Electromagnetic passive localization and tracking of moving targets in a WSN-infrastructured environment.

UNIVERSITY

OF TRENTO

DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE

38123 Povo – Trento (Italy), Via Sommarive 14

http://www.disi.unitn.it

ELECTROMAGNETIC PASSIVE LOCALIZATION AND TRACKING

OF MOVING TARGETS IN A WSN-INFRASTRUCTURED

ENVIRONMENT

F. Viani, P. Rocca, M. Benedetti, G. Oliveri, and A. Massa

January 2011

Moving Targets in a WSN-Infrastru tured Environment

F. Viani,P. Ro a, M. Benedetti, G.Oliveri, and A. Massa

Department of Information Engineeringand ComputerS ien e,

University of Trento,Via Sommarive 14,38123 Trento - Italy

Tel. +39 0461 882057,Fax +39 0461 882093

E-mail: andrea.massaing.unitn.it,

{federi o.viani, paolo.ro a, manuel.benedetti, gia omo.oliveri}disi.unitn.it

Moving Targets in a WSN-Infrastru tured Environment

F. Viani,P. Ro a, M. Benedetti, G.Oliveri, and A. Massa

Abstra t

In this paper, an innovative strategy for passive lo alization of trans eiver-free

obje ts is presented. The lo alization is yielded by pro essing the re eived signal

strengthdatameasuredinaninfrastru turedenvironment. Theproblemis

reformu-latedintermsofaninversesour eone,wheretheprobabilitymapofthepresen eof

an equivalent sour e modeling the movingtarget is looked for. Towardsthis end, a

ustomized lassi ation pro edurebasedon asupportve tor ma hineis exploited.

Sele ted, butrepresentative,experimentalresultsarereportedto assessthe

feasibil-ityofthe proposedapproa h andtoshowthe potentialities and appli abilityofthis

passive andunsupervised te hnique.

Key words: Obje ttra king,wirelesssensor networks,trans eiver-free obje ts, re eived

In the re ent years, therehas been awide and rapiddiusion of wirelesssensor networks

(

W SN

s)[1℄thankstothe availabilityofsu hlow-powerandpervasivedevi esintegrating on-board sensing, pro essing, and radio frequen y (

RF

) ir uitry for information trans-mission. Usually, short-range ommuni ations are at hand sin e the wireless nodes are

generally densely distributed and hara terized by low power onsumption to ensure a

long lifetime. Therefore,

W SN

shave been also protablyused for lo ationand tra king purposes. Insu haframework,themaineortshavebeendevotedtodevelopad-ho

sys-temsbasedondedi atedtransponders/sensors[2℄orassumingana tive targetequipped

with a transmitting devi e [3℄[4℄. Dierent properties of the re eived signal, su h as the

timeofarrival(

T OA

)andthedire tionofarrival(

DOA

),havebeensu essfullyexploited to address the lo alization problem [5℄[6℄. Other modalities to lo ate a tive targets are

based on the evaluation of the re eived signal strength (

RSS

) [7℄[8℄[9℄[10℄. This is an easily measurable quantity that has been also used to lo alize the wireless nodes of the

network through ee tive triangulation strategies [8℄. Moreover, the distan e between

nodes hasbeenestimatedthanks tosimpliedradio propagationmodels. Althougheasier

thanapassive lo alizationte hnique,themaindrawba koftheseapproa hesistheneed

of the target to be equipped with anad-ho devi e. Whethersu h a fa t an be

onsid-ered negligible when tra king either obje ts or animals (although the osts unavoidably

in rease), other problems arise when dealing with people (e.g., priva y) and espe ially

with non- ooperativesubje tsasfor elderly people. Moreover, su hwearable devi es an

undergo( asualorvoluntary)damagesthuslimitingthe reliabilityofthetra kingsystem.

