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This paper presents a technique for reducing speckle in Polarimetric Synthetic Aperture Radar (PolSAR) imagery using Nonlocal Means and a statistical test based on stochastic divergences. The main objective is to select homogeneous pixels in the filtering area through statistical tests between distributions. This proposal uses the complex Wishart model to describe PolSAR data, but the technique can be extended to other models. The weights of the location-variant linear filter are function of the p-values of tests which verify the hypothesis that two samples come from the same distribution and, therefore, can be used to compute a local mean. The test stems from the family of (h-phi) divergences which originated in Information Theory. This novel technique was compared with the Boxcar, Refined Lee and IDAN filters. Image quality assessment methods on simulated and real data are employed to validate the performance of this approach. We show that the proposed filter also enhances the polarimetric entropy and preserves the scattering information of the targets.
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Speckle Reduction in Polarimetric SAR Imagery
with Stochastic Distances and Nonlocal Means
1Leonardo Torres, 1Sidnei J. S. Sant’Anna, 1Corina C. Freitas, and
2Alejandro C. Frery?
1Instituto Nacional de Pesquisas Espaciais – INPE
Divis˜ao de Processamento de Imagens – DPI
Av. dos Astronautas, 1758, 12227-010
ao Jos´e dos Campos – SP, Brazil
2Universidade Federal de Alagoas – UFAL
Laborat´orio de Computa¸ao Cient´ıfica e An´alise Num´erica – LaCCAN
Av. Lourival Melo Mota, s/n, Tabuleiro dos Martins, 57072-900
Macei´o – AL, Brazil
Abstract. This paper presents a technique for reducing speckle in Po-
larimetric Synthetic Aperture Radar (PolSAR) imagery using Nonlocal
Means and a statistical test based on stochastic divergences. The main
objective is to select homogeneous pixels in the filtering area through
statistical tests between distributions. This proposal uses the complex
Wishart model to describe PolSAR data, but the technique can be ex-
tended to other models. The weights of the location-variant linear filter
are function of the p-values of tests which verify the hypothesis that two
samples come from the same distribution and, therefore, can be used
to compute a local mean. The test stems from the family of (h-φ) di-
vergences which originated in Information Theory. This novel technique
was compared with the Boxcar, Refined Lee and IDAN filters. Image
quality assessment methods on simulated and real data are employed to
validate the performance of this approach. We show that the proposed
filter also enhances the polarimetric entropy and preserves the scattering
information of the targets.
Keywords: Hypothesis testing, Information theory, Multiplicative noise,
PolSAR imagery, Speckle reduction, Stochastic distances, Synthetic Aper-
ture Radar
1 Introduction
Among the remote sensing technologies, Polarimetric Synthetic Aperture Radar
(PolSAR) has achieved a prominent position. PolSAR imaging is a well-developed
?The authors are grateful to CNPq, CAPES, FAPEAL and FAPESP for supporting
this research.
arXiv:1304.4634v1 [cs.IT] 16 Apr 2013
2 Torres, Sant’Anna, Freitas & Frery
coherent microwave remote sensing technique for providing large-scale two-di-
mensional (2-D) high spatial resolution images of the Earth’s surface dielectric
properties [20].
In SAR systems, the value at each pixel is a complex number: the amplitude
and phase information of the returned signal. Full PolSAR data is comprised of
four complex channels which result from the combination of the horizontal and
vertical transmission modes, and horizontal and vertical reception modes.
The speckle phenomenon in SAR data hinders the interpretation these data
and reduces the accuracy of segmentation, classification and analysis of objects
contained within the image. Therefore, reducing the noise effect is an important
task, and multilook processing is often used for this purpose in single- and full-
channel data. In the latter, such processing yields a covariance matrix in each
pixel, but further noise reduction is frequently needed.
According to Lee and Pottier [20], Polarimetric SAR image smoothing re-
quires preserving the target polarimetric signature. Such requirement can be
posed as: (i) each element of the image should be filtered in a similar way to
multilook processing by averaging the covariance matrix of neighboring pixels;
and (ii) homogeneous regions in the neighborhood should be adaptively selected
to preserve resolution, edges and the image quality. The second requirement,
i.e. selecting homogeneous areas given similarity criterion, is a common problem
in pattern recognition. It boils down to identifying observations from different
stationary stochastic processes.
Usually, the Boxcar filter is the standard choice because of its simple design.
However, it has poor performance since it does not discriminate different targets.
Lee et al. [17,18] propose techniques for speckle reduction based on the multi-
plicative noise model using the minimum mean-square error (MMSE) criterion.
Lee et al. [19] proposed a methodology for selecting neighboring pixels with sim-
ilar scattering characteristics, known as Refined Lee filter. Other techniques use
the local linear minimum mean-squared error (LLMMSE) criterion proposed by
Vasile et al. [37], in a similar adaptive technique, but the decision to select homo-
geneous areas is based on the intensity information of the polarimetric coherency
matrices, namely intensity-driven adaptive-neighborhood (IDAN).
C¸ etin and Karl [4] presented a technique for image formation based on regu-
larized image reconstruction. This approach employs a tomographic model which
allows the incorporation of prior information about, among other features, the
sensor. The resulting images have many desirable properties, reduced speckled
among them. Our approach deals with data already produced and, thus, does
not require interfering in the processing protocol of the data.
Osher et al. [25] presented a novel iterative regularization method for inverse
problems based on the use of Bregman distances using a total variation denoising
technique tailored to additive noise. The authors also propose a generalization for
multiplicative noise, but no results with this kind of contamination are show. The
main contributions were the rigorous convergence results and effective stopping
criteria for the general procedure, that provides information on how to obtain
an approximation of the noise-free image intensity. Goldstein and Osher [15]
Speckle Reduction in Polarimetric SAR Imagery with SD and NL-means 3
presented an improvement of this work using the class of L1-regularized opti-
mization problems, that originated in functional analysis for finding extrema of
convex functionals. The authors apply this technique to the Rudin-Osher-Fatemi
model for image denoising and to a compressed sensing problem that arises in
magnetic resonance imaging. Our work deals with full polarimetric data, for
which, to the best of our knowledge, there are no similar results that take into
account its particular nature: the pixels values are definite positive Hermitian
complex matrices.
Soccorsi et al. [28] presented a despeckling technique for single-look com-
plex SAR image using nonquadratic regularization. They use an image model, a
gradient, and a prior model, to compute the objective function. We employ the
full polarimetric information provided by the multilook scaled complex Wishart
Chambolle [5] proposed a Total Variation approach for a number of problems
in image restoration (denoising, zooming and mean curvature motion), but under
the Gaussian additive noise assumption.
Li et al. [21] propose the use of a particle swarm optimization algorithm and
an extension of the curvelet transform for speckle reduction. They employ the
homomorophic transformation, so their technique can be used either in ampli-
tude or intensity data, but not in complex-valued imagery, as is the case we
present here.
