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
aUniversity 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: olivier.besson@isae-supaero.fr (O. Besson), francois.vincent@isae-supaero.fr (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 matrixareunknown,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 andα
de-notesits unknown amplitude. Whenβ
=1 one obtains the con-ventionaladditivemodel.Whenβ
=1−α
thereplacementmodel isrecovered,andthemixedmodelcorrespondstoanarbitraryβ
.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 -distributionwithν
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+1ν
,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 backgroundwhiledenotes 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)
=Cβ
−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)/2p((ν+p−1)/2). It should beobserved that the columns of
y Z are only uncorrelated but not independent, as p(
y,z1,...,zn)
cannot be factored asp
(
y)
ni=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)−1S− 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)
=Cy−
μ
Z−μ
1T n(
y−μ
)
T(
Z−μ
1T n)
T− n+1 2 max p1
(
y,Z)
=Cβ
−py˜−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)
−1γ
−1]−p(ν+n+p)/2.SinceCisthesameunderH0andH1 itwillcanceloutintheGLRandtherefore theGLRdoesnotdependon
ν
.Thismeansthatν
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− xμ
T+μμ
T +ZZT−μ
1T nZT− Z1nμ
T+nμμ
T =(
n+1)
μμ
T−μ
(
x+Z1 n)
T−(
x+Z1n)
μ
T +xxT+ZZT =(
n+1)
μ
−x+Z1n n+1μ
−x+Z1n n+1 T +xxT+ZZT−(
x+Z1n)(
x+Z1n)
T n+1 =(
n+1)
μ
−x+Z1n n+1μ
−x+Z1n n+1 T +x ZIn+1− 1n+11Tn+1 n+1 x ZT (8) Consequently, min μ|
M(
μ
)
|
= x ZP⊥n+1x ZT (9)with P⊥n+1 the orthogonal projector on the null space of 1n+1.
Hence,wearriveat max μ, p0
(
y,Z)
=Cy ZP⊥ n+1 y ZT− n+1 2 max μ, p1
(
y,Z)
=Cβ
−py˜ ZP⊥ n+1 ˜ y ZT− n+1 2 (10)
Next,foranyvectorxandmatrixQ(notnecessarilyP⊥n+1),
x ZQx ZT=x ZQ
11 Q12 Q21 Q22x ZT =Q11xxT+ZQ21xT+xQ12ZT+ZQ22ZT =Q11 x+Q11−1ZQ21 x+Q11−1ZQ21 T +ZQ2.1ZT (11) whereQ2.1=Q22− Q21Q11−1Q12.Therefore, x ZQx ZT=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,wehave Q=
1 0T 0 In −(
n+1)
−1 1 1T n 1n 1n1Tn (13) sothatQ11=1−
(
n+1)
−1=n(
n+1)
−1 Q21=−(n+1)
−11nQ22=In−
(
n+1)
−11n1TnQ2.1=In− n−11n1Tn=P⊥n (14) It follows that Q11−1ZQ21=−n−1Z1n=−¯z and ZQ2.1ZT=ZP⊥nZT= ZZT− n¯z¯zT=S.Hence,theGLRisgivenby
GLR=
1+Q11(
y− ¯z)
TS−1(
y− ¯z)
(n+1)/2 minα,ββ
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 minα,ββ
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, lettingν
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.ThisyieldsGLR2AM/(n+1)= 1+ n n+1
(
y− ¯z)
TS−1(
y− ¯z)
minα1+ 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) TS−1t]2 tTS−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 shouldbeconductedwithrespecttoα
only,i.e.,GLRRM=
1+ n n+1S−1/2(
y− ¯z)
2(n+1)/2 minα(
1−α
)
p1+ n n+1 S−1/2(y−αt−(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,minimizationoverα
canbedoneanalytically, leavingonlya minimizationoverβ
: GLRMM= 1+ n n+1S−1/2(
y− ¯z)
2(n+1)/2 minββ
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 20P
fagain
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− 1000nm)
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
andarerespectivelythesamplemeanandsample 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
]withα
=0.05,fordifferentvalues ofν
ranging fromν
=5(heavy-taileddistribution) toν
=10,000 (nearlyGaussiandistributedbackground).Ascouldbe anticipated, ACUTEexhibits thebest performance since it corresponds tothe GLRT for the specific caseβ
=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 betweenβ
= 1−α
(replacement model) andβ
=1(additive model). InFig. 2, wedisplay the probabilityof falsealarm (Pfa) gain ofACUTEand SPADEwithrespect tomKELLY, i.e., 10log10Pf a(GLRAM)
Pf a(GLRRM/MM).This Pfa gain allows tomeasure the improvement(if positive)or theloss (whennegative)ofbothACUTEandSPADEwithrespecttomKELLY.
Inthisfigure
α
isfixedto0.05andtheprobabilityofdetectionisPd=0.5. Fig.2confirmsthatACUTE isslightlybetterthan SPADE whentheactualvalueof
β
iscloseto1−α
.However,assoonasβ
departsfrom1−α
SPADEshowsbetterperformancethanACUTE. Moreover,SPADEperformsbetterthanmKELLYformostvaluesofβ
unlessβ
comesverycloseto1,butinthiscasethelossofSPADE withrespecttomKELLYismarginal.Therefore,SPADEprovidesthe bestrobustness,withsmallperformancelosscomparedtothe op-timalsolutionwhateverthevalueofβ
.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-timalonlyforspecificvaluesofβ
.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+1μ μ
1TnH1:
y Z
=d Np,n+1α
t+μ μ
1Tn,
In+1
(A.1)
Thep.d.f.of(y,Z)isinthiscase
p0
(
y,Z)
=(
2π
)
− (n+1)p 2|
|
− n+1 2 × etr −1 2−1y−
μ
Z−μ
1T n(
y−μ
)
T(
Z−μ
1T n)
Tp1
(
y,Z)
=(
2π
)
− (n+1)p 2|
|
− n+1 2 × etr −1 2−1y−
μ
Z−μ
1T n(
y−μ
)
T(
Z−μ
1T n)
T(A.2)
It iswell-knownthat
|
|
−n+12 etr{
−12
−1S
}
achievesitsmaximumat
∗=
(
n+1)
−1S,andhence max|
|
−n+1 2 etr −1 2−1S= e n+1
−
n+1 2|
S|
−n+12 (A.3)Itfollowsthatmaxp0(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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