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https://oatao.univ-toulouse.fr/27001

https://doi.org/10.1016/j.sigpro.2020.107905

Vincent, François and Besson, Olivier Robust adaptive target detection in hyperspectral imaging. (2021) Signal

Processing, 181. 107905. ISSN 0165-1684

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Robust

adaptive

target

detection

in

hyperspectral

imaging

François Vincent

, Olivier Besson

ISAE-SUPAERO, 10 Avenue Edouard Belin, Toulouse 31055, France

Keywords: Hyperspectral imagery Robust Detection GLRT AMF

a

b

s

t

r

a

c

t

Oneofthemainissueindetectingatargetfromanhyperspectralimagereliesonproperlyidentifyingthe background.Manyassumptionsaboutitsdistributioncanbeadvocated,eveniftheGaussianhypothesis prevails.Nevertheless,thehugemajorityoftheresultingdetectionschemesassumethatthebackground distributionremainsthesamewhetherthetargetispresentornot.Inpractice,becauseofthespectral variabilityofthetargetand thenon-linearmixingwiththebackgroundradiance,thishypothesisisnot strictlytrue.Inthispaper,weconsiderthatanunknownbackgroundmismatchexistsbetweenthetwo hypotheses.Undertheassumptionthatthismismatchissmall,wederiveanapproximationofthe Like-lihoodRatiofortheproblemathand.ThisgeneralformulationisthenappliedtothecaseofGaussian distributedbackground,leadingtoarobustAdaptiveMatchedFilter.Thebehaviourofthisnew detec-torisanalysedandcomparedtopopulardetectors.Numericalsimulations,basedonrealdata,showthe possibleimprovementincaseoftargetsignaturemismatch.

1. Introduction

Anhyperspectralimage isatwo-dimensionalmap,whereeach pixeliscomposed ofhundreds ofspectral bands.Thisspectral in-formation allows to characterize the different materials present in the image and hence hyperspectral imaging has encountered a large field of applications, ranging from remote sensing to medicine [1–8].One of theseapplications consistsin detecting a target whose spectral signature could be known fromlaboratory experiments,orunknown.Inthislattercase,onetalksofanomaly detection.Inthispaper,wewillfocusonthefirstcase,namely tar-getdetection.

Thedifficultyofthisproblemliesinthefactthatthesignature ofinterest(SoI) t isburiedinabackground b withunknown statis-tics.Evenifitsdistributionwereknown,theparametersdescribing it (for instance meanand covariancematrix) are not known and mustbeestimatedfromtheavailabledata.Consequently,detection oftheSoIinapixelundertest(PUT) y requiresusingotherpixels (socalledtrainingsamples)tolearnthebackgroundpresentinthe PUT.Ithastobenoticedthatinthehyperspectraldomain,wedeal withrealpositivedataleadingtononzeromeansignals.

Certainlydueto itseasy handlingproperties,theGaussian as-sumptionprevailstomodelthebackground.Inthiscontext,many algorithms have been developed, such as the Adaptive Matched

Corresponding author.

E-mail addresses: francois.vincent@isae-supaero.fr , francois.vincent@isae.fr (F. Vincent), olivier.besson@isae-supaero.fr (O. Besson).

Filter (AMF) [9],Kelly’s Generalized Likelihood Ratio Test (GLRT)

[10],theOrthogonal SubspaceProjection(OSP) [11],or the Adap-tiveCoherent/CosineEstimator(ACE)[12],tonamethemost pop-ularones. However, withrealhyperspectral data,Gaussian distri-butionsrarelyoccur [13–16],andmorerepresentativebackground distributions have been considered. One ofthe most common is the EllipticallyContoured (EC)t-distributed modelthat allows to extendtheGaussian distribution toa broaderclass ofprobability densityfunctions(p.d.f.).Differentdetectorshavebeenderived un-dersuchanhypothesis,astheEC-GLRT[17]forinstance.

Nevertheless,althoughconsideringabroadclassofdistribution tomodel thebackground,thehuge majorityofthedecisiontests relyontheassumptionthatthebackgroundremainsthesame un-derthetwohypotheses.

H0:y=b (1)

H1:y=

α

t+b

where

α

istheso-calledfill-factor.