Wong and Fieguth [41] presented a novel approach for performing blind
decorrelation of SAR data. They use a similarity technique between patches
of the point-spread function using a Bayesian least squares estimation approach
based on a Fisher-Tippett log-scatter model. In a similar way, Sølbo and Eltoft [29]
assume a Gamma distribution in a wavelet-based speckle reduction procedure,
and they estimate all the parameters locally without imposing a fixed number
of looks (which they call “degree of heterogeneity”) for the whole image.
Buades et al. [3] proposed a methodology, termed Nonlocal Means (NL-
means), which consists in using similarities between patches as the weights
of a mean filter; it is known to be well suited for combating additive Gaus-
sian noise. Deledalle et al. [11] applied this methodology to PolSAR data using
the Kullback-Leibler distance between two zero-mean complex circular Gaus-
sian laws. Following the same strategy, Chen et al. [6] used the test for equality
between two complex Wishart matrices proposed by Conradsen et al. [8].
This paper proposes a new approach for speckle noise filtering in PolSAR
imagery: an adaptive nonlinear extension of the NL-means algorithm. This is
an extension of previous works [33,34], where we used an approach similar to
that of Nagao and Matsuyma [23]. Overlapping samples are compared based
on stochastic distances between distributions, and the p-values resulting from
such comparisons are used to build the weights of an adaptive linear filter.
The goodness-of-fit tests are derived from the divergences discussed by Frery
et al. [14] and Nascimento et al. [24]. The new proposal is called Stochastic
Distances Nonlocal Means (SDNLM), and amounts to using those observations
which are not rejected by a test seeking for a strong stationary process.
4 Torres, Sant’Anna, Freitas & Frery
This paper is organized as follows. First, we summarize the basic principles
that lead to the complex Wishart model for full polarimetric data. In Section 3 we
recall the Nonlocal Means method. Our approach for reducing speckle in PolSAR
data using stochastic distances between two complex Wishart distributions is
proposed in Section 4. Image Quality Assessment is briefly discussed in Section 5.
Results are presented in Section 6, while Section 7 concludes the paper.
2 The Complex Wishart Distribution
PolSAR imaging results in a complex scattering matrix, which includes intensity
and relative phase data [14]. Such matrices have usually four distinct complex
elements, namely SVV ,SVH ,SHV , and SHH, where Hand Vrefer to the horizon-
tal and vertical wave polarization states, respectively. In a reciprocal medium,
which is most common situation in remote sensing, SVH =SHV so the com-
plex signal backscattered from each resolution cell can be characterized by a
scattering vector Ywith three complex elements [36].
Thus, we have a scattering complex random vector Y= [SHH, SV H , SVV ]t,
where [·]tindicates vector transposition. In general, PolSAR data are locally
modeled by a multivariate zero-mean complex circular Gaussian distribution
that characterize the scene reflectivity [35,36], whose probability density function
f(Y;Σ) = 1
where | · | is the determinant, and the superscript ‘’ denotes the complex con-
jugate of a vector; Σis the covariance matrix of Y. This distribution is defined
on C3. The covariance matrix Σ, besides being Hermitian and positive definite,
has all the information which characterizes the scene under analysis.
Multilook processing enhances the signal-to-noise ratio. It is performed av-
eraging over Lideally independent looks of the same scene, and it yields the
sample covariance matrix Zgiven, in each pixel, by Z=L1PL
ι=1 YιY
Lis the number of looks.
Goodman [16] proved that Zfollows a scaled multilook complex Wishart
distribution, denoted by ZW(Σ, L), and characterized by the following prob-
ability density function:
fZ(Z0;Σ, L) = L3L|Z0|L3
where, for L3, Γ3(L) = π3Q2
i=0 Γ(Li), Γ(·) is the gamma function, tr(·)
is the trace operator, and the covariance matrix Zis given by
Σ=E{Y Y t}=
VV }
VV }
VV }
where E{·} denote expectation. Anfinsen et al. [2] removed the restriction L3.
The resulting distribution has the same form as in (1) and is termed the “relaxed”
Speckle Reduction in Polarimetric SAR Imagery with SD and NL-means 5
Wishart. We assume this last model, and we allow variations of Lalong the
The support of this distribution is the cone of positive definite Hermitian
complex matrices [12].
The parameters are usually estimated by maximum likelihood (ML) due to
its statistical properties. Let Zrbe a random matrix which follows a W(Σ, L)
law. Its log-likelihood function is given by
`r(Σ, L) = 3Llog L+ (L3) log |Zr| − Llog |Σ| − 3 log π
log Γ(Lq)Ltr(Σ1Zr),
resulting in the following score function:
3(log L+ 1) + log |Zr| − log |Σ| − P2
j=0 ψ0(Lj)tr(Σ1Zr),
where ψ0is the digamma function. Let {Z1,Z2,...,ZN}be an i.i.d. random
sample of size Nfrom the W(Σ, L) law. The ML estimator ( b
L) of its param-
eters is b
r=1Zrand the solution of
3 log b
log |Zr| − log |Z| −
The case L < q was also treated by Anfinsen et al. [2].
3 Nonlocal Means
The NL-means method was proposed by Buades et al. [3] based on the redun-
dancy of neighboring patches in images. In this method, the noise-free estimated
value of a pixel is defined as a weighted mean of pixels in a certain region. Un-
der the additive white Gaussian noise assumption, these weights are calculated
based on Euclidean distances which are used to measure the similarity between
a central region patch and neighboring patches in a search window. The filtered
pixel is computed as:
g(x, y) = Pu,vWf(x+u, y +v)w(u, v)
Pu,vWw(u, v),(3)
where w(u, v) are the weights defined on the search window W. The resulting
image gis, thus, the convolution of the input image fwith the mask w=
w0/Pw0(u, v). The factor w0(u, v) is inversely proportional to a distance between
the patches, and is given by
w0(u, v) = expn1
|f(ρu(k)) f(ρv(k))|2o,
6 Torres, Sant’Anna, Freitas & Frery
where h > 0 controls the intensity, in a similar way the temperature controls
the Simulated Annealing algorithm, f(ρu(k)) and f(ρv(k)) are the observations
in the k-th pixels of the patches centered in uand v, respectively. When h
the weights tend to be equal, while when h0 they tend to zero unless
f(ρu(k)) = f(ρv(k)). In the first case the filter becomes a mean over the search
window; in the last case, the filtered value will remain unaltered.
