Sub-pixel detection in hyperspectral imaging with elliptically contoured t-distributed background

an author's

https://oatao.univ-toulouse.fr/26072

https://doi.org/10.1016/j.sigpro.2020.107662

Besson, Olivier and Vincent, François Sub-pixel detection in hyperspectral imaging with elliptically contoured

t-distributed background. (2020) Signal Processing, 175. 107662-107667. ISSN 0165-1684

Short

communication

Sub-pixel

detection

in

hyperspectral

imaging

with

elliptically

contoure

d

t

-distribute

d

background

Olivier

Besson

a,∗

_,

_François

_Vincent

a

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

Keywords: Detection

Generalized likelihood ratio test Hyperspectral imaging Replacement model Student distribution

a

b

s

t

r

a

c

t

Detectionofatargetwithknownspectralsignaturewhenthistargetmayoccupyonlyafractionofthe pixelisanimportantissueinhyperspectralimaging.Werecentlyderivedthegeneralizedlikelihoodratio test(GLRT)forsuchsub-pixeltargets,eitherfortheso-calledreplacementmodelwherethepresenceof atargetinducesadecreaseofthebackground power,duetothesum ofabundancesequaltoone,or foramixedmodelwhichalleviatessomeofthelimitationsofthereplacementmodel.Inbothcases,the backgroundwasassumedtobeGaussiandistributed.Theaimofthisshortcommunicationistoextend thesedetectorstothebroaderclassofellipticallycontoureddistributions,morepreciselymatrix-variate

t-distributionswithunknownmeanandcovariancematrix.Weshowthatthegeneralizedlikelihood ra-tiotestsinthet-distributedcasecoincidewiththeirGaussiancounterparts,whichconfersthelatteran increased generalityfor application.The performanceas wellas the robustnessofthesedetectors are evaluatedthroughnumericalsimulations.

1. Problemstatement

Hyperspectralimaginghasbecomeanincreasinglypopulartool for remote sensing and scene information retrieval, whether for civil ormilitaryneeds andin a largenumber ofapplications, in-cludinganalysisofthespectralcontentofsoils,vegetationor min-erals,detectionofman-madematerials orvehicles,tonameafew [1,2].Oneofthechallengesofhyperspectralimagingistodetecta target-whosespectralsignatureisassumedtobeknown-withina backgroundwhosestatisticalpropertiesarenotfullyknown[3–5]. Depending onthe spatialresolution ofhyperspectral sensors and thesizeofthetarget,thelattermayoccupythetotalityoronlya fractionofthepixelundertest(PUT),inwhichcaseonespeaksof sub-pixeltargets.Inthelattercase,thetargetreplacespartofthe backgroundinthePUT,leadingtotheso-calledreplacementmodel [6,7].

Whateverthe case,full-pixel orsub-pixel targets,the problem can be formulated asa conventional composite hypothesis prob-lem[3–9]:givena vectory∈Rp_{-wherepdenotesthenumberof} spectral bandsused-whichrepresentsthereflectance inthePUT, is there a component along t-the signature of interest (SoI)-in addition to the background? Since the background statistics de-pend onunknown parameters (for instancemeanandcovariance

∗ _{Corresponding author.}

E-mail addresses: [email protected] (O. Besson), [email protected] (F. Vincent).

matrix) a set of trainingsamples Z∈Rp×n_{, hopefully free of the}

SoIt,is observedwhosestatisticsare assumedtomatchthoseof thebackgroundinthePUT.Thesetrainingsamplesaregatheredin thevicinityofthePUT(localdetection)oralongthewholeimage (globaldetection).

