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 andL. J. Guibas

Stanford University

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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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SelectionSelection

MSE(S) is given by:

Combinatorial optimization problem --

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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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MC Simulation ResultsMC Simulation Results

30 total cameras

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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

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Thank YouThank You