The NL-Means proposal and its extensions rely on computing the weights
of a convolution mask as functions of similarity measures: the closer (in some
sense) two patches are, the heavier the contribution of the central pixel to the
Deledalle et al. [10] analyze several similarity criteria for data which depart
from the Gaussian assumption. In particular, the authors consider the Gamma
and Poisson noises, because they are good image models. Deledalle et al. [9]
extended the NL-means method to speckled imagery using statistical inference
in an iterative procedure. The authors derive the weights using the likelihood
function of Gaussian and square root of Gamma (termed “Nakagami-Rayleigh”)
noises. This idea is extended by Deledalle et al. [11] to PolSAR data under the
complex Gaussian distribution, and by Su et al. [30] to multitemporal PolSAR
Chen et al. [6] presented a NL-Means filter for PolSAR data under the com-
plex Wishart distribution. The authors employ the likelihood ratio test of equal-
ity between two W(Σ, L) laws with the same number of looks L, as presented
by Conradsen et al. [8], to calculate the weights between the patches.
Our proposal addresses the problem in a more general fashion: we use the
p-value of a goodness-of-fit test between two samples. The tests are derived using
an Information Theory approach to compute stochastic divergences, which are
turned into distances and then scaled to exhibit good asymptotic properties:
they obey a χ2distribution. The weights are computed with a soft threshold
which incorporates all the observations which were not rejected by the test, and
some of the others. As presented in the next section, all these elements can be
easily generalized to other models, tests and weight functions.
4 Stochastic Distances Filter
4.1 The weights
In this paper we use a 7×7 search window. The shape and size of the neighboring
patches and the central patch are the same: squares of 3 ×3 pixels. The central
patch, with center pixel Z1, is thus compared with 24 neighboring patches, whose
center pixels are Zi,i= 2,...,25, as illustrated in Figure 1.
The estimate of the noise-free observation at Z1is a weighted sum of the
observations at Z2,...,Z25, being each weight a function of the p-value (p(1, i))
observed in a test of same distribution between two complex Wishart laws:
w(1, i) =
1 if p(1, i)η,
ηp(1, i)1 if η
2< p(1, i)< η,
0 otherwise,
Speckle Reduction in Polarimetric SAR Imagery with SD and NL-means 7
Fig. 1. Central pixel Z1and its neighboring Zi,i={2,...,25}with patches 3 ×3
pixels and filtering window 5 ×5 pixels.
where ηis the significance of the test, specified by the user. By definition,
p(1,1) = 1. This function is illustrated in Figure 2.
w(1, i)
η/2η1p(1, i)
Fig. 2. Weight function.
In this way we employ a soft threshold instead of an accept-reject decision.
This allows the use of more evidence than with a binary decision, as the one
used in [32,34] which was 1 if the sample was not rejected and 0 otherwise.
When all weights are computed, the mask of convolution coefficients is scaled
to add one.
This setup can be generalized in many ways, among them:
The support, i.e., the set of positions ido not need to be local; it can extend
The shape and size of the local windows.
8 Torres, Sant’Anna, Freitas & Frery
The weight function; we opted for the piecewise linear function presented in
equation (4) as a good compromise between generality and computational
The test which produces each p-value.
The rationale for using the weight function specified in equation (4) is the
following. If the p-values were used instead, the presence of a single sample with
excellent match to the central sample would dominate the weights in the mask,
forcing other samples that were not rejected by the test to be practically dis-
carded. For instance, consider the case p(1, i1) = 0.89, p(1, i2) = 0.05 and all
other p-values close to zero. Without the weight function, the nonzero weights
would be, approximately, 0.46 and 0.03, so the second observation would have
a negligible influence on the filtered value while the first, along with the cen-
tral value, dominate the result. Using the aforementioned expression, the three
weights would equal 1/3. This increases the smoothing effect without loosing the
discriminatory ability.
4.2 The statistical test
As defined in the previous section, filtering each pixel requires computing a
number of goodness-of-fit tests between the patch around the central pixel Z1
and the patches surrounding pixels Zi, 2 i25. This will be performed using
tests derived from stochastic distances between samples.
Denote b
θ1the estimated parameter in the central region Z1, and b
the estimated parameters in the remaining areas. To account for possible de-
partures from the homogeneous model, we estimate b
θ= ( b
L) by maximum
The proposal is based on the use of stochastic distances between the patches.
Consider that Z1and Zi, 2 i25, are random matrices defined on the
same probability space, whose distributions are characterized by the densities
fZ1(Z0;θ1) and fZi(Z0;θi), respectively, where θ1and θiare parameters. As-
suming that the densities have the same support given by the cone of Hermitian
positive definite matrices A, the h-φdivergence between fZ1and fZiis given
φ(Z1,Zi) = hZA
fZi(Z0;θi)fZi(Z0;θi) dZ0,
where h: (0,)[0,) is a strictly increasing function with h(0) = 0 and
h0(x)>0 for every x, and φ: (0,)[0,) is a convex function [26].
Choices of functions hand φresult in several divergences.
Divergences sometimes are not symmetric. A simple solution, described in
Frery et al. [14,24,13], is to define a new measure dh
φgiven by
φ(Z1,Zi) = Dh
φ(Z1,Zi) + Dh
Speckle Reduction in Polarimetric SAR Imagery with SD and NL-means 9
Distances, in turn, can be conveniently scaled to present good statistical prop-
erties that make them suitable as test statistics [26]:
θi) = 2mn
(m+n)h0(0)φ00(1) dh
where b
θ1and b
θiare maximum likelihood estimators based on samples size m
and n, respectively. When θ1=θi, under mild conditions Sh
θi) is asymp-
totically χ2
Mdistributed, being Mthe dimension of θ1. Observing Sh
the (asymptotic) p-value of the test is p= Pr(χ2
θi)), and the null
hypothesis θ1=θican be rejected at significance level ηif pη; details can
be seen in the work by Salicr´u et al. [26]. Since we are using the same central
sample for 24 nontrivial tests (notice that Sh
θ1) = 0).
Frery et al. [14,13] obtained several distances between W(Σ, L) distributions.
The test used in this paper was derived from the Hellinger distance, yielding:
θi) = 8mn
provided L3. If L < 3, the following expression can be used
θi) = 8mn
Σ1| | b
it was also derived by Frery et al. [14,13] for the case when L1=L2=L; in
this last case, the equivalent number of looks b
Lshould be estimated using all
the available data, for instance as presented by Anfinsen et al. [2]. The setup
presented in Section 4.1 leads to m=n= 9 since squared windows of side 3 are
The tests based on the Kullback-Leibler, Bhattacharyya and R´enyi of order
β∈ {.1, .5, .9}were also used. They produced almost exactly the same results
as of the Hellinger distance, at the expense of more computational load.
Although the distribution of the test statistics given in equation (5) is only
known in the limit when m, n → ∞ such that m/(m+n)λ(0,1), it has been
observed that the difference between the asymptotic and empirical distributions
is negligible in samples as small as the ones considered in this setup [13,14].
The filter obtained using the p-values produced by the test statistic given
in equation (6) can be applied iteratively. The complex Wishart distribution is
preserved by convolutions, and since the number of looks is estimated in every
pairwise comparison, the evolution of the filtered data is always controlled. This
also holds for any filter derived from h-φdivergences and the setup presented in
Section 4.1.
