- 1. ROBUST DETECTION USING THE SIRV BACKGROUND MODELLING FOR
HYPERSPECTRAL IMAGINGJ.P. Ovarlez1,3, S.K. Pang2, F. Pascal3, V.
Achard1 and T.K. Ng21 : FRENCH AEROSPACE LAB, ONERA, France,
[email protected], [email protected] : DSO
National Laboratories, Singapore, [email protected],
[email protected] : SONDRA, SUPELEC, France,
[email protected] SIRV MODELLING FOR DETECTION AND
ESTIMATION PROBLEMS Jean-Philippe OVARLEZ (ONERA &
SONDRA)1CONTEXT OF THE PROBLEMNOISE MODELLING WITH SIRV CHANGE
DETECTION
2. OUTLINE OF THE TALKProblems Description and Motivation,The
Spherically Invariant Random Process Modelling forHyperspectral
Imaging,Estimation in the SIRV Background,Detection in the SIRV
Background,Anomaly Detection and Target Detection Results
onExperimental Data,Conclusion.THE SIRV MODELLING FOR DETECTION
AND2 / 17 IEEE IGARSS2011 Vancouver, Canada ESTIMATION PROBLEMS
Jean-Philippe OVARLEZ (ONERA & SONDRA) 3. ground subspace
spanned by the columns of B or U Q[30]. The perfo PROBLEMS
DESCRIPTIONThe bar charts in Fig. 11 provide the range of the
de-tection statistic of the target and the maximum value
ofdetectionthe background detection statistics for various back- is
challengrounds. The target-background separation or overlap isthe
quantity used to evaluate target visibility enhance-limitatio
ANOMALY DETECTION IN HYPERSPECTRAL IMAGES ment. For example, it can
be seen that the ACE detector limited aperforms better than the OSP
algorithm for the six dataTo detect all that is different from the
background (Mahalanobis distance) -sets shown.Regulation of False
Alarm. Application to radiance images.straight line sh The expected
probability distribution of the detectiongood fit and thstatistics
under the target absent hypothesis can becompared to the actual
statistics using a quantile-quantile old for CFAR(Q-Q) plot. A Q-Q
plot shows the relationship between The previou DETECTION OF
TARGETS IN HYPERSPECTRAL IMAGESthe quantiles of the expected
distribution and the actualdata. An agreement between the two is
illustrated by atargets. To evtargets, we havstraight line. The Q-Q
plots in Fig. 12 illustrate the com- (3). SubpixelTo detect (GLRT)
targets (characterized by a given spectral signature p) -parison
between the experimental detection statistics todomly
chosenRegulation of False Alarm. Application to reflectance images
(after somethe theoretically predicted ones for the matched filter
al- of the backgrogorithms. The actual statistics for two different
back- sults shown inatmospherical corrections or others). grounds
is compared to the normal distribution. A bility as a func RXD
CDFand OSP detewith the size of 100a large set of dCauchyhas been
show Probability of Exceedance 101 ability using a Blocks Mixture
of t-Distributions formance [32] 102 When the sas N( , ), its M 103
distribution w Normal mean, we obta 104 Normal Mixedfor
nonnorma(x2(144)) Trees tance is not ch Grassfalse alarm foreight
blocks ob0 100 200 300 400 500 600 700 800 900 1000four by two
maMahalanobis Distancedictions basedF-distribution DSO data 2010
[Manolakis 2002]I 14. Modeling the statistics of the Mahalanobis
distance.description forTHE SIRV MODELLING FOR DETECTION
ANDdistribution. W3 / 17 IEEE IGARSS2011 Vancouver, Canada
ESTIMATION PROBLEMSmultivariate t d Jean-Philippe OVARLEZ (ONERA
& SONDRA)100 4. xT 1S(ST 1S) 1ST 1x H1which is the Euclidean
distance of the test pixel from>ACE , (13) xT 1x< the
background mean in the whitened space. We note CONVENTIONAL METHODS
OF DETECTIONH0 which can be obtained from Equation 11 by
remov-that, in the absence of a target direction, the detectoruses
the distance from the center of the background ing the additive
term N from the denominator.distribution.Many methodologies
forwhitening transformation classificationamplitude variability,by
the direc-images canIf we use the adaptivedetection and 1 and the
target subspacehyperspectral = For targets within S is specified we
have P be found in radar detection, community. We ofcan retrieve
all formulas for thez 1/ 2 xtion a single vector s. Then the the
detectors family commonly where =in1/radarthe square-root
decomposi- (intensity detectors are simplified to (angle
detector),used 2 1/2 is detection (AMF previous GLR detector), ACE
sub-spaces detectors, covariance matrix, the ACE can tion of the
estimated ...). ( sT 1x )2H1 y = D( x ) = T 1> , be expressed as
(s s)( + xT 1x ) < 1 2 H0 zT S(ST S) 1ST z zT PS z D ACE ( x )
=,where (= N,= 1) for the Kelly detector andAlmost all the
conventional ztechniques 1 =for1 2 = 1) for the ACE. detection was
zT z Tz( 0, anomaly Kellys algorithm and targets= 2 detection are
based on Gaussian assumptionreal-valued signals and has been
applied to 1/ 2S and P S(ST S) 1ST is the or- derived for and on
spatial homogeneity in where S multispectral target detection [20].