Other strategies on erned with trans eiver-free targets have been also presented in the

s ienti literature. State-of-the-artapproa hes are basedonDoppler radarsystems able

to estimate the distan e between the target and the sensor [11℄. As a matter of fa t,

moving targets an be tra ked through the analysis of the Doppler signature indu ed

by the obje t motion [12℄. Unfortunately, the arising performan e in real environments

thelo alizationandtra kingofpassiveobje tsstartingfromthemeasurementsofthe

RSS

indexes available at the nodes of a

W SN

. Sin e the transmission of information among the wirelessnodes isallowedby

RF

signals,thearisingele tromagneti radiations an be also protablyexploited to sense the surrounding environment. Indeed, any target lying

within the environment intera ts with the ele tromagneti waves radiated by the nodes.

Therefore, the measurements of the perturbation ee ts on the other re eiving nodes

is dealt with a suitable inversion strategy to determine the equivalent sour e modeling

the presen e of the target/s atterer generating the perturbation itself. By virtue of the

fa t that the number of nodes in a

W SN

an vary and the need to have a simple and exibletra king/lo alizationmethodallowingreal-timeestimates,alearning-by-examples

(

LBE

) strategy based on a Support Ve tor Ma hine (

SV M

) is used. Although only

re entlyappliedforthesolutionofele tromagneti inverse problems,ee tiveapproa hes

based onlearning-by-exampleste hniques havebeen alreadyproposed forthe estimation

of the dire tion of arrival of desired/undesired signals [15℄[16℄, the dete tion of buried

obje ts[17℄[18℄[19℄,and the earlybeast an erimaging[20℄ thanks totheire ien y and

versatility.

The outlineof the paperis as follows. The mathemati alissues on erned with the

pro-posedapproa haredetailedinSe t. 2wherethe

SV M

-basedmethodisdes ribed,aswell. InSe t. 3,representativeresultsfromawide setofexperimentsdealingwiththetra king

of single as well as multiple targets in both outdoor and indoor

W SN

deployments are shown. Eventually, some on lusions are drawn (Se t. 4).

2 Mathemati al Formulation

Let us onsider the two-dimensional(

2D

)s enario shown in Fig. 1(a). Theinvestigation

domain

D

is inhomogeneous and onstituted by a set of obsta les and movingtargets to

belo alized/tra ked alllying infree-spa e. The known host s enario(i.e., the target-free

domain) is des ribed by the obje t fun tion

τ

h

(r) = ε

h

(r) − 1 − j

σ

h

(r)

ωε

0

where

ω

is the

by the diele tri distribution

τ

o

(r)

,

r

∈ D

o

. The area under test is infrastru tured with

a

W SN

and

S

nodes are deployed at

r

s

,

s

= 1, ..., S

spatial lo ations. The

s

-th wireless

node radiates an ele tromagneti signal,

ξ

inc

s

(r)

(1)

, and the eld measured by the other

S

− 1

nodes and arisingfromthe intera tionsof thein identeldwiththe s enariounder

test isgiven by

ξ

_s

tot

(r

m

) = ξ

s

inc

(r

m

) +

Z

D

J

(r

′

_{) G}

0

(r

′

, r

m

) dr

′

(1)

where

G

0

is the free-spa e Green's fun tion[21℄ and

r

m

is the positionof the

m

-th (

m

=

1, ..., S − 1

) re eiving node. As a matter of fa t, the eld indu ed in

D

is equivalent to

that radiated in free-spa e by an equivalent urrent density

J

(r) = τ (r) ξ

tot

_(r)

,

r

∈ D

[22℄ modeling the presen e of whatever dis ontinuity of the free-spa e (i.e., both the

obsta les and the moving targets) where

τ

(r) = τ

o

(r)

if

r

∈ D

o

and

τ

(r) = τ

h

(r)

if

r

∈ D

h

= D − D

o

,

D

o

and

D

h

being the support of the targets and its omplementary area.

