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- 1. ROBUST DETECTION USING THE SIRV BACKGROUND MODELLING FOR HYPERSPECTRAL IMAGING J.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.fr THE SIRV MODELLING FOR DETECTION AND ESTIMATION PROBLEMS Jean-Philippe OVARLEZ (ONERA & SONDRA) 1 CONTEXT OF THE PROBLEM NOISE MODELLING WITH SIRV CHANGE DETECTION
- 2. OUTLINE OF THE TALK Problems Description and Motivation, The Spherically Invariant Random Process Modelling for Hyperspectral Imaging, Estimation in the SIRV Background, Detection in the SIRV Background, Anomaly Detection and Target Detection Results on Experimental Data, Conclusion. THE SIRV MODELLING FOR DETECTION AND 2 / 17 IEEE IGARSS‘2011 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 DESCRIPTION The bar charts in Fig. 11 provide the range of the de- tection statistic of the target and the maximum value of detection the background detection statistics for various back- is challen grounds. The target-background separation or overlap is the 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 a performs better than the OSP algorithm for the six data To 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 detection good fit and th statistics under the “target absent” hypothesis can be compared 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 IMAGES the quantiles of the expected distribution and the actual data. An agreement between the two is illustrated by a targets. To ev targets, we hav straight line. The Q-Q plots in Fig. 12 illustrate the com- (3). Subpixel To detect (GLRT) targets (characterized by a given spectral signature p) - parison between the experimental detection statistics to domly chosen Regulation of False Alarm. Application to reflectance images (after some the theoretically predicted ones for the matched filter al- of the backgro gorithms. The actual statistics for two different back- sults shown in atmospherical corrections or others). grounds is compared to the normal distribution. A bility as a func RXD CDF and OSP dete with the size of 100 a large set of d Cauchy has been show Probability of Exceedance 10−1 ability using a Blocks Mixture of t-Distributions formance [32] 10−2 When the s as N(µ , ), its M 10−3 distribution w Normal mean, we obta 10−4 Normal Mixed for nonnorma (x2(144)) Trees tance is not ch Grass false alarm for eight blocks ob 0 100 200 300 400 500 600 700 800 900 1000 four by two ma Mahalanobis Distance dictions based F-distribution DSO data 2010 [Manolakis 2002] I 14. Modeling the statistics of the Mahalanobis distance. description for THE SIRV MODELLING FOR DETECTION AND distribution. W 3 / 17 IEEE IGARSS‘2011 Vancouver, Canada ESTIMATION PROBLEMS multivariate t d Jean-Philippe OVARLEZ (ONERA & SONDRA) 100
- 4. xT ˆ 1S(ST ˆ 1S) 1ST ˆ 1x H1 which is the Euclidean distance of the test pixel from > ACE , (13) xT ˆ 1x < the background mean in the whitened space. We note CONVENTIONAL METHODS OF DETECTION H0 which can be obtained from Equation 11 by remov- that, in the absence of a target direction, the detector uses 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 can If we use the adaptive detection and 1 and the target subspacehyperspectral = For targets with in S is specified we have P be found in radar detection, community. We ofcan retrieve all formulas for the z ˆ 1/ 2 x tion a single vector s. Then the the detectors family commonly where ˆ =in1/radarthe square-root decomposi- (intensity detectors are simplified to (angle detector), used ˆ 2 ˆ 1/2 is detection (AMF previous GLR detector), ACE sub-spaces detectors, covariance