Computational information about our implementation is provided in A.
10 Torres, Sant’Anna, Freitas & Frery
5 Image Quality Assessment
According to Wang et al. [39] image quality assessment in general, and filter
performance evaluation in particular, are hard tasks and crucial for most image
processing applications.
We assess the filters using simulated and real input data by means of visual
inspection and quantitative measures. Visual inspection includes edges, small
features and color preservation. The measures are computed on the HH,HV
and VV intensity channels, and they amount to (a) the Equivalent Number of
Looks (ENL) in homogeneous areas; and (b) the Structural Similarity Index
Measure (SSIM), proposed by Wang et al. [40] for measuring the similarity be-
tween two images. (c) the Blind/Referenceless Image Spatial QUality Evaluator
(BRISQUE), proposed by Mittal et al. [22] to a holistic measure of quality on
no-reference images.
The equivalent number of looks b
Ncan be estimated by maximum likelihood
solving equation (2). The ENL although it is a measure of the signal to noise
ratio, and it must be considered carefully since it will repute highly blurred
data as excellent: for this measure, the best result is a completely flat (constant)
image, i.e., an image with no information whatsoever. The Boxcar filter usually
has a higher ENL than other proposals because it will always use all the pixels in
the filtering window and the result is a large loss of spatial resolution. Therefore,
the equivalent number of looks should be used with care and only to assess noise
reduction and not image quality in general.
The SSIM index is a structural information measure that represents the
structure of objects in the scene, regardless the average luminance and con-
trast. The SSIM index takes into account three factors: (I) correlation between
edges; (II) brightness distortion; and (III) distortion contrast. Let fand gbe the
original data and the filtered version, respectively, the SSIM index is expressed
SSIM(f, g) = Cov(f, g ) + C1
2fg +C2
where fand gare sample means, bσ2
fand bσ2
gare the sample variances, Cov(f, g )
is the sample covariance between fand g, and the constants C1, C2, C3are
intended to stabilize the index. Following [40], we used the following values:
C1= (K1L)2,C2= (K2L)2and C3=C2/2, where Lis the observed dynamic
range and both K1, K21; we used K1= 0.01 and K2= 0.03.
The SSIM is defined for scalar-valued images and it ranges in the [1,1]
interval, being the bigger value observed is the better result. It was computed
on each intensity channel as the means over squared windows of side 8, and the
reported value is the mean over the three channels. This index is a particular
case of the Universal Quality Index proposed by Wang and Bovik [38]. We also
applied this last measure to all the images here discussed, and the results were
in full agreement with those reported by the SSIM index.
The BRISQUE is a model that operates in the spatial domain and requires
no-reference image. This image quality evaluator does not compute specific dis-
Speckle Reduction in Polarimetric SAR Imagery with SD and NL-means 11
tortions such as ringing, blurring, blocking, or aliasing, but quantifies possible
losses of “naturalness” in the image. This approach is based on the principle
that natural images possess certain regular statistical properties that are mea-
surably modified by the presence of distortions. No transformation to another
coordinate frame (DFT, DCT, wavelets, etc) is required, distinguishing it from
previous blind/no-reference approaches. The BRISQUE is defined for scalar-
valued images and it ranges in the [0,100] interval, and smaller values indicate
better results.
6 Results
The proposed filter, termed “SDNLM (Stochastic Distances Nonlocal Means)
filter” was compared with the Refined Lee (Scattering-model-based), IDAN
(intensity-driven adaptive-neighborhood) and Boxcar filters. These filters act on
a well defined neighborhood (as our implementation of the SDNLM): a squared
search window of side of 5 ×5 pixels, and the former is adaptive (as is the
6.1 Simulated data
Sampling from the Wishart distribution The simulated data was obtained
mimicking real data from six classes on a phantom, a segmented image of size
496 ×496. The original data were produced by a polarimetric sensor aboard the
R99-B Brazilian Air Force aircraft in October 2005 over Campinas, S˜ao Paulo
State, Brazil. The sensor operates in the L-band, and produces imagery with
four nominal looks. The data were simulated in single look, i.e., in the lowest
signal-to-noise possible configuration. The covariance matrices are reported in B.
Figures 3(a) and 3(b) show, respectively, the phantom and simulated single-
look PolSAR image with with false color using |SHH |2in the Red channel, |SHV |2
in the Green channel, and |SVV |2in the Blue channel. The filtered versions with
the Boxcar, Refined Lee, IDAN and SDNLM filters are shown in Figures 3(c),
3(d), 3(e) and 3(f), respectively. The latter was obtained at the η= 90% confi-
dence level.
The noise reduction is noticeable in all the four filtered images. The speckled
effect was mitigated without loosing the color balance, but the Boxcar filter
produces a blurred image. The Refined Lee filter also introduces some blurring,
but less intense that the Boxcar filter. The IDAN filter also reduces noise, but
less effectively that the Refined Lee. The SDNLM filter is the one which best
preserves edges. This is particularly noticeable in the fine strip which appears
light, almost white, to the upper left region of the image. After applying the
Boxcar filter it appears wider than it is. The star-shaped bright spot in the
middle of the blue field looses detail after Boxcar, Refined Lee and IDAN filters
are applied, while the SDNLM noise reduction technique preserves its details
well. The Refined Lee filter introduces a noticeable pixellation effect which is
evident mainly in the brown areas.
12 Torres, Sant’Anna, Freitas & Frery
The Canny detector was applied to the HH band of all the available data
presented in Figure 3, and the results are presented in Figure 4. Even when
applied to the phantom, the Canny detector fails to produce continuous edges in
a few situations. In particular, it completely misses the small light features and a
few edges between regions; see Figure 4(a). Edge detection with this method is,
in practice, impossible using the original single-look data, cf. Figure 4(b) with the
other results, although a few large linear features are visible in the clutter of noisy
edges. As expected, the edges detected in the Boxcar filtered image are smooth,
but they neither grant continuity nor identify fine details; this information is
lost by the filter, see Figure 4(c). The edges detected in the image processed by
the IDAN filter are only marginally better than those observed in the original,
unfiltered, data; see Figure 4(e). Figures 4(d) and 4(f) are the edges detected on
the data filtered by the Refined Lee and SDNLM filters, respectively. Although
they look alike, it is noticeable that the latter preserves better the small details;
see, for instance, the star-shaped object to the center-right of the image. It
appears round in the former, while in the latter it is possible to identify minute
The phantom is available, so it is possible to make a quantitative assessment
of the results. Table 1 presents the assessment of the filters in the three intensity
channels HH,HV and VV of the data presented in Figure 3 using the Equivalent
Number of Looks (ENL) and the SSIM index.
Table 1. Image quality indexes in the images shown in Figure 3.