Equation(1) anbereformulatedinadierentfashionbydeningadierentialequivalent

urrent density

J

ˆ

(r)

radiatinginthe inhomogeneous host medium[21℄ [Fig. 1(b)℄. Sin e

the host mediumis a-priori known, the radiatedeld an be then expressed as

ξ

_s

tot

(r

m

) = ˆ

ξ

s

inc

(r

m

) +

Z

D

o

ˆ

J

(r

′

_{) G}

1

(r

′

, r

m

) dr

′

(2) where

ˆ

ξ

_s

inc

(r

m

) = ξ

s

inc

(r

m

) +

Z

D

τ

h

(r

′

_{) ξ}

tot

s,u

(r

′

) G

0

(r

′

, r

m

) dr

′

(3)

is the eld of the s enario without targets and equivalent to an in ident eld on the

targets,

J

ˆ

(r) = ˆ

τ

(r) ξ

tot

s,p

(r)

and

τ

ˆ

(r) = τ (r) − τ

h

(r)

is the dierential obje t fun tion.

In (3), the se ond term on the right side is the eld s attered from the host medium

without targets,

ξ

tot

s,u

being the ele tri eld related to

ξ

inc

s

in orresponden e with the

(1)

The s alar ase has been onsidered to simplify the notation. However, the extension to the

target-free s enario. Moreover,

G

1

is the inhomogeneous Green's fun tion for the

target-free onguration[21℄, whi hsatises the followingintegralequation

G

1

(r, r

′

) = G

0

(r, r

′

) +

Z

D

τ

h

(r

′

_{) G}

0

(r, r”) G

1

(r”, r

′

) dr”.

(4)

With the knowledge of

G

1

(i.e., the knowledge of the target-free s enario) the s attering

problem turns out to be the retrieval of the dierential sour e

J

ˆ

o upying the target

domain

D

o

. Thedete tionofthetargetpositionandthedenitionofthetargettraje tory

in

D

an bethen formulatedasthe denitionof the supportof the dierentialequivalent

sour e,whi hsatisestheinverse s atteringequation(2),startingfromthemeasurements

of

ξ

tot

_(r

m

)

,

m

= 1, ..., S −1

. Thisispossibleina

W SN

-infrastru turedenvironmentsin e

the nodes athandare simpleand heap devi esthat give anindire testimateof theeld

value through the

RSS

index. A ordingly, the

RSS

is measured at the

m

-th node

when the

s

-thnode is transmittingby onsideringboth the target-frees enario[

ξ

inc

s

(r

m

)

knowledge℄and the presen e oftargetswithin

D

[

ξ

tot

s

(r

m

)

knowledge℄and the dierential

eld

ξ

sct

m,s

= ξ

s

tot

(r

m

) − ˆ

ξ

s

inc

(r

m

)

ould be used for the inversion pro edure.

However, itisworthtotakeintoa ountthatthe powerradiatedbythe

W SN

nodes an

vary due to several fa tors (e.g., battery level of the

W N S

nodes, environmental

ondi-tions)thusblurring the dataa quisitionand, onsequently, ompli atingthe solutionof

the inverse problemathand. Toover omethis drawba k, the inversion isstatisti ally

re- ast asthedenition ofthe probabilitythatatargetislo atedinapositionof

D

starting

from the knowledge of

ξ

sct

m,s

,

s

= 1, ..., S

,

m

= 1, ..., S

,

m

6= s

. The arising lassi ation

problemis thensolved by means of asuitable

SV M

-based approa h. Su h astrategy

al-lows one to improvethe generalization apability of the lo alizationand tra kingsystem

sin e it is less sensitive to the instantaneousvariationsof the measurements by virtue of

the underlying probabilisti model. Moreover, it is also able to deal with s enarios not

onsidered inthe training phase as well as to perform the real time tra king of multiple

targets. More spe i ally, the proposed approa h works as follows. The region

D

where

the targetsare looked for is partitionedintoa grid of

C

ells entered at

r

c

,

c

= 1, ..., C

.

o upied (

χ

c

= 1

)whether a target(i.e.,the orresponding dierentialequivalentsour e)

is present or absent. Moreover, the probability that a target belongs to the

c

-th ell,

α

c

= P r { χ

c

= 1| (Γ, c)}

, isgiven by

α

c

=

1

1 + exp

n

pH

h

ϕ

(Γ, c)

i

+ q

o

, c

= 1, ..., C

(5) where

Γ =

n

ξ

sct

m,s

; s = 1, ..., S; m = 1, ..., S; m 6= n

o

, and

p

,

q

are unknown parameters to be determined [23℄. In (5), the fun tion

ϕ

(·)

is a non-linear mapping from the data of the originalinputspa e,

Γ

,toahigherdimensionalspa e ( alledfeature spa e)wherethe data are more easily separable through a simplerfun tion (i.e., the hyperplane