matrix, the ACE can tion of the estimated ...). ( sT ˆ 1x )2 H1 y = D( x ) = T 1 > , be expressed as (s ˆ s)( + xT ˆ 1x ) < 1 2 H0 ˜ ˜ ˜ ˜ zT S(ST S) 1ST z zT PS z ˜ D ACE ( x ) = , where ( = N, = 1) for the Kelly detector and• Almost all the conventional ztechniques 1 =for1 2 = 1) for the ACE. detection was zT z T z ( 0, anomaly Kelly’s algorithm and targets = 2 detection are ˜based on Gaussian assumptionreal-valued signals and has been applied to ˜ ˜ ˜ ˜ ˆ 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 pixel AMF Distance ^ threshold aΓ –1/2 s x2 ACE as z2 Angular Test pixel ^ z = Γ –1/2 x threshold ^ Γ Γ –1/2 x Adaptive x background z x1 whitening z1 Elliptical Spherical background background distribution distribution FIGURE 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, which uses an angular threshold. The test-pixel vector and target-pixel vector are shown after the removal of the background mean vector. [Manolakis 2002] All these techniques need to estimate the data covariance matrix 100 LINCOLN LABORATORY JOURNAL VOLUME 14, NUMBER 1, 2003 (whitening process). THE SIRV MODELLING FOR DETECTION AND 4 / 17 IEEE IGARSS‘2011 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 SOME x = 1/ 2 ( z) + µ , (21) trix. The Mahalanobis distance is distributed asRESULTS FOUND IN THE LITERATURE µ) ~ F , (22) where z ~ N(0, I) and is a non-negative random 1 ( x µ) C ( x T 1 K, variable independent of z. The density of the ECD is K• 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 heterogeneous in which is known as the characteristic probability den- of freedom. The integer v controls the tails of the intensity and/or Note that f be) only characterized by Gaussian statistic:Cauchy sity function of x. cannot ( and can be distribution: v = 1 leads to the multivariate specified independently. One of the most important properties of random 100 vectors with ECDs is that their joint probability den- sity is uniquely determined by the probability density Cauchy Blocks of the Mahalanobis distance Mixed Probability of exceedence 10–1 distributions 1 RXD-SCM ( K / 2) 1 f d (d ) = d hK ( d ) , 2 2K / 2 ( K /2) 10–2 Normal Hotelling T2 where (K /2) is the Gamma function. As a result, 10–3 Mixed the multivariate probability density identification Trees and estimation problem is reduced to a simpler Grass univariate one. This simplification provides the cor- 10–4 0 200 400 600 800 1000 nerstone for our investigations in the statistical char- RXD on DSO DATA Mahalanobis distance acterization of hyperspectral background data. If we know the mean and covariance of a multi- [Manolakis 2002] FIGURE 50. Modeling the distribution of the Mahalanobis variate random sample, we can check for normality distance for the HSI data blocks shown in Figure 33. The by 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 in Mahalanobis distance against a chi-squared distribu- studied in the hyperspectral estimate the tion. However, in practice we have to scientific community but one still uses .... Gaussian Figure 33 and correspond to trees, grass, and mixed (road and grass) materials. The dotted red curves represent the estimates ! mean and covariance from the available data by using family of heavy-tailed distributions defined by Equation 22. THE SIRV MODELLING FOR DETECTION AND 5 / 17 IEEE IGARSS‘2011 Vancouver, Canada 14, NUMBER 1, 2003 ESTIMATION PROBLEMS VOLUME LINCOLN LABORATORY JOURNAL 113 Jean-Philippe OVARLEZ (ONERA & SONDRA)
