Target detection in hyperspectral imaging combining replacement and additive models

an author's https://oatao.univ-toulouse.fr/28325

https://doi.org/10.1016/j.sigpro.2021.108212

Vincent, François and Besson, Olivier Target detection in hyperspectral imaging combining replacement and

additive models. (2021) Signal Processing, 188. 108212. ISSN 0165-1684

Target detection in hyperspectral imaging combining replacement and additive models

François Vincent

^∗

, Olivier Besson

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

Keywords:

Hyperspectral imaging Replacement model GLRT

Target detection

a b s t r a c t

Inhyperspectralimagingthe replacementmodelwhere atarget,ifpresent,partly replacesthedistur- banceis oftenadvocated.Inthispaper,weconsider asomehow morerealisticmodel whereonlythe low-rankbackgroundissubstitutedforthetargetwhilearesidualnoise,whichbelongstotheorthogo- nalcomplement,isunaffectedbythepresence/absenceofthetarget.Atwo-stepgeneralizedlikelkihood ratiotestisformulatedforsuchamodel.Furthermoreweshowthattheloglikelihoodcanbewellap- proximatedby aweighted combinationoftheloglikelihoods oftheFTMFand theAMF, and thatthe dimensionofthebackgroundsubspaceisthetuningparameterwhichenablestobalancebetweenthese twowell-knowndetectors.Acomparisonwithstandard techniquesonrealhyperspectraldatarevealsa goodperformanceofthenewdetectors.

1. Introduction

Compared tostandard imagingsystems, an hyperspectralsen- sor can retrieve a precise spectral signature, composed of hun- dredsnarrowbands,ineachpixel.Thisfeatureallowstorecognize thedifferentcomponentswithintheareaofinterest.Thereby,Hy- perSpectral Imaging (HSI)has encountered a large field of appli- cations, ranging from remote sensing to medicine [1–8]. One of the main applicationconsists indetection ofa target whose sig- natureisknown[9–13].Nevertheless,thespectralsignaturesmea- suredattheremotesensor(satelliteorairplane)cannotbedirectly comparedtoanytargetsignaturesdatabase[14].Indeed,manyef- fects are likely to modify the reflectance of the area ofinterest.

Among them, we can list the non-uniform sun illumination, the atmosphericabsorptionandscattering,butalsothesensortransfer function itself.Thereby,aradiometric correctionhasusuallytobe conductedbeforeprocessinganyHSI.

Tothisend,twokindsofcompensationcan beconducted.The firstoneconsistsininvertingradiometricmodels,suchasthepop- ular MODTRAN [15]. The second one tends to identify specific groundtargetswithknownreflectance,inordertoestimatetheco- efficientsofthesupposedlinearrelationshipbetweentheradiance measured atthesensorlevelandthereflectancecomingfromthe ground. The so-calledEmpirical Line Method(ELM) [15,16]isthe bestexampleofsuchatechnique.

∗Corresponding author.

E-mail address: [email protected] (O. Besson).

Hence,afterradiometriccompensation,thedatarepresentsthe reflectanceatgroundlevel.Asaconsequence,ifasub-pixeltarget withknown signaturetis presentina PixelUnder Test(PUT) y, it will mask a partof the background reflectance b, andthe so- called replacement model is usually advocated, namely y=

α

^t+ (¹−

α

)^b,where

α

^{representsthe} replacementfraction.Ifthisso- calledfillfactor

α

^{issmall,thesimplifiedandwidelyusedadditive}

modelcanbeused,namelyy=

α

^t+b.

Nevertheless, using such a popular replacement model raises differentissues.Indeed,thismodelassumesa perfectradiometric compensation.Thereby, thePUT yis a simplelinear mixture be- tweenofthetargetandthebackground,sothatwhenthetargetis afull pixeltarget (

α

=1),there isnomorestochastic partinthe PUT,whichisnottotallyrealisticforreallifemeasurements.More- over, as the background is usually composed of a small number ofcomponents,so-calledendmembers, thebackgroundbelongsto a low-rank subspace. Then, as the amplitudes of each endmem- ber, so-called abundances, are usually assumed to be stochastic, the background covariance matrix hasalso a low-rank structure, which isanother problem, as mostpopular detectors exploit the inverseofsuchamatrix,thatissingularinsuchacase.

