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A Complex Sprectrum Based SAR Image Resampling
Method With Restricted Target Sidelobes and Statistics
Preservation
Rémy Abergel, Saïd Ladjal, Florence Tupin, Jean-Marie Nicolas
To cite this version:
Rémy Abergel, Saïd Ladjal, Florence Tupin, Jean-Marie Nicolas. A Complex Sprectrum Based SAR
Image Resampling Method With Restricted Target Sidelobes and Statistics Preservation. 2017 IEEE
International Geoscience and Remote Sensing Symposium, Jul 2017, Fort Worth, United States.
�10.1109/IGARSS.2017.8128214�. �hal-01447608�
A COMPLEX SPECTRUM BASED SAR IMAGE RESAMPLING METHOD WITH
RESTRICTED TARGET SIDELOBES AND STATISTICS PRESERVATION
R
´
emy Abergel, Sa
¨
ıd Ladjal, Florence Tupin, and Jean-Marie Nicolas (first.last@telecom-paristech.fr)
LTCI, T
´
el
´
ecom ParisTech, Universit
´
e Paris Saclay, Paris, France
ABSTRACT
The aim of this work is to present a resampling scheme for SAR im-
ages that preserves spatial resolution and produces statistically accu-
rate images at the same time. Indeed, SAR images are, for reasons
due to their acquisition process, well sampled signals according to
the Shannon sampling theory. In the presence of strong responses,
that we will refer to as targets, a sinc-like function centered at the
target is smeared over the entire image and is particularly visible in
the range of tens of pixels surrounding the target. To mitigate this
phenomenon, the usual solution is to apply an apodization window
in the Fourier domain so as to change the cardinal sine impulse re-
sponse into a much rapidly decaying one. This approach has two
major drawbacks. It reduces the resolution of the image and intro-
duces inaccurate statistical dependency between pixels. We propose
to resample the image in an adaptive and robust way so that the target
smear is canceled and the new sampled image is completely faithful
to the underlying signal.
Index Terms— Shannon interpolation, complex spectrum, total
variation, targets, subpixellic image processing
1. INTRODUCTION
SAR images are provided by complex signal processing being at
the heart of the SAR technique (range compression, SAR synthe-
sis). The raw data received by the antenna before these operations
are usually not provided by space agencies. The provided Single
Look Complex data (SLC) are affected by two important factors
that can be seen in the complex Fourier spectrum of the image:
over-sampling and weighting of the azimuth and range spectrum [1].
These factors can change depending on the data provider even for
similar resolutions of the SLC images. For instance, the weight-
ing functions of TerraSAR-X and CSK images are different. These
processing have a strong impact on the appearance of the images
(spreading of the strong targets) and induce a correlation between
neighboring pixels, which can affect further processing like physi-
cal parameter estimation [2]. In this paper we investigate how the
complex spectrum information can be taken into account to improve
SAR images by unweighting images while limiting sidelobes of the
impulse response of strong targets, and we show how to produce an
image suitable both for visualization and further processing. The pa-
per is organized as follows. Section 2 introduces the notations and
gives a method to cancel apodization when the weighting function is
unknown. Section 3 presents the proposed resampling scheme and
is concluded by a proof of statistical accuracy of the result under a
reasonable model.
This work is supported by the ANR through the MIRIAM project.
(a) amplitude image (b) Fourier spectrum
Fig. 1. Example of TerraSAR-X image data. We display in (a)
the modulus of a SAR image u : Ω → C, and in (b) the modulus
of bu over a single period
b
Ω (high values are displayed in dark, low
values in bright, white meaning zero). The Fourier spectrum (b)
reveals the presence of zero-padding (bu is zero everywhere outside
of a rectangular sub-frequency domain bω (
b
Ω delimited by the red
dashed line), showing that the image u was oversampled.
2. FROM THE SAR IMAGE DATA TO THE PSEUDO-RAW
SPECTRUM
2.1. Pseudo-raw image and pseudo-raw spectrum
Let u : Ω → C denote a complex-valued SAR image with domain
Ω = I
M
× I
N
, where I
K
= {0, . . . , K − 1} . We denote by bu the
discrete Fourier transform (DFT) of u, which is the two-dimensional
and (M, N)-periodic signal defined by
∀(α, β) ∈ Z
2
, bu(α, β) =
X
(k,l)∈Ω
u(k, l) e
2iπ
(
αk
M
+
βl
N
)
. (1)
We note
b
I
K
=
−
K
2
,
K
2
∩ Z, then we set
b
Ω =
b
I
M
×
b
I
N
, which
represents the cannonical frequency domain associated to Ω and is
also a period of bu.
