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, jean-philippe.ovarlez@onera.fr, veronique.achard@onera.fr2 : DSO National Laboratories, Singapore, pszekim@dso.org.sg, nteckkhi@dso.org.sg3 : SONDRA, SUPELEC, France, frederic.pascal@supelec.frTHE 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)