Hence, if considering the SoI and

α

as deterministic, which isa widespread assumption, thismodel amountsto consider the same distributionunder the two hypotheses. The only difference betweenthetwohypothesesisameanshift.

Unfortunately,thisstrongassumptionisusuallynotmetinreal hyperspectraldetectionschemes.Indeed,thefirstreasonisrelated tothedataacquisitionitself.Theradianceresultingfromthe mix-ture of a background and a target are driven by non-linear ef-fects [18]. These effects are due to multi-reflections or masking effects. Thereby, the additive model from Eq.(1), is over

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fied. A morerepresentative model, namelythe so-called replace-ment model, can be used [1]. This model assumes that the tar-get, ifpresent,replaces a partofthebackground,asit willmask an areaproportionaltoits fill-factor

α

.Somedetectorshavebeen derived undersuch a model,as the FTMF [19]or ACUTE[20]. A moregeneralmodel,relaxingtheunitaryconstraintsonthe ampli-tudes, onlyassumesa linearmixingbetweenthebackgroundand the target [21].Butevenifthesetwo models aremore represen-tative,they donot takeintoaccountthemultiplelight reflections conductingtonon-linearmixing,andconsequentlyamore compli-catedbackgroundbehaviourwhenthetargetexists.

The second main reasonconductingto achange inthe distri-bution between the two hypotheses is related to mismatches in the target SoI.Indeed, thisso-calledspectral variability isdueto changesintheconditionsbetweenthelabmeasurementsandthe data acquisition [9,22]. Infact, the image isacquired by a differ-entinstrumentandata largerdistancefromthescene.Then,the rawradiancemeasurements havetobeconvertedintoreflectance data using a so-called radiometric compensation [23]. Different techniques canbe used [24,25],neverthelessall these compensa-tion techniques are based on models inducing possible SoI mis-matches. Then, when assuming the SoIasdeterministic and per-fectly known, thisspectral variability resultsin anoise measure-ment increase. Again,the consequenceis that thePUT exhibitsa differentdistributionbeyondasimplemeanshift.

Thereby, in this paper, we consider a more general detection model that may include a large class of mismatches, including those statedhere-above. In this model,the background distribu-tionmayvarybetweenthetwohypotheses.Morepreciselywe as-sumethatwhenthetargetispresent,thebackgroundundergoesa possibleperturbationwithunknownstatistics:

H0:y=b (2)

H1:y=

α

t+b+



b

where



b isthemismatch.

Conventionally, the distribution of b can be estimated from trainingsamples x k.Ontheotherhand,



b hasanunknown distri-bution,possiblylinkedto

α

.Thisunknownperturbationcanmodel a large class of possiblemismatches induced by non-linear mix-turesbetweenthetargetandthebackgroundorfromtarget signa-tureerrors.

Underthehypothesis that



b issmallcomparedto b ,we de-rivea robustLikelihood Ratio (LR),andshowthat itcan be writ-ten asthesumofthe standard LRanda correction factor.When applied to the popularcase where b is supposed to be Gaussian distributed, we derivean approximationof theGLRT. Thisrobust detector is shownto be the popular AMF, corrected by a simple additiveterm.

The remaining of this paper is organized asfollows. We first derivetheLikelihoodRatioforthegeneralcaseofanybackground distribution insection2.Then, wefocuson thepopularGaussian caseinSection 3,wherewe propose asimpleadditive correction fortheAMFdetectortoimproverobustness.InSection4,we com-pare the behaviour of the proposed robust AMF to popular de-tectors, thanks to their thresholdlimit plottedin a Matched Fil-ter Residual(MFR)diagram.Tofinish,wecomparethesedetectors using a Monte-Carlo simulation based on realdata, inSection 5. FinallyconcludingremarksendthispaperinSection6.

2. Robust likelihood ratio

As alreadysaid inthe introduction,we assume thatthe back-groundunder H1 writes b 1= b +



b ,where theprobability

den-sity function (pdf) of b , pb

(

.

)

is supposed to be known, andits parameterscanbeestimatedfromthetrainingsamples,thatisthe

standard hypothesis.Hence, forany



b pdf, p

(

.