H

). The hyperplane

H

is o-line dened throughout the training phase by exploiting the knowledgeof a set of

T

known examples where both the input data (

Γ

,

t

= 1, ..., T

)and the orresponding maps (

χ

t

= {χ

c,t

; c = 1, ..., C}

,

t

= 1, ..., T

) are available. Usually, a linear de isionfun tion is adopted

H

h

ϕ

(Γ, c)

i

= w · ϕ (Γ, c) + b, c = 1, ..., C

(6)

w

and

b

being anunknown normalve tor andabias oe ient,respe tively.The de ision fun tion parameters unequivo ally dene the de ision plane and are omputed in the

training phaseby minimizingthe following ost fun tion

Ψ (w) =

kwk

2

2

+

λ

P

T

t=1

C

(t)

+

T

X

t=1

C

₊

(t)

X

c=1

η

(t)

c+

+

λ

P

T

t=1

C

(t)

−

T

X

t=1

C

(t)

−

X

c=1

η

_c−

(t)

(7)

subje tto the separability onstraints

w

· ϕ (Γ, c) + b ≥ 1 − η

c+

(t)

, c

= 1, ..., C

w

· ϕ (Γ, c) + b ≤ η

(t)

_c−

− 1, c = 1, ..., C

(8)

where

λ

is a user-dened hyperparameter [24℄ that ontrols the trade-o between the training error and the model omplexity toavoid overtting. Moreover,

η

(t)

c+

and

η

(t)

c−

are the so- alled sla k variables related to the mis lassied patterns. They are introdu ed

The minimization of (7) is performed following the guidelines detailed in [17℄ and also

exploiting the so- alledkernel tri k method [23℄.

3 Experimental Validation

The feasibilityand theee tiveness oftheproposedapproa hhavebeenassessed through

an extensive experimental validation arried out in both indoor and outdoor s enarios

(Fig. 2). The nodes have been pla edat xed positions

r

s

= (x

s

, y

s

)

,

s

= 1, ..., S

, onthe

perimeterofthe investigationarea

D

. In allexperiments,theregion

D

hasbeenassumed

having the same size (

−20λ ≤ x ≤ 20λ

and

−12λ ≤ y ≤ 12λ

) whatever the s enario at

hand,

λ

being the free-spa e wavelength of the wireless signals transmittedby the nodes

(e.g.,

f

= 2.4 GHz

), and

S

= 6

Tmote Sky nodes have been used to obtained a suitable

trade-o between the omplexity ofthe sensor network(i.e., the numberof sensornodes)

and the e ien y of the system (i.e., the sampling rate) while guaranteeing a omplete

overageof

D

(i.e.,ea hsensornodeis onne tedatleasttoanothernodeofthenetworkin

ase of target-free s enario). Although the same topology has been adopted for outdoor

as well as indoor situations, two dierent trainings of the

SV M

-based approa h have

beenperformed sin ethe arisingele tromagneti phenomenasigni antlydier(e.g.,the

ele tromagneti interferen es). Otherwise, the alibration of training examples (

T

), the

separationhyperplane

H

(

λ

),andthedis retizationoftheinvestigationarea(

C

)hasbeen

performedonlyon e,namely forthe outdoor ase, sin e theformat ofthe data pro essed

by the

SV M

does not hange. More in detail, the following setup has been onsidered:

T

∈ [100, 700]

with step

∆T = 100

,

λ

= 10

i

,

i

= {0, 1, 2, 3}

, and

C

∈ [15, 960]

from a

rough dis retization with

C

= 5 × 3

ells of dimension

4λ × 4λ

to the nest one having

C

= 40 × 24

ells ofdimension

λ

× λ

. Thesevalues have been alibrated withreferen eto

single-target experiments by evaluating the behavior of the lo alizationerror dened as