The one-dimen- hyperspectral images. S Target pixelAMF Distance^
threshold a 1/2 sx2 ACEas z2 Angular Test pixel ^z = 1/2 x
threshold ^ 1/2 xAdaptivexbackgroundzx1whiteningz1
EllipticalSphericalbackgroundbackgrounddistributiondistributionFIGURE
26. Illustration of generalized-likelihood-ratio test (GLRT)
detectors. These are the adaptive matched filter Intensity Detector
(Matched Filer)Angle Detector (ACE)(AMF) detector, which uses a
distance threshold, and the adaptive coherence/cosine estimator
(ACE) detector, whichuses an angular threshold. The test-pixel
vector and target-pixel vector are shown after the removal of the
backgroundmean vector.[Manolakis 2002]All these techniques need to
estimate the data covariance matrix 100 LINCOLN LABORATORY
JOURNALVOLUME 14, NUMBER 1, 2003(whitening process).THE SIRV
MODELLING FOR DETECTION AND4 / 17 IEEE IGARSS2011 Vancouver, Canada
ESTIMATION PROBLEMS Jean-Philippe OVARLEZ (ONERA & SONDRA) 5. v
2 ated by degrees of freedom, and C is known as the scale ma-FIRST
COMMENTS AND ADEQUACY WITH SOMEx = 1/ 2 ( z) + , (21) trix. The
Mahalanobis distance is distributed asRESULTS FOUND IN THE
LITERATURE ) ~ F , (22) where z ~ N(0, I) and is a non-negative
random 1 ( x ) C ( xT1 K, variable independent of z. The density of
the ECD isK Hyperspectral data provided density of , (or where FK,v
is the F-distribution with K and v degrees uniquely determined by
the probability by DSO others!) are spatially heterogeneousin which
is known as the characteristic probability den-of freedom. The
integer v controls the tails of theintensity and/or Note that f be)
only characterized by Gaussian statistic:Cauchy sity function of x.
cannot ( and can bedistribution: v = 1 leads to the multivariate
specified independently.One of the most important properties of
random100 vectors with ECDs is that their joint probability den-
sity is uniquely determined by the probability densityCauchyBlocks
of the Mahalanobis distanceMixed Probability of exceedence101
distributions 1RXD-SCM ( K / 2) 1 f d (d ) =d hK ( d ) ,2 2K / 2 (
K /2)102NormalHotelling T2where (K /2) is the Gamma function. As a
result, 103Mixedthe multivariate probability density identification
Treesand estimation problem is reduced to a simplerGrassunivariate
one. This simplification provides the cor-1040200 400 600800
1000nerstone for our investigations in the statistical char- RXD on
DSO DATAMahalanobis distanceacterization of hyperspectral
background data. If we know the mean and covariance of a
multi-[Manolakis 2002] FIGURE 50. Modeling the distribution of the
Mahalanobisvariate random sample, we can check for
normalitydistance for the HSI data blocks shown in Figure 33. Theby
comparing the empirical distribution of the blue curves correspond
to the eight equal-area subimages Spherical Invariant Random
Vectors(SIRV) models have been started to be in Figure 33. The
green curves represent the smaller areas inMahalanobis distance
against a chi-squared distribu-studied in the hyperspectral
estimate thetion. However, in practice we have to
scientificcommunity but one still uses .... Gaussian Figure 33 and
correspond to trees, grass, and mixed (road and grass) materials.
