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ROBUST ANOMALY DETECTION IN HYPERSPECTRAL IMAGING

!

J. Frontera-Pons1, M.A. Veganzones2, S. Velasco-Forero3, F. Pascal1, J.P. Ovarlez1,4 and J. Chanussot2

!1: SONDRA Lab, SUPELEC, Gif sur Yvette, France, 2: GIPSA-lab, Grenoble-INP, Saint Martin d’Hyères, France 3: National University of Singapore, Singapore 4: ONERA, the French Aerospace Lab, Palaiseau, France,

1

IEEE International Geoscience And Remote Sensing Symposium, July 2014, Quebec, Canada

/19

OUTLINE OF THE TALK

2 IEEE International Geoscience And Remote Sensing Symposium, July 2014, Quebec, Canada

Problems description and motivation, !

The Elliptical Distribution modeling for Hyperspectral Imaging, !

Estimation of the Elliptical Distribution background parameters, !

Adaptive and Anomaly Detection in the Elliptical Distribution background, !

Detection and False Alarm regulation results on experimental data, !

Conclusions and Perspectives.

/19

PROBLEMS DESCRIPTION

• ANOMALY DETECTION IN HYPERSPECTRAL IMAGES To detect all that is « different » from the background (Mahalanobis distance) - Regulation of False Alarm. Application to radiance images.

!

• DETECTION OF TARGETS IN HYPERSPECTRAL IMAGES To detect (GLRT) targets (characterized by a given spectral signature p) - Regulation of False Alarm. Application to reflectance images (after some atmospherical corrections or others).

!

!

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ground subspace spanned by the columns of B or U Q[30].

The bar charts in Fig. 11 provide the range of the de-tection statistic of the target and the maximum value ofthe background detection statistics for various back-grounds. The target-background separation or overlap isthe quantity used to evaluate target visibility enhance-ment. For example, it can be seen that the ACE detectorperforms better than the OSP algorithm for the six datasets shown.

The expected probability distribution of the detectionstatistics under the “target absent” hypothesis can becompared to the actual statistics using a quantile-quantile(Q-Q) plot. A Q-Q plot shows the relationship betweenthe quantiles of the expected distribution and the actualdata. An agreement between the two is illustrated by astraight line. The Q-Q plots in Fig. 12 illustrate the com-parison between the experimental detection statistics tothe theoretically predicted ones for the matched filter al-gorithms. The actual statistics for two different back-grounds is compared to the normal distribution. A

straight line shown that the postulated model provides agood fit and therefore can be used to estimate the thresh-old for CFAR operation.

The previous results dealt with full-pixel or resolvedtargets. To evaluate detection performance for subpixeltargets, we have simulated subpixel targets using formula(3). Subpixel targets were simulated by adding a ran-domly chosen target pixel from the target pixel set to eachof the background pixels at a constant fraction. The re-sults shown in Fig. 13, show target-background separa-bility as a function of the target fill factor a for the ACEand OSP detectors. Clearly, target visibility improveswith the size of the target. A more detailed comparison ofa large set of detection algorithms is provided in [31]. Ithas been shown that taking into consideration target vari-ability using a subspace model can increase detection per-formance [32].

When the spectral observation vector x is distributedas N( , )µ ! , its Mahalanobis distance follows a chi-squareddistribution with L degrees of freedom. By removing themean, we obtain the anomaly detector (19). However,for nonnormal data the distribution of Mahalanobis dis-tance is not chi-squared. Fig. 14 shows the probability offalse alarm for the three sets shown in Fig. 9 as well aseight blocks obtained by partitioning this data cube into afour by two matrix. The figure also shows theoretical pre-dictions based on a chi-squared and a mixture of twoF-distributions. Evidently, the F-mixture provides a gooddescription for the body and the tails of the underlyingdistribution. We note that if the data follow an ellipticalmultivariate t distribution, the Mahalanobis distance fol-lows a univariate F distribution [33]. The multivariatenormal and t distributions is a special case of the family ofelliptically contoured distributions [33] specified by thedistribution f g T( ) | | {( ) ( )}/x x x= − −− −! !1 2 1µ µ . Theform of function g( ) leads to distributions with heavieror lighter tails than the normal.

The heavy tails in the univariate distribution of theMahalanobis distance imply heavy tails in the multivariatedistribution of the data. Therefore, heavy tails may appearnot only in the quadratic Mahalanobis distance, but inother linear and quadratic statistics employed in severalwidely used [34], [31] target detection techniques.

The family of symmetric α-stable (SαS) distributionsprovides a good model for data with impulsive behavior.They are characterized by a parameterα (characteristic ex-ponent) that takes values in the range 0 2< ≤α . The value

JANUARY 2002 IEEE SIGNAL PROCESSING MAGAZINE 41

NormalTrees

Grass

Mixed

BlocksMixture of t-Distributions

Cauchy

Normal

0 100 200 300 400 500 600 700 800 900 1000

10−4

10−3

10−2

10−1

100

Mahalanobis Distance

Prob

abilit

y of

Exc

eeda

nce

(x (144))2

! 14. Modeling the statistics of the Mahalanobis distance.