ρ

=

r



x

act

j

− x

est

j



2

+



y

act

j

− y

j

est



2

ρ

max

(9) where

r

act

j

=



x

act

j

, y

j

act



and

r

est

j

=



x

est

j

, y

j

est



target,

ρ

max

being the maximum admissiblelo ationerror. As for the test ase at hand,

it turns out that

ρ

max

=

q

X

2

D

+ Y

D

2

and

r

est

j

has been al ulated from the probability

map a ording to the followingrelationships

x

est

_j

=

P

C

c=1

α

c

x

c

P

C

c=1

α

c

y

_j

est

=

P

C

c=1

α

c

y

c

P

C

c=1

α

c

.

(10)

Figure3 gives thenormalized values of the lo ationindexes obtained fordierent

ombi-nationsofthe ontrolparameters. Ea hplot referstothevariationofa ontrolparameter

keeping onstantthe others(

T

opt

_{= 500}

,

λ

opt

_{= 100}

,

C

opt

_{= 60}

).

As far as the

SV M

training phase is on erned, the referen e measurements have been

rst olle ted in the target-free s enarios [i.e.,

τ

ˆ

(r) = 0 ⇒ ξ

sct

m,s

= 0

,

m, s

= 1, ..., S

,

m

6= s

℄. Su essively, the sets of

RSS

measurements [i.e.,

RSS

m,s

(t)

,

m, s

= 1, ..., S

,

m

6= s

,

t

= 1, ..., T

℄ have been olle ted with the target lo ated at

T

dierent positions,

r

j

= (x

j

, y

j

)

,

j

= 1, .., T

, randomly sele ted within

D

to over as uniformly as possible

the whole area under test. Con erning the required omputational time, the burden of

the training phase grows proportionally with the number of training samples and the

dis retization of

D

from a minimum of

3 × 10

2

_[s]

when

T

= 100

,

C

= 15

up to a

maximum of

10

4

_[s]

(i.e., almostthree hours) when

T

= 700

,

C

= 960

.

As regards the

SV M

test step, both single (

J

= 1

) and multiple (

J

= 2

) target

tra k-ing problems have been onsidered. Sin e o-the-shelf sensor nodes are used for these

experiments, they allow to obtain one

RSS

measurement ea h

5 × 10

−2

_[s]

. Therefore,

onsideringthe situationwhereea hnode hasto olle ta

RSS

measurementforallother

S

− 1

nodes, the maximum a quisition time is

2 [s]

. The system is then able to pro ess

the data and dene alo alizationmap

α

c

,

c

= 1, ..., C

, in

0.1 [s]

using a

3 GHz

PC with

2 GB

of RAM.

Therstexperimentdealswiththeoutdoortra kingofasinglehumanbeingmovinginside

D

. Figure4shows theprobabilitymapestimatedwhenthe targetisat

r

act

spe i ally, Figure 4(a) shows the probability map assuming that the same experiment

has been taken into a ount in the training phase. Dierently, the map in Fig. 4(b) has

beenobtained theexamplenot belongingtothetrainingdataset. It isworthnotingthat

the targetis orre tlylo alizedinboth mapssin ethe enter ofthe targetlieswithinthe

regionwithhigherprobability. The sameexperimenthas been su essively onsidered for

the indoors enario. The resultsofthe

SV M

-based lo alizationpro ess areshown inFig.

5. As fortheprevioustest, theresultswhenthe sameexamplehas been either onsidered

[Fig. 5(a)℄ or not [Fig. 5(b)℄in the training phase havebeen reported. As expe ted, the

valuesof the lo alizationerrors in rease whateverthe trainingbe auseof the omplexity

of the ele tromagneti intera tions arising from the presen e of the walls (i.e., multiple

ree tions) in indoor environments. Nevertheless, the region with high probability still

ontains the a tual position of the target thusdemonstratinga gooddegree of reliability

of the approa halsoin this ase.

