![Fig. 12. Quality of the speckle plus target decomposition. We synthesized a set C0 containing ten targets (x0, y0, A0) with random center position (x0, y0) ∼ U[0,100]×[0,100], and random complex amplitude A0 = eiϕ, such as ϕ ∼ U[0,2π]. The image S0(C0) containing those targets was corrupted by a stationary speckle with reflectivity 2σ2, by adding independently to the real and imaginary parts of S0(C0) a Gaussian noise with zero-mean and standard deviation σ. Then, we used Algorithm 4 (with parameters setting K = 25, NT = 20 and ε = 1) to compute from the image the set of detected targets C . The same experiment was repeated one thousand times for different noise level σ. On the second column of this Table, we display the average number of extracted targets, on the third column, the average mean square error (MSE) between S0(C0) and S0(C ), and on the last column, the corresponding PSNR (using PSNR = −10 · log10(MSE)).](/figures/fig-12-quality-of-the-speckle-plus-target-decomposition-we-3r1tg7zw.webp)
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Sub-pixellic Methods for Sidelobes Suppression and
Strong Targets Extraction in Single Look Complex SAR
Images
Rémy Abergel, Loïc Denis, Saïd Ladjal, Florence Tupin
To cite this version:
Rémy Abergel, Loïc Denis, Saïd Ladjal, Florence Tupin. Sub-pixellic Methods for Sidelobes Sup-
pression and Strong Targets Extraction in Single Look Complex SAR Images. IEEE Journal of
Selected Topics in Applied Earth Observations and Remote Sensing, IEEE, 2018, 11 (3), �10.1109/JS-
TARS.2018.2790987�. �hal-01570857v2�
SUB-PIXELLIC METHODS FOR SIDELOBES SUPPRESSION AND STRONG TARGETS EXTRACTION IN SINGLE LOOK COMPLEX SAR IMAGES 1
Sub-pixellic Methods for Sidelobes Suppression and Strong
Targets Extraction in Single Look Complex SAR Images
R
´
emy Abergel, Lo
¨
ıc Denis, Sa
¨
ıd Ladjal and Florence Tupin
Abstract—SAR images display very high dynamic ranges. Man-made
structures (like buildings or power towers) produce echoes that are several
orders of magnitude stronger than echoes from diffusing areas (vegetated
areas) or from smooth surfaces (e.g., roads). The impulse response of
the SAR imaging system is thus clearly visible around the strongest
targets: sidelobes spread over several pixels, masking the much weaker
echoes from the background. To reduce the sidelobes of the impulse
response, images are generally spectrally apodized, trading resolution for
a reduction of the sidelobes. This apodization procedure (global or shift-
variant) introduces spatial correlations in the speckle-dominated areas
which complicates the design of estimation methods. This paper describes
strategies to cancel sidelobes around point-like targets while preserving
the spatial resolution and the statistics of speckle-dominated areas. An
irregular sampling grid is built to compensate the sub-pixel shifts and
turn cardinal sines into discrete Diracs. A statistically grounded approach
for point-like target extraction is also introduced, thereby providing a
decomposition of a single look complex image into two components: a
speckle-dominated image and the point-like targets. This decomposition
can be exploited to produce images with improved quality (full resolution
and suppressed sidelobes) suitable both for visual inspection and further
processing (multi-temporal analysis, despeckling, interferometry).
Index Terms—SAR imaging, sub-pixel target detection, apodization,
sidelobe reduction, speckle, a contrario methodology.
I. INTRODUCTION
Synthetic aperture radar (SAR) images offer insight about the
back-scattering mechanisms at hand when radar pulses are emitted
towards a scene. In particular, two different types of mechanisms
generally occur: (i) man-made structures generate multiple bounces
that lead to both very high amplitude and well-localized echoes in the
SAR images; (ii) in vegetated areas, numerous echoes are produced,
these echoes interfere, leading to a speckle phenomenon. Hence, a
typical SAR image like the one shown in Fig. 1 (a) contains rather
homogeneous areas with fluctuations due to speckle phenomenon
(the larger the average intensity in the area, the larger are these
fluctuations) and the signature of man-made structures in the form of
intensities that are several orders of magnitude larger.
