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Optimal Placement and Optimal Placement and Selection of Camera Selection of Camera
Network Nodes for Target Network Nodes for Target LocalizationLocalization
A. O. Ercan, D. B. Yang, A. El Gamal and
L. J. Guibas
Stanford University
2
Low vs. High Data Rate Low vs. High Data Rate SensorsSensors
Recent work has focused on low data rate sensors, e.g. [Mainwaring’02]
Video cameras, which have very high data rate, are needed in many applications Security Surveillance Healthcare Traffic monitoring
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Security/surveillance Use expensive cameras Analog and wired Video is shipped to
monitors Observed by human
operators Not scalable Extremely hard to interpret,
and search data Slim chance of catching
anything!
Today’s Multi-Camera Today’s Multi-Camera InstallationsInstallations
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Many low cost nodes combining:
Sensing Processing Communication
Networked Scalable, easy to deploy Automated monitoring Main challenge: limited BW
and energy: Cannot send everything Cannot perform vision
algorithms at nodes
Imaging Sensor Networks Imaging Sensor Networks
8 mm
Agilent ADCM 2650
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SolutionSolution
Task-driven approach: Network performs a task or answers a query Simple local processing to reduce data Nodes collaborate to perform the task
Node selection: Measurements are highly correlated Select best subset of nodes for the task Reduces BW and energy usage greatly Makes the network scalable to many nodes
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Selection ProblemSelection Problem
Formulation: Given N sensor nodes (already placed) Use metric: Find best subset of size k , i.e.,
Previous work Sensor networks:
Information theoretic quantities [Chu’01], [Doucet’02], [Ertin’03], [Wang’04]
Coverage [Slijepcevic’01] Geometric quantities [Yang’04], [Isler’04] General utility functions [Byers’00], [Bian’06]
Computer vision and graphics: Viewpoint selection [Roberts’98], [Wong’99], [Vazquez’01]
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Task: Target LocalizationTask: Target Localization
Useful for: Tracking Surveillance Human-computer interaction Robotics
Navigation Controlling an end-effector to perform
delicate task
We focus on camera selection to minimize 2-D localization error 2-D location is most relevant in many
tasks
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OutlineOutline
Setup Local processing Camera Model Selection Metric Placement Selection Simulation Results
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SetupSetup
Cameras pointing horizontally, placed around a room
Positions and orientations of cameras are known to some accuracy
Prior statistics about the position of the object available
No occlusions
Prior for object to localize
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Local Processing Local Processing [Yang’04][Yang’04]
Simple background subtraction to detect objects Resulting bitmap is summed vertically and thresholded
Horizontal position is most relevant for 2D localization Reduces noise
Resulting bits is called “scan-line” Center of the scan-line is sent to cluster head
Scan-line
A few bytes!
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Camera Measurement ModelCamera Measurement Model
v1 and v2 independent, have zero means
Assume d >> prior , replace by (known) mean:
Projective model: Linear model
Object x
Camera position error
Read noise, camera angle error
Focal length
Perspective model:
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Selection Metric Selection Metric
Could use linear estimation to locate object
So, choose MSE of best linear estimate of location as metric for selection
Actual localization need not be performed using LE
Use MSE of LE for selection Query the selected set of
cameras for measurements Can utilize any localization
method suitable to non-linear camera model
cami
x1
zi
x2
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Assume diagonal object prior covariance
The MSE for the best LE reduces to:
MSE of Linear EstimateMSE of Linear Estimate
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PlacementPlacement
Only terms to consider
Assume: Centered prior
Circular Room
Cameras pointing to center
Minimize MSE over
N
2
1
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Symmetric CaseSymmetric Case
vi = v, = 1
Minimize:
Many optimal solutions, e.g., clusters of cameras doing locally optimal thing
Solution: N unit vectors arranged to sum to zero
and
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General CaseGeneral Case
Minimize:
Solution: N vectors of length summing to offset
from 0:
Similar to “inverse kinematics”
problem of robotics Solved using steepest descent
[Welman’93]
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SelectionSelection
Non-centered prior is OK Any room shape is OK Cameras already placed
and fixed Positions and orientations
are known to some accuracy
Prior for object to localize
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N
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SDP HeuristicSDP Heuristic
Drop the numerator Give weights to cameras
Solve dual problem using SDP [Poljak’95,Boyd’04] Plug dual optimal variables into the Lagrangian Find the set of weights that maximize it Set top k weights = 1 and rest to 0
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ConclusionsConclusions
Presented analytical approach for camera placement and selection for target localization in a camera network
Placement: globally optimal solution is found Selection: SDP outperforms other heuristics and
achieves close results to brute-force enumeration Selection approach suitable for implementation
in a large sensor network Simple local processing at each node Small amount of data shipped around Selection performed at each cluster head