1. Takehisa YAIRI RCAST, University of Tokyo
Session : VISION, GRAPHICS AND ROBOTICS
Map Building without Localization by Dimensionality Reduction Techniques
Takehisa YAIRI
RCAST, University of Tokyo

2. Outline Background Related Works Proposed Framework - LFMDR Experiment
Motivation, Purpose and Problem to consider
Related Works
SLAM, and Mapping with DR methods
Proposed Framework - LFMDR
Basic idea, Assumptions, Formalization
Experiment
Visibility-Only and Bearing-Only Mappings
Conclusion

3. Motivation Map building SLAM (Simultaneous Localization and Mapping)
An essential capability for intelligent agents
SLAM (Simultaneous Localization and Mapping)
Has been mainstream for many years
Very successful both in theory and practice
I like SLAM too, but I feel something’s missing..
Are the mapping and localization really inseparable ?
Are the motion and measurement models necessary ?
How about the aspect of map building as an abstraction of the world ?
Is there another map building framework ?

4. Purpose
Reconsider the robot map building from the viewpoint of dimensionality reduction and propose an alternative framework
Localization-Free Mapping by Dimensionality Reduction (LFMDR)
No localization, no motion and measurement models
Heuristics : Closely located objects tend to share similar histories of being observed by a robot
t=N
t=2
Reduce dimensionality,
while preserving locality
t=1

5. Map Building Problem to Consider
Feature-based map (i.e. Not topological, not occupancy-grid)
A map is represented by 2-D coordinates of objects
There EXIST motion and measurement models
But, they are not necessarily known in advance m
Positions of objects
Motion model
(State transition model)
Measurement model
(Observation model)
Move
Observation
State
Exist, but may be unknown

6. Related Works : SLAM [Thrun 02]
Problem : “Estimate m and x1:t from y1:t , given f and g”
Solutions:
Kalman Filter with extended state
Incremental maximum likelihood [Thrun, et.al. 98]
Rao-Blackwellized Particle Filter [Montemerlo, et.al. 02]
Motion and measurement models must be given
Estimations of map and robot position are coupled
Given:
Input:
Output:
Measurement data
Motion model
Map
Measurement model
Robot trajectory

7. Related Works : Dimensionality Reduction and Mapping (1)
Idea of using DR for robot map building is not new itself ..
[Brunskill & Roy 05]
PPCA to extract low-dimensional geometric features (line segments) from range measurements
[Pierce & Kuipers 97]
PCA to obtain low-level mappings between robot’s actions and perceptions (sensorimotor mapping)
Point features (High dimensional) DR
Line segments (Low dimensional)

8. Related Works : Dimensionality Reduction and Mapping (2)
Another existing idea is to estimate robot’s states (locations, poses) from a sequence of high dimensional observation data
Appearance manifolds [Ham, et.al. 05]
LLP + Kalman filter
Action respecting embedding [Bowling, et.al. 05]
SDE
Wifi-SLAM [Ferris, et.al. 07]
GP-LVM
Observation Space
Dimensionality
Reduction x2
State Space q x1

9. Related Works : Dimensionality Reduction and Mapping (cont.)
Observation Space
y(1)
y(M)
y(j)
Observation data (from time 1 to N)
Dimension of features
Time DR x2
State Space
Treat row vectors as data points
Estimate x1:N and g, from y1:N , given f q
trajectory x1

10. Proposed Framework : LFMDR (1) Assumptions
All objects are uniquely identifiable
Measurement model can be decomposed to homogeneous submodels for individual objects
Locations of at least 3 objects are known in advance (Anchor objects)
Decomposable
An observation about an object is roughly dependent only on its location, given the map and robot’s position
The second assumption may look too restrictive, but, ..

11. Proposed Framework : LFMDR (2) Interpretation as a DR Problem
Imagine a mapping between an object position and its history of observation
Observation Data Matrix
Mapping
Time
Observation History Space
(N-dimensional)
XY coordinates
(2-dimensional)
“If two objects are closely located, their histories of observation are similar ”

12. Proposed Framework : LFMDR (3) Illustration

13. Proposed Framework : LFMDR (4) Procedure
Explore the environment and obtain observation history data Y1:N
Break Y1:N into a set of column vectors {y(j)1:N}j=1,…,M
Apply a DR method to the vectors and obtain a set of 2-D vectors
Perform the optimal Affine transformation w.r.t anchor objects, and obtain final estimates

14. Features of LFMDR (1) (Comparison with SLAM)
Common
Based on state space model
Different
No assumption that motion and measurement models are known
Map is directly estimated without robot localization (localization-free mapping)
Off-line procedure
Larger amount of data required
Assumption of no missing data
Advantages
Disadvantages

15. Features of LFMDR (2) (Comparison with Other DR-based Approaches)
Comparison with [Brunskill & Roy 05]
Global vs. Local
Comparison with [Ham,et.al. 05] [Bowling,et.al. 05] [Ferris,et.al. 07]
Column vectors vs. Row vectors (i.e., Object positions vs. Robot positions) DR DR
v.s. DR DR
v.s.