Visualization of such high dynamic range images requires clipping
values above some threshold (as done to produce Fig. 1 (a)). Because
the strongest echoes are several orders of magnitude more intense
than that of the surrounding areas, the impulse response of the
complete SAR imaging system (comprised of the SAR sensor and
the digital synthesis of the SAR image) is visible. A trade-off must
then be found between resolution preservation (limited widening of
the impulse response) and attenuation of the sidelobes of the impulse
response.
Images delivered by spatial agencies underwent a Fourier apodiza-
tion process to prevent the strongest echoes to spread over several tens
of pixels in a cross shape (the typical SAR impulse response without
apodization). Beyond a resolution loss, noticeable by the reduced
ability to separate two close-by echoes, this apodization impacts
regions dominated by speckle. While, without neither apodization
R
´
emy Abergel, Sa
¨
ıd Ladjal and Florence Tupin are with LTCI, T
´
el
´
ecom
ParisTech, 75013 Paris, France (e-mails: firstname.lastname@telecom-
paristech.fr). Lo
¨
ıc Denis is with the Univ Lyon, UJM-Saint-Etienne, Institut
d’Optique Graduate School, Laboratoire Hubert Curien CNRS UMR 5516,
F-42023 Saint-Etienne, France (e-mail: loic.denis@univ-st-etienne.fr).
nor oversampling, fluctuations within speckle-dominated regions are
statistically independent, apodization (and over-sampling) introduces
spatial correlations in these fluctuations. Visually, these fluctuations
appear grainy, but, more importantly, these correlations impact sta-
tistical methods for speckle reduction. Almost all these methods
assume that speckle is not spatially correlated [1]. Even methods
that are relatively immune to speckle correlation (e.g., NL-SAR
[2]) behave better in the absence of these correlations. Two options
are generally considered to handle speckle correlations: (i) sub-
sampling the apodized image to reduce the spatial correlations,
and (ii) speckle whitening, by processing the unapodized (and, if
necessary, resampled at Nyquist rate) image [3], similarly as done
in image processing to denoise images suffering from correlated
noise [4]. None of these two approaches is very satisfactory: the first
approach involves a resolution loss while the second, by removing
the apodization, reinforces the impulse response sidelobes so that the
resulting image is corrupted by large cross-shaped streaks around
brightest pixels.
Sidelobes reduction in SAR imagery has been the subject of
several works. Beyond apodization (i.e., linear processing by spectral
weighting), two main approaches have been proposed to improve
images with very strong echoes such as those created by ships or
buildings: nonlinear spatially variant apodization methods and target
extraction techniques.
The aim of spatially variant apodization methods is to reduce side-
lobes without sacrificing the spatial resolution, i.e., without widening
the main lobe. Nonlinear processing methods were introduced in [5]
under the name Spatially Variant Apodization (SVA) to reach this
goal. The starting point is the observation that, by non-linearly com-
bining impulse responses from a family of spectral weighting func-
tions (e.g., cosine-on-pedestal weighting functions that include Hann
and Hamming apodizations as special cases) an improved impulse
response can be obtained. All impulse responses are maximum at the
location of the target, they then differ in terms of sidelobes amplitude
(reduced when the apodization is stronger) and main lobe width
(narrowest in the absence of apodization). By retaining at each pixel
the smallest amplitude among all amplitudes obtained by applying the
family of apodization functions, sidelobes are kept minimum while
preventing the widening of the main lobe. The choice of cosine-on-
pedestal apodization functions leads to a very efficient algorithm that
can be applied in the signal domain as a space-variant finite impulse
response filter with very compact response [5]. Several extensions of
SVA were later introduced, in particular to perform super-resolution,
see e.g. [6]. We show in this paper that, despite its attractive
computational efficiency, SVA suffers from several drawbacks: (i) it
modifies the statistics of speckle-dominated areas, (ii) point-like
targets are still spread over several pixels, (iii) a negative bias is
introduced (homogeneous regions appear to have a lower reflectivity
after SVA processing). A somewhat related method for sidelobes
reduction, also based on a filter bank, has been investigated in [7].