In thispaper, we propose to consider a more realistic model, whichalleviatestheabove-mentionedproblemsbyaddingaresid- ualpartntothereplacement model,namelyy=

α

^t+(¹−

α

)^b+ n. The existence of this residual additive noise as been raised in some papers showing that at best, one can reach a 5% re- flectance accuracy, using well-tuned radiometric compensation methods [17]. Indeed, there always remains a reflectance error

https://doi.org/10.1016/j.sigpro.2021.108212

due tonon-modelled effects such asspatialillumination changes or unknown target’s attitude. Then, it seems relevant to model thisresidual errorusingazero-meanGaussian distributedvector, namelyn∼N(⁰,C_n)^{.Thisway,themodelproposedinthispaper} becomesamixedreplacementandadditivemodel,andwewillsee here-afterhowitcangeneralizethem.

Basedonthismoregeneralandrealisticmodel,wewillderive, in thispaper, the generalizedlikelihood ratio test (GLRT) forthe problemathand.Tothisend, we firststate thehypotheses asso- ciated withthisproblem,inSection2,andtheexactGLRTformu- lation is derived inSection 3.Then, in orderto simplify thisex- pression, we propose to rewrite the problem athand in a more relevantwayinSection4.Thisnewformulationallowstoderivea closed-formandasymptoticallyefficientexpression ofthefillfac- tor. This simpler estimation conducts to a closed-from andmore explainableexpressionofthetest,thatcanberelatedtotheGLRT of the additive and the replacement models, namely the Adap- tiveMatchedFilter(AMF)[18]andtheFiniteTargetMatchedFilter (FTMF)[19].Tofinishwith,wecomparethenewdetectorstostate of the art algorithms, on a real HSI benchmarking, in Section 5. ConcludingremarksendthispaperinSection6.

2. Detectionproblemstatement

Asstatedintheintroduction,thetargetdetectionproblemcan be writtenasatwohypothesestest,basedonthefollowingmore realisticmixedadditiveandreplacementmodel:

H0 y=b+n

H₁ y=

α

^t+

(

¹⁻

α )

^{b+n (1)}

where both the background b and the residual noise n are as- sumed to be independent Gaussian distributed vectors, namely b∼N(

μ

,C_b) ^{and n}∼N(⁰,C_n)^{. As usually assumedin this con-} text,thetargetsignaturetissupposedtobeknown,andtheback- ground and noise can be learnt from secondary pixels assumed to be distributed as y under H0, namely z_k∼N(

μ

,C_b+Cn) ^for k=0,...,(^K−1)^.

As stated in the introduction, it is usually assumed that the background is composed of a small number of endmembers, so that C_b isa low-rankmatrixthat canbe writtenasC_b=Us^sUTs, wherethecolumnsofUsare thereigenvectorsand^{s isthedi-} agonal matrixcomposed ofthe reigenvalues ofC_b.Onthe other hand, C_n is assumed to be full-rank and can be written as C_n= Us^snU^T_s+Un^nUTn.As nis onlya residualerror, we willassume hereafter that ^sn isnegligible compared to s.This assumption amounts to neglect the energy of the projection of n onto the background subspace comparedto thebackground energy,which is a far less restrictive assumption than in the conventional re- placementmodelwherenissimplynull.

In thispaperwefocuson aso-calledtwo-step approach.That is,weassumethatboththebackgroundandnoiseparametersare known from the secondary pixels z_k. Indeed, as z_k∼N(

μ

,Cz= C_b+Cn)^{,wecaneasilyestimateboththemeanandthecovariance} matrix,inaMaximumLikelihood(ML)senseas

μ

ˆ =_K¹K−1

k=0z_kand Cˆz=K¹

K−1

k=0(^zk−

μ

)(^zk−

μ

)^{T. Then, using} eigen-decomposition, wecanestimateUs,Un,(s+^sn)^andnforagivenr.Adiscus- sionabouthowtochooseraswellasitsinfluenceonthedetector structurewillbedevelopedinSection4.

3. GLRTDerivation

Assuming that the residual noise energy is negligible in the background subspace (^sn^s)^{, as explained in the pre-} vious section, the data under H₁ are distributed as y∼

N

α

^t+(¹−

α

)

μ

,(¹−

α

)^2UssU^T_s +UnnU^T_n

. Then, the log like- lihoodwrites

L

( α

;y

)

−1

2[Nlog2

π

+log

((

¹−

α )

^2r

|

s

| )

+log

( |

n

| )

+

(

^y−

α

^t−

(

¹⁻

α ) μ )

^T

(

^Us

(

¹⁻

α )

²

^sUTs+Un

^nUTn

)

⁻¹

×

(

^y−

α

^t−

(

¹−

α ) μ )

^{] (2)}

or,moresimply

L

( α

;y

)

−rlog

(

¹−

α )

−

^ys−

α

^ts

²

2

(

¹−

α )

^{2 −} 1

2

^yn−

α

^tn

^2+const.

