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EECS 442 – Computer vision
Multiple view geometry
Affine structure from Motion
Reading:
[HZ]
Chapters: 6,14,18
[FP]
Chapter: 12
Some slides of this lectures are courtesy of prof. J. Ponce,
prof FF Li, prof S. Lazebnik &
prof. M. Hebert
 Affine structure from motion problem
 Algebraic methods
 Factorization methods
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View Full Document Structure from motion problem
x
1
j
x
2
j
x
3
j
X
j
M
1
M
2
M
3
Given m images of n fixed 3D points
•
x
ij
=
M
i
X
j
, i =
1
, … , m,
j =
1
, … , n
From the m
x
n correspondences
x
ij
, estimate:
•m projection matrices
M
i
•n 3D points
X
j
x
1
j
x
2
j
x
3
j
X
j
motion
structure
M
1
M
2
M
3
Structure from motion problem
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View Full Document Affine structure from motion
(simpler problem)
Image
World
Image
From the m
x
n correspondences
x
ij
, estimate:
•m projection matrices
M
i
(affine cameras)
•n 3D points
X
j
Finite cameras
p
q
r
R
Q
O
P
[ ]
X
T
R
K
x
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
1
0
0
y
0
x
s
K
o
y
o
x
α
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
1
0
T
R
0
1
0
0
0
0
1
0
0
0
0
1
K
M
3
x
3
M
Perspective projection matrix
Homography (in 3D)
Homography
(in 2D)
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View Full Document Affine cameras
[ ]
X
T
R
K
x
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
1
0
0
0
0
0
s
K
y
x
α
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
1
0
T
R
1
0
0
0
0
0
1
0
0
0
0
1
K
M
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
1
0
T
R
0
1
0
0
0
0
1
0
0
0
0
1
K
M
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
1
0
0
y
0
x
s
K
o
y
o
x
Projective case
Affine case
Parallel projection matrix
(points at infinity are mapped as points at infinity)
Orthographic Projection
WeakPerspective Projection
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
1
0
0
0
1
0
0
0
1
K
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
1
0
0
0
0
0
0
K
y
x
α
Scaling function of the distance
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View Full Document Orthographic Projection
WeakPerspective Projection
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
1
0
0
0
1
0
0
0
1
K
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
1
0
0
0
0
0
0
K
y
x
α
Affine cameras
[]
X
T
R
K
x
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
1
0
0
0
0
0
s
K
y
x
α
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
1
0
T
R
1
0
0
0
0
0
1
0
0
0
0
1
K
M
⎥
⎦
⎤
⎢
⎣
⎡
=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
=
×
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
×
=
1
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This note was uploaded on 10/26/2010 for the course EECS 442 taught by Professor Savarese during the Fall '09 term at University of Michigan.
 Fall '09
 Savarese

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