10−5

0 100 200 300 400 500 600 700 800 900 1000

10−4

10−3

10 2−

10−1

100

Threshold

Prob

abilit

y of

Fal

se A

larm

Class 2Mixed

Class 9α = 1.96

α = 1.95Trees

GrassNormal ( = 2)α α = 1.99α = 1.98

α = 1.97

! 15. Modeling the matched filter output statistics using stabledistributions.

The performance evaluation ofdetection algorithms in practiceis challenging due to thelimitations imposed by thelimited amount of target data.

[Manolakis 2002]DSO data 2010


RXD CDF

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CONVENTIONAL METHODS OF DETECTION!

• Many methodologies for detection and classification in hyperspectral images can be found in radar detection community. We can retrieve all the detectors family commonly used in radar detection (AMF (intensity detector), ACE (angle detector), sub-spaces detectors, ...). !

• Almost all the conventional techniques for anomaly detection and targets detection are based on Gaussian assumption and on spatial homogeneity in hyperspectral images.

• MANOLAKIS, MARDEN, AND SHAWHyperspectral Image Processing for Automatic Target Detection Applications

100 LINCOLN LABORATORY JOURNAL VOLUME 14, NUMBER 1, 2003

H

H0

1

:

:

x v

x Sa v

=

= +

target absent

target present ,�

implying x ~ N(0, �2��) under H0 and x ~ N(Sa, �2��)under H1. In other words, the background has thesame covariance structure under both hypotheses butdifferent variance. This variance is directly related tothe fill factor of the target, that is, the percentage ofthe pixel area occupied by the target object. The GLRapproach leads to the following adaptive coherence/cosine estimator (ACE) detector [18, 19]:

DACET T T

TH

HACE

( )ˆ ( ˆ ) ˆ

ˆ ,

x

x S S S S x

x x

=

<>� � � �

�

��

��

1 1 1 1

10

1� (13)

which can be obtained from Equation 11 by remov-ing the additive term N from the denominator.

If we use the adaptive whitening transformation

z x� �ˆ ,/�� 1 2

where ˆ ˆ ˆ/ /�� = 1 2 1 2 is the square-root decomposi-tion of the estimated covariance matrix, the ACE canbe expressed as

DACE

T T T

T

T

T( )

˜(˜ ˜) ˜,

˜x

z S S S S z

z z

z P z

z zS= =

�1

where ˜ ˆ /S S� �� 1 2 and P S S S SS˜(˜ ˜) ˜� �T T1 is the or-

thogonal projection operator onto the column spaceof matrix S. Since P PS S˜ ˜

2 = , we can write Equation13 as

DACE ( ) cos ,˜

xP z

z

S= =

2

22 �

which shows that, in the whitened coordinate space,y = DACE (x) is equal to the cosine square of the anglebetween the test pixel and the target subspace.

With the whitening transformation, the anomalydetector in Equation 12 can be expressed as

DAT( ) ,x z z=

which is the Euclidean distance of the test pixel fromthe background mean in the whitened space. We notethat, in the absence of a target direction, the detectoruses the distance from the center of the backgrounddistribution.

For targets with amplitude variability, we have P =1 and the target subspace S is specified by the direc-tion of a single vector s. Then the formulas for theprevious GLR detectors are simplified to

y DT

T TH

H= =

+ <>�

� �( )

( ˆ )

( ˆ )( ˆ ),x

s x

s s x x

��

��

1 2

11 2

10

1

� ��

where (�1 = N, �2 = 1) for the Kelly detector and(�1 = 0, �2 = 1) for the ACE. Kelly’s algorithm wasderived for real-valued signals and has been applied tomultispectral target detection [20]. The one-dimen-

FIGURE 26. Illustration of generalized-likelihood-ratio test (GLRT) detectors. These are the adaptive matched filter(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.

z = Γ –1/2 x

Γ –1/2 x

aΓ –1/2 s

ΓΓ

z2

x1z1

x2

x

AMF

as ACE

AngularthresholdAngular

threshold

Adaptivebackgroundwhitening

Sphericalbackground distribution

Sphericalbackground distribution

Ellipticalbackground distribution

Ellipticalbackground distribution

Test pixel

Target pixel

z

DistancethresholdDistancethreshold

^

^

^

All these adaptive techniques need to estimate the data covariance matrix (whitening process).

Intensity Detector (Matched Filer) Angle Detector (ACE)

[Manolakis 2002]


/195 IEEE International Geoscience And Remote Sensing Symposium, July 2014, Quebec, Canada

ADAPTIVE DETECTION IN HOMOGENEOUS GAUSSIAN BACKGROUND

−50 −40 −30 −20 −10 0 10

−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

Adaptive Matched Filter

Threshold (dBs)

log

10 P

fa

data

Theoretical

TWO-STEP ADAPTIVE MATCHED FILTER

Derived and valid only under Gaussian hypotheses, Their false alarm rate are independent of the covariance matrix: CFAR-matrix property in homogeneous Gaussian data.