(3) where

^{stands for the Euclidean norm of the vector, ys}= ⁻s¹²U^T_s(^y−

μ

)^{, t}s=⁻s¹²U^T_s(^t−

μ

)^{, y}n=⁻n¹²U^T_n(^y−

μ

) ^{and t}n= ⁻n¹²U^T_n(^t−

μ

)^{are thecentredandwhitenedversionsofthepro-} jectionsofboth thedataandthetarget ontothenoise andback- groundsubspaces.

In orderto derive the GLRT,we need to estimate the fillfac- tor

α

^{. Tothisend, we haveto maximize thislog likelihoodwith}

respectto

α

^.Differentiating(3)withrespectto

α

^{andsettingthe}

resulttozeroleadstothefollowing4thorderequation:

(

^tTntn

)(

¹−

α )

⁴+

(

^tTn

(

^yn−tn

))(

¹−

α )

³+r

(

¹−

α )

²

−

(

^tTs

(

^ys−ts

))(

¹−

α )

−

^ys−ts

^{2=0 (4)}

Then,

α

ˆML, the ML estimation of

α

^{can be obtained by finding}

theroots ofsuch an equation, butno simpleclosed-formformu- lation exists. It can be noticed that when r=0, ys=ts=0, and Eq.(4)reducestothefirstorderequationassociatedwiththestan- dardadditive model.Ontheother hand,whenr=N,yn=tn=0, and Eq. (4) reduces to that of the FTMF derivation [20], which comfortsthevalidityofEq.(4).

Then,theGLRTfortheproblemathandwrites

GLRT=L

( α

^ˆML;y

)

−L

(

⁰;y

)

⁽⁵⁾

Althoughoptimal,thisnonclosed-formexpressionisalsodifficult to analyse in a comprehensive way. In the next section, we will givean equivalentclosed-formexpression(eq.(15))forthisGLRT, thatismoreeasilyrelatedtobothAMFandFTMF.

4. ApproximateGLRTandinsights

Inspecting Eq. (3), we can see that L(

α

;y)=Lr(

α

;ys)+ La(

α

;yn)+const.where

Lr

( α

;ys

)

=−rlog

(

¹−

α )

−

^ys−

α

^ts

²

2

(

¹−

α )

^{2 (6a)}

La

( α

^;yn

)

⁼⁻¹₂

^yn−

α

^tn

^{2 (6b)}

correspondrespectivelytotheloglikelihoodofareplacementand an additive model.Indeed,when projectingthe datayonto both the background subspace Us andthe noise subspace Un, and af- ter thewhiteningsteps by⁻s¹² and⁻n¹²,we havethefollowing equivalentmodelforthedata:

ys=

α

^ts+

(

¹−

α )

^{bs (7a)}

yn=

α

^{tn+nn (7b)}

wherebs=⁻s¹²U^T_s(^b−

μ

)∼N(⁰,I) ^andnn=⁻n¹²U^T_nn∼N(⁰,I)^, andwehavetodecidewhether

α

=0ornot.

This new writingof the model is then equivalent to the ini- tialmodel,andthelikelihoodofobservingbothysandynremains 2

L(

α

;y)=Lr(

α

;ys)+La(

α

;yn)^{.Butthebenefitsofsucha subspace} decompositionistoeasilylinktheinitialproblemtomorefamiliar ones,namelyareplacementandanadditivemodel.

Now, in order to simplify the estimation of

α

^{, we can ex-}

ploit thisnatural splitting ofthe general model.To this end, we can refertotheEXtended InvariancePrinciple(EXIP)[21,22].This procedure allows to replace a challenging maximization by two simpler steps, while preserving the efficiency of the ML estima- tion,atleastasymptotically.Thisextension ofthepopularMLre- parametrizationtonon-bijectivefunctionshasbeenwidelyusedin many areas, sometimes without explicitly naming it [23–29]. Its principleisshortlydescribedinthefollowingsub-section.

4.1. EXIPprinciple

ThemainideabehindEXIPisare-parametrizationoftheprob- lem at hand whose goal is to simplify the minimization. Then, we can estimate the initial parameters by solving an optimally weightedleast-squares minimization problem. Moreprecisely, as- sumeonewantstominimizeafunctionV(

θ

)^(V(

θ

)=−L(

θ

)^,inour caseofinterest)wheresomestructureisimposedon

θ

^.Often,re-

laxingtheconstraintsbyreparameterizingtheproblemintermsof alessstructured vector

η

= f(

θ

)^{leadstoasimplesolution,say}

η

^ˆ.

