the trifocal tensor class 17
DESCRIPTION
The Trifocal Tensor Class 17. Multiple View Geometry Comp 290-089 Marc Pollefeys. Multiple View Geometry course schedule (subject to change). Scene planes and homographies. plane induces homography between two views. 6-point algorithm. x 1 ,x 2 ,x 3 ,x 4 in plane, x 5 ,x 6 out of plane. - PowerPoint PPT PresentationTRANSCRIPT
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The Trifocal TensorClass 17
Multiple View GeometryComp 290-089Marc Pollefeys
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Multiple View Geometry course schedule(subject to change)
Jan. 7, 9 Intro & motivation Projective 2D Geometry
Jan. 14, 16
(no class) Projective 2D Geometry
Jan. 21, 23
Projective 3D Geometry (no class)
Jan. 28, 30
Parameter Estimation Parameter Estimation
Feb. 4, 6 Algorithm Evaluation Camera Models
Feb. 11, 13
Camera Calibration Single View Geometry
Feb. 18, 20
Epipolar Geometry 3D reconstruction
Feb. 25, 27
Fund. Matrix Comp. Fund. Matrix Comp.
Mar. 4, 6 Rect. & Structure Comp.
Planes & Homographies
Mar. 18, 20
Trifocal Tensor Three View Reconstruction
Mar. 25, 27
Multiple View Geometry
MultipleView Reconstruction
Apr. 1, 3 Bundle adjustment Papers
Apr. 8, 10
Auto-Calibration Papers
Apr. 15, 17
Dynamic SfM Papers
Apr. 22, 24
Cheirality Project Demos
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Scene planes and homographies
plane induces homography between two views
0HFFH TT
H'eF
ve'Fe'H
Hee'
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6-point algorithm
6655 Hx'xHx'xe'
x1,x2,x3,x4 in plane, x5,x6 out of plane
Compute H from x1,x2,x3,x4
He'F
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Three-view geometry
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The trifocal tensor
Three back-projected lines have to meet in a single line
Incidence relation provides constraint on lines
Let us derive the corresponding algebraic constraint…
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Notations
iii lll
0]|[IP ]a|[AP' 4 ]b|[BP" 4
0
llPπ T
l'al'Al'P'π' T
4
TT
l"bl"Bl"P"π" T
4
TT
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Incidence
iii lll
0
llPπ T
l'al'Al'P'π' T
4
TT
l"bl"Bl"P"π" T
4
TT
2rank is ]π"π'[πM ,,
e.g. is part of bundle formed by ’ and ”
l"bl'a0l"Bl'Al]m,m,m[M T
4T
4
TT
321
321 βmαmm "lβb'lαa0 T4
T4
'la T4k
"lb T
4k "lB'la'lA"lbl TT4
TT4
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Incidence relation
"lBa'l'lAb"l"lB'la'lA"lbl T4
TT4
TTT4
TT4
"lba'l'lab"l T4
TT4
Tiiil "lba'l"lba'l T
4TT
4T
ii
T4
T4 babaT iii
"lT'l Tiil
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The Trifocal Tensor
"lT,T,T'll 321TT
Trifocal Tensor = {T1,T2,T3}
"lT'l,"lT'l,"lT'l 3T
2T
1T
Only depends on image coordinates and is thus independent of 3D projective basis
"lT'll' TTi 'lT"ll" TT
iAlso and but no simple relation
General expression not as simple asT
4T
4 babaT iii
DOF T: 3x3x3=27 elements, 26 up to scale 3-view relations: 11x3-15=18 dof
8(=26-18) independent algebraic constraints on T(compare to 1 for F, i.e. rank-2)
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l"'ll
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Homographies induced by a plane
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Line-line-line relation
Eliminate scale factor:
(up to scale)
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Point-line-line relation
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Point-line-point relation
note: valid for any line through x”, e.g. l”=[x”]xx”arbitrary
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Point-point-point relation
note: valid for any line through x’, e.g. l’=[x’]xx’arbitrary
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Overview incidence relations
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Non-incident configurationincidence in image does not guarantee incidence in space
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Epipolar lines
if l’ is epipolar line, then satisfied for arbitrary l”
inversely,
epipolar lines are right and left null-space of
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Epipoles
With points
becomes respectively
Epipoles are intersection of right resp. left null-space of
(e=P’C and e”=P”C)
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Extracting F
21Fgood choice for l” is e” (V3
Te”=0)
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Computing P,P‘,P“
?
ok, but not
specifically, (no derivation)
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matrix notation is impractical
Use tensor notation instead
0 jkikj
i Tllx
jkiT
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Definition affine tensor
• Collection of numbers, related to coordinate choice, indexed by one or more indices
• Valency = (n+m)• Indices can be any value between 1
and the dimension of space (d(n+m)
coefficients)
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Conventions
0iijbA
Einstein’s summation:(once above, once below)
ii
iji
ij bAbA
Index rule: jbA iij ,0
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More on tensors
• Transformations
iji
j xAx
ijji llA
(covariant)
(contravariant)
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Some special tensors
• Kronecker delta
• Levi-Cevita epsilon
(valency 2 tensor)
(valency 3 tensor)
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Trilinearities
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Transfer: epipolar transfer
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Transfer: trifocal transfer
Avoid l’=epipolar line
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Transfer: trifocal transferpoint transfer
line transfer
degenerate when known lines are corresponding epipolar lines
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Image warping using T(1,2,N)(Avidan and Shashua `97)
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Next class: Computing Three-View Geometry
building block for structure and motion computation