We display in Fig. 1 the modulus of a SLC TerraSAR-X image
u and the modulus of bu. We see on that particular example that bu is
non-zero on a rectangular sub-frequency domain bω =
b
I
m
×
b
I
n
⊂
b
Ω
(for some given integers m and n such as m ≤ M, n ≤ N), and zero
outside, showing that u was oversampled. Besides, it happens that
the non-zero part of the Fourier spectrum is in fact apodized, which
means that it resulted from a multiplication in the Fourier domain
by a frequency attenuating function. This function results from the
weighting affecting the antenna pattern and the weighting applied to
the data [1] which depends on the data provider.
In the following, we assume that for any (α, β) ∈
b
Ω, we have
bu(α, β) =
cu
0
(α, β) · γ(α, β) if(α, β) ∈ bω
0 otherwise,
(2)
where bω =
b
I
m
×
b
I
n
⊂
b
Ω, γ : bω → R
++
is a known frequency atten-
uating function and cu
0
: bω → C is called the pseudo-raw spectrum.
We call pseudo-raw image the complex-valued image u
0
: ω → C
with spatial domain ω = I
m
× I
n
obtained by taking the inverse
DFT of cu
0
. Since u
0
and u have different resolutions, the pseudo-
raw image will be compared to the image u
ω
: ω → C obtained by
removing the zero-padding from bu, that is, the image defined in the
Fourier domain by cu
ω
(α, β) = bu(α, β) for all (α, β) ∈ bω.
2.2. Practical estimation of the pseudo-raw spectrum
Now, let us focus on the inversion of (2), that is, on the computation
of the pseudo-raw spectrum cu
0
. When the subfrequency domain bω
and the frequency attenuating function γ are known (for instance
provided by the spatial agency who generated the image) the rela-
tion (2) can be easily inverted and we get
∀(α, β) ∈ bω, cu
0
(α, β) =
bu(α, β)
γ(α, β)
. (3)
When those pieces of information are unavailable or lost, as we as-
sume in all the experiments that we propose in this document, they
must be estimated. We make the reasonable assumption that the sub-
frequency domain bω can be easily retrieved by looking at the Fourier
spectrum bu, as it is the case in Fig. 1. In the case γ is not explic-
itly known, we propose to consider the following separable estimate
given by
∀(α, β) ∈ bω, γ(α, β) = a
γ
· γ
1
(α) · γ
2
(β), (4)
where a
γ
∈ R
++
is a normalization factor whose setting is dis-
cussed below, and γ
1
, γ
2
are the signals obtained by averaging |cu
ω
|
in both directions, that is, by setting for all (α, β) ∈ bω,
γ
1
(α) =
1
n
X
β∈
b
I
n
|cu
ω
(α, β)| , γ
2
(β) =
1
m
X
α∈
b
I
m
|cu
ω
(α, β)| ,
which yields two positively valued functions γ
1
:
b
I
m
→ R
++
and
γ
2
:
b
I
n
→ R
++
as soon as |cu
ω
| does not have a column or line
being identically-zero. The normalization factor a
γ
is computed in
order that |u
0
| and |u
ω
| assume the same maximum over ω, which
yields
a
γ
=
max
(k,l)∈ω
|u
0
0
(k, l)|
max
(k,l)∈ω
|u
ω
(k, l)|
, (5)
where u
0
0
is such that
c
u
0
0
(α, β) = cu
ω
(α, β)/(γ
1
(α) · γ
2
(β)) for all
(α, β) ∈ bω. Finally, u
0
can by computed using (3), (4) and (5).
In Fig. 2, we display a cropping of the amplitude images |u
ω
|
and |u
0
| computed from the image u of Fig. 1. As expected the
unweighted image is better localized (finer localization of the infor-
mation) at the price of a large spreading of strong targets in azimuth
and range directions.
