Table Of ContentMULTIPLEINSTANCEHYBRIDESTIMATORFORLEARNINGTARGETSIGNATURES
ChangzheJiaoandAlinaZare
ElectricalandComputerEngineering,UniversityofMissouri
ElectricalandComputerEngineering,UniversityofFlorida
ABSTRACT [1], training data is partitioned into sets of labeled bags (in-
7 Signature-based detectors for hyperspectral target detection steadofbeingindividuallylabeled).Apositivebagmustcon-
1 rely on knowing the specific target signature in advance. tain at least one true positive (target) data point and nega-
0
However, targetsignatureareoftendifficultorimpossibleto tivebagsarecomposedentirelyofnegativedata. Multiplein-
2
obtain. Furthermore, common methods for obtaining target stanceconceptlearningisabranchofMILthataimstolearn
n
signatures,suchasfromlaboratorymeasurementsormanual one or a set of concepts to describe the target class. For ex-
a
selection from an image scene, usually do not capture the ample,DiversityDensity(DD)[2]identifiesatargetconcept
J
discriminative features of target class. In this paper, an ap- thatisclosetotheintersectionofallpositivebagsandfarfrom
9
proach for estimating a discriminative target signature from negativeinstances. Theexpectationmaximizationversionof
] imprecise labels is presented. The proposed approach maxi- DD[3]improvesconvergence.PreviousmethodsforMILtar-
V
mizes the response of the hybrid sub-pixel detector within a getcharacterizationwereproposedin[4,5]. [4]combinesall
C multiple instance learning framework and estimates a set of positive bags into one large positive bag and, thus, discards
. discriminative target signatures. After learning target signa- bag-level label information. [5] does not assume a mixing
s
c tures,anysignaturebaseddetectorcanthenbeappliedontest model and, thus, may fail to take advantage of the mixing
[ data. Both simulated and real hyperspectral target detection modelifitisknown. Incontrast,theproposedmethodmaxi-
1 experiments are shown to illustrate the effectiveness of the mizestheresponseofthehybridsub-pixeldetector,preserves
v method. bag-levellabels,andassumesthelinearmixingmodel[6].
8
5 IndexTerms— targetdetection,conceptlearning,hyper-
2 spectral,multipleinstance,targetcharacterization,subpixel.
2. MULTIPLEINSTANCEHYBRIDESTIMATOR
2
0
. 1. INTRODUCTION Let X = [x1,··· ,xN] ∈ Rd×N be training data where d
1
is the dimensionality of an instance and N is the total num-
0
The wealth of spectral information in hyperspectral imagery
7 ber of training instances. The data is grouped into K bags,
providesfortheabilitytoperformsub-pixeltargetdetection.
1 B={B1,...,BK},withassociatedbinarybag-levellabels,
: Signature-based hyperspectral target detectors need to know L = {L ,...,L } where L ∈ {0,1}; n is the number of
v 1 K i i
i thetargetsignatureinadvance. However,inanumberofsce- instancesinbagBi andxij ∈ Bi denotesthejth instancein
X narios,obtaininganeffectivetargetsignatureisoftenachal-
bag B with instance-level label l ∈ {0,1}. When identi-
i ij
r lenging problem. For instance, in some situations analysts fying the label on certain bag or instance is important, pos-
a
mayhaveapproximatelocationsofsub-pixeltargetsandwant itive bags will be indicated as B+ with associated bag level
i
toestimatetheirtargetsignatureandsearchforthetargetma- label L+ containing instances x+ with instance-level labels
i ij
terial elsewhere. In these situations, manual selection of a
l+,j = 1,··· ,n+, where n+ is the number of instances in
targetsignaturefromanimagecubewouldnotonlybediffi- ij i i
positive bag B+. Similarly, B−, L−, x−, l− and n− rep-
cult(asonlyapproximatelocationsareknown)butmayresult i i i ij ij i
resent a negative bag, label for this negative bag, the jth in-
inselectingamixed(asopposedtopuretarget)pixels.