Filter ENL SSIM Index
Boxcar 15.696 5.768 25.111 0.083 0.038 0.083
Refined Lee 11.665 10.136 14.398 0.164 0.092 0.144
IDAN 2.164 3.171 1.977 0.199 0.137 0.188
SDNLM 80% 7.269 5.999 11.217 0.234 0.150 0.230
SDNLM 90% 8.786 6.578 13.559 0.181 0.101 0.177
SDNLM 99% 14.429 7.129 23.787 0.101 0.055 0.101
The ENL was estimated on homogeneous areas far from edges, so no smudg-
ing from other areas contaminated these values. As expected, the most intense
blurring produces the best results with respect to this criterion: the Boxcar filter
outperforms in two out of three bands and, when, it is not the best, the Refined
Lee filter is. Regarding the Equivalent Number of Looks, IDAN produces worse
results, but a good performance in SSIM index. The SDNLM filter improves
with respect to the ENL criterion when the significance level increases. In order
to make it competitive with the Boxcar, Refined Lee and IDAN filters, the ap-
plication to real data was done using η= 90%. Regarding the SSIM index, our
proposal consistently outperforms the other three filters and, as expected, the
Speckle Reduction in Polarimetric SAR Imagery with SD and NL-means 13
smallest the significance of the test the better the performance is with respect
to this criterion since the least the image is blurred.
Physical-based simulation Sant’Anna et al. [27] proposed a methodology for
simulating PolSAR imagery taking into account the electromagnetic character-
istics of the targets and of the sensing system. The simulated images are more
realistic than the ones obtained by merely stipulating the posterior distribu-
tion given the classes, in particular spatial correlation among pixels emerges and
mixture of classes in the borders as observed.
Each simulated pixel is a complex scattering matrix based on a phantom
image (an idealized cartoon model) with five distinct regions. Multifrequency
sets of single-look PolSAR images have been generated in the L-, C- and X-bands,
corresponding to 1.25 , 5.3and 9.6 GHz, respectively. The acquisition geometry
is that of an airborne monostatic sensor flying at 6,000 m of altitude and 35
grazing angle imaging a 290 ×290 m2area terrain. The 3.0 m spatial resolution
and 2.8 m pixel spacing were set in the range and the azimuth directions. The
data have 128 ×128 pixels; details in [27].
Figure 5 shows the simulated images in the L-, C- and X-bands (Figures 5(a),
5(f) and 5(k), resp.) and its filtered versions by the Boxcar (Figures 5(b), 5(g)
and 5(l), resp.), Refined Lee (Figures 5(c), 5(h) and 5(m), resp.), IDAN (Fig-
ures 5(d), 5(i) and 5(n), resp.) and SDNLM filters (Figures 5(e), 5(j) and 5(o),
resp.) with η= 90%.
The three main visual drawbacks of the Boxcar, Refined Lee and IDAN filters
are noticeable in these results, namely the excessive blurring introduced by the
former, and the pixellate effect produced by the latter. The SDNLM filter con-
sistently presents a good compromise between smoothing and edge preservation.
Table 2 presents the assessment of the filters in three intensity channels by
means of the ENL (computed in the central region, a homogeneous area), and
by means of the SSIM index. The best results are highlighted in boldface. The
results are consistent with those observed in Table 1: the Boxcar filter is the best
with respect to the ENL computed in homogeneous areas far from edges, but
the SDNLM filter outperforms the other three filters when a more sophisticated
metric is used: the SSIM index, which takes into account not only smoothness
but structural information. Regarding these data, the ideal significance level lies
approximately between η= 80% and η= 90%; these values provide a good
smoothing without compromising the structural information.
6.2 Real data
In the remainder of this section the data will be presented in false color using the
Pauli decomposition [20]. This representation of PolSAR data has the advantage
of being interpretable in terms of types of backscattering mechanisms. It consists
of assigning |SHH +SVV |2to the Red channel, |SHHSV V |2to the Green channel,
and 2|SHV |2to the Blue channel. This is one of many possible representations,
and it is noteworthy that the data are filtered in their original domain, before
being decomposed for visualization.
14 Torres, Sant’Anna, Freitas & Frery
Table 2. Image quality indexes in the images shown in Figure 5.
Filter ENL SSIM Index
Boxcar 19.294 21.952 23.072 0.067 0.077 0.115
Refined Lee 12.639 14.059 14.850 0.159 0.163 0.186
IDAN 4.278 4.400 4.655 0.239 0.205 0.237
SDNLM 80% 8.683 9.619 9.588 0.246 0.231 0.251
SDNLM 90% 9.737 11.128 10.780 0.120 0.187 0.220
SDNLM 99% 17.559 20.359 21.125 0.100 0.168 0.140
Boxcar 24.008 21.970 23.547 0.078 0.077 0.087
Refined Lee 14.305 12.947 12.434 0.177 0.165 0.144
IDAN 2.626 3.079 4.492 0.268 0.253 0.243
SDNLM 80% 10.290 9.217 9.429 0.271 0.265 0.244
SDNLM 90% 11.840 10.599 10.801 0.212 0.211 0.194
SDNLM 99% 22.061 19.576 21.045 0.110 0.110 0.114
Boxcar 18.553 23.125 23.747 0.080 0.102 0.154
Refined Lee 9.694 13.603 14.526 0.169 0.195 0.235
IDAN 1.945 2.151 2.986 0.259 0.259 0.273
SDNLM 80% 8.348 9.463 9.574 0.267 0.270 0.300
SDNLM 90% 9.371 11.049 11.620 0.210 0.222 0.261
SDNLM 99% 17.023 21.082 21.547 0.121 0.131 0.190
A National Aeronautics and Space Administration Jet Propulsion Laboratory
(NASA/JPL) Airborne SAR (AIRSAR) image of the San Francisco Bay was used
for evaluating the proposed filter, see
datasets.html. The original PolSAR image was generated in the L-band, four
nominal looks, and 10 ×10 m spatial resolution. The test region has 350 ×350
pixels, and is shown in Figure 6(b), along with a Google Map c
of the area
(Figure 6(a), see
The four filters employ a kernel of 5×5 pixels, and the patches in our proposal
are 3 ×3 windows. Figure 6(c) shows the effect of the Boxcar filter. Albeit the
noise reduction is evident, it is also clear that the blurring introduced eliminates
useful information as, for instance, the Presidio Golf Course: the curvilinear dark
features to the center of the image: a forested area. Figure 6(d) is the result of
applying the Refined Lee filter, which shows a good performance, but some
details in the edges are eliminated. In particular, the Mountain Lake, the small
brown spot to the center of the image, is blurred, as well as the blocks in the
urban area. The results of the IDAN filter and of our proposal with η= 90% are
shown in Figures 6(e) and 6(f), respectively. Both filters are able to smooth the
image in a selective way, but the SDNLM filter enhances more the signal-to-noise
ratio while preserving fine details than the IDAN filter.