TWO-STEP GLRT KELLY DETECTOR

MLE:

8>>>><

>>>>:

µSCM =1

K

KX

k=1

ck

MSCM =1

K

KX

k=1

(ck � µSCM ) (ck � µSCM )H

Complex Multivariate Gaussian PDF:

fc(c) =1

⇡m |M|exp��(c� µ)H M�1

(c� µ)�

Binary Hypotheses test⇢

H0 : c = b, c1, c2, . . . , cKH1 : c = b+Ap, c1, c2, . . . , cK

GLRT ACE DETECTOR

⇤(c) =|pH M�1

SCM (c� µSCM )|2

(pH M�1SCM p) (c� µSCM )H M�1

SCM (c� µSCM )

H1

?H0

� Pfa = (1� �)K�m+12F1 (K �m+ 1,K �m;K;�)

p known A unknown

⇤(c) =|pH M�1

SCM (c� µSCM )|2

pH M�1SCM p

H1

?H0

�

⇤(c) =|pH M�1

SCM (c� µSCM )|2

(pH M�1SCM p)

⇣N + (c� µSCM )H M�1

SCM (c� µSCM )⌘

H1

?H0

�

Pfa = 2F1

✓K �m,K �m+ 1;K;� �

K + 1

◆

[J. Frontera-Pons et al., IEEE Trans. on SP, in review]


ANOMALY DETECTION IN HOMOGENEOUS GAUSSIAN BACKGROUND

[I.S. Reed and X. Yu, 90]

Derived and valid only under Gaussian hypotheses, Its false alarm rate is independent of the covariance matrix: CFAR-matrix property in homogeneous Gaussian data.

GLRT RX ANOMALY DETECTOR: Mahalanobis Distance

(Hotelling’s T-squared distributed)



fc(c) =1

⇡m |M|exp��(c� µ)H M�1

(c� µ)�

A known, p unknown

⇤(c) = (c� µSCM )H M�1SCM (c� µSCM )

H1

?H0

�

4 CHAPTER 1. PRELIMINARY NOTIONS

400 500 600 700 800 900 1,000 1,1000

100

200

300

400

500

600

Background

Anomalies

Threshold

Band 30

Band80

Figure 1.1: Graphical interpretation in two-dimensional space for the ⇤KellyAD detector.

samples. Hence, N+1 vectors are available in the latter and ⇤KellyAD ⌃ does not represent anymorea benchmark structure.

The quadratic form in (1.15) corresponds to the Mahalanobis distance detailed in Mahalanobis(1936). It performs statistically as an outlier detector. When Gaussian assumption is valid, thequadratic form (x�µ)H ⌃�1 (x�µ) follows a �2 distribution for ⌃ and µ perfectly known. In casethe parameter ⌃ is replaced by its MLE, (??), the distribution of the quadratic form:

⇤KellyAD ⌃ = (x� µ)H ⌃�1

SCM (x� µ) ⇠ T 2 , (1.16)

becomes a Hotelling T 2 distribution and thus,

N �m+ 1

mN⇤KellyAD ⌃ ⇠ Fm,N�m+1 (1.17)

where Fm,N�m+1 is the non-central F -distribution with m and N � m + 1 degrees of freedomWeisstein (2010). For high values of N, (N > 10m), the distribution can be approximated by the�2 distribution.

As discussed above, when both covariance matrix and mean vector are unknown, they can bereplaced by their estimates leading to:

⇤KellyAD ⌃,µ = (x� µ)T ⌃�1

(x� µ)H1

?H0

� . (1.18)

The distribution of this detection test is given in the next Proposition.

K�mm (K+1) ⇤(c) ⇠ Fm,K�m

(When K tends to infinity, this test becomes chi-squared distributed)

/19

FIRST COMMENTS AND ADEQUACY WITH SOME RESULTS FOUND IN THE LITERATURE

Hyperspectral data are generally spatially heterogeneous in intensity and/or cannot be only characterized by Gaussian statistic:

Elliptical distribution models have started to be studied in the hyperspectral scientific community and can help in characterizing heterogeneous background but one generally uses .... Gaussian estimates !

RXD on Experimental data

• MANOLAKIS, MARDEN, AND SHAWHyperspectral Image Processing for Automatic Target Detection Applications

VOLUME 14, NUMBER 1, 2003 LINCOLN LABORATORY JOURNAL 113

FIGURE 49. Detection statistics in the target buffer area forthe ACE, AMF, and GLRT target detection algorithms. Allthree algorithms can detect the target (circled pixel) withfewer than forty false alarms in the entire cube of about200,000 pixels.

the maximum-likelihood estimates. When the num-ber of pixels N is greater than about 10K , the esti-mated Mahalanobis distance, for all practical pur-poses, follows a chi-squared distribution.