Rather than locally selecting the best suited apodization, spectral
super-resolution techniques (Capon [8] or APES [9]) are applied
to improve the localization of point targets. These spectral super-
resolution techniques require an estimation of the signal covariance.
SUB-PIXELLIC METHODS FOR SIDELOBES SUPPRESSION AND STRONG TARGETS EXTRACTION IN SINGLE LOOK COMPLEX SAR IMAGES 2
After decomposition of the original SLC image into small patches
(typically 32 ×32), many sub-aperture images are generated for each
patch and the covariance is estimated based on these sub-aperture
images. A super-resolved patch is then produced and the final super-
resolved image is obtained by mosaicking all processed patches. This
method is computationally expansive but produces super-resolved
images with strongly reduced sidelobes provided that the signal
covariance can be correctly estimated and inverted. However, bright
targets are not explicitly detected nor extracted, speckle-corrupted
areas appear to be affected by the process (the super-resolution
introduces strong spatial correlations), and extended targets tend to
be thinned by the super-resolution process. This method is probably
not applicable as a general-purpose sidelobe reduction step prior to
subsequent automated processing.
The second approach to sidelobe reduction is based on the de-
tection of the strongest targets in order to extract them from the
SAR image. The residual image then contains speckle-dominated
regions that can be further processed, for example by a speckle
reduction method. This is the approach followed by [10] and some
other denoising methods, see the discussion in section V.C of [1].
Target detection in SAR images is an extensively studied topic.
In particular, many works are devoted to identifying weak targets
or moving targets [11]–[15], under foliage targets [16], using po-
larimetric data [17]–[19] or interferometric data [20], [21]. In the
context of image contamination by the sidelobes of strong targets,
the difficulty does not lie in the detection of the target (since the
signal-to-noise ratio is very high), but on the necessity to perform sub-
pixel localization of the target to correctly account for the sidelobes.
Detection techniques based on CLEAN algorithm [22]–[24] and its
extension RELAX [25] are then well-adapted. They process a SAR
image iteratively, detecting an additional target at each iteration
from the residuals obtained by subtracting already detected target
signatures from the SAR image. Each time a target is found, the
amplitude and sub-pixel location of the target are estimated in the
maximum likelihood sense, i.e., such that the residuals are minimized.
Decomposition models of a SAR image into strong scatterers
and homogeneous regions have been proposed [26], [27]. These
decomposition approaches however do not take into account the
sidelobes of strong scatterers (they apply to images that are apodized).
Contributions: this paper extends the recent conference pa-
per [28] and introduces:
(i) an irregular resampling procedure that suppresses sidelobes of
strong targets; this procedure preserves the statistical properties
of speckled areas and prevents from spreading bright targets,
(ii) a criterion based on the statistical framework of a contrario
methods to detect targets in speckle, with an explicit control
over the number of false alarms in the image,
(iii) a decomposition method that can separate a SAR image into two
components: one with speckle-dominated areas, the other with
all the strong targets; these components can be further processed
in order to reduce speckle or to improve the resolution.
The paper is organized as follows. Section II describes the spectral
apodization and oversampling that are typical in single look complex
(SLC) SAR images. Section III introduces a criterion to identify the
sub-pixel translation that would suppress at best the sidelobes. This
criterion is then applied to obtain an irregular resampling scheme that
produces images with strongly reduced sidelobes. Section IV derives
a target detection method. This method leads to a decomposition
scheme into a speckle-dominated component and a target compo-
nent whose efficiency and practical interest is illustrated in several
experiments.
(a) amplitude image |u| (b) Fourier modulus |bu|
Fig. 1. An example of TerraSAR-X image data. We display in (a) the
modulus of a SLC TerraSAR-X image u with size M × N (by convention in
all this work, the horizontal axis represents the range direction), and in (b) the
modulus of its DFT bu (low values are displayed in bright and high values are
displayed in dark). The Fourier spectrum (b) vanishes outside a rectangular
sub-frequency domain bω with size m × n (delimited by the red dashed line),
showing that the image u is oversampled by a factor M/m in the range
direction, and by a factor N/n in the azimuth direction.
