SVM-based Classification Approach for Synthetic-Impulse Microwave Imaging–SVM Input Data

UNIVERSITY

OF TRENTO

DIPARTIMENTO DI INGEGNERIA E SCIENZA DELL’INFORMAZIONE

38050 Povo – Trento (Italy), Via Sommarive 14

http://www.disi.unitn.it

SVM-

BASED

C

LASSIFICATION

A

PPROACH FOR

S

YNTHETIC

-I

MPULSE

M

ICROWAVE

I

MAGING

–

SVM

I

NPUT

D

ATA

F. Viani, M. Benedetti, M. Donelli, L. Lizzi, A. Cresp, I. Aliferis,

C. Pichot, and A. Massa

October 2008

InformationandCommuni ation Te hnology Dept. Universityof Trento

ViaSommarive14,38050Trento,ITALY Phone+390461882057Fax +390461882093

E-mail: andrea.massaing.unitn.it

Contra t No. DIT-PRJ-08-005

Te hni al Report N. 3

SVM-based Classi ation Approa h for Syntheti -Impulse

Mi rowave Imaging

SVM Input Data F.Viani

(1)

,M.Benedetti

(1)

,M.Donelli

(1)

,L.Lizzi

(1)

, A.Cresp

(2)

,I.Aliferis

(2)

,C.Pi hot

(2)

, andA.Massa

(1)

(1)

DepartmentofInformationEngineering andComputerS ien e ELEDIAResear hGroup

UniversityofTrento,ViaSommarive14,38050Trento-Italy Tel.+39 0461882057,Fax. +390461882093

E-mail: andrea.massaing.unitn.it

(2)

Laboratoired'Ele tronique,AntennesetTélé ommuni ations(LEAT) UniversitédeNi e-SophiaAntipolis

250rueAlbertEinstein,06560Valbonne

Tél: +33(0)492942802,Fax: +33(0)492942812 E-mail: hristian.pi hotuni e.fr

1. Introdu tion 3

2. SVM InputData 3

Inthelast years,thelearningmethodologyhasbeeninspiredbytheoryofstatisti allearningleadingupto

solutionswithgoodperforman eandrmmathemati alproperties. Inthisframework,thetheoryofsupport

ve torma hine (SVM)isbasedontheintera tionbetweenoptimizationtheoryand kerneltheory[1℄.

Re ently, widely used ma hine learning algorithms have been su essfully applied in the framework of

wireless ommuni ationproblems[2℄ andinverses atteringproblems[3℄[4℄inorderto exploittheir

general-ization apabilities and real-time hara teristi s. Moreover,when a losesolution to the problem at hand

doesnot exist, SVM appears to be a good andidate to solve the optimization problem with a trial and

errorapproa h. Asfor theSyntheti -Impulse Mi rowaveImagingSystem (SIMIS)developed atLEAT, the

learning methodologyadopted for the dete tionof target position anbe onsidered asupervisedlearning

sin eitexploitsinput/outputexamplesthatarereferredtoasthetrainingdata. Whenanunderlying

fun -tionfrominputsto outputsexists,itisreferredtoasthetargetfun tion. Intheframeworkof lassi ation

theory,thisfun tion is alledde isionfun tion andgivesbinaryoutputsifabinary lassi ationproblemis

dealt with, otherwiseit givesanitenumberof ategories formulti- lass lassi ation. The omputational

timesavingprovidedbyanonlinebinary lassi ationapproa hjustiessomelimitationslikethequalitative

re onstru tionoftheobje tpositioninsteadofthequantitativeestimationoftheele tromagneti properties.

Within theintegrationof aSVM lassierand theSIMISforobje tsdete tionandmorein generalforthe

re onstru tionoftheinvetigationarea,themaingoal onsistsinthedenitionofariskmapofthepresen e

ofthetargets.

2 SVM Input Data

The lassi ationpro edureisbasedupontrainingandtestingdata. Inthetrainingsetea hinstan e onsists

inatargetvalue,thatisthelabelofthe lass,towhi hdatabelongs,andseveralattributes,namedfeatures.

Thelearningma hinerequiresthatalldatainstan esarerepresentedasave torofrealnumbers. Inorderto

denetheformatofthepro esseddata,letusbrieyintrodu etheSVMbasi s. Thetrainingsetis omposed

by

L

observations,ea hofthem onsistsofpairs

(x

i

, y

i

) , i = 1, ..., L

,where

x

i

∈ R

n

istheve torofattributes that determines theinput spa e and

y

i

∈ {1, −1}

is theasso iatedtarget value givenby atrusted sour e. Theve tors

x

i

, i

= 1, ..., L

aremappedinahigherdimensionalfeaturespa ewhereaseparatinghyperplane hastobefoundinorder tomaximizethemarginbetweenthetrainingdatathat belongto dierent lasses.

AsfarastheSIMISdatahavebeen on erned,theelds olle tedbythemeasurementsystembe omea

partoftheinputve tors

x

i

, i

= 1, ..., L

. InordertoexploittheUWBpropertiesofthe

M

exponentialtapered slots (ETS)antenna,theinputdata

I

i

arethetime-gateddierentialtime-domain

S

21

data[5℄obtainedby the appli ation of the inverse fourier transform. The position of the

n

-obje t, identied by its bary enter

X

obj

n

, Y

n

obj

, is also afeature that belongs to theinput ve tor. The adopted binary lassi ationrequires

that inputdata

x

i

arelabeledbothto thepositive lass

y

i

= +1

andnegative lass

y

i

= −1

. Towardsthis end, for ea h input data

I

i

,

K

training data are labeled with negative lass and, instead of the features related to the obje tposition,random spatial points

X

emty

k

, Y

empty

k

, k = 1, ..., K

where the obje t does notresideareusedasshownin Tab. I.

[1℄ V.N.Vapnik,Thenatureofstatisti allearningtheory,statisti sforengineeringandinformations ien e.

SpringerVerlag,2ndedition,1999.

[2℄ M.J. Fernandez-Getino Gar ia, J. L.Rojo-Alvarez, F. Alonso-Atienza, M. Martinez-Ramon, Support

ve torma hinesforrobust hannelestimationin OFDM,IEEE SignalPro ess.Lett.,vol.13,no.7,pp.

397-400,Jul. 2006.

[3℄ I.T.Rekanos,Neural-networkbasedinverse-s atteringte hniqueforonlinemi rowavemedi alimaging,

IEEETrans.Magneti s,vol.38,no.2,Mar.2002.

[4℄ A.Massa,A.Boni,andM.Donelli,A lassi ationapproa hbasedonSVMforele tromagneti

subsur-fa esensing, IEEETrans. Geos i.RemoteSens.,vol.43,no.9,Sept.2005.

[5℄ V. Chatelee, A. Dubois, I. Aliferis, J.-Y. Dauvigna , C. Pi hot, M. J. Yedlin, Real data mi rowave

imaging and time reversal, IEEE Antennas Propagat. So . Int. Symp., Homolulu, Hawaii, USA, pp.

y

i

x

i

+1



X

₁

obj

, Y

₁

obj



I

_i

+1



X

₂

obj

, Y

₂

obj



I

_i

. . . . . . . . .

+1



X

_N

obj

, Y

_N

obj



I

_i

−1

X

₁

emty

, Y

₁

empty

I

_i

−1

X

₂

emty

, Y

₂

empty

I

_i

. . . . . . . . .

−1

X

_K

emty

, Y

_K

empty

I

_i

Tab. I-Inputdataformatofa

i

-th inputdata

I