Recentlyin[10]weaddressedsub-pixeldetectionusingthe re-placement model under a Gaussian background, and we derived the plain generalized likelihood ratio test (GLRT) by maximizing thejointdistributionof(y,Z)withrespecttoallunknown param-eters.Moreover,motivatedbysomelimitationsofthereplacement model,especiallythefact thatthefillingfactorofasub-pixel tar-getmay not be inpracticeas largeas expected, we alsoderived theGLRT fora mixedmodel where presence ofa target induces a partial replacement of the background [11]. These two detec-tors assume a Gaussian background. However, evidence of non-Gaussianity of hyperspectral data has been brought [12,13] and thereforeitisofinteresttoextenddetectorsoriginallydevisedfor Gaussianbackgroundtothebroaderclassofellipticallycontoured (EC)distributions[14,15].Theaimofthiscommunicationisthusto extendourrecentGLRTsfromtheGaussiancasetothematrix vari-atet-distributedcase.Wewillshowthat theGLRTscoincidewith theirGaussian counterparts. Thepaperisorganizedasfollows.In Section 2, we consider each of the three models andderive the correspondingGLRTs.ThelatterareevaluatedinSection3on sim-ulateddatabutwherethetarget spectralsignature,themeanand covariancematrixofthebackgroundareobtainedfromreal hyper-spectralimages.

2. GLRTformatrixvariatet-distributedbackground

Asstatedbefore,letusassumethatwewishtodecidewhether agivenvector ycontainsasignatureofinteresttinthe presence of disturbance z whose mean value

μ

and covariance matrix



areunknown,andletusassume thatasetoftrainingsampleszi,

i₌1_,...,n are available which share the same distribution as z. ThesesamplescanbecollectedaroundthePUToralongthewhole image. We simply assume herethat n> p. Therefore, we would liketosolvethefollowingproblem:

H0:y=z; zi d

=z,i=1,...,n

H1:y=

α

t+

β

z; zi=d z,i=1,...,n (1) where =d means “has the same distribution as”. In (1), t corre-spondstotheassumedspectralsignatureofthetarget and

α

de-notesits unknown amplitude. When

β

=1 one obtains the con-ventionaladditivemodel.When

β

=1−

α

thereplacementmodel isrecovered,andthemixedmodelcorrespondstoanarbitrary

β

.

In orderto derive theGLRT,we need to specifythejoint dis-tributionofyandZwhereZ=



z1 z2 ... zn



.Assaidinthe introduction,we assume that



y Z



follows a matrix-variate t -distributionwith

ν

degreesoffreedom so that we need tosolve thefollowingcompositehypothesistestingproblem:

H0:



y Z



=d Tp,n+1

(

ν

,M0,

(

ν

− 2)



,In+1

)

H1:



y Z



=d Tp,n+1



ν

,M1,

(

ν

− 2

)



,



β

2 ₀T 0 In



(2) where M0=



μ μ

1T n



, M1=



α

t+

βμ μ

1T n



, 1n is a n × 1 vectorwithallelementsequaltoone,

μ

standsforthemeanvalue ofthe backgroundwhile



denotes its covariance matrix.In (2),

T

()

standsforthematrixvariatet-distribution[16,17]sothat the probabilitydensityfunction(p.d.f.)oftheobservationsundereach hypothesisisgivenby p0

(

y,Z

)

=C

|



|

− n+1 2





_I p+



−1

ν

− 2



y−

μ

Z−

μ

1T n



×



y−

μ

Z−

μ

1T n



T





− ν+n+p 2 p1

(

y,Z

)

=C

β

−p

|



|

− n+1 2





_I p+



−1

ν

− 2



˜ y−

μ

Z−

μ

1T n



×



y˜−

μ

Z−

μ

1T n



T





− ν+n+p 2 (3) with y˜₌

β

−1

(

y₋

α

t

)

andC₌ p((ν+n+p)/2) πp(n+1)/2_p((ν+p−1)/2). It should be

observed that the columns of



y Z



are only uncorrelated but not independent, as p

(

y_,z1,...,zn

)

cannot be factored as

p

(

y

)

n_i=1p

(

zi

)

.