In image quality assessment, the SSIM requires a reference image, as was the
case of sections 6.1 and 6.1, therefore this index is not applied with ease on real
Speckle Reduction in Polarimetric SAR Imagery with SD and NL-means 15
images. Table 3 presents the result of assessing the filters in the three intensity
channels by means of the ENL in the large forest area, a homogeneous area, and
of the BRISQUE index. Again, the results are consistent with what was observed
before: the mere evaluation of the noise reduction by the ENL suggests the
Boxcar filter as the best one, but the natural scene distortion-generic BRISQUE
index is better after applying the SDNLM filter in all intensity channels.
Table 3. Image quality indexes in the real PolSAR image.
Filter ENL BRISQUE Index
Real data 3.867 4.227 4.494 58.258 70.845 61.593
Boxcar 14.564 25.611 18.946 36.498 37.714 36.792
Refined Lee 11.491 20.415 15.407 44.997 51.547 49.412
IDAN 2.994 3.732 3.923 28.823 34.853 34.691
SDNLM 80% 7.263 11.532 8.299 27.841 27.256 33.541
SDNLM 90% 8.177 12.404 9.013 28.622 35.622 35.016
SDNLM 99% 10.828 18.379 13.075 31.026 33.881 36.881
6.3 Effect of the filters on the scattering characteristics
Polarimetric target decompositions aim at expressing the physical properties
of the scattering mechanisms. Among them, the entropy-based decomposition
proposed by Cloude and Pottier [7] is widely used in PolSAR image classification.
It extracts two parameters from each observed covariance matrix: the scattering
entropy H[0,1], and α[0,90], an indicator of the type of scattering. The
(H, α) plane is then divided into nine regions which provide both a classification
rule and an interpretation for the observed data.
Figure 7 shows the effect of the filters on the entropy of the data. Three classes
of entropy are barely visible corresponding, in increasing brightness, to the sea,
urban area an forest. The difference between the classes is small. All filters
enhance the discrimination ability of the entropy, cf. figures Figures 7(b), 7(c),
7(d) and 7(e) with Figure 7(a). Again, the best preservation of detail is obtained
by the IDAN and SDNLM techniques that, among others, retain the information
of the Presidio Golf Course. Additionally, our proposal is the one that best
preserves the low entropy small spots within the urban area, a feature typical of
the high variability of these areas due to their heterogeneous composition.
Samples from the sea, the urban area, and the forest, identified in red, blue
and magenta, respectively, in Figure 6(b), were taken. The (H, α) value of each
point from these samples is presented in Figure 8 before and after applying the
filters. The improvement of using filters is notorious when comparing Figure 8(a)
with figures 8(b), 8(c), 8(d) and 8(e). While the original data is mixed, specially
16 Torres, Sant’Anna, Freitas & Frery
the samples from urban and forest areas, after applying the filters the data tend
to group in clusters.
The sea samples are confined to Zone 9 in all datasets, and all filters have
the effect of reducing their variability. While the Boxcar and Refined Lee filters
produce very similar clusters of data, the IDAN and SDNLM filters reduce both
the entropy and the αcoefficient, but still within the zone of low entropy surface
scatter, making the sample much more distinguishable from the rest of the data.
The samples from urban (in blue) and forest (in magenta) areas have different
mean values of entropy and α. The former occupy mostly zones 4 (medium
entropy multiple scattering) and 5 (medium entropy vegetation scattering), while
the latter span mostly zones 2 (high entropy vegetation scattering) and 5. While
both are present in Zone 5, they seldom overlap; the forest samples have higher
values of α. Comparing these two samples in the images filtered by the Refined
Lee and SDNLM techniques, one notices that they would produce very similar
classifications. The SDNLM produces clusters with more spread than the Refined
Lee, but not at the expense of mixing different classes.
In this manner, the filters preserve the scattering properties of the samples, a
central feature of every speckle smoothing technique for PolSAR data, according
to Lee and Pottier [20].
6.4 Effect of iterations number in filtering
As previously discussed, SDNLM can be iterated since the properties upon which
it is based are preserved by convolutions. Figure 9 presents the original image
for reference (Figure 6(b)), and the result of applying each technique (Boxcar,
Refined Lee, IDAN and SDNLM with η= 80% in each row) one, three and five
times (first, second and third column, respectively).
The most notorious new result stems from comparing the IDAN and SDNLM
filters. Three iterations are enough for the former to smudge the original data,
and with five iterations the blurring it produces is comparable with that of the
Refined Lee and Boxcar filters. The SDNLM filter, even after five iterations, still
preserves most of the spatial information.
Figure 10 presents the (H, α) scatter plot of the samples before and after
iterating the filters one, three and five times. Each new iteration adds cohesion to
the clusters, whatever the filter employed. Regarding the SDNLM, the difference
between one and three iterations is noticeable.
Table 4 presents the quantitative analysis of the resulting images. Again, the
Boxcar procedure yields the best noise reduction in smudge-free areas in most
of the situations, followed by the Refined Lee filter. The SDNLM filter tuned
somewhere between 80% and 90% of significance produces the best BRISQUE
Table 5 presents the values of mean, variance and ENL estimator on the
samples from the three regions of interest (sea, urban and forest), in the cross-
polarized band (HV ). The fist line in Table 5 presents the values observed in
the original (unfiltered) image. The best values are highlighted in bold; being
Speckle Reduction in Polarimetric SAR Imagery with SD and NL-means 17
Table 4. Image quality indexes in the real PolSAR image.
Filter ENL BRISQUE Index
Real data 3.867 4.227 4.494 58.258 70.845 61.593
Boxcar 14.564 25.611 18.946 36.498 37.714 36.792
Refined Lee 11.491 20.415 15.407 44.997 51.547 49.412
IDAN 2.994 3.732 3.923 28.823 34.853 34.691
SDNLM 80% 7.263 11.532 8.299 27.841 27.256 33.541
SDNLM 90% 8.177 12.404 9.013 28.622 35.622 35.016
SDNLM 99% 10.828 18.379 13.075 31.026 33.881 36.881
Boxcar 24.107 54.348 37.501 78.379 63.823 79.307
Refined Lee 24.451 45.088 47.423 62.421 42.882 67.016
IDAN 30.880 40.220 41.640 55.811 52.470 67.016
SDNLM 80% 15.502 32.595 21.686 39.500 39.801 42.866
SDNLM 90% 17.279 35.896 23.478 41.800 42.371 45.177
SDNLM 99% 21.892 50.069 33.347 55.824 52.824 56.844
Boxcar 29.950 73.504 50.811 81.585 71.628 82.365
Refined Lee 36.142 72.948 91.634 70.908 51.728 77.271
IDAN 31.310 41.050 42.340 65.906 64.803 66.400
SDNLM 80% 20.069 52.620 31.233 49.357 46.318 50.872
SDNLM 90% 23.793 61.817 36.548 48.803 47.177 50.430
SDNLM 99% 27.798 72.209 46.450 66.251 58.663 65.944
the best mean the closest to the original value, and the best standard deviation
the smallest one.