From the multitude of ECDs discussed in statisticsliterature, the elliptically contoured t-distribution [2]has been shown to provide a good model for manyhyperspectral data sets [32]. This distribution is de-fined by

tK

K K

T

K

( ; , , )[( )/ ]

( / )( )

( ) ( ) ,

/ /x C

C

x C x

µµ

µµ µµ

��

� ��

�

�

=+

� + � ��

��

�

��

+

�

�

2

2

11

2 1 2

1 2

where E(x ) = µµ , Cov(x ) = vv�2 C , � is the number of

degrees of freedom, and C is known as the scale ma-trix. The Mahalanobis distance is distributed as

1 1

KFT

K( ) ( ) ~ ,,x C x� ��µµ µµ � (22)

where FK ,v is the F-distribution with K and v degreesof freedom. The integer v controls the tails of thedistribution: v = 1 leads to the multivariate Cauchy

The random vector x ~ ( , , )EC µµ �� h can be gener-ated by

x z= +�� µµ1 2/ ( ) ,� (21)

where z ~ N(0, I ) and � is a non-negative randomvariable independent of z. The density of the ECD isuniquely determined by the probability density of �,which is known as the characteristic probability den-sity function of x. Note that f�(�) and �� can bespecified independently.

One of the most important properties of randomvectors with ECDs is that their joint probability den-sity is uniquely determined by the probability densityof the Mahalanobis distance

f dK

d h dd KK

K( )( / )

( ) ,/

( / )= �1

2 222 1

�

where �(K /2) is the Gamma function. As a result,the multivariate probability density identificationand estimation problem is reduced to a simplerunivariate one. This simplification provides the cor-nerstone for our investigations in the statistical char-acterization of hyperspectral background data.

If we know the mean and covariance of a multi-variate random sample, we can check for normalityby comparing the empirical distribution of theMahalanobis distance against a chi-squared distribu-tion. However, in practice we have to estimate themean and covariance from the available data by using

FIGURE 50. Modeling the distribution of the Mahalanobisdistance for the HSI data blocks shown in Figure 33. Theblue curves correspond to the eight equal-area subimagesin Figure 33. The green curves represent the smaller areas inFigure 33 and correspond to trees, grass, and mixed (roadand grass) materials. The dotted red curves represent thefamily of heavy-tailed distributions defined by Equation 22.

ACE AMF GLRT

0 200 400 600 800 100010–4

Mahalanobis distance

BlocksCauchy

Normal

MixedTrees

Grass

Mixed distributions

Pro

babi

lity

of e

xcee

denc

e

10–3

10–2

100

10–1

[D. Manolakis 02]


Hotelling T2

RXD-SCM

�2

/19

is a m-dimensional random complex vector with the scatter matrix and the mean vector, usually called density generator, is assumed to be known.

Powerful statistical model that allows: to encompass the Gaussian model but also to extend it (K, Weibull, Fisher, Cauchy, Alpha-Stable, Generalized Gaussian, etc.), to take into account the heterogeneity of the background power with the texture, to take into account possible correlation existing within the m-channels of observation.

Complex Elliptically Contoured Distributions [D. Kelker 70, Olilla 03]:

For a given set of spatial pixels of the hyperspectral image, (equal to the covariance, up to a scalar factor) characterizes the correlation structure 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 and therefore can models the heterogeneity of the spatial background.

ELLIPTICAL/SIRV DISTRIBUTION FOR HYPERSPECTRAL BACKGROUND MODELLING

⌃ µ


⌃

Subclass of Spherically Invariant Random Vector: Compound Gaussian Process [Yao 73]

fc(c) = |⌃|�1hc

�(c� µ)H ⌃�1 (c� µ)

�

c

hc

c =

p⌧ x fc(c) =

1

⇡m |⌃|

Z 1

0

1

⌧mexp

✓� (c� µ)H ⌃

�1(c� µ)

⌧

◆p(⌧) d⌧+µ

x ⇠ CN (0,⌃) ⌧ ⇠ p(⌧)

⌧

Statistical modelling and associated estimators in signal processing Complex elliptical distributions

Statistical modelling and associated estimators in signal processing

MLE

8>>>>>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>>>>>:

= SCM for Gaussian distributions

b⌃SCM =1N

NX

n=1

znz

Hn

⇡ FPE for Compound-Gaussian distributions

b⌃FPE =mN

NX

n=1

cn.cHn

c

Hnb⌃�1

FPE cn

= depend on g for elliptical distributions

b⌃ML =1N

NX

i=1

� g0(zHib⌃�1

ML zi )

g(zHib⌃�1

ML zi )zi z

Hi

Elliptical distributions provide a more general model, but the MLE can not be used if g is unknown.

Estimators which are robust all over elliptical distributions, must be used.