- 6. SIRV FOR HYPERSPECTRAL BACKGROUND MODELLING Spherically Invariant Random Vector : Compound Gaussian Process [Yao 1973] Z ✓ ◆ p +1 1 cH M 1 c c= ⌧x pm (c) = exp p(⌧ ) 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 IGARSS‘2011 Vancouver, Canada ESTIMATION PROBLEMS Jean-Philippe OVARLEZ (ONERA & SONDRA)
- 7. SIRV DETECTORS Detectors developed in the SIRV context SIRV texture p() modelling with Padé 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]. < > λ. }−1 y) H1 " ,-./01#&1$#230&*0.%3412&256078/90*�(5**$#$:20;</=% !" Normalized Matched Filterf >+?%% @A(5%2#5B?25&: ;2?($:2A2 2 pH M 1 c C$5B?02$D2?#$ !"$&#$256 H1 ? &! !" ⇤(c) = Hn (p M 1 p) (cH M 1 c) H0 )*+) &# !" c: cell under test p: spectral steering vector of the target) &$ !" (c) is SIRV CFAR " ! # $ % !" !" !" !" !" !"#$%"&( ! Texture-CFAR property for the GLRT-LQ but needs to know the true covariance MMATRIX M IS GENERALLYIGARSS‘2011 Vancouver, Canada THE SIRV MODELLING FOR DETECTION AND 7 / 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 Filter 2 ˆ p M 1y H H0 < (y) = ⇣ ⌘⇣ ⌘ > ˆ yH M 1 y ˆ pH M 1 p H1 These detectors are SIRV-CFAR only for some particular estimates of M ! Some well known estimates K ˆ 1 X MSCM = ck cH ˆ M ??? K k=1 K k m X ck cH ˆ MN SCM = k K cH ck k=1 k [Gini-Conte, 2002] THE SIRV MODELLING FOR DETECTION AND 8 / 17 IEEE IGARSS‘2011 Vancouver, Canada ESTIMATION PROBLEMS Jean-Philippe OVARLEZ (ONERA & SONDRA)
- 9. CHOICE OF THE COVARIANCE MATRIX ESTIMATE The Sample Covariance Matrix SCM may be a «poor» estimate of the SIRV Covariance Matrix M because of the texture contamination: K K ˆ 1 X 1 X MSCM = ck cH k = k xk xH k K K k=1 k=1 K 1 X 6= xk xH k K k=1 The Normalized Sample Covariance Matrix (NSCM) may be a good candidate of the SIRV Covariance Matrix M: K K ˆ m X ck cH k m X xk xHk MN SCM = = K cH ck k=1 k K xH xk k=1 k h i ˆ 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 / 17 IEEE IGARSS‘2011 Vancouver, Canada ESTIMATION PROBLEMS Jean-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 ˆ m X ck cH k Fixed Point (FP): MF P = K ˆ 1 cH M F P ck k=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 di↵´rents estimateurs ´tudi´s e e e Les di↵´re e Wishart distributed with mK/(m+1) degrees of freedom, Les di↵´rents estimateurs ´tudi´s e e e Les di↵´rents estimateurs ´tudi´s e e e Les di↵´rents estimateurs e Existence and unicity of the solution des donn´es {x , i = 1...N} de taille m, ils v´riﬁent tous l’´quation Les di↵´rentsare proven. eThe solution can be reached by recurrence e Obtenus ` partir a estimateurs ´tudi´s e e e i e Mk=f(Mk-1) whatever the starting point Mdes(ex:ees b0i=I, X1 de b m, ils v´riﬁent tous l’´quation a M , =M {x h=MNSCM), i Obtenus ` partir 0 donn´ M =i 1 1...N} M x ) x x e N Obtenus ` partir des donn´es { a e u(x taille e (1) H i 1 i H i i Robust to outliers, strong targets or scatterers inNthehu(xH M 1 x )i x xH cells. b M= 1 X reference N b i=1 (1) b M i i i u est une fonction de pond´ration N xi xH . e des i i i=1 u est une fonction de pond´rat e u est une fonction de pond´ration des xi xH . eThe Fixed Point belongs to the family of M-estimators (Robust Statistics [Huber Exemples : Maronna 1964, i Exemples :1976, Yohai 2006]) in the more general Exemplescontext of Elliptically Random Process: ) = La SCM : : L’estimateur d’Huber L’estimateur FP : u(r m r u(r ) = 1 (M-estimateur) : ⇢ La SCM : m La SCM : L’estimateur /e si r <= e K d’Huber L’estimateur FP : u(r ) = u(r ) = 1 u(r ) = (M-estimateur) si r > e u(.) choice r u(r ) = 1 ⇢ K /r : 1 K X ⇣ ⌘ u(r ) = K /e si r <= e ˆ M= u cH ˆ M 1 ck ck cH K /r si r > e k k Huber FP SCM K k=1 THE SIRV MODELLING Mahot (ONERA, SONDRA) M. FOR DETECTION AND JDDPHY 2011 6 / 16 10 / 17 IEEE IGARSS‘2011 