II. PSEUDO-RAW IMAGE AND PSEUDO-RAW SPECTRUM
Let u : Ω → C be a SLC discrete image with size M × N and
spatial domain Ω = I
M
× I
N
, noting 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 complex-valued signal defined
by
∀(α, β) ∈ Z
2
, bu(α, β) =
X
(k,`)∈Ω
u(k, `) e
−2iπ
(
kα
M
+
`β
N
)
. (1)
Generally, the study of bu is restricted to the period
b
Ω =
b
I
M
×
b
I
N
,
noting
b
I
K
=
−
K
2
,
K
2
∩ Z. The domain
b
Ω is called the canonical
frequency domain (or reciprocal grid) associated to Ω.
The SLC SAR images are, for reasons due to their acquisition
process, band-limited and well-sampled signals. This is illustrated
in Fig. 1, in which we display the modulus of a SLC TerraSAR-X
image u (Fig. 1 (a)) and the modulus of its DFT bu (Fig. 1 (b)).
Indeed, the restriction of bu to
b
Ω is zero-valued everywhere outside
a rectangular frequency domain bω :=
b
I
m
×
b
I
n
⊂
b
Ω of size m × n,
showing that the pixel spacing was adjusted during the sampling
process to (over) satisfy the Shannon-Nyquist criterion. Besides, and
as described in [29], the non-zero part of the Fourier spectrum is in
fact apodized, which means 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 γ : bω → R
+
\ {0} is a frequency attenuating function which
depends on the data provider, and we name cu
0
: bω → C the pseudo-
raw spectrum. The complex-valued image u
0
: ω → C with spatial
domain ω := I
m
×I
n
is hereafter called the pseudo-raw image. Since
the images u and u
0
have different pixel sizes, the pseudo-raw image
u
0
will always be compared to the subsampled image u
ω
: ω → C
obtained by resampling u at the Nyquist frequency, i.e. the image
defined in the Fourier domain by
∀(α, β) ∈ bω, cu
ω
(α, β) =
|ω|
|Ω|
· bu(α, β) , (3)
where |ω| = m · n and |Ω| = M · N denote the cardinality of the
sets ω and Ω.
In the particular case of TerraSAR-X [29], the apodization γ is,
up to a multiplicative factor a
γ
∈ R that we introduce to ensure
SUB-PIXELLIC METHODS FOR SIDELOBES SUPPRESSION AND STRONG TARGETS EXTRACTION IN SINGLE LOOK COMPLEX SAR IMAGES 3
(a) apodized image u
ω
(b) pseudo-raw image u
0
oversampling of u
ω
oversampling of u
0
Fig. 2. Comparison between apodized and pseudo-raw images. On the
first row, we display the modulus of a portion of the apodized image u
ω
and the pseudo-raw image u
0
. Without apodization (pseudo-raw image u
0
),
the spatial resolution is higher but strong targets lead to sidelobes that mask
out the surrounding structures. The magnifications (zooming with factor ten
by Shannon interpolation) shown on the second row illustrate this trade-off
between widening of the central lobe and attenuation of the sidelobes (the
images are displayed using a common grayscale).
that both amplitude images |u
0
| and |u
ω
| have the same maximum,
a 2D-separable Hamming function,
∀(α, β) ∈ bω, γ(α, β) = a
γ
· γ
m
(α) · γ
n
(β) , where
γ
K
(ξ) = λ + (1 − λ) cos
2πξ
K
, and λ = 0.6 , (4)
so that the unapodized pseudo-raw spectrum cu
0
can be easily
recovered from the initial spectrum bu by inverting (2). We refer the
reader to [3], [28] for more details about the computation of the
pseudo-raw image u
0
from u (in particular in the case when the
frequency attenuating function γ is unknown).