MM (SONDRA, ONERA, SATIE) Robust Covariance Matrix Estimation in Signal Processing 6/12/12 10 / 44


ROBUST M-ESTIMATORS FOR ELLIPTICAL DISTRIBUTIONSIn the framework of ECD, complex M-estimators of location and scatter [P.J. Huber 64, R.A. Maronna 76] are defined as the joint solutions of two implicit equations:

µ =

KX

n=1

u1(tn)cn

KX

n=1

u1(tn)

M =1

K

KX

n=1

u2(t2n)(cn � µ)(cn � µ)H

where . tn =�(cn � µ)HM�1(cn � µ)

�1/2

,

are two weighting functions acting on the quadratic form, i.e. Mahalanobis distance, The choice of results in different estimates for the covariance matrix and the mean vector, Existence and uniqueness of the solution have been proven provided that satisfy given conditions [Maronna 1976], Robust to outliers, to the presence of strong targets or high impulsive samples in the reference cells, Generalization of Maximum Likelihood Estimators if the density generator hy(.) is known:

u1, u2

u1, u2

u1, u2

u2

�t2�= u1(t) = �h0

y

�t2�/hy

�t2�


THE FIXED POINT ESTIMATORSFor an unknown but deterministic texture parameter, the Maximum Likelihood Estimate (MLE) of the Covariance M (approached MLE in the SIRV context), called the Fixed Point MFP (FP), is the solution of the following implicit equations [Tyler 1987]:

Fixed Point (FP):

This estimator does not depend on the elliptical distribution density generator, Robust to outliers, strong targets or scatterers in the reference cells, The Fixed Point Matrix Estimator is consistent, unbiased, asymptotically Gaussian and is, for a fixed number K of secondary data, Wishart distributed with mK/(m+1) degrees of freedom.

[Pascal et al. 2008]

These two quantities can be jointly reached by a convergent iterative algorithm

u2(x) =m

x

µFP =

1

K

KX

k=1

⇣(ck � µFP )

H M�1FP (ck � µFP )

⌘�1/2ck

1

K

KX

k=1

⇣(ci � µFP )

H M�1FP (ck � µFP )

⌘�1/2MFP =

m

K

KX

k=1

(ck � bµFP ) (ck � bµFP )H

(ck � bµFP )H M�1

FP (ck � bµFP )


THE HUBER’S M-ESTIMATORSLes di↵erents estimateurs etudies

Les di↵erents estimateurs etudies

Obtenus a partir des donnees {xi , i = 1...N} de taille m, ils verifient tous l’equation

bM =1N

NX

i=1

hu(xH

ibM�1xi)

ixix

Hi (1)

u est une fonction de ponderation des xixHi .

Exemples :

La SCM :u(r) = 1

L’estimateur d’Huber(M-estimateur) :

u(r) =

⇢K/e si r <= e

K/r si r > e

L’estimateur FP : u(r) = mr

M. Mahot (ONERA, SONDRA) JDDPHY 2011 6 / 16

The Huber’s M-Estimators [P.J. Huber 64] is defined with the following two functions u1(.) and u2(.):




bM =1N

NX

i=1

hu(xH

ibM�1xi)

ixix

Hi (1)


Exemples :

La SCM :u(r) = 1


u(r) =

⇢K/e si r <= e

K/r si r > e






bM =1N

NX

i=1

hu(xH

ibM�1xi)

ixix

Hi (1)


Exemples :

La SCM :u(r) = 1


u(r) =

⇢K/e si r <= e

K/r si r > e



FP SCMHuber

u(.) choice

Extreme values of outside the interval are attenuated (Fixed Point behavior), Normal values below are uniformly kept (SCM behavior), The parameter can be adjusted to choose the percentage of data treated as Gaussian, The Huber estimate is consistent, unbiased, asymptotically Gaussian and is, for a fixed number K, Whishart distributed with a

slight increase K of degrees of freedom ( very close to 1).

t2n [0, k2]k2

k

5

An M-estimator of scatter (respectively location) can beinterpreted as an implicit weighted covariance matrix(respectively mean vector) estimator. In general, a robustweighting function should decrease to zero. This means thatsmall weights are given to those observations that are highlyoutlying in terms of the quadratic form ti. Observationswhich stay far-off from the background model will havea smaller contribution to the parameters estimation. Forexample, the SCM gives unit weight (u2(t) = u1(t) = 1)to all observations, and hence is naturally non-robust. Whendealing with heavy tailed clutter models, as in HSI, the useof robust estimates decreases the impact of highly impulsivesamples and possible outliers in the reference cells.One of the most important property of M -Estimators is thatthey behave asymptotically in elliptically complex distributeddata, up to a known scalar factor, as SCM in Gaussiancontext. This latter property has been derived in [?], [?], [?]and is recalled here (A FAIRE PAR FRED pour M et H(M)expliquer d’ou sort le m/(m+1))

1) Fixed Point Estimators: The Fixed Point approach, ac-cording to the definition proposed by Tyler in [?], satisfy thefollowing equations:

ˆµFP =

NX

i=1

zi⇣(zi � ˆµFP )

Hˆ

M

�1FP (zi � ˆµFP )

⌘1/2

NX

i=1

1

⇣(zi � ˆµFP )

Hˆ

M

�1FP (zi � ˆµFP )

