The Archimedes principle applied to separation of uniformly distributed sources

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The Archimedes principle applied to separation of

uniformly distributed sources

Jean Barrère, Gilles

Chabriel-To cite this version:

Jean Barrère, Gilles Chabriel-. The Archimedes principle applied to separation of uniformly

dis-tributed sources. Proceedings of the 2nd Physics in Signal and Image Processing International

Sym-posium (PSIP’2001), 2001, Marseille, France. �hal-01826178�

UNIFORMLY DISTRIBUTED SOURCES

Jean Barrere - Gilles Chabriel

MS/GESSY-ISITV-UniversitedeToulonetduVar

Av. GeorgesPompidou,BP56-83162LAVALETTEDUVARCEDEX(FRANCE)

Fax: 330494142598-chabriel@isitv.univ-tln.fr-barrere@isitv.univ-tln.fr

ABSTRACT

Inthiswork,weareinterestedintheseparationofNsourcesignalsrecordedsimultaneouslybyNreceivers.Wepresentinthis

workamethodbasedonananalogybetweentheresearchofindependentaxesofanhypercube(geometricalrepresentation

ofamixingofuniformsources)andtheresearchofequilibriumstatesofweighingsystemsubmittedtodiscontinuedgravity

elds. Themethodonlyuse oneorderstatisticsand is able totreat alargeamount ofsources. Presentlythis methodis

limitedtouniformlyorsymmetricallydistributedsources.

1. INTRODUCTION

1.1. Position ofthe problem

Let'sconsiderN independentsourceswithuniformprobabilitydistribution,simultaneouslyreceivedonN sensors.

Themixingprocessischaracterizedbythefollowingequation:

y (t)=Mx(t); (1) inwhichx( t)=[x 1 ( t);x 2 (t);::: ;x N (t) ] T

isthevectorofzero-meanstatisticallyindependentunknownsources

andy(t)=[y 1 (t);y 2 (t);:::;y N ( t)] T

istheobservationvector. In(1), MistheNN unknownmixingmatrix,

assumedtobefull columnrank.

The identication problem consists in estimating aseparatingmatrix Ssuch as: SM =DP , where D is an

regulardiagonalmatrix. Pisapermutation matrixwhichhasonenonzeroentryineachrowandcolumn.

TheproductofSwiththeobservationvectorleadsto:

z(t)=DPx(t): (2)

Thevectorz(t) representsthesourcevectorx(t) exceptforone permutationandascalingfactor.

Wewill recallin the main paperhowtheuse ofsecond order statisticspermitsto reduce the problem tothe

researchofanunitarymixing matrix(whiteningoftheobservations). TheFigure1representsatwodimensional

geometricalillustrationof the spacesources(a), the mixing space(b), the actionof thewhitening procedure (c)

andtheestimatedsourcesinthewhiteningspace(d).

Sources space

Observations space

"Whitened"

Observations space

Estimated

Sources space

(a)

(b)

(c)

(d)

x

2

x

1

z

2

z

1

Figure1: 2Ddierentsspaces

Sointhefollowing,withoutanylossofgenerality,wewillonlyconsiderthecasewhereMisanunitarymatrix

and thesourcesare powerunit. Theproblem reduces in ndingthe unitarymatrixSi.e. theunitarytransform

Let'sconsider thefollowingidealtwodimensionalexperiment: anhomogeneoussquare thinplate is immersed in

anhomogeneousliquid. Theplateisrigidand itsdensityishalf thedensityoftheliquid. Theplateissubmitted

to twokindof dierentforces: volumeforces(surface force in thetwodimensional case)dueto thegravityeld,

andsurfaceforces(lineforceinthetwodimensionalcase)duetothepressureofliquidactingonthebound ofthe

immersedsolid. Wedenoteby

! F

g

theresultantofgravityforces,Oitspointofapplicationand

! F

p

theresultant

ofsurfaceforces,Citspointofapplication. Theequilibriumoftheplateisobtainedwhenthesumoftheresultant

is null and when the momentum of forces is null i.e. when

!

OC and

! F

g

are linearly dependent. The Figure 2

illustratessuchexperiment.

F

p

C

(a)

F

_g

O

(c)

F

g

F

_p

F

p

F

g

O

(b)

C

C

O

Figure2: ArchimedesPrinciple

Wecanconsiderpressureforcesasvolumeforcesactingonanequivalentvolumeofliquidcorrespondingtothe

immersed partofthesolidand C is itsthecenter ofgravity. Thisphenomenon iswell known astheArchimedes

principle. Becauseofhomogeneity,theequilibriumpositionsgiveustheaxisofsymmetryoftheplate. Forattractive

forces,thestableequilibriumstateisobtainedwhenthedistancejOCjisminimum(seegure(2.c))unstablewhen

thedistance jOCj is maximum(see gure(2.b)). Forrepulsiveforces, thestability stateswill beinverted. This

basicapplicationofstaticcanbeeasilyextendtoNdimensionalhomogeneoussystems. Following,wewillinspired

fromittoconstructaniterativemethodextractingthestablestateofequilibriumoftheN-dimensionalhypercube

ofobservationsimmersed inadiscontinuousvectoreld.