Vancouver, Canada ESTIMATION PROBLEMS M. Mahot (ONERA, SONDRA) Jean-Philippe OVARLEZM. Mahot (ONERA, SONDRA) (ONERA & SONDRA) JDDPHY 2011 6 / 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: ˆ 1 cH M F P ck k ⌧k = ˆ 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 ˆ 1 µ)H MF P (ck µ) RXDSCM = (ck ˆ 1 µ)H MSCM (ck µ) THE SIRV MODELLING FOR DETECTION AND 11 / 17 IEEE IGARSS‘2011 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 or positive 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 µ leads to [16]: Z +1 ✓ ◆ 1 (c µ)H M 1 (c µ) K pm (c) = m |M| exp 1 (ck p(⌧ ))d⌧ k µ)H ⇥ µ (c ⇥ 0 (⇡ ⌧ ) Mˆ F P =⌧ K ˆ 1 (ck µ)H MF P (ck µ) ⇥ ⇥ k=1Joint MLE jolutions are [Bilodeau 1999]: and K ck K ˆ µ)H M 1 ˆ m X (ck µ) (ck µ)H b b k=1 (ck ⇥ FP (ck ⇥ µ) MF P = ⇥ µ= . K ˆ 1 (ck µ)H MF P (ck µ) b b K 1 k=1 (ck ˆ µ)H M ⇥ 1 (ck ⇥ µ) k=1 FP These two estimates given by implicit equations (Fix These two quantities can be jointly reached by computed using a recursive ap Equation) can be easily iterative process In the section dealing with applications to experime THE SIRV MODELLING FOR DETECTION AND 12 / 17 ESTIMATION PROBLEMSperspectral data, we will use the GLRT-FP ˆ (MF P IEEE IGARSS‘2011 Vancouver, Canada ⇥ Jean-Philippe OVARLEZ (ONERA & SONDRA)
- 13. FIRST RESULTS FOR ANOMALY DETECTION (DSO DATA) Local Covariance Matrix estimate approachRXDF P = (ck b ˆ 1 µ)H MF P (ck b µ) RXDSCM = (ck b ˆ 1 µ)H MSCM (ck b µ) [Reed and Yu, 1990] THE SIRV MODELLING FOR DETECTION AND 13 / 17 IEEE IGARSS‘2011 Vancouver, Canada ESTIMATION PROBLEMS Jean-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 (ck b µk ) ˆ 1 MF P (ck b µk ) N RXDF P = (ck b H µk ) (ck b µk ) THE SIRV MODELLING FOR DETECTION AND 14 / 17 IEEE IGARSS‘2011 Vancouver, Canada ESTIMATION PROBLEMS Jean-Philippe OVARLEZ (ONERA & SONDRA)
- 15. FALSE ALARM REGULATION FOR THE DETECTION SIRV-CFAR TEST 2 ˆ 1 b p MF P (c µ) H H1 ˆ ⇤(MF P , µ) = ⇣ b ⌘⇣ ⌘ ? ˆ 1 b ˆ 1 pH MF P p (c µ)H MF P (c b µ) H0 UTD−FP Pfa−threshold Experimental Data 0 0 OGD −0.5 −0.5 UTD G Data −1 Theoretical −1 −1.5 −1.5 −2 −2 log10(Pfa) log10(Pfa) −2.5 −2.5 −3 with complex data −3 with real data −3.5 −3.5 −4 −4 −4.5 −4.5 −5 −5 −60 −50 −40 −30 −20 −10 −30 −20 −10 0 10 20 Threshold (dB) Threshold (dB) ACE - FP AMF - SCM m ! K m+1 m m m Pf a = (1 )m + 1 2 F1 K m + 2, K m + 1; K + 1; m+1 m+1 m+1 THE SIRV MODELLING FOR DETECTION AND 15 / 17 IEEE IGARSS‘2011 Vancouver, Canada ESTIMATION PROBLEMS Jean-Philippe OVARLEZ (ONERA & SONDRA)
- 16. SOME RESULTS FOR ARTIFICIAL TARGETSDETECTION (DSO DATA) Random target 4 Pf a = 4.6 10 ACE and Fixed Point Uniform target AMF and SCM THE SIRV MODELLING FOR DETECTION AND 16 / 17 IEEE IGARSS‘2011 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 of anomaly detectors (RXD) and target detectors (ACE),• SIRV modelling is a very nice theoretical tool for the hyperspectral context that can match and control the heterogeneity and non- Gaussianity of the images,• Jointly used with powerful and robust estimates, hyperspectral detectors may provide better performances, with nice CFAR properties.• The SIRV methodology has already been widely used successfully for other topics such as radar detection, STAP, SAR classification [Formont et al., Vasile et al. IGARSS’11], SAR change detection, target detection in SAR images, INSAR, POLINSAR. It can also help in understanding hyperspectral detection and classification problems. THE SIRV MODELLING FOR DETECTION AND 17 / 17 IEEE IGARSS‘2011 Vancouver, Canada ESTIMATION PROBLEMS Jean-Philippe OVARLEZ (ONERA & SONDRA)

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