⌘1/2

, (10)

and

ˆ

MFP =

m

N

NX

i=1

(zi � ˆµFP ) (zi � ˆµFP )H

((zi � ˆµFP )H

ˆ

M

�1FP (zi � ˆµFP ))

, (11)

which are the particular cases of (??) and (??) for u1(t) = t�1

and u2

�t2�

= mt�2. They are specified by fixed pointequations and can be easily computed using a recursivealgorithm. If the limit of the algorithm exists, it must be asolution. Although, the convergence of the procedure has notbeen proven, the empirical behavior is suitable. The majordifficulty is found when there is at least one sample belongingto the secondary data which has the same value as the mean,or some samples with values close to the mean. In that case,the algorithm will probably diverge. Huber type M-estimatordescribed below could be used instead, as it overcomes thepossible algorithm divergence.

For the scale-only problem, the Fixed Point covariancematrix estimate has been widely investigated in statistics andsignal processing literature [?]. We refer to [?] for a detailedperformance analysis.The main results on the statistical properties of ˆ

MFP arerecalled for elliptical distribution framework (and µ assumedto be known):

• ˆ

MFP is a consistent and unbiased estimate of M;

• its asymptotic distribution is Gaussian and its covariancematrix is fully characterized in [?];

• its asymptotic distribution is approximately as the asymp-totic distribution of a Wishart matrix with mN/(m+ 1)

degrees of freedom.

Remark that the distribution of ˆ

MFP does not depend on thespecific elliptical distribution. In order to establish consistencyand asymptotic normality, the population distribution can notbe too heavily concentrated around the center. Consistencyand asymptotic distribution of ˆ

MFP are demonstrated forthe joint location-scatter estimation in the real case in [?].For identification purposes, one may define a normalizationconstraint on the matrix estimate, e.g. Tr( ˆMFP ) = m whereTr(.) denotes the trace operator. In the SIRV framework, theFixed Point covariance matrix estimator has been obtainedaccording to the MLE approach in [?]. Hence, consideringthe texture ⌧ as a deterministic and unknown parameter,the scatter matrix estimate in (??) is the true MLE but itcorresponds to the approximate MLE when ⌧ is a randomvariable.

The problem of both location and scatter estimation leadsto consistent estimates. Although we do not discuss herethe proof for consistency, the empirical behavior claims forconsistent assumption. The empirical consistency has beenanalyzed over K-distributed simulated data of dimensionm = 10 and shape parameter ⌫ = 0.1. On Fig. ??, the x-axisrepresents the number of N secondary data beginning at 2m;and on the y-axis the relative error of the estimator.

2) Huber type M-estimators: The Huber’s M-estimators [?]are defined in the complex case [?] when taking in (??) and(??) the following weighting functions:

u1(t) = min

✓1,

k

t

◆u2

�t2�=

1

�min

✓1,

k2

t2

◆

where k > 0 and � depend on an adjustable parameter 0 <q < 1 according to

q = F2m

�2 k2

�(12)

� = F2m+2

�2 k2

�+ k2

1� q

m

where Fm(.) is the cumulative distribution function of a �2

distribution with m degrees of freedom. Thus, Huber-estimatescan be obtained as the solutions of the following implicitequations:

ˆµHub =1

N

NX

i=1

zi 1tik

+

k

N

NX

i=1

zi⇣(zi � ˆµHub)

Hˆ

M

�1Hub (zi � ˆµHub)

⌘1/21ti>k ,

(13)

where and where Fm (.) is the Chi2 CDF with m degrees of freedomq = F2m

�2 k2

�and � = F2m+2

�2 k2

�+ k2

1� q

m

Huber’s M-Estimators is a mix between FP and SCM

k2

�1�1




bM =1N

NX

i=1

hu(xH

ibM�1xi)

ixix

Hi (1)


Exemples :

La SCM :u(r) = 1


u(r) =

⇢K/e si r <= e

K/r si r > e



k2


ADAPTIVE DETECTION IN ELLIPTICAL BACKGROUND

This two-step GLRT test is homogeneous of degree 0: it is independent of any particular Elliptical distribution: CFAR texture and CFAR Matrix properties,

Under homogeneous Gaussian region, it reaches the same performance than those of the detector built with the SCM estimate.

ADAPTIVE NORMALIZED MATCHED FILTER (ACE) BUILT WITH ANY M-ESTIMATES

where the parameter is very close to 1 but depends on the M-estimator:

Ex: for the Fixed Point, =(m+1)/m

⇤(c) =|pH M�1 (c� µ)|2

(pH M�1p)⇣(c� µ)H M�1 (c� µ)

⌘H1

?H0

�.

0 2 4 6 8 10 12 14 16−5

−4.5

−4

−3.5

−3

−2.5

−2

−1.5

−1

−0.5

0

Threshold (γ)

log

10 P

fa

ANMF

Fixed Point

Theoretical

SCM

Huber

FP Huber

SCM

Pfa = (1� �)K�1�1

�m+12F1

⇣K�1�1

�m+ 2, K�1�1

�m+ 1; K�1�1

+ 1;�⌘

Synthetized K-distributed data

�1

�1


!