The reason why the spatial agencies introduce an apodization is
to attenuate the sidelobes of the strong point target responses (highly
present in urban areas) which are visible due to the SAR impulse
response. Indeed, an unresolved and isotropic target with constant
reflectivity over the radar electromagnetic band can be modeled after
the synthesis of the SAR image by a 2D separable cardinal sine
whose sidelobes remain particularly visible in the vicinity of the
target center, as we observe in Fig. 2 (b). Unfortunately, the reduction
of those sidelobes using apodization also results in a degradation of
the spatial resolution (see Fig. 2 (a)) and the choice of using the
Hamming apodization (4) in the case of TerraSAR-X is presented
in [29] as a trade-off between sidelobe reduction and deterioration of
the resolution.
III. AN IRREGULAR RESAMPLING SCHEME FOR COMPLEX
PSEUDO-RAW SAR IMAGES
In this section, we complete with more details and experimen-
tal results our previous work presented in [28], and we discuss
the strengths and weaknesses of the proposed irregular resampling
scheme.
A. The cardinal sine impulse response model
In stripmap mode, the SAR imaging system (acquisition + SAR
synthesis) exhibits an approximate separability in range and azimuth,
with rectangular spectra in both dimensions. Consequently, the im-
pulse response of the system is very well approached by a two-
dimensional product of cardinal sine functions,
∀(x, y) ∈ R
2
, sinc(x, y) =
sin (πx)
πx
·
sin (πy)
πy
, (5)
with the continuity preserving condition
sin (0)
0
= 1. Notice that
in (5), the coordinates (x, y) are scaled in order to set the range and
azimuth bandwidths both equal to 2π (i.e., the horizontal and vertical
dimensions of the spectrum support represented by the red-dashed
rectangle in Fig. 1 (b)). Under this model, the continuous signal
U
c
0
: R
2
→ C, before sampling, can be modeled as the convolution
between the continuous latent scene and the impulse response. It
follows that U
c
0
is a band-limited signal which can be reconstructed
exactly (neglecting sensor noise), according to the Shannon-Whittaker
Sampling Theorem [30], [31], provided an infinite number of its
samples {U
c
0
(k · δ
r
, ` · δ
az
)}
(k,`)∈Z
2
are observed at regularly spaced
locations, with steps δ
r
≤ 1 along the range direction and δ
az
≤ 1
along the azimuth direction. The pseudo-raw image u
0
corresponds to
the finite sampling of U
c
0
at the critical sampling step, δ
r
= δ
az
= 1,
while the initial image u corresponds to the sampling of U
c
0
using
the sub-critical steps δ
r
= m/M < 1 and δ
az
= n/N < 1. In the
following, we will denote by U
0
: R
2
→ C the Shannon interpolate
of u
0
(U
0
is different from U
c
0
since u
0
is made of only a finite
number of samples), which can be computed as U
0
= U
r
0
+ i · U
i
0
,
noting U
r
0
and U
i
0
the (real-valued) Shannon interpolates of the real
and imaginary parts of u
0
(see for instance Definition 2 in [32] for
the explicit interpolation formula).
In the rest of this section, we drop the double indexes for the spatial
coordinates in order to simplify the equations. The contribution to
the pseudo-raw image of an unresolved target (that is, a target with a
spatial extension that is negligible with respect to the SAR resolution)
located at sub-pixel location k
0
+ δ is:
∀k ∈ ω, u
0
(k) = A sinc(k − (k
0
+ δ)) + u
∗
0
(k) , (6)
where A ∈ C denotes target’s amplitude, k
0
∈ ω, δ ∈
−
1
2
,
1
2
2
, and
u
∗
0
is the signal in the absence of the target. When δ 6= 0, the position
of the target center does not coincide with the sampling grid and for
large values of |A|, the signal u
∗
0
is dominated by the oscillations of
the cardinal sine function in the vicinity of k
0
.