The dotted red curves represent theestimates !mean and covariance
from the available data by using family of heavy-tailed
distributions defined by Equation 22.THE SIRV MODELLING FOR
DETECTION AND5 / 17 IEEE IGARSS2011 Vancouver, Canada 14, NUMBER 1,
2003 ESTIMATION PROBLEMSVOLUME LINCOLN LABORATORY JOURNAL 113
Jean-Philippe OVARLEZ (ONERA & SONDRA) 6. SIRV FOR
HYPERSPECTRAL BACKGROUNDMODELLING Spherically Invariant Random
Vector : Compound Gaussian Process [Yao 1973] Z p +11 cH M 1 cc=
xpm (c) = expp( ) d 0( )m |M| x is a multivariate complex circular
zero-mean Gaussian m-vector (speckle) with covariancematrix M with
identifiability consideration tr(M)=m, is a positive scalar random
variable (texture) well defined by its pdf p(). For a given set of
spatial pixels of the hyperspectral image, M characterizes the
correlation existing within the spectral bands, Conditionally to
the pixel, the spectral vector is Gaussian. The texture variable
characterizes here the variation of the norm of each vector from
pixels to pixels.Powerful statistical model that allows: to
encompass the Gaussian model, to extend the Gaussian model (K,
Weibull, Fisher, Cauchy, Alpha-Stable, ...), to take into account
the heterogeneity of the background power with the texture, to take
into account possible correlation existing on the m-channels of
observation, to derive optimal or suitable detectors. THE SIRV
MODELLING FOR DETECTION AND 6 / 17 IEEE IGARSS2011 Vancouver,
CanadaESTIMATION PROBLEMSJean-Philippe OVARLEZ (ONERA & SONDRA)
7. SIRV DETECTORS Detectors developed in the SIRV context SIRV
texture p() modelling with Pad Approximants [Jay et al., 2000],g
detection test is the GLRT-LQ [Conte, Gini, Normalized Matched
Filter [Picinbono 1970, Scharf 1991], GLRT-LQ [Gini, Conte,1995],
H0 Bayesian estimation (BORD) of the SIRV texture p() [Jay et al.,
2002]. . }1 y) H1
",-./01#&1$#230&*0.%3412&256078/90*(5**$#$:20;=% !"
Normalized Matched Filterf >+?%%@A(5%2#5B?25&:;2?($:2A22 pH
M 1 cC$5B?02$D2?#$!"$$256H1 ? &! !" (c) = Hn(p M 1 p) (cH M1 c)
H0 )*+) !"c: cell under testp: spectral steering vector of the
target) &$ !"(c) is SIRV CFAR"! # $%!"!" !"!"!" !"#$%"&(
!Texture-CFAR property for the GLRT-LQbut needs to know the true
covariance MMATRIXM IS GENERALLYIGARSS2011 Vancouver, CanadaTHE
SIRV MODELLING FOR DETECTION AND7 / 17 IEEE ESTIMATION
PROBLEMSN.Jean-Philippe OVARLEZ (ONERA & SONDRA) 8. ADATIVE
DETECTION IN SIRV BACKGROUNDNew detectors called Adaptive Detectors
can be derived by replacing in the NMFa good estimate of the
covariance matrix (two step GLRT). ACE : Adaptive Coherence
Estimator ANMF : Adaptive Normalized Matched Filter2 p M 1yH H0 yH
M 1 y pH M1 pH1These detectors are SIRV-CFAR only for some
particular estimates of M !Some well known estimates K1 XMSCM = ck
cH M ??? Kk=1K kmX ck cHMN SCM=kKcH ckk=1 k [Gini-Conte, 2002] THE
SIRV MODELLING FOR DETECTION AND 8 / 17 IEEE IGARSS2011 Vancouver,
CanadaESTIMATION PROBLEMSJean-Philippe OVARLEZ (ONERA & SONDRA)
9. CHOICE OF THE COVARIANCE MATRIX ESTIMATEThe Sample Covariance
Matrix SCM may be a poor estimate of the SIRVCovariance Matrix M
because of the texture contamination:K K 1 X 1 XMSCM= ck cHk=k xk
xHkK K k=1k=1K1 X 6= xk xHkKk=1The Normalized Sample Covariance
Matrix (NSCM) may be a good candidate of theSIRV Covariance Matrix
M:KKm X ck cH km X xk xHkMN SCM == KcH ck k=1 kKxH xkk=1 kh i This
estimate does not depend on the texture but it is biased (E MN SCM
and M share the same eigenvectors but have different eigenvalues,
with the same ordering) [Bausson et al. 2006].THE SIRV MODELLING
FOR DETECTION AND 9 / 17IEEE IGARSS2011 Vancouver, Canada
ESTIMATION PROBLEMSJean-Philippe OVARLEZ (ONERA & SONDRA) 10.