)

, thepdf of b 1

canbewrittenas

pb1

(

u

)

=

(

pb∗ p

)(

u

)

= 

p

(

z

)

pb

(

u− z

)

dz=E[pb

(

u− z

)

]

(3)

Now, using the approximation derived in the Appendix, for any given u ,wehave E[pb

(

u− z

)

]pb

(

u

μ



)

+12Tr



2p b

(

z

)

z

zT

|

uμC



(4)

whereTr

{}

standsforthetraceofthematrixbetweenbraces,

μ

 and C denoterespectivelythemeanandcovariancematrixofthe unknown background mismatch



b , and [2pb(z)

zzT

|

uμ] denotes

theHessian of pb

(

z

)

evaluatedat

(

u

μ



)

.It should bepointed

outthat thisapproximationdoesnotrequirecompleteknowledge ofthepdfof



b ,butonlyofitsmeanandits covariancematrix, whichisanappealingfeature.

Hence, forany PUTmeasurement y ,the Likelihood Ratio (LR) associatedwiththedetectionproblemfromEq.(2)canbewritten inthefollowingform,assoonasthebackgroundmismatch



b is small: LRr= pb1

(

y

α

t

)

pb

(

y

)

 pb

(

y

α

t

μ



)

pb

(

y

)

+ Tr

{

2pb(z) zzT

|

yαtμC

}

2pb

(

y

)

(5) = pb

(

y

α

t

μ



)

pb

(

y

)



1+1 2 Tr

{

2pb(z) zzT

|

yαtμC

}

pb

(

y

α

t

μ



)



(6)

Moreover,



b beingsomeperturbationaround b ,weassume that

μ

=0,sothatwesimplyhave LRrLR



1+1 2Tr

{

2p b(z) zzT

|

yαt pb

(

y

α

t

)

C

}



(7) whereLR= pb(yαt)

pb(y) istheLRinthestandardcase,i.e.whenthere

isnobackground mismatch(



b =0). Hence,we can seethatfor any zero-mean mismatch distribution, we can approximate the modifiedLR by the sumof theLR withoutmismatch anda cor-rectiveterm,dependingonthepdfof b andthecovariancematrix of



b .

3. Robust adaptive matched filter

In this section, we now investigatethe casewhere the back-grounddistributionunderH0isGaussian, b ∼ N

(

μ

,C

)

.Letusstart

withthenominalcase,torecalltheAMFderivation.

Ifnomismatchispresent(b 1=b ),thestandardLRwrites LRG= pb

(

y

α

t

)

pb

(

y

)

=

e−12(yαtμ)TC−1(yαtμ)

e−12(yμ)TC−1(yμ)

(8)

Assuming that the background parameters are known from the trainingsamples,theonlyunknown parameteris

α

, whose Max-imumLikelihood (ML) writes

α

ˆ= tTC−1(yμ)

tTC−1t .Then, the logarithm

oftheGeneralizedLR(GLR)isshowntobe:

logGLRG=log



pb

(

y− ˆ

α

t

)

pb

(

y

)



=1 2

|

tTC−1

(

y

μ

)

|

2 tTC−1t = 1 2AMF (9)

wherewerecognizethepopularAMF= |tTC−1(yμ)|2

tTC−1t .

(4)

Fig. 1. Typical MFR plot.

Now, in the mismatched case (b 1=b ), the Hessian in

Eq.(7)writes

2p b

(

z

)

z

zT

|

yαt=C− 1 2

C−12

(

y

α

t

μ

)(

y

α

t

μ

)

TC−12− I

C−12 ×pb

(

y

α

t

)

(10)

sothatthemodifiedLRbecomes

LRG r LRG ·

1+12Tr

C −12

(

y

α

t

μ

)(

y

α

t

μ

)

TC −12 − I

C −12C C −12



(11)

First, we havetoestimate

α

.As thebackground mismatchunder

H1 is supposed to be small,we can usethe ML estimation of

α

obtainedhereabove inthenominalcase, asa firstorder approx-imation, namely

α

ˆ=tTC−1(yμ)

tTC−1t . Hence, taking the logarithm and

substituting

α

,theGLRwrites: logGLRG r logGLRG+log ·

1+1 2Tr

C −12

(

y − ˆ

α

t

μ

)(

y − ˆ

α

t

μ

)

TC −12 − I

C −12C C −12



(12)

or,thankstoEq.(9) 2logGLRG r AMF+2log

1+1 2Tr



uuT− I



C−1 2CC− 1 2



(13) where u =P t cy c, with t c=C −1 2t and y c=C − 1 2

(

y

μ

)

, the whitenedversionsof t and

(

y

μ

)

respectively.