3. ADAPTATIVE RESAMPLING SCHEME FOR THE
PSEUDO-RAW IMAGE
3.1. Model
We illustrate in Fig. 3 that the bright targets observed on the pseudo-
raw image can be very well approached by a two-dimensional cardi-
nal sine function defined by (as given by the SAR processing [1]):
∀(x, y) ∈ R
2
, sinc(x, y) =
sin(πx)
πx
·
sin(πy)
πy
, (6)
modulus of u
ω
modulus of u
0
(pseudo-raw)
Fig. 2. Comparison between apodized and pseudo-raw images.
Removing the apodization from u
ω
yields the pseudo-raw image u
0
,
we display here the modulus of a subpart of those two images. We
can see that the pseudo-raw image u
0
shows a better level of details
than u
ω
, as well as a more precise localization of the information.
However, in the presence of strong targets, the signal is polluted by
horizontal and vertical patterns.
with the continuity-preserving condition
sin(0)
0
= 1. An interpre-
tation of this phenomenon is that the target is sufficiently narrow to
be transformed, by the acquisition process, to the impulse response,
yielding the cardinal sine function. When the position of the tar-
get does not coincide with the sampling grid, an oscillatory pattern
peculiar to the sinc function pollutes the values of the pixels in the
vicinity of the target, yielding (we drop the double indexes for the
spatial coordinates in this section) an observed signal of the type
∀k ∈ ω, u
0
(k) = A sinc(k − (k
0
+ δ)) + u
∗
0
(k) , (7)
where A ∈ C denotes the target amplitude, k
0
+ δ (with k
0
∈ ω,
δ ∈ [−
1
2
,
1
2
]
2
) the subpixellic position of the target, and u
∗
0
the signal
that would be acquired in the absence of the target.
The obvious solution to this problem is to resample the image
on a grid such that the coordinates of the target are integers, thus
suppressing the side lobes contributions. In order to do that, one has
to estimate the subpixellic position of the target and then perform
a translation of the image by the vector t = −δ. Indeed, noting
U
0
: R
2
→ C the Shannon interpolate of u
0
which is computed
as U
0
= U
r
0
+ i · U
i
0
, where U
r
0
and U
i
0
denote the (real-valued)
Shannon interpolates of the real and imaginary parts of u
0
(see for
instance [3] for the explicit definition), the translated signal is v
0
:
k 7→ U
0
(k + δ) and satisfies
∀k ∈ ω, v
0
(k) = U
∗
0
(k + δ) +
A if k = k
0
0 otherwise,
(8)
where U
∗
0
denotes the Shannon interpolate of u
∗
0
. We see that, con-
trary to u
0
, the resampled signal v
0
is not polluted anymore by the
oscillations of the cardinal sine.
Since in practice, there may and will be numerous targets in a
single image, a global translation will not be sufficient to accom-
modate all the targets of the image. For that reason, we propose to
devise a local scheme to compute, at each position in the image, the
translation that reduces the interference of the target-induced car-
dinal sine. Indeed, contrary to [4, 5], we made the choice to not
explicitly detect targets to keep the process as robust as possible.
3.2. Local displacement vector field
Our approach consists in computing from u
0
a dense field of dis-
placements T = (T
x
, T
y
) : ω → [−
1
2
,
1
2
] × [−
1
2
,
1
2
], and resample
(a) pseudo-raw image (b) horizontal translation (c) 2-D translation
816 818 820 822 824 826 828
horizontal axis (x)
-5000
0
5000
10000
15000
real part of the complex signal
cardinal sine function
horizontal section of (a)
horizontal section of (b)
Fig. 3. Resampling targets at the subpixellic scale. We display
in (a) the modulus of the pseudo-raw image u
0
in the vinicity of a
strong target. We display in (b) and (c) the modulus of the image
u
0
resampled by means of a translation of vector t = (0.3, 0) and
t = (0.3, 0.1) respectively, yielding more localized signals. We
display in the second row a plot of the real part of an horizontal
section of u
0
, yielding the red dashed curve, and we do the same
for the resampled signal displayed in (b), yielding the blue dotted
curve. The green plain curve represents a pure cardinal sine, that
is a function of the type x 7→ A sinc(x) where A ∈ R. We see
that the samples of both red and blue signals match very well with
a sampling of the cardinal sine, the same observation holds for the
imaginary parts of the considered signals.
the image u
0
over the locally translated grid, which means to com-
pute the image v
0
: ω → C defined by
∀(k, l) ∈ ω, v
0
(k, l) = U
0
(k − T
x
(k, l), l − T
y
(k, l)) . (9)
In practice, the two components T
x
and T
y
of T will be estimated
independently, we describe here the computation of T
x
, that of T
y
being totally similar.