COVARIANCE MATRIX ESTIMATION IN SIRV BACKGROUNDFor an unknown but
deterministic texture parameter, the Maximum Likelihood Estimate
(MLE) ofthe Covariance M (approached MLE in the SIRV context),
called the Fixed Point MFP (FP), is thesolution of the following
implicit equation [Conte-Gini 2002]: K m X ck cHk Fixed Point (FP):
MF P =K 1 cH M F P ckk=1 k [Pascal et al. 2006] This estimate does
not depend on the texture, The Fixed Point is consistent, unbiased,
asymptotically Gaussian and is, at a fixed number K, Les dirents
estimateurs tudiseee Les diree Wishart distributed with mK/(m+1)
degrees of freedom,Les dirents estimateurs tudisee e Les dirents
estimateurs tudiseee Les dirents estimateurse Existence and unicity
of the solution des donnes {x , i = 1...N} de taille m, ils vrient
tous lquationLes direntsare proven. eThe solution can be reached by
recurrence eObtenus ` partir aestimateurs tudisee e ie Mk=f(Mk-1)
whatever the starting point Mdes(ex:ees b0i=I, X1 de b m, ils
vrient tous lquationaM , =M {x h=MNSCM), iObtenus ` partir 0 donn M
=i 1 1...N} M x ) x x e N Obtenus ` partir des donnes { aeu(x
taillee(1) H i 1 i H i i Robust to outliers, strong targets or
scatterers inNthehu(xH M 1 x )i x xH cells. bM= 1 X reference Nbi=1
(1) b M i i iu est une fonction de pondration N xi xH .e des i
ii=1u est une fonction de pondrateu est une fonction de pondration
des xi xH .eThe Fixed Point belongs to the family of M-estimators
(Robust Statistics [Huber Exemples : Maronna1964,iExemples :1976,
Yohai 2006]) in the more general Exemplescontext of Elliptically
Random Process: ) = La SCM : :Lestimateur dHuber Lestimateur FP :
u(r mru(r ) = 1(M-estimateur) : La SCM : mLa SCM : Lestimateur /e
si r eu(.) choiceru(r ) = 1 K /r :1KXu(r ) = K /e si r ek kHuberFP
SCMKk=1 THE SIRV MODELLING Mahot (ONERA, SONDRA) M. FOR DETECTION
ANDJDDPHY 2011 6 / 16 10 / 17IEEE IGARSS2011 Vancouver,
CanadaESTIMATION PROBLEMS M. Mahot (ONERA, SONDRA) Jean-Philippe
OVARLEZM. Mahot (ONERA, SONDRA) (ONERA & SONDRA) JDDPHY 20116 /
16 11. TEXTURE ESTIMATION IN SIRV BACKGROUNDFor an unknown but
deterministic texture parameter, the Maximum Likelihood Estimate of
the texture atpixel k is given by: 1 cH M F P ckkk = mThis quantity
plays exactly the same role as the Polarimetric Whitening Filter
[Novak and Burl 1990 -Vasile et al. 2010] for reducing the speckle
in Polarimetric SAR images. It can also be seen as anextended
Mahalanobis distance between ck and the background.RXDF P = (ck 1
)H MF P (ck )RXDSCM = (ck 1 )H MSCM (ck )THE SIRV MODELLING FOR
DETECTION AND 11 / 17 IEEE IGARSS2011 Vancouver, Canada ESTIMATION
PROBLEMS Jean-Philippe OVARLEZ (ONERA & SONDRA) 12. and its
easy implementation and practical use. Eq. (SOME PROBLEMS SOLVED IN
HYPERSPECTRALi s. T ously implies that MF P is independent of the
results of the statistical properties of MF P are reCONTEXT MF P is
a consistent and unbiased estimate of M; itsThe hyperspectral data
are real and totic distribution is represent radiance orpositive as
they Gaussian and its covariance mreflectance. fully characterized
in [15]; its asymptotic distributio same as the asymptotic
distribution of a Wishart mat A mean vector has to be included in
the SIRVdegrees of freedom. estimated m N/(m + 1) model and to be
jointly with the covariance matrix, When the noise is not centered,