stanceinthisbag,labelforthejthinstanceandtotalnumber
In this paper, we address this problem by proposing a
ofinstancesinthisbag. Thenumberofpositiveandnegative
methodtocharacterizeatargetsignaturefromimpreciselyla-
bagsaredenotedasK+ andK−,N+ andN− representthe
beled hyperspectral imagery. Specifically, we model the hy-
totalnumberofpositiveandnegativeinstances,respectively.
perspectraltargetestimationtaskasamultipleinstancecon-
cept learning problem. In multiple instance learning (MIL) Thus N = N+ + N− = (cid:80)Ki=+1n+i + (cid:80)Ki=−1n−i and B+,
B− representtheunionofallpositiveinstancesandnegative
This material is based upon work supported by the National Science
instances,respectively.
Foundation under Grant IIS-1350078-CAREER: Supervised Learning for
Given this notation, the proposed multiple instance hy-
IncompleteandUncertainDataandascholarshipfromChinaScholarship
Council(No.201206960005). brid estimator (MI-HE) aims to maximize the probability of
the labels of the bags as in (1). Since, in MIL, each pos- whereD=(cid:2)D+ D−(cid:3)∈Rd×(T+M),D+ =(cid:2)d+,··· ,d+(cid:3)
1 T
itive bag must contain at least one positive instance, we can isthesetofT targetsignaturesandD− = (cid:2)d−,··· ,d−(cid:3)is
1 M
thensubstitutetheprobabilityforabagtobepositivewiththe thesetofM backgroundsignatures,βisascalingparameter;
probability of the most likely point in the bag to be a target, α+ andα+b arethesparserepresentationofx+ givenentire
i.e., max Pr(x+ = +|B+). Since each negative bag is ij ij ij
j∈n+i ij i signaturessetDandbackgroundsignaturessetD−, respec-
assumedtoonlycontainnon-targetinstances,thentheproba-
tively. Specifically,solvingforasparseαgivenadictionary
bilityforanegativebagtobenegativecanberepresentedby
setDismodeledastheLassoproblem[8,9]shownin(8):
thejointprobabilityofallinstancesinthisbagtobenegative,
1
J = K(cid:89)+Pr(L+ =+|B+)K(cid:89)−Pr(L− =−|B−). (1) αˆ =argmin2(cid:107)x−Dα(cid:107)22+λ(cid:107)α(cid:107)1, (8)
1 i i i i
where λ is a scaling vector to control the sparsity of α.
i=1 i=1
= K(cid:89)+maxPr(l+ =+|B+)K(cid:89)−(cid:89)n−i Pr(l− =−|x−).(2) H(ISerTeAw)e[1a0d]ofpotrtshoelvitienrgattihveesspharirnskeacgoed-ethsrαes.holdingalgorithm
ij i ij ij
i=1j∈n+i i=1j=1
Algorithm1MI-HEalgorithm
Eq. (2)containsamaxoperationthatisdifficulttoopti- 1: InitializeD0,iter =0
mize numerically. Some algorithms in the literature adopt a
2: repeat
noisy-OR model instead of a max [2, 3]. However, exper-
3: fort=1,··· ,T do
imental results show that the noisy-OR is non-smooth and 4: Solving(cid:8)α+(cid:9)N ,(cid:8)α−(cid:9)N byISTA
generally needs to be optimized repeatedly with many dif- i i=1 i i=1
ferent initializations to avoid local optima. In the proposed 5: Updatedtbyoptimizing(4)usinggradientdescent
approach,weadoptthegeneralizedmeanasanalternative, 6: dt ← (cid:107)d1t(cid:107)2dt
7: endfor
J2 =K(cid:89)+n1+ (cid:88)n+i Pr(li+j =+|B+i )pp1 K(cid:89)−(cid:89)n−i Pr(li−j =−|x−ij), 89:: forSkol=vin1g,·(cid:8)·α·+i,(cid:9)MNi=1d,o(cid:8)α−i (cid:9)Ni=1byISTA
i=1 i j=1 i=1j=1 10: Updatedk byoptimizing(4)usinggradientdescent
(3)
fwrohmereapm∈in[−to∞a,+m∞ax],isreasppaercatmiveetleyr.thWatevathrieenstohpetiompiezreatitohne 11: dk ← (cid:107)d1k(cid:107)2dk
12: endfor
negative logarithm of J and add a scaling factor, ρ < 1, to
2 13: iter ←iter+1
thesecondtermtocontroltheinfluenceofthenegativebags
14: untilStoppingcriterionmeets
(whenN− (cid:54)=N+)asshownin(4).