WenowderivetheGLRTfortheproblemin(2).Letusstartby consideringthe following function f(



) where S issome positive definitematrix: f

(

)

=

|



|

−n+1 2



_I p+

(

ν

− 2)−1



−1S



− ν+n+p 2 =

|



|

ν+p−1 2





+

(

ν

− 2)−1_S



− ν+n+p 2 (4)

Differentiationoflogf(



)yields

∂

logf

(

)

∂



=

ν

+ p− 1 2



−1₋

ν

+n+p 2

(



+

(

ν

− 2

)

−1S

)

−1. (5)

Settingthisderivativeoftozero,wecanseethatf(



)achievesits maximumat



∗=

₍

_ν

(

ν

_{− 2}+p

₎₍

− 1)_n+1S

₎

=

γ

S (6) Itfollowsthat max  p0

(

y,Z

)

=C 



_



_y−

_μ

_Z−

_μ

₁T n



₍

y−

μ

)

T

(

Z−

μ

1T n

)

T





− n+1 2 max  p1

(

y,Z

)

=C 

_β

−p







_y˜−

_μ

_Z−

_μ

₁T n



₍

y˜−

μ

)

T

(

Z−

μ

1T n

)

T





− n+1 2 (7)

withC=C

γ

−p(n+1)/2_[1+

₍

_ν

_{− 2}

₎

−1

_γ

−1_]−p(ν+n+p)/2_.SinceC_isthe

sameunderH0andH1 itwillcanceloutintheGLRandtherefore theGLRdoesnotdependon

ν

.Thismeansthat

ν

doesnotneedto beknownsinceitsestimationisnotactuallyrequiredtoobtainthe GLR.

Now,foranyvectorx,

M

(

μ

)

=



x−

μ

Z−

μ

1T n



x−

μ

Z−

μ

1T n



T =

(

x−

μ

)(

x−

μ

)

T₊

₍

_Z−

_μ

₁T n

)(

Z−

μ

1Tn

)

T =xxT₋

_μ

_xT_{− x}

_μ

T₊

_μμ

T +ZZT₋

_μ

₁T nZT− Z1n

μ

T+n

μμ

T =

(

n+1

)

μμ

T₋

_μ

₍

_x+Z1 n

)

T−

(

x+Z1n

)

μ

T +xxT_+ZZT =

(

n+1

)

μ

−x+Z1n n+1



μ

−x+Z1n n+1



T +xxT_+ZZT₋

(

x+Z1n

)(

x+Z1n

)

T n+1 =

(

n+1

)

μ

−x+Z1n n+1



μ

−x+Z1n n+1



T +



x Z



In+1− 1n+11Tn+1 n+1



x Z



T (8) Consequently, min μ

|

M

(

μ

)

|

=







x Z



P⊥_n+1



x Z



T





(9)

with P⊥_n+1 the orthogonal projector on the null space of 1n+1.

Hence,wearriveat max μ, p0

(

y,Z

)

=C 



_



_{y Z}



_P⊥ n+1



y Z



T





− n+1 2 max μ, p1

(

y,Z

)

=C

β

−p



_



_{y˜ Z}



_P⊥ n+1



˜ y Z



T





− n+1 2 (10)

Next,foranyvectorxandmatrixQ(notnecessarilyP⊥_n+1),



x Z



Q



x Z



T=



x Z

Q

11 Q12 Q21 Q22



x Z



T =Q11xxT+ZQ21xT+xQ12ZT+ZQ22ZT =Q11



x+Q₁₁−1ZQ21



x+Q₁₁−1ZQ21



T +ZQ2.1ZT (11) whereQ2.1=Q22− Q21Q11−1Q12.Therefore,







x Z



Q



x Z



T





=



ZQ2.1ZT



×

1+Q11



x+Q₁₁−1ZQ21

T



ZQ2.1ZT

−1



x+Q₁₁−1ZQ21



(12)

Comingback tothecaseQ₌P⊥_n+1=In+1−

(

n+1

)

−11n+11Tn+1,we

have Q=



1 0T 0 In



−

(

n+1

)