The SDNLM filter is the best at preserving the original mean values, and
with reduced standard deviation. Our proposal does not provide the best vari-
ability reduction, a behavior that may be associated with the preservation of fine
structures and a smaller loss of spatial resolution, as can be noted in Figure 9.
The equivalent number of looks behaves consistently with what was observed in
previous examples.
7 Conclusions
The use h-φdivergences, a tool in Information Theory, led to test statistics (with
a known and tractable asymptotic distribution) able to check if two samples
cannot be described by the same complex Wishart distribution, the classical
model for PolSAR data. Using one of these test statistics, namely the one based
on the Hellinger distance, we devised a convolution filter whose weights are
function of the p-value of tests which compare two patches of size 3 ×3 in a
search window of size 5 ×5 pixels.
The filter obtained in this manner (SDNLM – Stochastic Distances Nonlocal
Means) was compared with the Boxcar, Refined Lee and IDAN filters in a variety
18 Torres, Sant’Anna, Freitas & Frery
of PolSAR imagery: data simulated from the complex Wishart law over a realistic
phantom using parameters observed in practice, data simulated from the elec-
tromagnetic properties of the scattering over a simplified cartoon model, and a
real PolSAR image over San Francisco, CA. The quantitative assessment verified
the equivalent number of looks (a measure of noise reduction) over smudge-free
samples, the structural SSIM index, and the BRISQUE index used appropriately
on no-references images (real images or blind). The Boxcar filter promotes the
strongest noise reduction in these conditions, but at the expense of obliterating
small details. The Refined Lee and IDAN filters are competitive, but produce a
pixellated effect and their SSIM index is worse than the produced by the SDNLM
filter in all instances. We noted this same feature with the BRISQUE index ap-
plied to real data and the index values remain stable even during iteration of
the SDNLM filter, which does not happen with other filters assessed.
A qualitative assessment was also made checking how the polarimetric en-
tropy is affected by the filters. We noticed that all the filters enhance it but,
in particular, our proposal performs the most refined enhancement since it pre-
serves very small details which are characteristic of complex urban areas.
The effect of the filters and of applying them iteratively was also verified
in the (H, α) plane. All filters produced more and more compact clusters of
observations in this plane as more iterations were applied. The SDNLM filter
yielded the best separation of the sea sample, while the other two were treated
at least as well as they were by the other filters.
The SDNLM filter has three tuning parameter: (i) filtering window size,
(ii) the size of the patches, and (iii) the significance of the test. We provide a
range of suggested values for the latter, and show good results with an economic
choice for the two former. The filter can also be applied iteratively if more
smoothing is requested and, provided an adequate statistical model, it can also
be applied to other types of data.
We conclude that our proposal is a good candidate for smoothing PolSAR im-
agery without compromising either small details or the scattering characteristics
of the targets.
The test statistics are invariant with respect to permutations of the sample;
directional features will be considered in forthcoming works. Future research
includes the proposal of quality measures for PolSAR imagery, and the use of
tests based on entropies [12].
The authors are grateful to CNPq, Capes and FAPESP for the funding of this re-
search, and to Professor Jos´e Claudio Mura (Divis˜ao de Sensoriamento Remoto,
Instituto Nacional de Pesquisas Espaciais, Brazil) for enlightening discussions
about the PolSARpro toolbox and PolSAR image decomposition.
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A Computational information
Computing the ML estimator of the equivalent number of looks b
Lgiven in equa-
tion (2) and the test statistic based on the Hellinger distance, c.f. equation (6),
are part of the computational core of this proposal. Each weight requires com-
puting these quantities. It is noteworthy that they involve only two operations
on complex matrices: the determinant and the inverse. Using the fact that the
matrices are Hermitian and positive definite, it is possible to reduce drastically
the number of operations required to calculate these two quantities. Specialized
accelerated function in the Rprogramming language [31] were developed with
the speed of the filters in mind.
The time required to filter a 128 ×128 pixels image with one iteration is
of about 75 s in an Intel R
CoreTM i7-3632QM CPU 2.20 GHz, with software
developed in Rversion 2.14.1 running on Ubuntu 12.04. Rwas the choice because
of its excellent accuracy with respect to similar platforms [1].
B Observed covariance matrices
In the following we present the observed covariance matrices that were used to
simulate the data presented in Figure 3(b).
22 Torres, Sant’Anna, Freitas & Frery
7.60830 0.74901 2.29165i1.38157 + 8.39200i
24.8580 5.90346 0.45011i
128.592 12.1941 7.12246i39.1107 + 18.7954i
336.959 8.49716 11.8210i
29.6303 4.86985 + 1.55848i3.41851 + 1.43502i
86.8985 2.03628 8.24319i
14.0576 0.25731 1.48967i4.36926 + 9.41493i
60.5614 4.92951 2.16850i
4.89301 0.52225 0.62765i1.38866 + 5.29889i
12.1149 3.30897 0.85846i
18.7013 0.81235 1.72513i1.26677 + 6.08878i
32.8094 3.01618 1.67916i
Speckle Reduction in Polarimetric SAR Imagery with SD and NL-means 23
(a) Phantom (b) Simulated single-look data
(c) Boxcar filter (d) Refined Lee filter
(e) IDAN filter (f) SDNLM filter
Fig. 3. Original single-look PolSAR simulated data and filtered versions after one it-
24 Torres, Sant’Anna, Freitas & Frery
(a) Phantom edges (b) Edges of the simulated HH band
(c) Boxcar filter edges (d) Refined Lee filter edges
(e) IDAN filter edges (f) SDNLM filter edges
Fig. 4. Edges detected by the Canny filter applied to the HH polarization channel of
the original and filtered images.
Speckle Reduction in Polarimetric SAR Imagery with SD and NL-means 25
(a) Simulated
(b) Boxcar fil-
(c) Refined Lee
(d) IDAN filter (e) SDNLM fil-
(f) Simulated
(g) Boxcar fil-
(h) Refined Lee
(i) IDAN filter (j) SDNLM fil-
(k) Simulated
(l) Boxcar filter (m) Refined
Lee filter
(n) IDAN filter (o) SDNLM fil-
Fig. 5. Original physical-based single-look images and their filtered versions after one
26 Torres, Sant’Anna, Freitas & Frery
(a) Map of the area (b) AIRSAR L-band data
(c) Boxcar filter (d) Refined Lee filter
(e) IDAN filter (f) SDNLM filter
Fig. 6. Pauli decomposition of the original AIRSAR image over San Francisco and its
filtered versions.