Then,onecangetanasymptoticallyequivalentsolutionon

θ

^from

theintermediateestimate

η

ˆ ^as:

θ

ˆ=argmin

θ

( η

^ˆ−f

( θ ))

^TW

( η

^ˆ−f

( θ ))

⁽⁸⁾

where W=E

∂

^2V

( η )

∂ η∂ η

^T

(9)

The goalofEXIPisto replaceachallengingmaximization bytwo simplerones,whilepreservingasymptoticallytheperformance.

4.2. FillfactorestimationusingEXIP

In our case ofinterest, referring to Eq.(7), we first relax the constraints between the two sub-models, defining

η

=[

α

^r

α

^a]T,

where

α

^{risthefillfactorforysand}

α

^{aisadifferentfillfactorfor}

yn.Thisway,wecanseparately estimatethefillfactorforthe re- placementmodelononehand,andfortheadditive modelonthe other hand,as they should be the sameaccording to model (7). Then,we willrefinetheestimationof

α

^{thankstothemeanleast}

square optimization(8).Thefillfactorsforeach modelareshown tobe[20]:

α

ˆ^r=1−tTs

⁽

^y₂^s_N^−ts

⁾

⁻

(

^tTs

(

^ys−ts

))

²+4N

(

^ys−ts

)

^T

(

^ys−ts

)

2N

(10)

α

ˆ^a=t_t^Tn_T^yn

ntn

where

α

ˆ^randis

α

^ˆaare respectivelythefillfactors derivedinthe FTMF[19]andtheAMF[18]detectors.

Nowtointegratebackthelinksbetween

α

^rand

α

^a,namely

η

= f(

α

)=

α

^1,andgetanasymptotically efficientestimationof

α

^,we

haveto solvethefollowingleastsquaresproblem, thankstoEXIP procedure:

α

ˆEXIP=argmin

α

( η

^ˆ−

α

¹

)

^TW

( η

^ˆ−

α

¹

)

⁽¹¹⁾

where

W=−E

∂

^2L

( η

;y

)

∂ η∂ η

^T

=

⎛

⎝

^−E

∂^2Lr(α^r;ys)

∂αr²

0

0 −E

∂^2La(αa;yn)

∂α²a

⎞

⎠

Fig. 1. SFMF understanding vs the background rank r.

(12) and

η

ˆ=[

α

ˆ^r

α

^ˆa]T.ItisstraightforwardtoshowthatWwrites W=

wr 0 0 wa

(13)

wherewr=₍^t₁^Ts₋^ts_α⁺_ˆ^2r

r)² andwa=t^T_ntn.

Thereby,thesolutionofEq.(11)isdirect,andgives

α

ˆEXIP=wr

α

ˆ^r+wa

α

ˆ^a wr+wa

(14)

Thisasymptotically efficientestimationof

α

^{dependsontherank}

of the background subspace r that can be estimated from the samplecovariancematrixofthe secondarypixels. Indeed,weas- sume that the eigenvalues of ^{n are smaller than those of s,} asn representsonly aresidual error.Then, differenttechniques can be used to estimate the two subspaces, such asthe popular AkaikeInformationCriterion(AIC)[30],forinstance.Aspresented inSection 5, we consider herea subspace splittingsimply based onanenergycriterion.

Ontheotherhand,rcanalsobeconsideredasatuningparam- etertochoosebetweenafullreplacementmodelifr=Nandafull additivemodelifr=0.Indeed,inthefirstcase,thenoisesubspace isnullandEq.(7)resumestoareplacementmodel,andw_a=0so that

α

ˆEXIP=

α

ˆ^{r. It is the opposite inthe second case, where the} simpler additive modelprevails on all the space, and

α

ˆEXIP=

α

ˆ^a, aswr=0.

Tofinish with, asthecriterion to be maximizedis a loglike- lihoodfunction, it canbe noticed that w⁻_r¹ andw⁻_a¹ are alsothe Cramer-RaoBounds(CRB)forthefillfactorsinareplacementand anadditivecase.