2. ALGORITHM

2.1. Brief description

Wedescribehereonestep(fromiterationk toiterationk+1)oftheiterativemethodproposed. Letconsiderthe

mixingspaceofN uniformlydistributedpowerunit centredsourcesasillustratedinFigure3.a. Eachpointofthis

spacey k ( t)=  y k 1 (t);y k 2 (t);:::;y k N (t)  T

issubmittedtoadiscontinuousvectoreldpsuchas:

p(y k (t))= ( [1;0;:::;0] T if y k 1 (t)0 [0;0;:::;0] T if y k 1 (t)<0 (3)

andacontinuousvectoreldg :

g (y k

(t))=[ 0:5;0;:::;0] T

(4)

Because ofuniformdistributionofsources,theresultantoftwoeldsare: F

p

= F

g

=Efpg.

Thepointsofapplicationoftheseresultantsare: C

k =Efy k = y k 1

0gforthediscontinuouseld,andO =Efy

k g

forthecontinuouseld.

Let'sdenoteu

1

thevectorsuchasu

1 = ! OC k j ! OC k j

. UsinganGram-Schmidtorthogonalizationonu

1

andasetofN 2

anylinearlyindependentvectors,weconstructasetofN 1ortho-normalizedvectoru

i

; i=2;:::N.

ItresultsaNNunitarymatrix:

U k 1 =[u 1 ;:::;u N ]:

Applying thetransform

 U k 1  T

to the mixing space y

k

we obtainannew balanced mixing space versusthe eld

forcein presenceasdescribedinFigure 3.b:

y k +1 =  U k 1  T y k k +1

y

1

>0

p

1

=1

p

1

=0

F

p

F

g

F

g

F

p

(a)

(b)

Figure3: Onestep

i.e.U 1

isthean identity matrix. Becauseof uniformity, after convergencewehaveseparationofonesource from

theothers: y n (t)=[z 1 (t);y n 2 (t);:::;y n N (t)] T

TheprocedurecanbeappliedagainontheN 1remainingobservations. Theseparationwillbeachievewhen

all sourceshavebeenextracted. A proof ofconvergence will begiven for thetwo-dimensionalcase in the whole

paper.

2.2. Acceleration ofconvergence

ThemethodcanbestronglyaccelerateifweuseintheGram-Schmidtprocedure asetofvectorjointlyevaluated

fromN dierentdiscontinuouseldvectors:

p j (y k (t))= 8 < : 1;:::;j;:::;N indexofsource [0;:::;1;:::;0] T if y k j (t)0 [0;:::;0;:::;0] T if y k j (t)<0 (5) 3. EVALUATION -APPLICATION

Thevaluationofperformanceofseparationismeasuredbyanindexintroducedin[3],constructedonthetheglobal

matrixG=SM,normalizedbythenumberofnon-diagonaltermsinthis matrix:

ind(G)= 1 2N(N 1) 2 4 X i 0 @ X j jG i;j j 2 max l jG i;l j 2 1 1 A + X j 0 @ X i jG i;j j 2 max l jG l;j j 2 1 1 A 3 5

Thisnon-negativeindextakesitsvaluesbetween0and1. Asmallvalueindicatestheproximityofthedesired

solutions.

Thefollowing table gives an ideaof the sensibility of the method versusthe data number for a mixing of 3

sources. Eachlinewas obtainedforaset of1000dierentrealizationsandmixingmatrices.

DataNumber Mean(ind) Std(ind) Numberof

nonseparatedcases

2048 0.00400 0.0030 5

4096 0.00120 0.0010 1

8192 0.00045 0.0034 0

0

50

100

150

200

250

300

350

400

0

0.05

0.1

0.15

0.2

0.25

Iterations

Performance

80

10

50

30

Figure4: Performancesfordierentsourcenumbers

4. CONCLUSIONS-EXTENSIONS

The method proposed can be considered as an extension of geometric ones, see [1], [2], by the addition of an

exterioreld vectoractingonthespaceobservation. Itcan beinterpretedalso asclassicalsecondorder statistics

onesappliedtononlinearlteredobservations. Theseparationhasbeenpresentedonuniformlydistributedsources.

Becausethemethod extractssymmetryaxes,itcanbeextended tosymmetricallydistributed sources. Ofcourse,

many other eld can be tested in order to generalize the separating process. In case of n-valuated sources the

existenceofmeta-stablecongurationsimpliessomemodicationsofthealgorithm.

REFERENCES

[1] C.G. Puntonet, A.Prieto, C.Jutten, M.Rodrigez-Alvarez,andJ. Ortega. Separation of sources: a geometry

basedprocedurefor reconstructionofn-valuedsignals, inSignalProcessing,Vol.46n3,pp267-284,1995.

[2] C.G.Puntonet, C.A.Mansour,andC.Jutten. Geometrical algorithms forblind separation ofsources, in Actes

duXV

 eme

colloqueGRETSI,pp273-275,Sept.1995.

[3] E.MorauandO.Macchi. High-ordercontrastsfor self-adaptativesourceseparation, inInternationalJournalof