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Numerical Optimization2016.12.12
김태원
Feature based method1,0kL
1,1kL
1,2kL
1,3kL
1kQ
0, 1kp
1, 1kp
2, 1kp
3, 1kp
1k wO O
kQ
0,kp
1,kp
2,kp
3,kp
kO
1
k
k kT T
• Real case
2
1 0k
1 expk
k kT
, 1,
, 1,
2
, ,
, |
ˆ |
ˆargmin
k k
k
i k k k i
i k k i
i k i k
p Q L Q K R t
p K R t L
p p
Direct method
2 is pixel coordinates
is intensity
is depth
i
ref i
ref i
I
D
p R
p R
p R
ip
ref
iP
i iT P P
ip
2
, ,ref i i ref i
i
E I I D p p p
2
2
2
,ref i i
i
ref i i
i
ref i i
i
E I I
E I I
E I I
p P
p P
p p
ref iD p
Cost functions
, 1,
, 1,
2
, ,
*
, |
ˆ |
ˆ
argmin
k k
k
i k k k i
i k k i
k i k i k
k k
p Q L Q K R t
p K R t L
E p p
E
• Feature based method • Direct method
Find pairMotion
estimationMotion
estimation
2
, 1 , 1argminN
k
i k k i k
i
L T L
* argmin E
2
, ,ref i i ref i
i
E I I D p p p
5
Goal of Numerical Optimization
* argmin E
E
*
Search Problem
* argmin E
E
0
0E
Two Approaches to Numerical Minimization
1. Gradient Search
2. Gradient-free search
Gradient• Gradient
E
0
0
E
0
3
2
20
3
3
E
E
E
ex)
1 2 3 4 5 6
1 2 3 4 5 6
T
E E E E E E E
• Jacobian
0
2 2 2 2 2 2
1 2 3 4 5 6
1 2 3 4 5 6
0
2 2 2 2 2 2
1 1 1 1 1 1
2 2 2 2 2 2
T
E
E
E
ex)
1
1
E
= Steepest Descent
Example
2 25 10 25
2 10
E
E
1 ( )
0.1
k k kJ
0
0 0
10E
1
1
0 0.1( 10)
1
1
1 1
8E
2
2
1 0.1( 8)
1.8
2
2 1.8
6.4E
3
3
1.8 0.1( 6.4)
2.44
0 1 2 3
12
Multi Dimension Example 1
• Update rule1 ( )k k kJ
13
Multi dimension example 2
• Update rule1 ( )k k kJ
14
Multi dimension example 3
• Update rule1 ( )k k kJ
15
Gradient descent problem
• Update rule
1 ( )k k kJ
1 2 1( ) ( )kk k kJJ
1 ( )k k kJ
17
18
19
Local Minima Problem
Two Types Gradient-Free Search
• Find Global Minima
• Just Gradient-Free
23
24
25
26
27
28
29
30
Paper study
• Feature based method
• Direct method
1. Real-time Depth Enhanced Monocular Odometry
2. Lidar Odometry and Mapping in Real-time
4. Large-Scale Direct SLAM with Stereo Cameras
3. LSD-SLAM
5. Semi-Direct Visual Odometry for a fisheye-stereo camera
Tutorial• Frame to frame motion estimation (16.11.14)• Numerical optimization (16.12.12)• Graph SLAM• Loop closure detection