Letusnow onsiderasingletargetmovingoutdoorinside

D

alongthestraightlineshown

inFig. 6. The

RSS

valueshavebeenmeasuredat

6

dierenttimeinstants,butitisworth

to point out that the a quisition time an be further shortened to rea h an almost

real-time tra king. Thesamples ofthe lo alizationmapsand the estimatedpath are reported

in Fig. 7 and Fig. 6, respe tively. As it an be observed, there is a good mat hing

betweenthe a tualpathandtheestimatedone assessingtheee tivenessofthe approa h

in real-time pro essing, as well. The same analysis has been arried out for the indoor

ase. Although the moving target is quite arefullylo alized,the result in Figure 8 and

the lo ation indexes in Tab. I onrm the higher omplexity of tra king the target as

ompared to the outdoor ase.

Inordertodealwiththe tra kingofmultipletargets,the

SV M

lassierhas beentrained

with a mixed data-set ontaining examples with one (

T

1

examples with

J

= 1

) and two

(

T

2

exampleswith

J

= 2

)targets. Sin e

T

= T

1

+ T

2

exampleshavebeenusedalsoforthe

single-targettraining,someexperimentshavebeen arriedout toanalyzethedependen e

smaller set of single-target examples has been used for the trainingphase (i.e.,

T

1

< T

2

).

Vi e versa, a larger number of example is needed for an ee tive lo alizationof the two

targets as pointed out by the maps in Fig. 10 and quantied by the lo ation indexes

in Tab. II. Su h a behavior was expe ted sin e the number of dierent ombinations

with twotargets ishigher if ompared tothe single-target ase. Therefore,

T

1

= 150

and

T

2

= 350

examples have been su essively used for the training phase of the following

tra king experiments.

As representative examples, two dierent situations with

J

= 2

have been dealt with.

In the former, one target (

j

= 1

) is moving within

D

while the other (

j

= 2

) remains

immobile inthe same position. Instead, both targets are moving in the se ond example.

The a tual traje tory and the estimated one are shown in Fig. 11 and Fig. 12,

respe -tively. Whateverthe exampleathand,aquite arefulindi ationonthepositionandpath

followed by the targets has been obtained as further onrmed by the average values of

the lo alizationerrors(outdoor:

ρ

1

= 0.070

,

ρ

2

= 0.061

-indoor:

ρ

1

= 0.101

,

ρ

2

= 0.070

).

4 Con lusions

In this work, the lo alizationand tra king of passive targetshave been addressed by

ex-ploiting the

RSS

values available at the nodes of a

W SN

. The problem at hand has

been reformulated into an inverse sour e one aimed at re onstru ting the support of an

equivalent sour e generatinga perturbation of the wireless links amongthe

W SN

nodes

equal to that due to the presen e of targets within the monitored area. The inversion

has beenfa ed with alearning-by-examplesapproa h basedona

SV M

lassierdevoted

to determine amap of the a-posteriori probability that adierentialequivalentsour e is

presentwithintheinvestigationdomain. Experimentalresultshaveassessed the

ee tive-ness and reliability of the proposed approa h in dealing with the tra king of single and

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[2℄ Z. Li, W. Dehaene, and G. Gielen, A 3-tier UWB-based indoor lo alizationsystem

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(a)

(b)

Fig. 1 -Equivalent Tra king Problem - Sket h of (a) the tra kings enario and (b) the

X

D

Y

D

x

y

(x

c

, y

c

)

(x

s

, y

s

)

D

W SN node

(a)

(x

s

, y

s

)

(x

act

j

, y

j

act

)

W SN node

(b)

Fig. 2 -Problem Geometry - Plots of (a) the outdoor and (b) the indoorenvironments

2

4

6

8

10

12

14

16

18

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Localization Error,

ρ

[x10

2

]

Normalized Parameter Value

T/T

_max

- [T

_max

=700]

λ/λ

_max

- [λ

_max

=1000]

C/C

_max

- [C

_max

=960]

Fig. 3 -Calibration -Lo alizationerror as afun tion of the

SV M

ontrolparameters:

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

(a)