A mean vector has to be included in the model and estimated jointly with the scatter matrix, In order to keep the analytic Pfa/thresholfd relationship derived for complex data, the real HS data has been

transformed into complex ones by a linear Hilbert filter and then be decimated by a factor 2 (principle of analytic signals)

The hyperspectral data are real and positive as they represent radiance or reflectance.

Original data set (Hymap data) False Alarm Regulation

Figure 1. Color rendering of self test hyperspectral image.

Figure 2. Screenshot of home page for Target Detection Blind Test website.

,,��

7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

�4

�3.5

�3

�2.5

�2

�1.5

�1

�0.5

0

Threshold �

log

10(PFA)

ANMF(SCM-SMV) theo. eq.(14)ANMF(SCM-SMV) MCANMF(FP estimates) theo. eq.(??)ANMF(FP estimates) MC

Fig. 5: PFA versus threshold for the ANMF under a K-distribution for m = 10 and N = 50 when(1) the SCM-SMV are used (red and black curves)(2) the FP estimates are used (yellow and green curves)

Fig. ??. The required SNR is slightly higher for the ANMFtest.

C. Real Hyperspectral Data

VI. CONCLUSION

We have detailed the class of complex elliptical distri-butions as a general model for background characterizationin Hyperspectral Imaging. Elliptical distributions account forheterogeneity and long tail distributions present in real hy-perspectral data. Once established that hyperspectral data cannot fit a multivariate Normal distribution, the use of theGaussian maximum likelihood estimates (SCM and SMV) donot provide the optimal parameter estimation. We propose theuse of robust estimates for the mean vector and the covariancematrix. We have described the M-estimators, notably the FPand the Huber type approach. The joint estimation of bothparameters is a new challenging problem that opens manyunknowns and it will be further investigated. We introduce theuse of these estimates on classical detection methods. For falsealarm regulation purposes, we have derived the theoreticalrelationship to set the proper threshold for a fixed probabilityof false alarm. Finally, we have validated the theoreticalanalysis over simulations and given some results on a realhyperspectral image. We conclude that the robust estimationtools presented in this paper offer a versatile alternative toGaussian estimates. We remark that proposed M-estimators inGaussian environment are capable of reaching the same resultsas the SCM and SMV. On the other hand, they outperformthe classical estimation methods in case of non-Gaussianimpulsive noise. This adaptability and their robustness makethem suitable estimates in most scenarios.

PLACEPHOTOHERE

Michael Shell Biography text here.

John Doe Biography text here.

Jane Doe Biography text here.

Non-Gaussian region

FIRST SET OF RESULTS FOR FALSE ALARM RATE REGULATION WITH ACE-FP


Original data set

Non-Gaussian region

Extracted region : ‣100 x 100 pixels, ‣5 bands, ‣Sliding Window: 19x19

Selected bands

SECOND SET OF RESULTS FOR FALSE ALARM RATE REGULATION WITH ACE-FP


ANOMALY DETECTION IN HETEROGENEOUS AND NON-GAUSSIAN BACKGROUND

Derived and valid for any Elliptical Contoured Distributions, Its false alarm rate unfortunately depends on texture statistic

of the data

EXTENDED GLRT RX ANOMALY DETECTOR: Mahalanobis Distance



p unknown4 CHAPTER 1. PRELIMINARY NOTIONS

400 500 600 700 800 900 1,000 1,1000

100

200

300

400

500

600

Background

Anomalies

Threshold

Band 30

Band80

Figure 1.1: Graphical interpretation in two-dimensional space for the ⇤KellyAD detector.

samples. Hence, N+1 vectors are available in the latter and ⇤KellyAD ⌃ does not represent anymorea benchmark structure.

The quadratic form in (1.15) corresponds to the Mahalanobis distance detailed in Mahalanobis(1936). It performs statistically as an outlier detector. When Gaussian assumption is valid, thequadratic form (x�µ)H ⌃�1 (x�µ) follows a �2 distribution for ⌃ and µ perfectly known. In casethe parameter ⌃ is replaced by its MLE, (??), the distribution of the quadratic form:

⇤KellyAD ⌃ = (x� µ)H ⌃�1

SCM (x� µ) ⇠ T 2 , (1.16)

becomes a Hotelling T 2 distribution and thus,

N �m+ 1

mN⇤KellyAD ⌃ ⇠ Fm,N�m+1 (1.17)

where Fm,N�m+1 is the non-central F -distribution with m and N � m + 1 degrees of freedomWeisstein (2010). For high values of N, (N > 10m), the distribution can be approximated by the�2 distribution.