−1



1 1T n 1n 1n1Tn



(13) sothat

Q11=1−

(

n+1

)

−1=n

(

n+1

)

−1 Q21=−(n+1

)

−11n

Q22=In−

(

n+1

)

−11n1Tn

Q2.1=In− n−11n1Tn=P⊥n (14) It follows that Q₁₁−1ZQ21=−n−1Z1n=−¯z and ZQ2.1ZT=ZP⊥nZT= ZZT_{− n¯z¯z}T_{=S.Hence,theGLRisgivenby}

GLR=



1+Q11

(

y− ¯z

)

TS−1

(

y− ¯z

)



(n+1)/2 min_α,β

β

p_[1+Q 11

(

y˜− ¯z

)

TS−1

(

y˜− ¯z

)

](n+1)/2 = [1+ n n+1

(

y− ¯z

)

TS−1

(

y− ¯z

)

](n+1)/2 min_α,β

β

p_[1+ n n+1

(

y−βαt− ¯z

)

TS−1

(

y−βαt− ¯z

)

](n+1)/2 (15)

A few important observationscan be maderegarding this re-sult. First,forallthreemodels,theGLRs in(15)coincide withtheir Gaussian counterparts. Forthe additive modela proof is givenin AppendixA.Amoreintuitive waytofigureoutthisequivalenceis torealizethat theexpressionoftheGLRin(15)doesnotdepend on

ν

andthat, letting

ν

grow to infinity, one should recoverthe GLRforGaussiandistributeddata.Asforthereplacementandthe mixedmodels,theexpressionin(15)isexactlythatoftheACUTE andSPADEdetectorsof[10]and[11]respectively,wheretheGLRTs forthereplacementmodelandthemixedmodelarederivedunder theGaussianassumption.Therefore,thelatterarestillGLRTsfora muchbroaderclassofdistributionsthaninitiallyexpected.

LetusalsobrieflycommentontheimplementationoftheGLRT. For the additive model

β

=1, and the minimization problemin (15) isa simple linearleast-squares problem forwhicha closed-formsolutioncanbeobtained.Thisyields

GLR2_AM/(n+1)= 1+ n n+1

(

y− ¯z

)

TS−1

(

y− ¯z

)

min_α1+ n n+1

(

y− ¯z−

α

t

)

TS−1

(

y− ¯z−

α

t

)

= 1+ n n+1

(

y− ¯z

)

TS−1

(

y− ¯z

)

1+ n n+1

(

y− ¯z

)

TS−1

(

y− ¯z

)

−n+1n [(y−¯z) T_S−1t]2 tT_S−1t ≡ n n+1[

(

y− ¯z

)

TS−1t]2 [1+ n n+1

(

y− ¯z

)

TS−1

(

y− ¯z

)

][tTS−1t] (16)

The GLRin(16)generalizesKelly’s detectortothe caseof a non-centered Student distributed background. Note that (16) differs fromKelly’sdetectorbythe _n+1n factor.

Asforthereplacementmodel,

β

₌1₋

α

andtheminimization shouldbeconductedwithrespectto

α

only,i.e.,

GLRRM=

1+ n n+1



S−1/2

(

y− ¯z

)



2



(n+1)/2 min_α

(

1−

α

)

p

₁₊ n n+1 S−1/2_(y−α_t−(1−α)_¯z)2 (1−α)2



(n+1)/2 (17)

Asshownin[10],thissimplyamountstofindingthe(unique) pos-itiverootofa2nd-orderpolynomial.Finally,forthemixedmodel where

β

isarbitrary,one hasa2-D minimization problem. How-ever,minimizationover

α

canbedoneanalytically, leavingonlya minimizationover

β

: GLRMM=

1+ n n+1



S−1/2

(

y− ¯z

)



2



(n+1)/2 min_β

β

p



1+ n n+1  P⊥ S−1/2tS −1/2_(y−β¯z)_ 2 β2



(n+1)/2 (18)

Again the solution is obtained as the unique positive root of a second-order polynomialequation [11].Thereforeforboththe re-placement modeland themixedmodel, all unknown parameters canbeobtainedinclosed-form.

