International audienceThis paper considers the problem of detecting changes in mul-tivariate Synthetic Aperture Radar image time series. Classical methodologies based on covariance matrix analysis are usually built upon the Gaussian assumption, as well as an unstructured signal model. Both of these hypotheses may be inaccurate for high-dimension/resolution images, where the noise can be heterogeneous (non-Gaussian) and where all channels are not always informative (low-rank structure). In this paper, we tackle these two issues by proposing a new detector assuming a robust low-rank model. Analysis of the proposed method on a UAVSAR dataset shows promising results 

Breloy, Arnaud

Ginolhac, Guillaume

Mian, Ammar

Ovarlez, Jean-Philippe

English

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Robust Low-rank Change Detection for SAR Image
Time Series
Ammar Mian, Arnaud Breloy, Guillaume Ginolhac, Jean-Philippe Ovarlez
To cite this version:
Ammar Mian, Arnaud Breloy, Guillaume Ginolhac, Jean-Philippe Ovarlez. Robust Low-rank Change
Detection for SAR Image Time Series. IGARSS 2019, Jul 2019, YOKOHAMA, Japan. ￿hal-02345330￿
ROBUST LOW-RANK CHANGE DETECTION FOR SAR IMAGE TIME SERIES
A. Mian∗,†, A.Breloy‡, G. Ginolhac∗,† and J.-P. Ovarlez∗,¶
∗ SONDRA, CentraleSupe´lec, Universite´ Paris-Saclay, F-91190, Gif-sur- Yvette, France
† LISTIC, Universite´ Savoie Mont-Blanc, F-74944, Annecy le Vieux, France
‡ LEME, Universite´ Paris-Nanterre, IUT Ville d’Avray, LEME-EA4416, France
¶ DEMR, ONERA, Universite´ Paris-Saclay, F-91123, Palaiseau, France
ABSTRACT
This paper considers the problem of detecting changes in mul-
tivariate Synthetic Aperture Radar image time series. Classi-
cal methodologies based on covariance matrix analysis are
usually built upon the Gaussian assumption, as well as an
unstructured signal model. Both of these hypotheses may
be inaccurate for high-dimension/resolution images, where
the noise can be heterogeneous (non-Gaussian) and where all
channels are not always informative (low-rank structure). In
this paper, we tackle these two issues by proposing a new
detector assuming a robust low-rank model. Analysis of the
proposed method on a UAVSAR dataset shows promising re-
sults.
Index Terms— Change detection; Synthetic aperture
Radar; Low Rank; Compound Gaussian;
1. INTRODUCTION
Analysis of Synthetic Aperture Radar (SAR) Image Time Se-
ries (ITS) has become a popular topic of study since it has
many practical applications for Earth monitoring such as dis-
aster assement or land-cover analysis. Developing reliable
methodologies for Change Detection (CD) in SAR-ITS is thus
an active topic of research. The CD problem is challenging
due to the lack of ground truths, which does not allow to ap-
ply supervised methods from the image processing literature.
Moreover, it is well known that SAR images are subjected
to speckle noise, which makes traditional optical approaches
unreliable. Under these conditions, unsupervised methodolo-
gies, often based on statistical tools, have been popular ap-
proaches in recent decades [1].
The CD problem can be seen as designing a distance.
Among popular methodologies, Coherent Change Detection
(CCD) [2] and the log-ratio operator [3] have received no-
ticeable attention. However, these methodologies are limited
to pairs of one-dimensional images, while modern sensors al-
low obtaining multidimensional ones (using e.g., polarimetric
The work was partially supported by PHOENIX ANR-15-CE23-0012
and MARGARITA ANR-17-ASTR-0015 grants of the French National
Agency of Research.
or spectro-angular channels [4, 5]). Exploiting this diversity
allows an improvement of performance in terms of CD appli-
cations for SAR-ITS.
For multivariate data, the covariance matrix has been
shown to be a relevant feature so as to assess changes [6].
Assuming a complex Gaussian model, [7] has considered
statistical information theory to design a distance, while [8]
have adapted covariance homogeneity tests from the statis-
tical literature [9], such as the Generalized Likelihood Ratio
Test (GLRT). These methodologies allow to reach good per-
formance, but suffer nonetheless from two issues encountered
in high-dimension/resolution images:
i) The Gaussian model has been shown to be inaccurate
in recent radar clutter analysis [10] due to the inherent
heterogeneity of these images. In order to be robust
to this non-Gaussianity, [11] proposed various GLRTs,
assuming a Compound Gaussian distribution.
ii) Standard detectors are derived assuming unstructured
covariance matrices, while the signal of interest usu-
ally lies in a low-dimensional subspace (e.g., only one
polarisation among HH, HV and VV).
In this scope, [12] proposed to extend some GLRT approaches
to Low-Rank (LR) structured covariance matrices.
To enjoy the best of both worlds, this paper proposes a
new detector based on both robust and LR model: we derive a
GLRT for Compound Gaussian distributed observations that
have a LR structured covariance matrix. This proposed de-
tector is then applied for CD on a SAR-ITS UAVSAR dataset
and exhibits promising results.
2. GENERIC FRAMEWORK
2.1. Data: We consider a multidimensional ITS, that means
each pixel at a given date corresponds to a vector of data
of dimension p. These p channels can correspond to a po-
larimetric diversity (p = 3), or to another kind of diversity
such as a spectro-angular one, obtained through wavelet
transforms [5]. The CD is applied using a local window ar-
round the pixel of interest. Locally, the data set is denoted
p
channels
t = 1
. . .
t = T
x11 =
xy
z
 xT1 =
xy
z

