1. Weighted Line Fitting Algorithms for Mobile Robot Map Building and Efficient Data Representation
Sam Pfister, Stergios Roumeliotis, Joel Burdick
Mechanical Engineering, California Institute of Technology
Overview:
Motivation
Problem Formulation
Experimental Results
Conclusion, Future Work

2. Motivation Problem Formulation Goal : Efficient data representation
Raw Point
Data
Fit Line
Motivation
Goal : Efficient data representation
Improved data compression
Effective data correspondence
Increased robustness to outliers and noise
Problem Formulation
Geometric Representation
Weighted Line Fitting
Correspondence and Merging

3. Candidate Geometric Representations
End points : [x1 y1 x2 y2]
- 4D representation
Point + Orientation : [x y ]
- 3D representation
(x1, y1)
(x2, y2)
(x,y,)
Polar Form : [R ]
- 2D representation
Slope Intercept : [m b] y=mx+b
- 2D representation R  m 1 b

4. Selected Geometric Representation S1 S2
Polar line form
Minimal representation : L = [R,]
Endpoints maintained as scalar value pairs : S1, S2
Uncertainty maintained as x2 covariance matrix : PL = R 
Combined RR,  bounds
RR bounds
  bounds R  R 

5. Weighted Line Fitting : Motivation
Least Squares Fit vs. Weighted Fit
Fit Line
Noisy Points
True Line
Line Fit Simulation
Single Run
Fit Line
Noisy Points
True Line
Line Fit Simulation
Single Run
Fit Lines
Noisy Points
True Line
Monte Carlo Simulation
100 Runs
Fit Lines
Noisy Points
True Line
Monte Carlo Simulation
100 Runs

6. Weighted Line Fitting : Formulationdk s sd
Robot
Pose k Qk
Laser Range
Scan Data
Initial Point Grouping
- Hough Transform (Duda & Hart [72])
Point Uncertainty Modelling
- Zero mean gaussian assumption
- Laser rangefinder uncertainty parameters determined experimentally

7. Weighted Line Fitting : Formulation
Point Error Formulation : k
Point Error Formulation : Pk

8. Maximum Likelihood Estimation
Likelihood of obtaining errors {k} given line
Non-linear Optimization Problem
Position displacement estimate obtained in closed form
Orientation estimate found using series expansion approximation

9. Line Correspondence and Merging
Line Correspondence : 2 Test
Line Merge
Can merge non-overlapping line segments

10. Hallway Data

11. Results :
Kalman Filter Lab Run 1

12. Results :
KF Lab Run 2

13. Conclusions and Future Work
Developed general approach for working with line segments in a probabilistic framework
Showed that accurate error modelling can significantly improve line segment extraction accuracy and can enable robust line segment correlation.
Future Work:
Method can likely be extended for use in image processing applications as well as applications using other other range sensors (radar, ultrasound, etc.)
requires specific sensor error models