1. Pursuit Evasion Games and Multiple View Geometry
René Vidal
UC Berkeley
Part I: Motivation, 10 minutes, 9 slides -> 8
Part II: Previous work, 35 minutes, 21 slides -> 20
Part II: Future work, 15 minutes, 8 slides -> 10

2. Current Research Robotics and Control Computer Vision
Pursuit-Evasion Games (ICRA’01-CDC’01-TRA’02)
Vision-Based Landing (ICRA’02)
Decidable Classes of DTHS (HSCC’00-CDC’01)
Computer Vision
Multi-body Structure from Motion (IFAC’02)
Multiple View Geometry (sub. IJCV’01)
Planar motions with Small Baselines (ECCV’02)
Optimal Motion Estimation (ICCV’01)
Camera Self-Calibration (ECCV’00)

3. Pursuit-Evasion Games
Jin Kim, David Shim, Cory Sharp, Omid Shakernia
Shankar Sastry

4. Pursuit-Evasion Game Scenario
Evade!

5. PEG: Map Building (Hespanha CDC’99)
Given measurements build a probabilistic map
Measurement step: model for sensor detection
Prediction step: model for evader motion

6. PEG: Pursuit Policies (Hespanha CDC’99)
Greedy Policy
Pursuer moves to the reachable cell with the highest probability of having an evader
The probability of the capture time being finite is equal to one (Hespanha CDC ’99)
The expected value of the capture time is finite (Hespanha CDC ’99)
Global-Max Policy
Pursuer moves to the reachable cell which is closest to the cell in the whole map with the highest probability of having an evader
May not take advantage of multiple pursuers (all the pursuers could move to the same place)

7. PEG: Control Architecture (ICRA’01)
position of evader(s)
position of obstacles
strategy planner
position of pursuers
map builder
communications network
evaders
detected
obstacles
pursuers
positions
Desired pursuers
positions
tactical
planner
trajectory
regulation
tactical planner & regulation
actuator
positions
[4n]
lin. accel.
& ang. vel.
[6n]
inertial
[3n]
height over terrain [n]
obstacles detected
evaders detected
vehicle-level sensor fusion
state of helicopter & height over terrain
obstacles detected
control signals
[4n]
agent dynamics
actuator
encoders
INS
GPS
ultrasonic altimeter
vision
Exogenous
disturbances
terrain
evader

8. PEG: Vision System

9. PEG: Experimental Results (Summer’00)

10. PEG: Architecture Implementation (CDC’01)
Navigation Computer
Serial
Vision Computer
Strategic Planner
Helicopter Control
GPS: Position
INS: Orientation
Camera Control
Color Tracking
UGV Position Estimation
Communication
UAV Pursuer
Map Building
Pursuit Policies
Communication
Runs in Simulink
Same for Simulation and Experiments
TCP/IP
Serial
Robot Micro Controller
Robot Computer
Robot Control
DeadReck: Position
Compass: Heading
Camera Control
Color Tracking
GPS: Position
Communication
UGV Pursuer
UGV Evader

11. PEG: Experimental Results (Spring’01)

12. PEG: Experimental Results (Spring’01)
Add simulation video before

13. Pursuit Policy: Sensing, Intelligence, Speed
Greedy
Global-max
Visibility Region
Forward View
Omni-directional View
Evasion Policy
Random
Global-min
Evader speed
Evaluated policies against different vision capabilities
Trapezoidal (narrow FOV) vs. OMNI-directional (wide FOV)
Both vision systems covered same number of cells
Narrow field of view
Can see farther into distance
Can sweep a larger area simply by rotation
Omni-directional (wide angle)
Can see in all angles
Rotation does not help see more

14. PEG: Pursuit-Policy v.s. Vision System

15. PEG: Evader Speed v.s. Intelligence

16. Vision Based Landing

17. Landing: (Sharp & Shakernia ICRA’01)

18. PEG: Summary
Proposed a probabilistic framework and a hierarchical control architecture for pursuit-evasion games
Implemented architecture with UGVs and UAVs in real-time
Sensor fusion, helicopter control, vision-based detection and vision-based landing
Evaluated strategies v.s. speed, sensing and intelligence
Global-max outperforms greedy in a real scenario
Forward view outperforms Omni-view Vision
It also works for intelligent evaders

19. Multiple View Geometry
The Multiple View Matrix
Yi Ma (UIUC), Jana Kosecka (GMU), Shankar Sastry (UCB)

20. Multiple View Geometry (MVG)

21. MVG: Anatomy of cases (state of the art)
surface
curve
line
point
theory
algorithm
practice
Euclidean
affine
projective
2 views
3 views
4 views
m views
algebra
geometry
optimization

22. MVG: A need for unification
Euclidean
surface
curve
line
point
2 views
3 views
4 views
m views
theory
algorithm
practice
affine
projective
algebra
geometry
optimization
rank deficiency of
Multiple View Matrix

23. MVG: Formulation Homogeneous coordinates of a 3-D point
Homogeneous coordinates of its 2-D image
Projection of a 3-D point to an image plane
Either Euclidean or Projective

24. MVG: Classical Approach
Given corresponding images of points recover motion, structure and calibration from
This set of equations is equivalent to

25. MVG: The Multiple View Matrix
WLOG choose frame 1 as reference
Theorem: [Rank deficiency of Multiple View Matrix]
(generic)
(degenerate)

26. MVG: Bilinear and Trilinear Constraints
Multiple View Matrix implies bilinear constraints
Multiple View Matrix implies trilinear constraints
Constraints among more than three views are algebraically dependent (quadrilinear in particular)

27. MGV: Line Features
Point Features
Line Features

28. MGV: Planar Features Point Features Line Features
Besides multilinear constraints, it simultaneously gives homography:

29. Multiple View Landing (ICRA’02)

30. MGV: Multiple View Planar Algorithm Results

31. Multiple View Matrix: Summary
Points
M implies bilinear and trilinear constraints
Quadrilinear constraints do not exist
Lines
All indep. constraints are among 3 views
Some are trilinear, some are nonlinear
Mixed points and lines
There are multilinear constraints among 2-3 views
There are nonlinear constraints among 3-4 views

32. Current Work: Multi-Body SFM
Geometry of multiple moving objects
Given a set of corresponding image points, obtain:
Number of evaders
Evader to which each point belongs to (segmentation)
Depth of each point
Motion of each pursuer and evader
Orthographic case (Costeira-Kanade’95)
Multiple moving points (Shashua-Levin’01)

33. Current Work: Multi-Body SFM [IFAC’02]
Costeira-Kanade in perspective
Form optical flow matrices
Obtain number of objects
Segment the points

34. Multi-body SFM: Results

35. Conclusions Computer Vision for Real-Time Control
Pursuit Evasion Games
Vision Based Landing
A new approach to Multiple View Geometry
Simple: just linear algebra
Unifying: Euclidean, projective, 2 views, 3 views, multiple views, points, lines, curves, surface?
New Constraints: There are nonlinear constraints