Given a locality parameter K ∈ N (we took K = 25 in all
our experiments) and a position (k
0
, l
0
) ∈ ω, we associate to each
translation t
x
∈ [−
1
2
,
1
2
] the mono-dimensional signal v
t
x
∈ C
2K+1
corresponding to the restriction of the horizontally translated signal
(x, y) 7→ U
0
(x − t
x
, y) to the horizontal 2K + 1 neighborhood of
the pixel (k
0
, l
0
), that is,
∀p ∈ [−K, K] ∩ Z, v
t
x
(p) = U
0
(k
0
− p − t
x
, l
0
) .
We propose to select among all candidate translations the one
that minimizes a particular cost function t
x
7→ J(v
t
x
) where
J : C
2K+1
→ R, yielding
T
x
(k
0
, l
0
) = argmin
−
1
2
≤t
x
<
1
2
J(v
t
x
) . (10)
We considered three different choices for J, denoted below as J
1
,
J
2
and J
3
.
(i) J
1
(v
t
x
) = − max
p∈[−K,K]
|v
t
x
(p)|. The idea underlying this
choice is that when a target exists in the vicinity of (k
0
, l
0
),
the appropriate translation is found by looking for the signal
v
t
x
having the highest maximal modulus (as it is for instance
the case for the blue dotted curve in Fig. 3).
(ii) J
2
(v
t
x
) = TV
d
(v
r
t
x
) + TV
d
(v
i
t
x
), noting v
r
t
x
and v
i
t
x
the
real and imaginary parts of v
t
x
, and TV
d
the discrete total
variation operator defined by
∀w ∈ R
2K+1
, TV
d
(w) =
X
−K≤p<K
|w(p + 1) − w(p)| .
The idea is that, when sampled on the appropriate grid, the
discrete total variation of a target-induced cardinal sine is min-
imal, whereas it is always higher for all non integer displace-
ments of the grid (the red dashed curve in Fig. 3 is more oscil-
latory than the blue dotted curve and exhibits a higher discrete
total variation).
(iii) J
3
(v
t
x
) = TV
d
mask
(v
r
t
x
) + TV
d
mask
(v
i
t
x
), TV
d
mask
being de-
fined for any w ∈ R
2K+1
by
TV
d
mask
(w) =
X
−K≤p<K
p6∈{p
0
−1,p
0
}
|w(p + 1) − w(p)| ,
where p
0
denotes the position where |w| is maximal. Com-
pared to the previous criterion, we choose to mask the contri-
bution of the brightest pixel to the total variation so that the
TV
d
mask
of a pure cardinal sine function sampled on the ap-
propriate grid is zero.
Since our numerical expriments revealed that the third choice led
to the most satisfying results, it was systematically used in all the
experimental results displayed below. The computation of the re-
sampled image v
0
defined by (9) from the pseudo-raw image u
0
is
summarized in Algorithm 1, and some experimental results are dis-
played and commented in Fig. 4 and Fig. 5.
Algorithm 1: pseudo-raw image resampling
Input: a pseudo-raw image u
0
: ω → C, a locality
parameter K, a discrete set of N
T
candidate translations
T = −
1
2
+
1
N
T
· {0, . . . , N
T
− 1}, and a cost function J (in
all our experiments, we took K = 25, N
T
= 20 and J = J
3
defined in Section 3.2).
Output: a resampled pseudo-raw image v
0
: ω → C, and a
translation map T : ω → T × T.