3. Numericalsimulations

In the present section, we will compare the detectors devel-oped above.The additive model GLRAM in(16)will be referred to

as mKELLY in the sequel as it consists in a slight modification of Kelly’s original GLRT [18]. The replacement model GLRRM in

10-4 10-3 10-2 10-1 100

Probability of false alarm

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Probabi li ty of detecti o n

Student background (ν =5), replacement model, α = 0.05

GLRAM(mKELLY)

GLRRM(ACUTE)

GLRMM(SPADE)

10-4 10-3 10-2 10-1 100

Probability of false alarm

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P ro b a b ilit y o f d etectio n

Student background (ν =20), replacement model, α = 0.05 GLRAM(mKELLY)

GLRRM(ACUTE)

GLRMM(SPADE)

10-4 10-3 10-2 10-1 100

Probability of false alarm

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Probabi li ty of detecti o n

Student background (ν =10000), replacement model, α = 0.05 GLRAM(mKELLY)

GLRRM(ACUTE)

GLRMM(SPADE)

0.95 0.96 0.97 0.98 0.99 1

Value of

β

-15 -10 -5 0 5 10 15 20

P

fa

gain

w

ith

resp

ect

to

m

K

E

L

L

Y

(decib

els)

Student background (

ν =20), α = 0.05

GLRRM(ACUTE) GLRMM(SPADE)

Fig. 2. P fa gain versus βfor P d = 0 . 5 .

(17)and the modified model GLRMM in (18) will be referred as

toACUTE [10]and SPADE[11]. Theselast two detectorshave al-readybeenassessedagainstrealdatadrawnfromtheRIT[19]and Viareggio[20]experiments.Herein,weevaluatetheirperformance aswell as their robustness on simulated yetrealistic data. More preciselyweconsideranimageacquiredinViareggio(Italy)inMay 2013froman aircraftflyingat1200meters.Theoriginal dataisa [450× 375]pixels map intheVisible NearInfraRed(VINR)band

(

400_{− 1000}nm

)

withaspatialresolutionofabout0.6meters.The sceneiscomposedofparkinglots,roads,buildings,sportfieldsand pinewoods. Different kindsof vehicles aswell ascolored panels servedasknowntargets.Foreach ofthesetargets,a spectral sig-natureobtained fromgroundspectroradiometermeasurements is available.Herein,weconsiderttobethe signatureoftheV5 tar-get,

μ

and



arerespectivelythesamplemeanandsample covari-ancematrixobtainedfromthewholeViareggioimage.Ithastobe noticedthattheserawradiancedatahavefirstbeenconvertedto reflectancemeasurementsusinganELMmethod[21,22].The num-berof spectral bands used is p=32 andthe number oftraining samplesisn=60.Thebackgroundissimulatedusingat distribu-tionwith

ν

degreesoffreedom.

Fig.1plotstheReceiverOperationCurve(ROC)obtainedforthe replacementmodel[

β

=1−

α

]with

α

=0.05,fordifferentvalues of

ν

ranging from

ν

=5(heavy-taileddistribution) to

ν

=10,000 (nearlyGaussiandistributedbackground).Ascouldbe anticipated, ACUTEexhibits thebest performance since it corresponds tothe GLRT for the specific case

β

=1−

α

. However, SPADE is shown to incur a small degradation compared to this optimal detector. Onthe contrary, mKELLY exhibitsa significant performance loss, mostlybecause E

{

z

}

=E

{

zk

}

, a fact that is not accounted for in theadditivemodel,contrarytotheothertwodetectors.