Therefore,wecanseethatassumingaGaussianbackground un-derH0,andforsmallmismatchesunderH1,theGLRisthepopular

AMF, plus a correction. Thisadditive term tends to measure the gap betweenthe supposed background covariance matrix C and the actual one underH1.Indeed, u isan estimation ofthe

back-groundunderH1,whitened bythesupposedcovariancematrix C ,

sothat

(

uu T− I

)

isakindofmismatchmeasurement.Inthesame

way, C −12C C −12 isalsoanormalizedmeasurementofthisgap. Unfortunately, this expression of the correction is difficult to use in practice,except ifone has an idea of C . In orderto ad-dressthisissue,wenowderiveanapproximationofthiscorrection inordertogetamoreconvenientexpression.

The correction depends on T=Tr

{



uu T− I



C −1

2C C −12

}

that canbewrittenasfollows

T=Tr

{

C˜

}



uTC˜ u Tr

{

C˜

}

− 1



(14)

Fig. 2. Typical MFR plot.

Fig. 3. Complete RGB view of the Viareggio test scene.

whereC ˜=C −12C C −12 isthecovariancematrixofthebackground mismatch whitened by C . Diagonalising this covariance matrix, ˜

C =



ku ku Tk, the quadratic term here above writes

uTC˜u Tr{C˜} =



(

k

k

)(

u

T

ku

)

2.Thereby,itcorrespondstoaweightedsumofthe

energyof u .Unfortunately, inmostcases,we donot haveaccess to C  andtheweightsare unknown.Tocircumventthisproblem, we simply propose to use a non-weighted sum to estimate this energy,namely uTC˜u

Tr{C˜} 

uTu

N .Thisapproximationamountsto

con-siderthatalltheeigenvaluesofC ˜areequal,namelythat C =



C . In other words, there is no approximation when the covariance matrix ofthe mismatch is proportional to the covariance matrix ofthebackground C .Thishypothesissometimesappearsinthe lit-eraturewhenconsideringtargetsignaturemismatches,evenifthe rationalebehinditisnotobvious[26].Nevertheless,this hypothe-sisincludesthepopularcasewherebothbackgroundandthe mis-matcharesupposedtobewhite.

On the other hand, Tr

{

C ˜

}

=Tr

{

C −21C C −12

}

=Tr

{

C C −1

}

=

Tr

{

E





b



b TC −1



}

=E





b TC −1



b



representstheexpectationof

theenergyofthemismatch,whitenedby C .Assumingthat



b and b aredecorrelated, we canestimate thisenergyasthe difference betweenthe energyof b 1 andtheenergyof b ,bothwhitened by C .Thefirstonecanbeestimatedby u Tu ,as u isan estimationof

(5)

Fig. 4. Receiver Operating Characteristic for V 5 and V 6 targets, for r = 0 (no mismatch).

Fig. 5. Receiver Operating Characteristic for V 5 and V 6 target, for r = 0 . 2 .

thebackgroundunderH1,whitenedby C ,whereasthesecondone

issimplyequaltoN,inaverage. Substituting both C ˜ and uTC˜u

Tr{C˜} by their estimation in Eq.(14)wehave TN



uTu N − 1



2 (15)

Then,fromEq.(13),wecandefinearobustAMFas

AMFr=AMF+2log



1+N 2



uTu N − 1



2



(16)

At thisstage,we canmake afew commentsontheproposed ro-bust AMF. Whereas the standard AMF only measures the energy ofthedataprojectedonthetargetsubspace,theproposedscheme alsomeasures theremaining energy,projectedonthebackground subspace. It checks ifthisenergy corresponds to the background

energyestimatedfromthetrainingsamples,andmeasuresthegap. Then,itcorrectstheAMFwithrespecttothisestimatedmismatch energy.Otherpopulardetectorsalsousethedataenergyprojected onthebackgroundsubspace toimproveperformance,likeACE or Kelly’s GLRT,as we will see hereafter. In the following part, we compare our robust AMF with thesepopular detectors regarding their behaviour withrespect to thisresidual energy. Tothisend, we plot both the data and the detectors threshold on the so-calledMatched FilterResidual(MFR) diagrams.These informative 2D plots show both the energyprojected on the target subspace (MF)andtheenergyprojectedonthesubspaceorthogonal tothe target(R).

