Multi-target Detection in Sensor Networks

Xiaoling Wang

ECE691, Fall 2003

Target Detection in Sensor Networks

Single target detection Energy decay model: Constant false-alarm rate (CFAR)

Multiple target detection Blind source separation (BSS) problem

Targets are considered as the sources “Blind”: there is no a-priori information on

the number of sources the probabilistic distribution of source

signals the mixing model

Independent component analysis (ICA) is common technique to solve the BSS problem

d

EE sourceobs

source

BSS in sensor networks

BSS problem involves Source number estimation Source separation

Assumptions Linear, instantaneous mixture model Number of sources = number of observations This equality assumption is not the case in sensor

networks due to the large amount of sensors deployed

Source Number Estimation

Source number estimation:

Available source number estimation algorithms Sample-based approach: RJ-MCMC (reversible-

jump Markov Chain Monte Carlo) method Variational learning Bayesian source number estimation

)|(maxarg Xmm

HPm

Bayesian Source Number Estimation (BSNE) Algorithm

allH

mmm HPHp

HPHpHP

)()|(

)()|()|(

X

XX

)|)((log)( mHtpmL x2))()((

2log

2

1)

2log()(

2

1))((log tAtAAmnt

T

axa

aaaxx

xX

dRAtpRAtp

RAtpRAp

nn

tnn

)(),,,|)((),,|)((

),,|)((),,|(

: sensor observation matrixX

: hypothesis of the number of sourcesmH

: source matrixS

: mixing matrix,A SX A: unmixing matrix,W and TT AAAW 1)( XS W

: latent variable,a Xa W and )(aS : non-linear transformation function

: noise, with variancenR

: marginal distribution of)( a/1

m

j

j mnanmn

1

2

]log)log(2

)2

log(2

[

Detailed derivation

Centralized vs. Distributed Schemes

Centralized scheme: long observed sequences from all the sensors are available for source number estimation

Centralized processing is not realistic in sensor networks due to: Large amount of sensor

nodes deployed Limited power supply on

the battery-powered sensor node

Distributed scheme: Data is processed locally Only the local decisions are

transferred between sensor clusters

Advantages of distributed target detection framework: Dramatically reduce the

long-distance network traffic

Therefore conserve the energy consumed on data transmissions.

Distributed Source Number Estimation Scheme

Sensor nodes clustering

The distributed scheme includes two levels of processing: An estimation of source number is obtained from each

cluster using the Bayesian method The local decisions from each cluster are fused using the

Bayesian fusion method and the Dempster’s rule of combination.

Distributed Hierarchy

Unique features of the developed distributed hierarchy M-ary hypothesis testing Fusion of detection

probabilities Distributed structure

Structure of the distributed hierarchy

Posterior Probability Fusion Based on Bayes Theorem

)(

)()|(

)(

)()|()|(

21

21

L

mmLmmm p

HPHp

p

HPHpΗP

XXX

XXX

X

XX

ii K

k ki

K

kk

iL

j j

i

dKE

Kp

p

12

11

111

)(

)(

X

X

)|()(

)(

)(

)()|()|(

111

1im

L

iL

j j

iL

i i

L

i mmim Hp

p

p

p

HPHpHp X

X

X

X

XX

Since ,21 LXXXX

where

LXXX ,,, 21 Since are independent, ,0)( jip XX for ji Therefore,

L

jiji

mji

L

imimL HpHpHp

1,121 )|()|()|( XXXXXX

L

imL Hp

1

)|(X

Dempster’s Rule of Combination

Utilize probability intervals and uncertainty intervals to determine the likelihood of hypotheses based on multiple evidence

Can assign measures of belief to combinations of hypotheses

jiHallH jgif

jiHHallH jgif

m

gf

mgf

HPHP

HPHPHP

,

,

)|()|(1

)|()|()|(

XX

XXX

Performance Evaluation of Multiple Target Detection

Sensor laydown

Target types

Results Comparison: Log-likelihood and Histogram

Results Comparison: Kurtosis, Detection Probability, and Computation Time

Discussion

The distributed hierarchy with the Bayesian posterior probability fusion method has the best performance, because: Source number estimation is only performed within each

cluster, therefore, the effect of signal variations are limited locally and might contribute less in the fusion process

The hypotheses of different source numbers are independent, exclusive, and exhaustive set which is in accordance with the condition of the Bayesian fusion method.

The physical characteristics of sensor networks are considered, such as the signal energy captured by each sensor node versus its geographical position

Derivation of the BSNE Algorithm

aaaxx

xX

dRAtpRAtp

RAtpRAp

nn

tnn

)(),,,|)((),,|)((

),,|)((),,|(

allH

mmm HPHp

HPHpHP

)()|(

)()|()|(

X

XX

Choose ),tanh()( aa then since aaa dd /)(log)(

/1))(cosh(

)(

1)( aaZ

where ,)1/log())(log( bcaZ cba ,, are constants.

Suppose noise on each component has same variance, ,/1 then

}))((2

exp{1

),,,|)(( 2axax AtH

Atp where

2/)2

(1 n

H

(1)

Assume the integral in (1) is dominated by a sharp peak at ,

aa

Laplace approximation of the marginal integral,

}))((2

exp{|))((

|)4

)((1

)(}))((2

exp{1 22/1

2

222/

2

ax

a

axaaaax At

At

HdAt

Hm

then by using

Therefore,

2))()((2

||log2

1)

2log()(

2

1))((log),,|)((log tAtAAmntAtp T

axax

where )()( tWt xa

Then dAAtpAPtp ),,|)(()(),|)(( xx

dAtAtAtmn

}))()((2

exp{)()2

))((( 2)(2

1

axa

where 2/1||

)()(

AA

APA

T

(2)

Assume the density function of A is sharply peaked at

A and use Laplace approximation,

}))()((2

exp{|))()((|

)()

4()

2))(((),|)(( 2

2/12

2)(

2

1

tAttAt

Attp

mnmn

axax

ax

}))()((2

exp{

))((

)()

4()

2))((( 2

2/1

1

22

)(2

1

tAt

a

At

m

j

nj

mnmn

axa

Assume ,)( mnAP ,||||2

A

Then )|)((log)( mHtpmL x

2))()((2

log2

1)

2log()(

2

1))((log tAtAAmnt

T

axa

m

j

j mnanmn

1

2

]log)log(2

)2

log(2

[

and ),tanh()( aa

Using the maximum-likelihood estimation, 0),,|)((

Atp x gives

2))()((1

tAtmn

ax