as in hyperspectr The real data can be transformed into complex
ones by a linear Hilbert filter. ing, the joint estimation of M and
leads to [16]:Z +1 1 (c )H M 1 (c ) K pm (c) =m |M| exp1(ck p( ))d
k )H (c 0( ) M F P =K 1(ck )H MF P (ck ) k=1Joint MLE jolutions are
[Bilodeau 1999]:and K ck K )H M 1 m X(ck ) (ck )Hbb k=1 (ck FP (ck
) MF P = =.K 1(ck )H MF P (ck )b b K1k=1 (ck )H M 1 (ck )
k=1FPThese two estimates given by implicit equations (Fix These two
quantities can be jointly reached by computed using a recursive
apEquation) can be easily iterative processIn the section dealing
with applications to experime THE SIRV MODELLING FOR DETECTION
AND12 / 17 ESTIMATION PROBLEMSperspectral data, we will use the
GLRT-FP (MF PIEEE IGARSS2011 Vancouver, Canada Jean-Philippe
OVARLEZ (ONERA & SONDRA) 13. FIRST RESULTS FOR ANOMALY
DETECTION(DSO DATA) Local Covariance Matrix estimate approachRXDF P
= (ck b 1 )H MF P (ckb )RXDSCM = (ck b 1)H MSCM (ck b )[Reed and
Yu, 1990] THE SIRV MODELLING FOR DETECTION AND13 / 17 IEEE
IGARSS2011 Vancouver, CanadaESTIMATION PROBLEMSJean-Philippe
OVARLEZ (ONERA & SONDRA) 14. CONSIDERATIONS ON MAHALANOBIS
DISTANCEMahalanobis Distance (RXD) built with SCM or FP still
depends on the texture ofthe cell under test ! A solution may be to
seek for a candidate which is invariantwith the texture. Example,
Mahalanobis distance built on the normalized cellunder test:H (ckbk
) 1 MF P (ck bk )N RXDF P = (ck b H k ) (ck bk )THE SIRV MODELLING
FOR DETECTION AND 14 / 17 IEEE IGARSS2011 Vancouver, Canada
ESTIMATION PROBLEMS Jean-Philippe OVARLEZ (ONERA & SONDRA) 15.
FALSE ALARM REGULATION FOR THE DETECTION SIRV-CFAR TEST2 1bp MF P
(c ) H H1(MF P , ) = b ? 1 b 1pH MF P p (c )H MF P (c b )H0UTDFP
Pfathreshold Experimental Data0 0 OGD0.50.5 UTD GData 1
Theoretical11.51.5 22 log10(Pfa) log10(Pfa)2.52.5 3with complex
data3 with real data3.53.5 444.54.5 556050 4030 2010 30 2010 0 1020
Threshold (dB) Threshold (dB) ACE - FPAMF - SCM m !Km+1 mm m Pf a =
(1 )m + 12 F1 K m + 2, Km + 1; K + 1;m+1m+1 m+1 THE SIRV MODELLING
FOR DETECTION AND15 / 17 IEEE IGARSS2011 Vancouver,
CanadaESTIMATION PROBLEMS Jean-Philippe OVARLEZ (ONERA &
SONDRA) 16. SOME RESULTS FOR ARTIFICIAL TARGETSDETECTION (DSO DATA)
Random target4Pf a = 4.6 10 ACE and Fixed Point Uniform target AMF
and SCMTHE SIRV MODELLING FOR DETECTION AND 16 / 17 IEEE IGARSS2011
Vancouver, Canada ESTIMATION PROBLEMS Jean-Philippe OVARLEZ (ONERA
& SONDRA) 17. CONCLUSIONS Hyperspectral images like radar or
SAR images can suffer from non-Gaussianity or heterogeneity that
can reduce the performance ofanomaly detectors (RXD) and target
detectors (ACE), SIRV modelling is a very nice theoretical tool for
the hyperspectralcontext that can match and control the
heterogeneity and non-Gaussianity of the images, Jointly used with
powerful and robust estimates, hyperspectraldetectors may provide
better performances, with nice CFAR properties. The SIRV
methodology has already been widely used successfully for
othertopics such as radar detection, STAP, SAR classification
[Formont et al.,Vasile et al. IGARSS11], SAR change detection,
target detection in SARimages, INSAR, POLINSAR. It can also help in
understanding hyperspectraldetection and classification
problems.THE SIRV MODELLING FOR DETECTION AND 17 / 17 IEEE
IGARSS2011 Vancouver, Canada ESTIMATION PROBLEMS Jean-Philippe
OVARLEZ (ONERA & SONDRA)