4. Insights

Just like for the AMF, the majority of popular detectors are basedonthewhitenedandcentredversionsofboththedataand 4

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Fig. 6. P f a gain for P d = 0 . 5 , for V5 target, with α= 0 . 1 and KN = 9 .

the target signature, namely y c and t c.As noticedpreviously,the

standard AMF simply consistsinprojecting thedata onto the1D

targetsubspaceandinmeasuringtheenergy,asAMF=



P tcy c



2= y T

cP tcy c, where P tc= tctTc tT

ctc is the projection onto the target sub-space. Nevertheless,theglobal energy,



y c



2,or theresidual

en-ergy,namelyR=



P tcy c



2 =y T cP tcy c=

(



y c



2 − AMF

)

,where P ⊥tc= I − Ptc istheprojectionontotheso-callednoisesubspace,arealso informative. Indeed, when thetarget amplitudeis unknown, this last quantityisan estimationoftheLoglikelihoodofthedata,up to a scaling factorand an additive constant. Hence plottingdata ina2Dspace, wherethex-axis istheAMFandthey-axisisR,is convenientandvaluablebothfordataanddetectorsanalysis.

SuchrealdataMFRanalysesexistintheliterature[27–29]and show thatthescatter plotsforall thepixelsaredistributedalong linesconvergingtotheoriginoftheplot,eachlinecorresponding to a given material on theground. Fig.1 showsa representative situation with just3 pixels classes(for instance, forest in green, roads in blue,andbuildings inred), considering thebuildings as thetargetfortheAMFaxis.

Observing such a MFR plot, itis clear that the standard AMF couldnotgetgooddetectionperformance,asitonlythresholdsthe x-axis, asshownon Fig.2.Toimprovedetection, oneshould also usethey-axis.Thatiswhatpopulardetectionschemesdo,suchas Kelly’s GLRTorACE. Indeed, thesetwo detectors share thesame formulation,namely

AMF

(

a+1

KyTcyc

)

(17)

where a=0 forACE, anda=1 forKelly, K being thenumber of trainingsamples.

In both cases, comparing this quantity to a threshold

γ

amounts to comparing AMF to a limit dependingon the square normofthedata,



y c



2,sothatthedecisionlimitisdefinedby

AMF =

γ

a+1 Ky T cyc

(18)

Now, as



y c



2=AMF+R, the thresholds of these two detectors

drawlinesintheMFRplot,whoseequationare

R=AMF



K

γ

− 1



− Ka (19)

As represented on Fig. 2, the threshold line corresponding to ACE goesthroughtheoriginoftheplot,asa=0.Inthiscase,

an-otherintuitive waytoseethatthethresholdregionisaline,isto refertothestandardinterpretationofACEasthesquarecosineof the anglebetween y c and t c.Thresholding ACE amounts to limit

thisangleas shownonFig. 2.In thecaseof Kelly, thethreshold is still a line,but shifted along the AMF axis,depending on the numberofsecondarydataK.

Now,referring to the definitionof u inEq. (13),we haveR=



u



2,so that thethreshold limitof therobust AMF, AMFr inthe

MFRplotisdefinedas AMF=

γ

− 2log



1+N 2

R N− 1

2



(20)

Then, this threshold limit is characterized by a second order polynomial equation in the MFR plot, as represented on Fig. 2. The threshold limit corresponding to the larger AMF being ob-tained when R=N. In the case of a white noise

(

C =

σ

2I

)

, this

point corresponds to the case where the residual energy equals the noise energy estimated from the training samples, namely



P t

(

y

μ

)



2=

σ

2N. Hence, as can be seen on Fig. 2, the AMF r

hasapproximatelythe samebehaviour asACE or Kellywhen the residual energy is smaller than that estimatedfrom the training samples.

In the opposite case, the behaviour of AMFr is not the same.