As discussed above, when both covariance matrix and mean vector are unknown, they can bereplaced by their estimates leading to:

⇤KellyAD ⌃,µ = (x� µ)T ⌃�1

(x� µ)H1

?H0

� . (1.18)

The distribution of this detection test is given in the next Proposition.

fc(c) = |⌃|�1hc

�(c� µ)H ⌃�1 (c� µ)

�

⇤(c) = (c� µFP )H M�1

FP (c� µFP )H1

?H0

�


Results obtained with artificial targets

Original image (Forest Region) Target Spectrum

50 x 50 pixels, 126 spectral bands

FIRST SET OF RESULTS FOR ANOMALY DETECTION


FIRST SET OF RESULTS FOR ANOMALY DETECTION

Better detection for close targets with FP estimates (same Pfa).

RXDSCM = (c� µSCM )H M�1SCM (c� µSCM ) RXDFP = (c� µFP )

H M�1FP (c� µFP )


(a) MUSE data cube (b) Classical RX detector (c) RX detector built the FP estimates

Fig. 3. Classical and Fixed-point anomaly detection in a hyperspectral image of 300⇥ 300 in 3578 channels. See details in thetext of Section 5.

7. REFERENCES

[1] S. Matteoli, M. Diani, and G. Corsini, “A tutorial overview ofanomaly detection in hyperspectral images,” IEEE Aero. andElectr. Systems Mag., vol. 25, no. 7, pp. 5 –28, july 2010.

[2] D.W.J. Stein, S.G. Beaven, L.E. Hoff, E.M. Winter, A.P.Schaum, and A.D. Stocker, “Anomaly detection from hyper-spectral imagery,” IEEE Signal Processing Magazine, vol. 19,no. 1, pp. 58 –69, jan 2002.

[3] I.S. Reed and X. Yu, “Adaptive multiple-band cfar detection ofan optical pattern with unknown spectral distribution,” Acous-tics, Speech and Signal Processing, IEEE Transactions on, vol.38, no. 10, pp. 1760–1770, 1990.

[4] D. Manolakis and D. Marden, “Non gaussian models forhyperspectral algorithm design and assessment,” in Geo-science and Remote Sensing Symposium, 2002. IGARSS’02.2002 IEEE International. IEEE, 2002, vol. 3, pp. 1664–1666.

[5] F. Gini and M. V. Greco, “Covariance matrix estimation forCFAR detection in correlated heavy tailed clutter,” Signal Pro-cessing, special section on SP with Heavy Tailed Distributions,vol. 82, no. 12, pp. 1847–1859, December 2002.

[6] J. Frontera-Pons, M. Mahot, JP Ovarlez, F. Pascal, SK Pang,and J. Chanussot, “A class of robust estimates for detec-tion in hyperspectral images using elliptical distributions back-ground,” in Geoscience and Remote Sensing Symposium(IGARSS), 2012 IEEE International. IEEE, 2012, pp. 4166–4169.

[7] Douglas Kelker, “Distribution theory of spherical distributionsand a location-scale parameter generalization,” Sankhya: TheIndian Journal of Statistics, Series A, pp. 419–430, 1970.

[8] D.E. Tyler, “A distribution-free M -estimator of multivariatescatter,” The Annals of Statistics, vol. 15, no. 1, pp. 234–251,1987.

[9] F. Pascal, P. Forster, J.-P. Ovarlez, and P. Larzabal, “Perfor-mance analysis of covariance matrix estimates in impulsivenoise,” IEEE Trans.-SP, vol. 56, no. 6, pp. 2206–2217, June2008.

[10] Prasanta Chandra Mahalanobis, “On the generalized distancein statistics,” Proceedings of the National Institute of Sciences(Calcutta), vol. 2, pp. 49–55, 1936.

[11] Eric W Weisstein, CRC concise encyclopedia of mathematics,CRC press, 2010.

[12] Melanie Mahot, Frederic Pascal, Philippe Forster, and Jean-Philippe Ovarlez, “Asymptotic properties of robust complexcovariance matrix estimates,” 2013.

[13] “Official website of the muse project ”http://muse.univ-lyon1.fr/”,” .



7. REFERENCES
















7. REFERENCES














MUSE images: 300 x 300 pixels,

3578 spectral bands

Problem of detecting galaxies in HS MUSE (Multi Unit Spectroscopic Explorer) data (465- 930 nm)

)

Classical RX Detector

Enhanced RX Detector

SECOND SET OF RESULTS FOR ANOMALY DETECTION

RXDSCM = (c� µSCM )H M�1SCM (c� µSCM ) RXDFP = (c� µFP )

H M�1FP (c� µFP )

/19

CONCLUSIONS AND PERSPECTIVES

• 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 (AMF, ANMF),

• Elliptically Contoured Distributions modeling is a very powerful theoretical tool in the hyperspectral context that can match and circumvent the heterogeneity and non-Gaussianity of the images,

• Jointly used with powerful and robust estimates, the proposed hyperspectral detectors may provide better performances and a more accurate false alarm regulation. Moreover, they keep the same performance than the conventional Gaussian detectors for homogeneous and Gaussian data.


CONCLUSIONS

SOME PERSPECTIVES BEING ADDRESSED

• Regularized Robust Covariance Matrix Estimation, • Strong connexion with Random Matrix Theory, • Change Detection problems,

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