A straightforward solution to this problem consists in resampling
the image u
0
over a translated grid in which the coordinates of the
target center are integers, or equivalently, to translate the image u
0
by the sub-pixellic translation vector t = −δ. Indeed, the translated
image 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,
(7)
where U
∗
0
denotes the Shannon interpolate of u
∗
0
. The resampled
image v
0
is not anymore polluted by the cardinal sine oscillations
present in u
0
, and the image v
0
corresponds to a resampling of
u
∗
0
, except at the position k = k
0
where the target appears. This
phenomenon is clearly illustrated on a portion of a TerraSAR-X
image in Fig. 3, where the profiles extracted from Fig. 3 (a) and
Fig. 3 (b) both correspond to a sampling over two translated grids of
the cardinal sine function. The noticeable difference between the two
sampling grids is that the one used for the blue-dotted line contains
the target center so that all the target energy is concentrated into a
single sample and the effects of the sidelobes are suppressed.
Unfortunately, when the image u
0
contains more than one target,
a global translation is in general not sufficient to accommodate all
the targets at the same time. For that reason, we introduced in [28]
SUB-PIXELLIC METHODS FOR SIDELOBES SUPPRESSION AND STRONG TARGETS EXTRACTION IN SINGLE LOOK COMPLEX SAR IMAGES 4
(a) pseudo-raw image (b) horizontal translation (c) 2D translation
814 816 818 820 822 824 826 828
horizontal axis (range)
-4000
-2000
0
2000
4000
6000
8000
10000
12000
14000
16000
real part of the pseudo-raw signal
cardinal sine function
horizontal section of (a)
horizontal section of (b)
Fig. 3. Profile of a strong target response at the sub-pixellic scale. We
display in (a) the modulus of the pseudo-raw image in the vicinity of a strong
target and we display in (b) and (c) the modulus of the pseudo-raw image
resampled over a translated grid, with translation vector t = (0.3, 0) and
t = (0.3, 0.1) respectively, yielding more localized target support. In the
second row, the real part of a range profile of (a) is represented by the red-
dashed line, while the corresponding profile in (b) is represented by the blue-
dotted line. The actual sampling of each curve is indicated using some colored
plus and cross marks. The green plain curve represents a pure cardinal sine
function of type x 7→ A
r
sinc(x−x
0
), for some appropriate values of x
0
∈ R
(sub-pixellic range coordinate of the target center) and A
r
∈ R (real part of
the target amplitude). Some similar profiles are observed when we consider
the imaginary part of the signal, as well as when we extend the study to the
azimuth direction (as can be verified in (c)).
an irregular resampling scheme to locally reduce the influence of the
sidelobes of the strongest targets.
B. Irregular resampling scheme
The irregular resampling scheme consists in computing from the
pseudo-raw image u
0
a dense displacement field T = (T
x
, T
y
) :
ω →
−
1
2
,
1
2
×
−
1
2
,
1
2
, that can be used to irregularly resample
u
0
into the image v
0
: ω → C defined by
∀(k, `) ∈ ω , v
0
(k, `) = U
0
(k − T
x
(k, `), ` − T
y
(k, `)) . (8)
What we expect from the local displacement T is the following.
(i) In the vicinity of a strong target, the local displacement should
cancel the effects of the sidelobes.
(ii) The cancellation of the sidelobes should be as robust as possible
to target mixing situations.
(iii) In the regions where no strong target is present, the local
displacement field should preserve the speckle statistics, so that
posterior processings (such as denoising, segmentation, etc.)
remain possible using standard methods with no modification.
The two components T
x
and T
y
of the displacement field T will
be computed independently, which is motivated by the separability
in range and azimuth of the system’s impulse response, but also by
the drastic reduction of the algorithmic complexity that it offers. We
recall below how the computation of the displacement field in range
T
x
is done, that of T
y
being totally similar (by simply exchanging
the two coordinates).
Let K ∈ N denote a locality parameter and set ∆ω = [−K, K]∩Z.