Indeed,whentheresidualenergyislarger thanthetraining sam-plesone,thethresholdreducestoimprovedetectionwithrespect to the datamodel from Eq.(2). Infact, according to thismodel, anynoiseenergyvariationwithrespecttotheestimatedonefrom thetrainingsamplesisacluefortheH1hypothesis.Obviously,this

possibledetectionimprovementcanalsoincrease thedetectionof confuserelements.Indeed,improvingdetectionaccordingtomodel

(2),goesagainsttheso-calledselectivityofthedetector.

Insituationswhere possibleunwantedtargetsexit,one hasto supplementthedetectionschemebyaso-calledFalseAlarm Miti-gation(FAM)step.Thesetwo-stepprocedureshavebeenaddressed inthe hyperspectral literature [27–30], butalsoin the radar one

[31].Themainideabehindthisconceptistoimprovethe selectiv-ityissuesbyaddingaseconddetectionstagebasedontheresidual energy.Indeed,asnoticedpreviously,theresidualenergyisrelated to thelikelihood ofthe data withrespect toa given target type. Then, the possible false alarms existing after the first stage can berejectedusinganappropriatethresholdontheresidualenergy. Thisstep amounts toaddinga second thresholdon the y-axisof theMFRplot.Inthehyperspectralcontext,thisprocedurehasbeen firstdevelopedtotackletheselectivityissuesoftheAMF,withthe MatchedFilterwithFalseAlarmMitigation(MF-FAM)[27]as rep-resented on Fig. 2.This algorithm is very similar to the popular MixtureTuned Matched Filter(MTMF)approach from ENVI®soft-wareenvironment.Inthislastcase,theresidualenergyisreplaced byanequivalentcriterion,calledinfeasibility[30].

5. Performance evaluation

Inordertoassessthevalidityoftheproposed robustdetector, wenowconducta Monte-Carlosimulationbasedonareal exper-iment, namelytheairborneViareggio 2013trial[32].This bench-markinghyperspectraldetectioncampaigntookplaceinViareggio (Italy), in May 2013, with an aircraft flying at 1200 meters. The open data consist in a [450× 375] pixels map composed of 511 samplesintheVisibleNearInfraRed(VINR)band

(

400− 1000nm

)

. Thespatialresolutionoftheimageisabout0.6meters.

Differentkindsofvehiclesaswellascolouredpanelsservedas knowntargets.Foreach of thesetargets, aspectral signature ob-tainedfromgroundspectroradiometermeasurements isavailable. Moreover,ablackandawhitecover,servingascalibrationtargets, were also deployed. As can be seen onFig. 3,the scene is com-posedofparkinglots,roads,buildings,sportfieldsandpinewoods.

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Fig. 7. P f a gain for P d = 0 . 5 , for V5 target, with α= 0 . 1 and KN = 9 .

Asforthemajorityofhyperspectraldetectionschemes,thefirst step ofthe processingaims atconverting theraw measurements into a reflectance map, namely removing all atmospheric effects andnon-uniformsunillumination.Tothisend,we usethe Empir-ical Line Method(ELM) [23,25], considering the black and white calibrationpanels.Thenaspectralbinning[33]isperformedto re-ducethevectorsizedimensiontoN=32.

Based onthis map,we willconduct a Monte-Carlosimulation byinsertingasynthetictarget t ,initiallynotpresentintheimage, intoarandomlychosenpixel y ofthemap(y

(

α

t +y

)

,where

α

is thetarget amplitude).The target signatureissubjectto uncer-taintiesandiscomposedofthesumoftheassumedtarget signa-ture¯tandarandomzero-meannormallydistributedwhitevector t ,whoseenergyisr timestheenergyof¯t:

t=¯t+t (21)

where t ∼ N

(

0 ,r¯tTN¯tI

)

.

Figs. 4 and 5 represent the Receiver Operating Characteristic (ROC) of different detectors, for both the so-called V5 and V6

targets and for r=0 (no mismatch) and r=0.2, respectively. In both cases,weusea 17× 17windowto estimatethe background mean and the sample covariance matrix. The target’s amplitude are

α

=0.1 for V5 and

α

=0.2 forV6, as this last one is more

difficulttodetect.