4.3. Subspacefittingmatchedfilter

Giventhissimplerestimationfor

α

^{,theGLRTnowwrites} SFMF=L

( α

^ˆEXIP;y

)

−L

(

⁰;y

)

=[Lr

( α

^ˆEXIP;ys

)

^−Lr

(

^0;ys

)

^]+[La

( α

^ˆEXIP;yn

)

^−La

(

^0;yn

)

^] (15) where we recall that L_r and L_a are the Log likelihoods for the replacement and additive part, respectively, and that

α

ˆEXIP→

α

ˆ^r when r→N, and

α

ˆEXIP→

α

ˆ^{a when r}→0, and that Lr(

α

,y) ^or La(

α

,y)^{are0foreachofthesetwocases.Hence,theproposedtest} alsotends toward FTMFor AMFdependingon thechoice on the covariance matrix rank r, namely SFMF→[Lr(

α

^ˆr;y)−Lr(⁰;y)^]= FTMFwhenr→NandSFMF→[La(

α

^ˆa;y)−La(⁰;y)^]=AMF when r→0.Then,rcanbeseenasatuningparametertofavourreplace- mentoradditivity,asshownonFig.1.Thatisthereasonwhywe choosetonamethisnewdetectorasSubspaceFittingMatchedFil- ter(SFMF).

Moreover,FTMFandAMFareknowntohavedifferentbenefits.

FTMFis knowntobe moreselectivethan AMF[31,32].Then ina complicatedscenario,wherepossibletarget-likebackgroundcom- ponentsmayexist,usingtheFTMFwillbebetterthanAMF.Onthe contrary,AMFismorerobustandcanimprovedetectioninamore simplescenario.TheproposedSFMFshouldkeepthesetwoadvan- tages,asitisthebestassociation ofthosetwopopulardetectors, withrespecttotherealisticmodelproposedinthispaper.

Fig. 2. RGB view of the Viareggio test scene.

Combining two detectors withdifferent features is not a new approach andithas beenstudied intheradar field [33] butalso inHSIfieldwiththeFalseAlarmMitigation(FAM)techniques[34–

38],forinstance. Nevertheless,unlike in our case, thesekinds of techniques use the same data asan input for the two detectors andone has tochoose differentthresholdsandcombine thetwo teststotakeafinaldecision.

5. Performanceevaluation

We now propose to compare the two kinds of detectors in- troduced in this paper to the state of the art ones, on a real- life benchmarking. Tothis end, we consider the airborneViareg- gio2013trial[39]thattookplaceinViareggio(Italy),inMay2013, withan aircraftflying at1200 meters.The image iscomposed of [450×375] pixels with511 samplesin the Visible Near InfraRed (VINR)band(⁴⁰⁰−1000nm)^.

Differentkindsofvehiclesaswellascolouredpanelsservedas known targets.Foreach ofthesetargets, a spectral signatureob- tained fromgroundspectroradiometermeasurements isavailable.

As can be seen on Fig.2, thescene is composed ofparking lots, roads,buildings,sportfieldsandpinewoods.

Inaddition,calibrationtargetsareusedtoconductanEmpirical Line Method(ELM)[15,16]toconverttherawmeasurementsinto reflectance. Then a spectral binning [40] is performed to reduce thevectorsizedimensiontoN=64.

In this benchmarking, we compare both SFMF and the GLRT based on the exactsolution for

α

^{(Eq.(5)) with AMF[18], FTMF}

[19],obviously,butalsowithKelly[41–43],ACE[44–46],themod- ified FTMFfrom [47]and SPADE[47]. We usethe so-calledfalse alarm scorestocomparethesealgorithms,namelythe numberof pixels havingtheir detector’soutput strictly higherthan the one forthetargetpixel.Thisnumbercanbeseenasafalsealarmnum- berwithanoptimalthresholding.

We focus on two kindsof targets, namelya panel (P₂) anda vehicle(V₆),ascanbeseenonFig.2.Thosetwotargetsareafew meterslengthandthespatialresolutionoftheimage isabout0.6 meters, sothat we chosea guardwindow sizeof9×9pixels, in order to avoidthe presence oftarget signaturein the covariance matrix estimation window. At last, the rank r is estimated from thesecondarypixelscovariancematrixbythresholdingtheeigen- values. Indeed,the residual noise energy beingquite small, after the radiometric compensation, we chose the background so that

Fig. 3. Covariance matrix background subspace rank estimation on D 2 F12 H2 image ( 19 ×19 window).

its energy islarger than 0.99 ofthe total energy. Fig.3 gives an exampleofthe estimatedrankon thewhole image fora19×19 window.As canbe seen,thisrankvariesbetween1and52with respect to the homogeneityof the area of interest in the image.