Initialization: precompute the horizontally and vertically
translated signals v
x
t
= U
0
(ω − (t, 0)), v
y
t
= U
0
(ω − (0, t))
for all t ∈ T.
for (k, l) ∈ ω do
∆ω
x
← (k + [−K, K] ∩ Z) × {l}
∆ω
y
← {k} × (l + [−K, K] ∩ Z)
t
∗
x
← argmin
t
x
∈T
J(v
x
t
x
(∆ω
x
))
t
∗
y
← argmin
t
y
∈T
J(v
y
t
y
(∆ω
y
))
T (k, l) ← (t
∗
x
, t
∗
y
)
v
0
(k, l) ← U
0
(k − t
∗
x
, l − t
∗
y
)
(a) pseudo-raw image u
0
(b) resampled image v
0
close-up view of (a) close-up view of (b)
Fig. 4. Resampling the pseudo-raw image. We display in (a)
and (b) the modulus of the pseudo-raw image u
0
and the modulus of
the resampled image v
0
computed using Algorithm 1. In the second
row, we display some close-up views of the images (a) and (b). The
blue arrows indicate the values of the computed translation fields T
at each pixel location. One can see that the resampled image v
0
ex-
hibits a similar level of details than u
0
but is free of target induced
cardinal sine phenomenon.
3.3. Statistical properties of the resampled image
We investigate the statistical properties of the resulting image and we
show that, under a reasonable assumption, our sampling scheme pro-
duces a signal that is completely faithful to the underlying signal. We
make the assumption that the scene is the superposition of a signal
stemming from a bright target of amplitude A and a fully-developed
speckle. Under this model, the term u
∗
0
(k) in (7) corresponds to the
sampling of a band-limited Gaussian white second-order stationary
process [6]. This means that the correlation between samples dis-
tant by an integer value is zero. The process being Gaussian, the
zero correlation implies independence. A consequence of the band-
limited property of the SAR-signal is that the original discrete image
allows one to recover the true values of U
0
(x) for real x by means
of Shannon interpolation of the available discrete signal. When ex-
amining the criteria J
1,2,3
proposed in Section 3.2, one can see that
for the entire line the optimal value of t
x
is the same for all pixels,
provided the computation window of the criterion is large enough.
Thus, provided that our estimated t
x
equals to δ the final discrete
result of our resampling will be, according to (8), U
∗
0
(k + δ) except
at pixel k = k
0
(the target appears here) which are integer distant
samples from the underlying fully-developed speckle and hence i.i.d
Gaussian variables. We reduced the statistical accuracy of the result-
ing signal to the one of our estimator of t
x
. One can show that this
estimator is unbiased and has a variance depending on the ratio be-
tween the power of the u
∗
0
process and A. In Fig. 6, we show how the
output image v
0
both respects the theoretical Gaussian distribution
of the samples and dramatically decreases the correlation between
neighboring pixels, supporting our claim of statistical accuracy of
the resampled image.
(a) apodized multi-look image (b) resampled multi-look image
Fig. 5. Multi-Look comparison between apodized data and re-
sampled pseudo-raw data. We display in (a) the multi-look real-
valued image u
ML
ω
= (
1
26
P
26
k=1
|u
k
ω
|
2
)
1/2
computed from a set of
26 views {u
k
ω
}
1≤k≤26
, and in (b) the multi-look image computed
from the corresponding set of resampled images {v
k
0
}
1≤k≤26
. The
image (b) exhibits a better level of details and a better localization
of the information than the image (a), as for instance in the areas
delimited by the colored rectangles, where the strong targets can be
more easily separated in (b) than in (a).
-300 -200 -100 0 100 200 300
0
1
2
3
4
5
6
7
8
×10
-3
empirical probability density function
Gaussian fitting (mean=1.8,std=64)
-300 -200 -100 0 100 200 300
0
1
2
3
4
5
6
7
8
×10
-3
empirical probability density function
Gaussian fitting (mean=0.25,std=64)
apodized image (corr = 0.49) resampled image (corr = 5 · 10
−3
)
Fig. 6. First order statistics of a region with constant radiome-
try. We display here the empirical distribution of a 50 × 60 patch
of the real part of the apodized image u
ω
and the resampled image
v
0
where the signal corresponds to pure speckle. In both cases, the
empirical probability density function is very well approached by a
Gaussian function. Pixels of u
ω
are highly correlated (0.49) while
those of v
0
exhibit a one hundred times smaller correlation (we com-
puted corr(w) =
1
kwk
2
P
K
x
−2
k=0
P
K
y
−1
l=0
w(k +1, l)·
w(k, l), where
w : I
K
x
× I
K
y
→ C denotes the considered patch).
4. REFERENCES
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ture Radar, EPFL Press, 2008.
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ager,
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[3] R. Abergel and L. Moisan, “The Shannon Total Variation,”
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