Consensus-based Distributed Estimation in Camera Networks

- A. T. Kamal, J. A. Farrell, A. K. Roy-ChowdhuryUniversity of California, Riverside

-akamal@ee.ucr.edu

ICIP 2012

Contents

• Problem Statement• Motivation for using Distributed Schemes• Challenges in Distributed Estimation in Camera

Networks• Our solution• Results

Problem Statement

Our goal is to estimate the state of the targets using the observations from all the cameras in a distributed manner.

C1C5

C3C2 C4

T1

T4

T3

T5

T2

Motivation for using Distributed Schemes

Issues using centralized or fully connected architectures:• High communication & processing power

requirements.• Intolerant of node failure.• Complicated to install.

Centralized

Partially connectedFully connected

Network architectures for multi-camera fusion

• Distributed schemes are scalable for any given connected network

Sensing Model

𝑥 𝑗,

𝐶 𝑖

𝒛 𝑖𝑗=𝑯𝑖

𝑗 𝒙 𝑗+𝝂 𝑖𝑗

Sending Model:

Parameter Vector: can be position, pose, appearance feature etc. of a target

1

2

3

4

5

4 1.5

3.5

3.5

2.5

… 3

… 3

… 3

… 3

... 3

Average Consensus: Review

Average Consensus Algorithm

Example of Average Consensus

𝑧1=¿

𝑧 2=¿

𝑧 3=¿𝑧 4=¿

𝑧5=¿

𝑧𝑖 (𝑘+1 )=𝑧𝑖 (𝑘 )+𝜖 ∑𝑗∈𝒩𝑖

(𝑧 𝑗 (𝑘 )−𝑧𝑖 (𝑘))

lim𝑘→∞

𝑧 𝑖(𝑘)=∑𝑗=1

𝑁

𝑧 𝑗 (0)

𝑁

Each nodes converges to the global average

R. Olfati-saber, J. A. Fax, and R. J. Murray, “Consensus and cooperation in networked multi-agent systems,” in Proceedings of the IEEE, 2007

𝑓𝑜𝑟 𝑘=0 :∞

𝑒𝑛𝑑

Challenges in Distributed Estimation in Camera Networks

C1 C5

C3C2C4

T1

Challenges:• Each node may not observe the target

(i.e. difference between vision graph and comm. graph)

• The quality (noise variance) of measurementsat different nodes may be different.

• Network sparsity makes the above challenges severe.

We propose a distributed estimation framework which:• Does not require the knowledge of the vision

graph.• Weights measurements by noise variances.• Network sparsity does not affect the estimate it

converges to.

Distributed Maximum Likelihood Estimation (DMLE)

𝑥 (𝑘)𝑧𝑖 ,𝑅 𝑖

𝐶𝑖

𝑦 𝑖(0) ,𝑊 𝑖 (0)

𝑦 𝑛(0) ,𝑊𝑛 (0)

𝑦𝑚(0) ,𝑊𝑚(0)�̂�𝑖❑ ,𝐶𝑜𝑣 ( �̂� 𝑖

❑)

𝐶𝑚

𝐶𝑛

Information MatrixWeighted Measurement

𝑦 𝑖(1) ,𝑊 𝑖(1)

How is does DMLE solve the challenges?

• Weighted-average consensus

• Converges to the optimal ML estimate

(not affected by network sparsity.)

• Presence/absence and quality of measurement is captured in .(, for no node measurement)

Experimental Evaluation

C1C5

C3C2 C4

Error StatisticsGround TruthObservationsAvg. ConsensusDMLE

Legend:

**

Conclusion

This work was partially supported by ONR award N00014091066 titledDistributed Dynamic Scene Analysis in a Self-Configuring Multimodal Sensor Network.

• We have proposed a distributed parameter estimation method generalized for• Limited observability of nodes• Variable quality of measurements and• Network sparsity

that approaches the performance of the optimal centralized MLE.

• Future Work: Dynamic State Estimation (Distributed Kalman Filtering)

Incorporation of prior information and state dynamics (“Information Weighted Consensus - IEEE Decision and Control Conference, Dec 2012”)

Thank you

http://www.ee.ucr.edu/~akamal/

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