1 - Ovarlez - IGARSS 2011 Hyperspectral.pdf

ROBUST DETECTION USING THE SIRV
BACKGROUND MODELLING FOR
HYPERSPECTRAL IMAGING
J.P. Ovarlez1,3, S.K. Pang2, F. Pascal3, V. Achard1 and T.K. Ng2

1 : FRENCH AEROSPACE LAB, ONERA, France, jean-philippe.ovarlez@onera.fr, veronique.achard@onera.fr
2 : DSO National Laboratories, Singapore, pszekim@dso.org.sg, nteckkhi@dso.org.sg
3 : 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

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.
2 / 17 IEEE IGARSS‘2011 Vancouver, Canada
ESTIMATION PROBLEMS
Jean-Philippe OVARLEZ (ONERA & SONDRA)

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
ESTIMATION PROBLEMS multivariate t d
Jean-Philippe OVARLEZ (ONERA & SONDRA) 100

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).
ESTIMATION PROBLEMS

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 as
RESULTS 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.

5 / 17 IEEE IGARSS‘2011 Vancouver, Canada 14, NUMBER 1, 2003
ESTIMATION PROBLEMS VOLUME LINCOLN LABORATORY JOURNAL 113

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 covariance
matrix 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.
ESTIMATION PROBLEMS

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$#230&*0.%3412&256078/90*&#0(5**$#$:20;</=%
!"

Normalized Matched Filter
f >+?%%
@A(5%2#5B?25&:
;2?($:2A2
2
pH M 1 c
C$5B?''02$D2?#$
!"$&#$256 H1
?
&!
!"

⇤(c) = H
n (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 M
MATRIX M IS GENERALLYIGARSS‘2011 Vancouver, Canada
7 / 17 IEEE
ESTIMATION PROBLEMS
N. Jean-Philippe OVARLEZ (ONERA & SONDRA)

ADATIVE DETECTION IN SIRV BACKGROUND
New detectors called Adaptive Detectors can be derived by replacing in the NMF
a «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]
ESTIMATION PROBLEMS

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].
ESTIMATION PROBLEMS

COVARIANCE MATRIX ESTIMATION IN SIRV
BACKGROUND
For 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 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 donnés {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 donnés {
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 .
e
The 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

ESTIMATION PROBLEMS M. Mahot (ONERA, SONDRA)
Jean-Philippe OVARLEZM. Mahot (ONERA, SONDRA)
(ONERA & SONDRA) JDDPHY 2011 6 / 16

TEXTURE ESTIMATION IN SIRV BACKGROUND
For an unknown but deterministic texture parameter, the Maximum Likelihood Estimate of the texture at
pixel k is given by:
ˆ 1
cH M F P ck
k
⌧k =
ˆ
m
This 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 an
extended 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)
ESTIMATION PROBLEMS

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 re
CONTEXT ⇥
MF P is a consistent and unbiased estimate of M; its
The hyperspectral data are real and totic distribution is represent radiance or
positive as they Gaussian and its covariance m
reflectance. 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=1
Joint 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
12 / 17 ESTIMATION PROBLEMSperspectral data, we will use the GLRT-FP ˆ (MF P
IEEE IGARSS‘2011 Vancouver, Canada ⇥

FIRST RESULTS FOR ANOMALY DETECTION
(DSO DATA)
Local Covariance Matrix estimate approach

RXDF 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]

ESTIMATION PROBLEMS

CONSIDERATIONS ON MAHALANOBIS DISTANCE
Mahalanobis Distance (RXD) built with SCM or FP still depends on the texture of
the cell under test ! A solution may be to seek for a candidate which is invariant
with the texture. Example, Mahalanobis distance built on the normalized cell
under test: H
(ck b
µk ) ˆ 1
MF P (ck b
µk )
N RXDF P =
(ck b H
µk ) (ck b
µk )

ESTIMATION PROBLEMS

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

ESTIMATION PROBLEMS

SOME RESULTS FOR ARTIFICIAL TARGETS
DETECTION (DSO DATA)

Random target
4
Pf a = 4.6 10

ACE and Fixed Point Uniform target AMF and SCM

ESTIMATION PROBLEMS

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.

ESTIMATION PROBLEMS

1 - Ovarlez - IGARSS 2011 Hyperspectral.pdf

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1 - Ovarlez - IGARSS 2011 Hyperspectral.pdf