15: return D
We now must define Pr(l+ = +|B+) and Pr(l− =
ij i ij
−|x−). As in [7], each instance is modeled as a sparse lin-
ij For points from negative bags, following (6), we model
ear combination of target and/or background signatures D, the reconstruction error of points x− ∈ B+ as a zero mean
ij i
x ≈ Dα , where α is the sparse vector of weights. Each
j j j Gaussiandistributionwithunknownvariance,
positivebagcontainsatleastoneinstancewithtarget:
Pr(l− =−|x−)=exp(cid:0)(cid:107)x− −D−α−(cid:107)2(cid:1), (9)
ij ij ij ij
ifL =1,∃x ∈B+s.t.
i j i whereα− isthesparserepresentationofx− givenD−. The
T M ij ij
x =(cid:88)α d++(cid:88)α d−+ε ,α (cid:54)=0, (5) objective function (4) is then optimized by gradient descent
j jt t jk k j jt withsparsecodingasoutlinedinAlg. 1
t=1 k=1
whereεj isanoiseterm.AnegativelylabeledbagB−i should 3. EXPERIMENTS
notcontainanytarget:
MI-HE was applied to simulated data generated from four
M
(cid:88) spectraselectedfromtheASTERspectrallibrary[11]shown
ifL =0,∀x ∈B−,x = α d−+ε . (6)
i j i j jk k j in Fig. 1. Specifically, the Red Slate, Verde Antique, Phyl-
k=1
liteandPyroxenitespectrafromtherockclasswith211bands
Giventhismodel,weintroducethehybridsubpixeldetec- andwavelengthsrangingfrom0.4µmto2.5µmwereusedas
tortoestimatetheprobabilityinstancesfrompositivebagsare endmembers to generate hyperspectral data. Red Slate was
positive. Specifically,theprobabilityforx+ inB+isatarget labeled as the target endmember. [4] provides a precise de-
ij i
pointisdefinedas: scriptionofhowthesimulateddatawasgenerated.
Three sets of highly-mixed noisy data with varied mean
(cid:32) (cid:33)
(cid:107)x+ −Dα+(cid:107)2
Pr(l+ =+|B+)=exp −β ij ij , (7) target proportion value (ptmean) were generated. Specifi-
ij i (cid:107)x+ −D−α+b(cid:107)2 cally, this synthetic data has 15 positive and 5 negative bags
ij ij
(cid:88)K+ 1 1 (cid:88)n+i (cid:88)K−(cid:88)n−i
−lnJ =− ln Pr(l+ =+|B+)p−ρ lnPr(l− =−|x−), (4)
p n+ ij i ij ij
i=1 i j=1 i=1j=1
1.2
Red Slate
Verde Antique 1
1 PPhyryollxiteenite 00..1146 MeEFMI-UH-DMEDI 0.8
0.8 0.12 Truth
Reflectance0.6 ecnatcelfeR00..000.681 DP00..46 MMeFII--UHHMEEI(((AHACDCE)E))
0.4 0.04 0.2 eEFMU-DMDI((HADC)E)
0.02 mi-SVM
0.2 0 0
0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1
Wavelength (7m) PFA
0 0.5 1 1.5 2 2.5 (a) Estimatedtargetspectra (b) ROCcurves
Wavelength (µm)
Fig.1. Signaturesusedtogeneratesimulateddata Fig.2.ComparisonofMI-HE,eFUMI,EM-DDandmi-SVM
onsyntheticdatawithmeantargetproportion0.3
Table1. ListofBackgroundEndmembersforSyntheticData
BagNumber BagLabel BackgroundEndmember
1-5 + VerdeAntique,Phyllite,Pyroxenite Table2. SimulatedHyperspectralDataExperiments. Results
6-10 + Phyllite,Pyroxenite listedinMedianAUCover5Runs
