Affine Structure from Motion

1
Affine Structure from Motion

• Frank Dellaert
• Slides all my own this time ?

2
Structure and Motion
3
Book Sequence
4
Book Structure Motion
5
2 Problems !

• Correspondence
• Optimization

6
A Correspondence Problem
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An Optimization Problem

• Find the most likely structure and motion ?

8
Optimization
Image i
Image i
uik
9
Affine Cameras

• Easy projection equation

10
Geometric Intuition !!

• From first lecture on cameras !

(0.6,2.5,1) ?
Z
Y
X
11
Geometric Intuition

• Projection matrix form

Z
Y
X
12
Translation of image i ?

• Take derivative wrpt to ti

13
World origin trick

• Move scene so centroid origin

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New Problem !

• Just subtract centroid from all measurements

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Matrix Form

• Projection in one image (i)

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Matrix Form

• Projection in two images

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Matrix Form

• Projection all points in all images

n
16
13
14
M1
1
15
2
3
11
12
4
5
6

24
M2
25
26
21
22
23
2m
34
M3
35
36
31
32
33
44
M4
45
46
41
42
43
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Geometric Intuition !!

• Joint Image of point 8-dimensional

Z
Y
X
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Subspace Constraints

• Does it really span 8-dim space ??

Z
Y
X
20
Joint Camera Mapping
83
R3
R8
Mapping from 3D to 8D space (for 4 cameras)
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But Range is only 3D !!!
83
R3
R3
R5
5-dimensional null-space !
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What does SVD look like ?
AU S VT
3x3
83
3x3
R3
R3P2
R5
Rank 3, economy SVD
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And measurement matrix ?
Also Rank 3 !!!
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SVD Approach to SFM

• Form measurement matrix
• Do rank 3 economy size SVD
• Voila

Motion
Structure
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Problem

• Where to stick the extra 3x3 ?

Motion
Structure
3x3
3x3
83

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Euclidean Upgrade

• Uncalibrated not so easy
• Calibrated cameras
• We know they are rotation matrices
• Impose constraints
• unit length 2m
• orthogonal m
• Solve for Q

3x3
3x3
27
House Sequence
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Epilogue (no pun intended)

• Facorization is better algorithm to estimate
  affine geometry than linear FA
• In RANSAC use factorization
• Sample set of 4 correspondences
• Form measurement matrix
• Do SVD
• Calculate FA from M, or, better
• Calculate points with