16. Experiment Applied to 2 different situations Common settings:
[Case 1] Visibility-only mapping
[Case 2] Bearing-only mapping
Common settings:
2.5[m]x2.5[m] square region
50 objects (incl. 4 anchors)
Exploration with random direction change and obstacle avoidance
Evaluation
Mean Position Error (MPE)
Mean Orientation Error (MOE)
Averaged over 25 runs
WEBOTS simulator
Triangle Orientation A A
Difference C B B C
A-B-C
A-C-B

17. DR Methods Linear PCA SMACOF-MDS [DeLeeuw 77]
(a) Equal weights, (b) kNN-based weighting
Kernel PCA [Scholkopf,et.al. 98]
(a) Gaussian, (b) Polynomial
ISOMAP [Tennenbaum,et.al. 00]
LLE [Roweis&Saul 00]
Laplacian Eigenmap [Belkin&Niyogi 02]
Hessian LLE [Donoho&Grimes 03]
SDE [Weinberger, et.al. 05]
Parameters (k, s2, d) were tuned manually

18. Case 1 : Visibility-Only Mapping Description
Building a map using only visibility information
i.e., Whether each object is visible (1) or not (0)
An assumption in this simulation:
An object is visible if its horizontal visual angle of non-occluded part is larger than 5 deg

19. Case 1 : Visibility-Only Mapping Visibility Measurements
Observation history vector of an object
Visibility Observation Data
(Binary matrix)
Column
Normalization
Object ID
Time
Euc. norm
Compensate variety of the frequencies the objects are observed
Observation
History Space

20. Case 1 : Visibility-Only Mapping Maps After 2000 Time Steps
LPCA
KPCA (Gaussian, s2=0.5)
SMACOF (k=5)
Isomap(k=6)
LLE (k=8)
LEM (k=6)
HLLE (k=8)
SDE (k=7)

21. Case 1 : Visibility-Only Mapping Mean Position Errors

22. Case 1 : Visibility-Only Mapping Final Map Errors
DR methods
Opt. param.
MPE [m]
Rnk
MOE[%]
LPCA(CMDS) -
1.055 10
18.19 8
SMA(UNWGT) -
0.421 7
5.86 6
SMA(WGT)
K=5
0.206 4
4.83
KPCA(GAUS)
s2 = 0.5
0.926 8
23.29 9
KPCA(POLY)
d=8
0.953 9
27.03 10
ISOMAP
K=6
0.177 2
4.11
LLE
K=8
0.241 5
5.4
LEM
K=6
0.352 6
8.17 7
HLLE
K=8
0.192 3
4.24
SDE
K=7
0.138 1
3.65

23. Case 2 : Bearing-Only Mapping Description
Building a map only with bearing measurements
Motivated by recent popularity of Bearing-Only SLAM
Assuming all objects are always visible (No missing observation)
(Relative direction angles to objects)
Bearing angles

24. Case 2 : Bearing-Only Mapping Bearing Measurements
Original Bearing Data
Object ID
Use a unit directional vector instead of bearing angle 1 2 j M 1
q1,1
q2,1 :
qj,1
qM,1
q1,2
q2,2
qj,2
qM,2
q1,N
q2,N
qj,N
qM,N
Time 2 -p p N
Discontinuity
Unit directional vectors 1 2 j M
Observation History Space
cosq1,1
cosq2,1 :
cosqj,1
cosqM,1
sinq1,1
sinq2,1
sinqj,1
sinqM,1
cosq1,2
cosq2,2
cosqj,2
cosqM,2
sinq1,2
sinq2,2
sinqj,2
sinqM,2
cosq1,N
cosq2,N
cosqj,N
cosqM,N
sinq1,N
sinq2,N
sinqj,N
sinqM,N
Time 1
2N-dimensional 2 N

25. Case 2 : Bearing-Only Mapping Maps After 2000 Time Steps
LPCA
SMACOF (k=8)
Isomap (k=9)
LLE (k=8)
LEM (k=7)
SDE (k=7)

26. Case 2 : Bearing-Only Mapping Mean Position Errors

27. Case 2 : Bearing-Only Mapping Final Map Errors
DR methods
Opt. param.
MPE [m]
Rnk
MOE[%]
LPCA(CMDS) -
0.168 5
2.33
SMA(UNWGT) -
0.101 4
1.38 3
SMA(WGT)
K=8
0.0609 1
KPCA(GAUS)
s2=1.0
3.47 9
49.2
KPCA(POLY)
d=2
0.605 8
9.15
ISOMAP
K=9
0.0979 3
1.83 4
LLE
K=8
0.173 6
3.03
LEM
K=7
0.367 7
8.46
HLLE - NA
SDE
K=7
0.0741 2
1.36
(*) It might imply the distribution approaches to linear

28. Conclusion
Reconsidered robot map building from the viewpoint of dimensionality reduction
Proposed a new framework named LFMDR
Motion and measurement models are not required
Not need to estimate robot’s poses (localization-free)
However, larger amount of data is needed
Tested on two types of sensor measurements
Visibility information, and Bearing angles
Compared a variety of DR methods

29. Future Works Relaxation of restrictions Scalability On-line algorithm
Missing measurements
Data association problem
Scalability
Mapping of a larger number of objects
On-line algorithm
Tracking of moving objects
Multi-sensor fusion
e.g. mapping with bearing and range measurements