For any position (k
0
, `
0
) ∈ ω, we will compute T
x
(k
0
, `
0
) using
T
x
(k
0
, `
0
) = argmin
−
1
2
≤t≤
1
2
J(U
0
(k
0
− t + ∆ω, `
0
)) , (9)
where J (defined below in (10)) is a cost function which is designed
to promote the choice of the best horizontal translation according
to (i), (ii) and (iii), and U
0
(k
0
−t+∆ω, `
0
) is the mono-dimensional
discrete signal of size 2K+1 obtained by restricting the 2D periodical
signal U
0
: R
2
→ C to the discrete set (k
0
− t + ∆ω) × {`
0
}. The
cost function J is a variant of the discrete total variation operator,
whose role is to promote the choice of the translation yielding the
mono-dimensional discrete signal U
0
(k
0
− t + ∆ω, `
0
) that is the
least oscillatory (as is the case of the blue-dotted signal in Fig. 3,
which is much less oscillatory than the red-dashed one). It is defined
by
∀v : ∆ω → C , J(v) = TV
d
mask
(v
r
) + TV
d
mask
(v
i
) ,
where TV
d
mask
(s) =
X
−K≤p<K
p6∈{p
0
(s)−1,p
0
(s)}
|s(p + 1) − s(p)| , (10)
noting v
r
and v
i
the real and imaginary parts of v, and p
0
(s) the
position where |s| is maximal. The resulting irregular resampling
algorithm is given in the box titled Algorithm 1.
Algorithm 1: Irregular resampling scheme proposed in [28]
Inputs: a pseudo-raw image u
0
: ω → C, a locality parameter
K, a number N
T
of tested translations such that the set of all
candidate (horizontal or vertical) translations is
T = −
1
2
+
1
N
T
· {0, . . . , N
T
− 1}.
Outputs: a displacement field T = (T
x
, T
y
) : ω → T × T and
the irregularly resampled pseudo-raw image v
0
: ω → C that
corresponds to the resampling of u
0
over the irregular grid
ω − T := {(k − T
x
(k, `), ` − T
y
(k, `), (k, `) ∈ ω}.
Initialization: precompute the horizontal and vertical
translations of u
0
, that is, compute for all t ∈ T,
u
x
t
= U
0
(ω − (t, 0)) and u
y
t
= U
0
(ω − (0, t)).
Iterations: for each (k, `) ∈ ω, compute T
x
(k, `), T
y
(k, `) and
v
0
(k, `) as follows
1
t
x
← argmin
t∈T
J(u
x
t
(k + [−K, K] ∩ Z, `))
t
y
← argmin
t∈T
J(u
y
t
(k, ` + [−K, K] ∩ Z))
T (k, `) ← (t
x
, t
y
)
v
0
(k, `) ← U
0
(k − t
x
, ` − t
y
)
(11a)
(11b)
(11c)
(11d)
1
In (11a) and (11b), we use a periodic convention for the images u
x
t
and u
y
t
when their evaluation outside of ω is needed. In (11d), the value U
0
(k−t
x
, `−
t
y
) can be efficiently computed by evaluating the Shannon interpolation of the
mono-dimensional signal ` 7→ u
x
t
x
(k, `) at the subpixellic position `−t
y
(this
operation simply involves the inner product between the mono-dimensional
discrete signal and a cardinal sine function).
First, we illustrate the performance of this resampling method
in Fig. 4, where Algorithm 1 is used to resample two pseudo-raw
images: a satellite image (TerraSAR-X in stripmap mode, copyright
DLR, LAN-1706 project acquired on the region Auvergne-Rh
ˆ
one-
Alpes, South of France) and an airborne image (RAMSES sensor,
ONERA). We show on this experiment that the spatial resolution is
preserved by the resampling procedure while suppressing sidelobes
that were visible in the unapodized image.
When the complex amplitude in a given resolution cell results from
the interferences from several echoes of comparable amplitude, strong
intensity fluctuations are observed in the SAR image (the so-called
speckle phenomenon [33]). If several acquisitions of the same scene
are available, the speckle can be reduced by temporal averaging (a.k.a.
multi-looking). More precisely, if {a
k
}
1≤k≤N
looks
is a sequence of
N
looks
complex-valued images, the associated multi-look image is the
real-valued image a
ML
= (1/N
looks
·
P
N
looks
k=1
|a
k
|
2
)
0.5
. In Fig. 5, we
compare the multi-look image computed from an apodized sequence