Fig. 1. Illustration of data for p = 3, N = 9. The pixels
highlighted in black correspond to the local observations.
{xtk}(k,t)∈[[1,K]]×[[1,T ]], corresponds to the concatenation of all
channels for pixel at date t and spatial index k, as described
in Figure 1.
2.2 GLRT for CD: For a given time t, the data is assumed
to follow a given distribution of parameter θt, leading to the
likelihood denoted L( {xtk}k∈[[1,K]] | θt ). The parameters θt
characterise the local data at each date. Hence, if there is a
change, the parameter is expected to vary.
For the sake of clear exposition, we will focus on T =
2 (CD between two acquisitions), which can be straightfor-
wardly extended to T > 2. The CD problem can be formu-
lated as a binary hypothesis test:{
H0 : θ1 = θ2 (no change),
H1 : θ1 6= θ2 (change). (1)
In order to derive a metric of decision, the Generalized Like-
lihood Ratio Test (GLRT), is considered. This test consists in
computing the following quantity:
Λˆ =
max
θ1,θ2
∏2
t=1 L
( {xtk}k∈[[1,K]] | H1; {θ1, θ2} )
max
θ1
∏2
t=1 L
( {xtk}k∈[[1,K]] | H0; θ1 ) , (2)
where {θ1, θ2} (resp. θ1) corresponds to the parameters
of the distribution of the observations under H1 (resp. H0).
Hence, to develop efficient detectors, the problem remains to
select an assumed distribution (and corresponding parame-
ters) that accurately reflects the behavior of the data. Addi-
tionally, depending on the assumptions, the evaluation of the
GLRT may lead to complex optimization problems.
3. GLRTS ON COVARIANCE MATRICES
3.1. Gaussian CD [6]: Assuming Gaussian distributed sam-
ples, the CD can be performed by testing a change in the co-
variance matrix. The corresponding GLRT, denoted ΛˆG cor-
responds to (2) with the following distribution/parameters:
xtk ∼ CN (0,Σt) and θtG = {Σt} (3)
this test has a closed-form expression and is well studied in
the statistical literature [8].
3.2. LR-Gaussian CD [12]: Radar signals usually lie in
lower dimensional subspaces, leading to a LR structured co-
variance matrix. The Gaussian GLRT that accounts for this
prior knowledge, denoted ΛˆLRG, can be formulated as (2)
with distribution/parameters:
xtk ∼ CN (0,ΣtR + σ2I) and θtLRG = {ΣtR} (4)
where ΣtR is the rank R signal covariance matrix and σ
2I
is the covariance matrix of the thermal noise. More details
about this GLRT (setting R, σ2, computation...) can be found
in [12] and section 5.2 of this paper.
3.3. Compound Gaussian CD [11]: For heterogeneous im-
ages, the Gaussian assumption may be a poor approxima-
tion of the underlying physics. In order to be robust to lo-
cal power disparities, we can rely on the Compound Gaussian
(CG) model (also referred to as a mixture of scaled Gaus-
sian), which can accurately fit the empirical distribution of
high-resolution data [10]. This model corresponds to a Gaus-
sian one, where each realization is scaled by a local power
factor τ referred to as texture (assumed unknown determinis-
tic in this work). Hence, a corresponding GLRT for change
detection, denoted ΛˆCG, can be formulated as (2) with distri-
bution/parameters:
xtk ∼ CN (0, τ tkΣt) and θtCG = {Σt, {τ tk}k∈[[1,K]]} (5)
i.e., we test if both the covariance matrix Σt and the textures
{τ tk} change between acquisitions. The computation of this
quantity involves fixed-point equations that can be computed
numerically. A study of this approach can be found in [11].
4. PROPOSED DETECTOR
In order to enjoy the improvement brought by both non-
Gaussian and structure assumptions, we propose the follow-
ing detector:
4.1. LR-Compound Gaussian CD: Assuming samples dis-
tributed as CG with a LR structured covariance matrix, the
proposed GLRT, denoted ΛˆLRCG, corresponds to (2) with dis-
tribution/parameters:
xtk ∼ CN (0, τ tk(ΣtR + σ2I))
θtLRCG = {ΣtR, {τ tk}k∈[[1,K]]}
(6)
where ΣtR is the rank R signal covariance matrix and σ
2I is
the covariance matrix of the thermal noise. Again, we test if
both the covariance matrix and the textures change between
acquisitions. The computation of this quantity involves opti-
misation techniques similar to ones used in [11] and [12]. The
Fig. 2. UAVSAR Dataset used in this study. Left: April 23, 2009. Middle: May 15, 2011. Right: Ground Truth.