First of all, in the nominal case, where no mismatch exists (r=0), all the detectors exhibit approximately the same perfor-mance, with a slight loss for the proposed AMFr. On the other

hand,whenthereisamismatchonthetarget’ssignature,the ro-bustAMFoutperformstheotherdetectors.Wecannoticethatboth ACEandKellyexhibit alargerlossthanthestandardAMF.Indeed, the existenceofamismatchonthetarget signatureincreasesthe residualpart(R)ofthedata.Then,referringtoFig.2,itisobvious thatbothACEandKellywillbethemoreaffectedinthissituation. Wehighlightherethecompromisebetweenselectivityand robust-ness.ThatiswhytheFAMtechniques,usingfirstarobust scheme toavoidnon-detection,followedbyaselectiveschemetosortthe targetsandreducefalsealarm,areworthyofinterest.

Now,we willsuccessivelyanalyse theinfluenceofthe fill fac-tor

α

andofthelevelofuncertaintiesr.Tothisend, wecompute the gain provided by AMFr compared to AMF interms of

proba-bility offalse alarm (Pfa) for a givenPd, i.e., 10log10 Pf a(AMF) Pf a(AMFr), for

Pd=0.5.ThisPfagain allowstomeasuretheimprovement(if pos-itive) ortheloss(when negative)oftherobustAMF withrespect

tothe standardone. Weconsiderthe so-calledV5 target forboth

theseanalysis.Fig.6representsthisfalsealarmgainwithrespect to

α

.OnthisfigurewekeepthesameconfigurationasinFig.4.We roughlyobservealinearperformanceincreaseasthetarget ampli-tudeincreases.

Tofinishwith,wenowmakevarythelevelofuncertaintieson t ,namelyr, inFig.7.As expected, weobservethat the improve-ment proposed by the robust AMF improves asthe level of un-certaintiesincreases,withlessthan1db loss whenthetarget sig-natureisperfectlyknown(r=0). Therefore,assoonasthetarget signatureisnotperfectlyknown,theproposed robustAMFhasto beconsidered.

6. Conclusions

Inthispaper,weconsideredthedetectionofatargetinan hy-perspectralimage when thebackgrounddistribution mayslightly varybetweenthe nullhypothesis andthe alternateone, duee.g., tospectralvariability.UndersuchanassumptionwederivedtheLR foranybackgrounddistribution,andthe GLRTunder thepopular Gaussiancase.Thislastdetectorisshowntobeasimplecorrection oftheAMF.Weprovidedinsightsintothebehaviourofthisrobust schemeandshoweditsgoodperformancethroughnumerical sim-ulations.

Declaration of Competing Interest

Theauthorsdeclarethattheyhavenoknowncompeting finan-cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

Appendix A. Proof of (4)

Inthisappendix,wederivetheprooftoEq.(4).Foranyrandom vector u ,wecan write u =E

{

u

}

+



u ,where



u isa zeromean vectorwiththesamecovariancematrixas u ,namely C u.Then,we

canwritethefollowingTaylorseriesexpansionaroundE

{

u

}

:

f

(

u

)

= f

(

E

{

u

}

+



u

)

f

(

E

{

u

}

)

+

f

(

u

)

uT

|

E{u}



u +12



uT



2f

(

u

)

u

uT

|

E{u}





u (A.1)

Takingtheexpectationofthisexpressionleadsto

E

{

f

(

u

)

}

f

(

E

{

u

}

)

+1 2E





uT[

2f

(

u

)

u

uT

|

E{u}]



u



(A.2) =f

(

E

{

u

}

)

+1 2Tr

{

[

2f

(

u

)

u

uT

|

E{u}]Cu

}

(A.3) CRediT authorship contribution statement

François Vincent: Conceptualization, Methodology, Validation, Formal analysis, Software, Writing - original draft, Writing - re-view & editing. Olivier Besson: Methodology, Validation, Writing -review&editing.

References

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(8)

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Figure

Fig.  3. Complete RGB view of the Viareggio test scene.
Fig.  4. Receiver Operating Characteristic for V  5  and V  6  targets, for r = 0 (no mismatch)
Fig.  6. P  f a  gain for P  d  = 0 . 5 , for V5 target, with  α = 0 . 1 and  K N  = 9
Fig.  7. P  f a  gain for P  d  = 0 . 5 , for V5 target, with  α = 0 . 1 and  K N  = 9

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