However,themeanrankvalueremains low(r¯=8.2)withrespect tothecovariancematrixdimensionN=64.

Fig.4representsan exampleofsome detectorsoutputs inthe surroundings of P₂ target, which is located in the center of the maps.As expected, we can seethat AMF presentsa quite strong peak at the target location, but also possible false alarms peaks corresponding to target-like pixels. This last point isclearly visi- bleonKelly’s output.FTMFlowersthesetarget-likepossible false alarms, butthetarget’speak is alsoreduced. The two GLRTpro- posedinthispaperofferaverygoodcompromisebetweendetec- tion performance andselectivity (falsealarm mitigation),asthey bothhaveahighpeakatthetargetlocation,whilemaintainingthe levelofthetarget-likespixelquitelow.

ThisresultcanalsobeobservedinFig.5wherethefillfactors estimatedby thedifferent methodsare representedforthe same localarea ofinterest.We can seethat AMFfinds

α

0.65atthe targetposition,whichislowerthantherealvalueof

α

^,aswecan

expectafullpixeltargetinthecentralpixel.Thisunder-estimation iscertainlylinkedtotargetsignaturevariabilityandincompletera- diometriccompensation,asmentionedinthispaper.AMFalsoes- timatesstrongtarget-likeitemswith

α

^{upto0.4innon-targetpix-}

els.Ontheotherside,thesepeaksarereducedwithFTMF,butthe target’speakisalsoreducedto

α

=0.42.BoththeMLandtheEXIP estimationof

α

^{,usingthemodelproposedinthispaper,areamix}

between thesetwo extremes, andincrease the gap between the target and the target-like pixels. This fillfactor estimation being thecentral point forthetest performance,we can expecta good behaviour ofthe two associated detectors,as canbe seen in the followingtables.

Tables 1 and2 presentthe false alarm scores forboth the P₂ andtheV₆targetsanddifferentwindowsizes.First ofall,we can observeasubstantialgapinperformancebetweenthetwotargets, V₆ being moredifficult to detect.Moreover,we canalso observe a differenceintherankingofthemethodsbetweenthetarget type.

Indeed,ACE is the bestalgorithm on theV₆ target, whereas it is worseon theP₂ target.Hence, asitisalreadyknown,theperfor- manceofeachmethodiscloselyrelatedtothekindsofimageand target. Nevertheless, the two proposed algorithms seems to per- form well in all the cases, as they always rank among the best 4

Fig. 4. detector’s output near the target position.

Fig. 5. fill factor estimations output near the target position.

Table 1

False Alarms score for P 2target in the D 1 F 12 H2 Viareggio open data image .

Window Size SFMF Eq. (5) AMF FTMF Kelly ACE Modif. FTMF SPADE

17 ×17 1 1 3 7 3 7 3 3

19 ×19 0 0 2 5 2 4 2 2

21 ×21 0 0 0 4 0 2 0 0

Table 2

False Alarms score for V 6target in the D 2 F 12 H2 Viareggio open data image .

Window Size SFMF Eq. (5) AMF FTMF Kelly ACE Modif. FTMF SPADE

17 ×17 4 2 36 241 4 0 37 14

19 ×19 4 2 33 123 10 0 40 21

21 ×21 4 1 29 103 10 0 35 18

ones. The exact GLRT for the mixedmodel proposed inthis pa- per (Eq.(5)) performsslightlybetterthantheSFMF,buttheprice tobe paidisatosolvea4thorderpolynomialequation,whereas SFMFisclosed-form.

6. Conclusions

In thispaper, weconsidered targetdetectioninan hyperspec- tral imaging context.We introduced a morerealistic modelthan theconventionalreplacementmodel,andderivedthecorrespond- ing GLRT.This model isshown to be approximately a mixedre- placement and additive model,so that we proposed a simplified versionoftheGLRTthatcanbeeasilylinkedtoboththeFTMFand theAMF, whichholdrespectivelyforthereplacementandforthe additive model.Asaconsequence,thisproposed detector,namely SFMFinheritsthe benefitsofthesetwo detectors,namelyagood selectivitywhilepreservingdetectionperformance.

DeclarationofCompetingInterest

Theauthorsdeclarethattheyhavenoknowncompetingfinan- cialinterestsorpersonalrelationshipsthatcouldhaveappearedto influencetheworkreportedinthispaper.

CRediTauthorshipcontributionstatement

François Vincent: Conceptualization, Methodology, Software, Writing -original draft. OlivierBesson: Methodology, Validation, Formalanalysis,Writing-review&editing.

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