x
ji
ji
ji xx
xx
Recovering metric and affine properties from images
• Parallelism• Parallel length ratios
• Angles • Length ratios
The line at infinity
l
1
0
0
1t
0ll
A
AH
TTA
The line at infinity l is a fixed line under a projective transformation H if and only if H is an affinity
Note: not fixed pointwise
Affine properties from images
projection rectification
APA
lll
HH
321
010
001
0,l 3321 llll T
Affine rectificationv1 v2
l1
l2 l4
l3
l∞
21 vvl
211 llv 432 llv
Distance ratios
badd :c,b:b,a
T0,1v' H
TTT 1,,1,,1,0 baa
c,b,a H
The circular points
0
1
I i
0
1
J i
I
0
1
0
1
100
cossin
sincos
II
iseitss
tssi
y
x
S
H
The circular points I, J are fixed points under the projective transformation H iff H is a similarity
The circular points
“circular points”
0233231
22
21 fxxexxdxxx 02
221 xx
l∞
T
T
0,,1J
0,,1I
i
i
TT 0,1,00,0,1I iAlgebraically, encodes orthogonal directions
03 x
Conic dual to the circular points
TT JIIJ* C
000
010
001*C
TSS HCHC **
The dual conic is fixed conic under the projective transformation H iff H is a similarity
*C
Note: has 4DOF
l∞ is the nullvector
*C
Angles
22
21
22
21
2211cosmmll
mlml
T321 ,,l lll T321 ,,m mmmEuclidean:
Projective: mmll
mlcos
**
*
CC
CTT
T
0ml * CT (orthogonal)
Length ratios
sin
sin
),(
),(
cad
cbd
Metric properties from images
vvv
v
'
*
*
**
TT
TT
T
TT
T
K
KKK
HHCHH
HHHCHHH
HHHCHHHC
APAP
APSSAP
SAPSAP
TUUC
000
010
001
'*
Rectifying transformation from SVD
UH
Metric from affine
000
0
3
2
1
321
m
m
m
lllTKK
0,,,, 2221211
212
21122122111 T
kkkkkmlmlmlml
Metric from projective
0c,,,,, 332332133122122111 5.05.05.0 mlmlmlmlmlmlmlmlml
0vvv
v
3
2
1
321
m
m
m
lllTT
TT
K
KKK
Pole-polar relationship
The polar line l=Cx of the point x with respect to the conic C intersects the conic in two points. The two
lines tangent to C at these points intersect at x
Correlations and conjugate points
A correlation is an invertible mapping from points of P2 to lines of P2. It is represented by a 3x3 non-singular matrix A as l=Ax
Conjugate points with respect to C(on each others polar)
0xy CT
Conjugate points with respect to C*
(through each others pole)
0ml * CT
Projective conic classificationTUDUC -TCUUD 1
321321321 ,,diag,,diag,,diag ssseeesssD0or 1ie
Diagonal Equation Conic type
(1,1,1) improper conic
(1,1,-1) circle
(1,1,0) single real point
(1,-1,0) two lines
(1,0,0) single line
0222 wyx
0222 wyx
022 yx
022 yx
02 x
Affine conic classification
ellipse parabola hyperbola
Chasles’ theorem
A B
C
DX
Conic = locus of constant cross-ratio towards 4 ref. points
Iso-disparity curves
X0X1
C1
X∞
C2
Xi Xj
ii
i
i
1
1:
1
01
1
01
11
:
11
10
11
0
Fixed points and lines
ee H (eigenvectors H =fixed points)
ll TH (eigenvectors H-T =fixed lines)
(1=2 pointwise fixed line)
Next course:Projective 3D Geometry
• Points, lines, planes and quadrics
• Transformations
• П∞, ω∞ and Ω ∞