(b)

Fig. 4 - Single-target lo alization - Outdoor S enario -Probabilitymaps of the

investigation region

D

obtained when the test data (a)belongsand (b)does not belong

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

(a)

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

(b)

Fig. 5 - Single-target lo alization - Indoor S enario -Probabilitymaps of the

investigation region

D

obtained when the test data (a)belongsand (b)does not belong

-12

-8

-4

0

4

8

12

-20

-16

-12

-8

-4

0

4

8

12

16

20

y/

λ

x/λ

Real

Estimated

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

(a)

(b)

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

(c)

(d)

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

(e)

(f )

Fig. 7 - Single-target tra king- Outdoor S enario - S reenshotsof the probabilitymap

-12

-8

-4

0

4

8

12

-20

-16

-12

-8

-4

0

4

8

12

16

20

y/

λ

x/λ

Real

Estimated

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

(a) (b)

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

( ) (d)

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

(e) (f)

Fig. 9 -Single-target lo alization - Outdoor S enario (

T

1

∈ [0, 500]

,

T

2

∈ [0, 500]

,

λ

= 100

,

C

= 60

) - Probabilitymaps of the investigation region

D

when using (a)

100

%

T

1

and

0

%

T

2

, (b)

80

%

T

1

and

20

%

T

2

,( )

60

%

T

1

and

40

%

T

2

, (d)

40

%

T

1

and

60

%

T

2

,

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

(a) (b)

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

( ) (d)

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

−20

20

12

0.0

1.0

−12

x/

λ

y/

λ

Pr

{χ

=+1|

Γ

}

(e) (f)

Fig. 10 -Multiple-targets lo alization - Outdoor S enario (

T

1

∈ [0, 500]

,

T

2

∈ [0, 500]

,

λ

= 100

,

C

= 60

) - Probabilitymaps of the investigation region

D

when using (a)

100

%

T

1

and

0

%

T

2

, (b)

80

%

T

1

and

20

%

T

2

,( )

60

%

T

1

and

40

%

T

2

, (d)

40

%

T

1

and

60

%

T

2

,

-12

-8

-4

0

4

8

12

-20

-16

-12

-8

-4

0

4

8

12

16

20

y/

λ

x/λ

Target 1 - Real

Target 1 - Estimated

Target 2 - Real

Target 2 - Estimated

Fig. 11 -Multiple-targets tra king - Outdoor S enario - A tualand estimated

-12

-8

-4

0

4

8

12

-20

-16

-12

-8

-4

0

4

8

12

16

20

y/

λ

x/λ

Target 1 - Real

Target 1 - Estimated

Target 2 - Real

Target 2 - Estimated

Fig. 12 -Multiple-targets tra king - Outdoor S enario - A tualand estimated

Outdoor

Indoor

T ime Instant

ρ

ρ

× ρ

max

[λ]

ρ

ρ

× ρ

max

[λ]

1

0.071

3.32

0.209

9.76

2

0.070

3.30

0.131

6.09

3

0.060

2.78

0.115

5.38

4

0.057

2.67

0.048

2.23

5

0.045

2.09

0.089

4.15

6

0.074

3.46

0.140

6.53

Average Error

: ρ

0.063

2.94

0.122

5.69

Tab. I -Single-target tra king - Lo alizationerrors for the outdoor and the indoor

Single T arget

M ultiple T arget

j

= 1

j

= 1

j

= 2

ρ

ρ

× ρ

max

[λ]

ρ

ρ

× ρ

max

[λ]

ρ

ρ

× ρ

max

[λ]

(a)

0.044

2.07

0.217

10.12

0.158

7.37

(b)

0.059

2.77

0.196

9.14

0.135

6.31

(c)

0.093

4.34

0.151

7.02

0.074

3.44

(d)

0.150

6.98

0.149

6.96

0.062

2.91

(e)

0.262

12.23

0.063

2.93

0.106

4.94

(f )

0.357

16.67

0.031

1.46

0.063

2.93

Tab. II - Multiple-targets lo alization - Outdoor S enario - Lo alizationerrors forthe