11-15 + Pyroxenite P
Algorithm tmean
16-20 − Phyllite,Pyroxenite 0.3 0.5 0.7
MI-HE(HD) 0.944 0.981 0.996
MI-HE(ACE) 0.974 0.997 0.999
eFUMI(ACE) 0.810 0.843 0.843
with each bag has 500 points. If it is positively labeled,
eFUMI(HD) 0.471 0.467 0.504
thereare100targetpointscontainingmeantarget(RedSlate)
EM-DD(ACE) 0.841 0.876 0.998
proportionp . Theparameterp thatcontrolsthe
tmean tmean
mean target proportion value was set to 0.3, 0.5 and 0.7 re- mi-SVM 0.834 0.884 0.891
spectively to vary the level of target presence from weak to
high. Gaussianwhitenoisewasaddedsothatsignal-to-noise
ratioofthedatawassetto30dB. Tohighlighttheabilityof
algorithms eFUMI, EM-DD and mi-SVM [12]. For MI-HE
MI-HE to leverage individual bag-level labels, we use dif-
and eFUMI, both the structured-background adaptive coher-
ferent subsets of background endmembers to build synthetic
ence/cosine estimator (ACE) [13, 14] and Hybrid Sub-pixel
dataasshowninTab. 1.
Detector(HD)[6]wereapplied; forEM-DD,onlyACEwas
MI-HE was compared to eFUMI and EM-DD. As we
appliedsinceEM-DDdoesnotlearnasetofbackgroundsig-
mentioned before, eFUMI combines all positive bags into
natures simultaneously. The results reported are the median
one big positive bag. Given the aforementioned synthetic
resultsoverfiverunsofthealgorithmonthesamedata.
data, eFUMI confuses both Red Slate and Verde Antique as
For experiments on real hyperspectral target detection
targetsignaturessinceVerdeAntiqueismissinginthetrain-
data, theMUUFLGulfportHyperspectraldatasetwasused.
ing negative bags (and eFUMI does not preserve bag-level
This data set was collected over the University of Southern
labels). However, the proposed MI-HE is able to learn the
Mississippi-GulfparkCampusandcontains325×337pixels
targetsignaturecorrectly. Fig. 2(a)showsthetargetsignature
estimated by MI-HE and comparison algorithms eFUMI [4]
and EM-DD [3]. We can see that MI-HE is able to learn
a target signature that matches the groundtruth, but eFUMI Table3. GulfportBrownDetection. ResultslistedinMedian
mistakesVerdeAntiqueastargetsignatureandEM-DDlearns NAUCover5RunsatFAR=1×10−3m2
onenoisy,non-puretargetsignature. Algorithm Train:Flight1,Test:Flight3 Train:Flight3,Test:Flight1
MI-HE(HD) 0.552 0.792
For quantitative evaluation, receiver operating curves
MI-HE(ACE) 0.442 0.699
(ROC) from detection on testing data generated using the eFUMI(ACE) 0.437 0.746
eFUMI(HD) 0.417 0.765
samemethodareshowninFig. 2(b). Tab. 2showstheareaof
EM-DD(ACE) 0.420 0.749
probability of detection (PD) under the curve (AUC), where mi-SVM 0.353 0.333
we can see proposed MI-HE outperforms the comparison
Brown 5. REFERENCES
Dark Green
FVG
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