0 2 4 6 8 10
0.00
0.10
0.20
0.30
i
λ
i
Fig. 3. Repartition of eigenvalues mean over the ITS.
cumbersome and technical calculus is left for a forthcoming
paper1. The following section will present an application of
this detector for CD in SAR-ITS.
5. STUDY ON REAL UAVSAR DATASET
5.1. Description of data
To assess the performance of the proposed method, a pair
of two images from UAVSAR SanAnd 26524 03 Segment 4
dataset2 has been chosen since a ground truth has been es-
tablished in [7] by using comparison with optical data. The
images presented in Figure 2, correspond to full-polarisation
data with a resolution of 1.67m in range and 0.6m in azimuth.
Since the scatterers present in this scene exhibit an interesting
spectro-angular behaviour, each polarisation of these images
has been subjected to the wavelet transform presented in [5],
allowing to obtain images of dimension p = 12.
5.2. Selection of rank and noise level
To compute the proposed detector, the rank R must be es-
timated beforehand. Several approaches exist in the litera-
ture [13] for its estimation. In this paper, we consider a sim-
ple approach by considering the distribution of the mean of
eigenvalues over the ITS plotted in Figure 3. For this dataset,
R = 3 appears to be an interesting value to separate signal
1A python code for the proposed methodology is available at
https://github.com/AmmarMian/Robust-Low-Rank-CD
2Available at https://uavsar.jpl.nasa.gov/.
Fig. 4. Output of detectors (normalised) with (p = 12, N =
25). Top-Left: ΛˆG. Top-Right: ΛˆLRG. Bottom Left: ΛˆCG.
Bottom Right: ΛˆLRCG (proposed).
from noise components. Notably, this rank gathers 81% of
the total variance. The noise variance σ2 is estimated locally
with the mean of the (p − R) lowest eigenvalues computed
with an SVD of the SCM of all samples {xtk} in the patch.
5.3. Results
As a mean to assess the effectiveness of combining LR struc-
ture with a robust model, it is compared to the following de-
tectors: i) the classic Gaussian statistic proposed in [6] (Sec-
tion 3.1); ii) the LR Gaussian statistic of [12] (Section 3.2);
iii) the CG statistic proposed in [11] (Section 3.3). Figure
4 presents the outputs of each detector for a window size of
5× 5. It appears that the LR detectors outputs (right column)
contain less visual false alarms compared to the non-LR ones,
which is expected since the most relevant channels are used
to compute the CD, making it less sensitive to noise.
Figure 5 shows the Receiver Operator Curve (ROC) for
the results of Figure 4. The proposed method allows ob-
taining the best performance of detection for any given false
0.2 0.4 0.6 0.8 1.0
0.5
0.6
0.7
0.8
0.9
1.0
PFA
P
D
ΛˆG [6]
ΛˆLRG [12]
ΛˆCG [11]
ΛˆLRCG (proposed)
Fig. 5. Probability of detection PD versus the the probability
of false alarm PFA with (p = 12, N = 25)
2 4 6 8 10 12 14
0.5
0.6
0.7
0.8
Window size
P
D
ΛˆG [6]
ΛˆLRG [12]
ΛˆCG [11]
ΛˆLRCG (proposed)
Fig. 6. PD versus the size of window at PFA = 5% with
(p = 12, R = 3)
alarm rate, which is an interesting result. The gain is most
apparent with regards to the LR Gaussian detector which per-
forms poorly for false alarms rate greater than 10%. This is
explained by the fact that an LR structure in Gaussian context
results in the loss of some signal power while in our model,
the texture parameters account for the entirety of this power.
Finally, Figure 6 shows the evolution of the performance
in terms of PD at PFA = 5% when increasing the size of the
window used to compute the detectors. Increasing the win-
dow improves the results at the cost of a resolution loss. The
interest of LR methods is well demonstrated here: they al-
low to obtain good detection results with a lower size of win-
dow compared to their non-LR counterparts. The proposed
method exhibits the best performance, which was to be ex-
pected given that it has been derived using a model more ap-
propriated to the data.
6. REFERENCES
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Robust Low-rank Change Detection for SAR Image Time Series

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