HAL Id: hal-02798974
https://hal.archives-ouvertes.fr/hal-02798974
Submitted on 5 Jun 2020
HAL
is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire
HAL, estdestinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Sub-pixel detection in hyperspectral imaging with elliptically contoured t-distributed background
Olivier Besson, François Vincent
To cite this version:
Olivier Besson, François Vincent. Sub-pixel detection in hyperspectral imaging with elliptically contoured t-distributed background. Signal Processing, Elsevier, 2020, 175, pp.107662-107667.
�10.1016/j.sigpro.2020.107662�. �hal-02798974�
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
Shortcommunication
Sub-pixel detection in hyperspectral imaging with elliptically contoure d t -distribute d background
Olivier Bessona,∗,François Vincenta
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.Weshowthatthegeneralizedlikelihoodra- 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,vegetationormin- 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-pixeldetectionusingthere- placement model under a Gaussian background, and we derived the plain generalized likelihood ratio test (GLRT) by maximizing thejointdistributionof(y,Z)withrespecttoallunknownparam- eters.Moreover,motivatedbysomelimitationsofthereplacement model,especiallythefact thatthefillingfactorofasub-pixeltar- 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 extendourrecentGLRTsfromtheGaussiancasetothematrixvari- atet-distributedcase.Wewillshowthat theGLRTscoincidewith theirGaussian counterparts. Thepaperisorganizedasfollows.In Section 2, we consider each of the three models andderive the correspondingGLRTs.ThelatterareevaluatedinSection3onsim- ulateddatabutwherethetarget spectralsignature,themeanand covariancematrixofthebackgroundareobtainedfromrealhyper- spectralimages.
https://doi.org/10.1016/j.sigpro.2020.107662
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=αt+βz; 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. When β=1 one obtains the con- ventionaladditivemodel.Whenβ=1−αthereplacementmodel 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+1
ν,M1,(ν−2),
β2 0T
0 In
(2) where M0=
μ μ1Tn
, M1=
αt+βμ μ1Tn
, 1n is a n × 1 vectorwithallelementsequaltoone,μstandsforthemeanvalue
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+12
Ip+ν−−12
y−μ Z−μ1Tn
×
y−μ Z−μ1Tn
T−
ν+n+p 2
p1(y,Z)=Cβ−p||−n+12
Ip+ν−−12
y˜−μ Z−μ1Tn
×
˜
y−μ Z−μ1Tn
T−
ν+n+p
2 (3)
with y˜=β−1(y−αt) andC= πp(n+1p)((ν/2+pn((ν+p+)p/2−)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)ni=1p(zi).
WenowderivetheGLRTfortheproblemin(2).Letusstartby consideringthe following function f() where S issome positive definitematrix:
f()=||−n+12 Ip+(ν−2)−1−1S−ν+n2+p
=||ν+p−12 +(ν−2)−1S−ν+n2+p (4) Differentiationoflogf()yields
∂logf()
∂ =ν+p−1
2 −1−ν+n+p
2 (+(ν−2)−1S)−1. (5)
Settingthisderivativeoftozero,wecanseethatf()achievesits maximumat
∗= (ν+p−1)S
(ν−2)(n+1) =γS (6)
Itfollowsthat max p0(y,Z)=C
y−μ Z−μ1Tn
(y−μ)T (Z−μ1Tn)T −
n+1 2
max p1(y,Z)=Cβ−p
˜
y−μ Z−μ1Tn
(y˜−μ)T (Z−μ1Tn)T −
n+1 2
(7)
withC=Cγ−p(n+1)/2[1+(ν−2)−1γ−1]−p(ν+n+p)/2.SinceCisthe
sameunderH0andH1 itwillcanceloutintheGLRandtherefore theGLRdoesnotdependonν.Thismeansthatνdoesnotneedto
beknownsinceitsestimationisnotactuallyrequiredtoobtainthe GLR.
Now,foranyvectorx, M(μ)=
x−μ Z−μ1Tn
x−μ Z−μ1Tn
T
=(x−μ)(x−μ)T+(Z−μ1Tn)(Z−μ1Tn)T
=xxT−μxT−xμT+μμT
+ZZT−μ1TnZT−Z1nμT+nμμT
=(n+1)μμT−μ(x+Z1n)T−(x+Z1n)μT
+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 ZT
(8)
Consequently, minμ |M(μ)|=
x Z
P⊥n+1
x ZT (9)
with P⊥n+1 the orthogonal projector on the null space of 1n+1. Hence,wearriveat
maxμ, p0(y,Z)=C
y Z
P⊥n+1
y ZT−
n+1 2
maxμ, p1(y,Z)=Cβ−p
˜ y Z
P⊥n+1
˜ y ZT−
n+1
2 (10)
Next,foranyvectorxandmatrixQ(notnecessarilyP⊥n+1),
x Z
Q
x ZT
=
x ZQ11 Q12
Q21 Q22 x ZT
=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 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,we have
Q=
1 0T 0 In
−(n+1)−1
1 1Tn 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−nz¯z¯T=S.Hence,theGLRisgivenby
GLR=
1+Q11(y−z¯)TS−1(y−z¯)(n+1)/2
minα,ββp[1+Q11(y˜−z¯)TS−1(y˜−z¯)](n+1)/2
= [1+n+n1(y−z¯)TS−1(y−z¯)](n+1)/2
minα,ββp[1+n+1n (y−βαt−z¯)TS−1(y−βαt−z¯)](n+1)/2 (15)
A few important observationscan be maderegarding thisre- 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
GLR2AM/(n+1)= 1+nn+1(y−z¯)TS−1(y−z¯) minα1+n+1n (y−z¯−αt)TS−1(y−z¯−αt)
= 1+n+n1(y−z¯)TS−1(y−z¯) 1+n+1n (y−z¯)TS−1(y−z¯)−n+1n [(y−tzT¯)ST−S1−1tt]2
≡
n
n+1[(y−z¯)TS−1t]2
[1+n+n1(y−z¯)TS−1(y−z¯)][tTS−1t] (16)
The GLRin(16)generalizesKelly’s detectortothe caseof anon- centered Student distributed background. Note that (16) differs fromKelly’sdetectorbythe n+1n factor.
Asforthereplacementmodel,β=1−αandtheminimization shouldbeconductedwithrespecttoαonly,i.e.,
GLRRM=
1+n+n1S−1/2(y−z¯)2(n+1)/2
minα(1−α)p
1+n+n1S−1/2(y−(1α−t−α)(12−α)z¯)2
(n+1)/2 (17)
Asshownin[10],thissimplyamountstofindingthe(unique)pos- itiverootofa2nd-orderpolynomial.Finally,forthemixedmodel whereβ isarbitrary,one hasa2-D minimization problem. How- ever,minimizationoverα canbedoneanalytically, leavingonlya minimizationoverβ:
GLRMM=
1+n+n1S−1/2(y−z¯)2(n+1)/2
minββp
1+n+1n
P⊥S−1/2tS−1/2(y−βz¯)2 β2
(n+1)/2 (18)
Again the solution is obtained as the unique positive root of a second-order polynomialequation [11].Thereforeforboththere- 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
Probabilityofdetection
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
Probabilityofdetection
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
Probabilityofdetection
Student background (ν=10000), replacement model,α= 0.05
GLRAM(mKELLY) GLRRM(ACUTE) GLRMM(SPADE)
Fig. 1. ROC for the replacement model β= (1 −α).