WenowassesstherobustnessofACUTEandSPADE.More pre-cisely, we study their performance when

β

varies between

β

= 1−

α

(replacement model) and

β

=1(additive model). InFig. 2, wedisplay the probabilityof falsealarm (P_fa) gain ofACUTEand SPADEwithrespect tomKELLY, i.e., 10log10

P_{f a}(GLR_AM)

P_{f a}(GLR_RM/MM).This Pfa gain allows tomeasure the improvement(if positive)or theloss (whennegative)ofbothACUTEandSPADEwithrespecttomKELLY.

Inthisfigure

α

isfixedto0.05andtheprobabilityofdetectionis

P_d=0_.5. Fig.2confirmsthatACUTE isslightlybetterthan SPADE whentheactualvalueof

β

iscloseto1−

α

.However,assoonas

β

departsfrom1−

α

SPADEshowsbetterperformancethanACUTE. Moreover,SPADEperformsbetterthanmKELLYformostvaluesof

β

unless

β

comesverycloseto1,butinthiscasethelossofSPADE withrespecttomKELLYismarginal.Therefore,SPADEprovidesthe bestrobustness,withsmallperformancelosscomparedtothe op-timalsolutionwhateverthevalueof

β

.

4. Concludingremarks

Inthis communication, we considered thedetectionof a sub-pixel targetin hyperspectral imagingwhenthe backgroundis no longer Gaussian butt-distributed.The GLRTs fora generalmixed model,including the standardadditive caseandthe replacement one, were derived, generalizingthe Gaussian versions previously derived.Forthethreespecific valuesof

β

consideredinthe liter-ature, it wasshown that the GLRTs remain the same andhence thedetectorsinitiallyproposed underaGaussian frameworkhave moregenerality thanexpected. Moreover, theydo notdepend on the unknown degree of freedom ofthe t-distribution. Numerical simulations showedthatSPADEprovides averygood trade-off as itisalwaysclosetoorbetterthanKellyandACUTEwhichare op-timalonlyforspecificvaluesof

β

.

DeclarationofCompetingInterest

The authors acknowledge that there is no conflict of interest regardingthepaperentitled“Sub-pixel detectioninhyperspectral imagingwithellipticallycontouredt-distributedback-ground”. AppendixA. GLRfortheadditivemodelandGaussian distributedbackground

Inthis appendix,we derive the GLRTforGaussian distributed backgroundandfortheadditivemodel.Wethusconsiderthe fol-lowingdetectionproblem

H0:



y Z



=d Np,n+1



μ μ

1Tn



H1:



y Z



=d Np,n+1



α

t+

μ μ

1Tn



,



In+1

(A.1)

Thep.d.f.of(y,Z)isinthiscase

p0

(

y,Z

)

=

(

2

π

)

− (n+1)p 2

|



|

− n+1 2 × etr



−1 2



−1



_y−

_μ

_Z−

_μ

₁T n



₍

y−

μ

)

T

(

Z−

μ

1T n

)

T



p1

(

y,Z

)

=

(

2

π

)

− (n+1)p 2

|



|

− n+1 2 × etr



−1 2



−1



_y−

_μ

_Z−

_μ

₁T n



₍

y−

μ

)

T

(

Z−

μ

1T n

)

T



(A.2)

It iswell-knownthat

|



|

−n+12 etr

{

−1

2



−1S

}

achievesitsmaximum

at



_∗=

(

n+1

)

−1S,andhence max 

|



|

−n+1 2 _etr



−1 2



−1_S



₌



e n+1

−

n+1 2

|

S

|

−n+12 _(A.3)

Itfollowsthatmax_p0(y,Z)andmaxp1(y,Z)areproportionalto

(7) whichholds for Studentdistributions. From there, everything followsandtheGLRsforStudentorGaussiandistributionsarethe sameandaregivenby(15).

CRediTauthorshipcontributionstatement

Olivier Besson: Conceptualization, Methodology, Validation, Writing -original draft, Writing -review & editing,Visualization. FrançoisVincent:Conceptualization,Methodology,Software, Writ-ing-originaldraft,Writing-review&editing.

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