Speckle Reduction in Polarimetric SAR Imagery with SD and NL-means 27
(a) Entropy of the Original data
(b) Entropy of the Boxcar filter (c) Entropy of the Refined Lee filter
(d) Entropy of the IDAN filter (e) Entropy of the SDNLM filter
Fig. 7. Entropy in the AIRSAR L-band data.
28 Torres, Sant’Anna, Freitas & Frery
(a) Original data
(b) Boxcar filtered data (c) Refined Lee filtered data
(d) IDAN filtered data (e) SDNLM filtered data
Fig. 8. Scatter plot in the (H , α) plane of samples from the AIRSAR L-band image.
Speckle Reduction in Polarimetric SAR Imagery with SD and NL-means 29
(a) Boxcar filter I= 1 (b) Boxcar filter I= 3 (c) Boxcar filter I= 5
(d) Refined Lee filter I= 1 (e) Refined Lee filter I= 3 (f) Refined Lee filter I= 5
(g) IDAN filter I= 1 (h) IDAN filter I= 1 (i) IDAN filter I= 1
(j) SDNLM filter I= 1 (k) SDNLM filter I= 3 (l) SDNLM filter I= 5
Fig. 9. Pauli decomposition of images filtered one, three and five times with each
30 Torres, Sant’Anna, Freitas & Frery
(a) AIRSAR L-band data
(b) Boxcar filter I= 1 (c) Boxcar filter I= 3 (d) Boxcar filter I= 5
(e) Refined Lee filter I= 1 (f ) Refined Lee filter I= 3 (g) Refined Lee filter I= 5
(h) IDAN filter I= 1 (i) IDAN filter I= 3 (j) IDAN filter I= 5
(k) SDNLM filter I= 1 (l) SDNLM filter I= 3 (m) SDNLM filter I= 5
Fig. 10. Scatter plot in the (H, α) plane of samples from the AIRSAR L-band image
after one, three and five iterations I
Speckle Reduction in Polarimetric SAR Imagery with SD and NL-means 31
Table 5. The values of mean, variance and ENL estimator on different regions of interest in HV polarization
Filtered Sea Urban Forest
Versions bµbσ2ENL bµbσ2ENL bµbσ2ENL
Real data 90.206 33.918 7.073 200.442 50.929 15.490 161.019 53.336 9.114
Boxcar 108.820 8.627 158.408 233.589 18.342 162.192 156.474 33.925 21.274
Refined Lee 113.101 12.838 77.618 232.879 18.931 151.322 164.650 40.189 16.784
IDAN 109.408 22.930 22.766 220.916 32.563 46.026 171.703 38.077 20.334
SDNLM 80% 105.013 15.401 46.495 215.326 38.632 31.068 162.737 37.397 18.936
SDNLM 90% 105.386 14.050 56.261 216.555 36.698 34.822 158.836 42.379 14.047
SDNLM 99% 107.814 9.622 125.548 221.442 32.516 46.381 156.288 40.129 15.168
Boxcar 109.314 4.930 491.591 242.176 12.331 385.696 150.560 37.035 16.527
Refined Lee 121.183 8.237 216.422 245.534 10.511 545.675 172.347 29.176 34.895
IDAN 119.340 11.558 106.616 235.508 17.974 171.684 169.539 39.354 18.559
SDNLM 80% 108.954 5.798 353.120 229.476 21.345 115.584 154.418 34.461 20.079
SDNLM 90% 109.039 5.154 447.608 229.900 22.228 106.976 155.870 33.690 21.406
SDNLM 99% 108.369 4.943 480.580 234.861 16.855 194.159 151.152 36.871 16.806
Boxcar 108.369 3.703 856.241 244.805 10.522 541.263 145.121 39.278 13.651
Refined Lee 124.033 7.368 283.382 249.903 8.389 887.365 175.836 25.191 48.723
IDAN 121.134 9.301 169.603 240.984 12.865 350.900 174.783 40.405 18.712
SDNLM 80% 107.843 3.898 765.393 234.505 17.174 186.439 150.619 35.711 17.789
SDNLM 90% 107.614 3.602 892.750 234.600 17.160 186.911 148.879 37.514 15.750
SDNLM 99% 107.199 3.693 842.776 239.963 13.178 331.599 146.762 36.236 16.404
... Buades et al. [2,3] proposed an adaptive convolutional approach: the Non-Local Means (NLM), in which "Non-Local" refers to the use of large convolutional windows. This class of filters has produced remarkable results in the denoising of natural [15,20], Synthetic Aperture Radar -SAR [6,7,23,31], and medical images [28], among other applications. ...
... Each p-value may be used directly as a weight w but, as discussed by Torres et al. [31], such a choice introduces a conceptual distortion. Consider, for instance, the samples z (1) and z (2) . ...
... Torres et al. [31] proposed using a piece-wise linear function that maps all values above η to 1, a linear transformation of values between η/2 and η, and zero below η/2. This work proposes a smooth activating function with zero first-and second-order derivatives at the inflection points. ...
... In essence, the weights measure the similarity between two patches of data; the more similar they are, the stronger will be the influence of the observation on the filter processing. Wu et al. [8], Torres et al. [7], and Gomez et al. [4] have highlighted the importance of taking into account the statistical properties of * Thanks to Natural Science Basic Research Plan in Shaanxi Province of China for funding 2019JQ-093, and Key Laboratory of Intelligent Perception and Image Understanding of Ministry of Education, Xidian University for Open funding IPIU2019004. the image for computing such weights. ...
... Torres et al. [7] proposed using a transformation of the p-values of a test statistic that verifies the hypothesis that the two data patches were produced by the same distribution. It was possible to apply this approach to PolSAR data because Nascimento et al. [6] had obtained test statistics under the Multivariate Scaled Complex Wishart distribution that is apt at describing them. ...
... The performance of InSAR and CD methods are impaired by the level of noise that affects the used set of SAR images. Several approaches have been developed to discriminate the valuable signals from the noise and filter out artefacts from SAR/InSAR products [52][53][54][55][56][57][58][59][60][61][62][63][64]. Most operate independently on every SAR image product (i.e., a single SAR image or single SAR interferogram) by exploiting exclusively spatial information. ...
Full-text available
Sequences of multi-temporal synthetic aperture radar (SAR) images are routinely used for land-use land-change (LULC) applications, allowing the retrieval of accurate and up-to-date information on the state of the Earth’s surface and its temporal variations. Change detection (CD) methods that rely on the exploitation of SAR data are, generally, made of three distinctive steps: (1) pre-processing of the SAR images; (2) comparison of the pairs of SAR images; and (3) the automatic extraction of the “changed areas”, employing proper thresholding algorithms. Within this general framework, the reduction in speckle noise effects, which can be obtained by applying spatial multi-looking operations and ad hoc noise filters, is fundamental for the better detecting and classifying of changed regions. Usually, speckle noise filters are singularly and independently applied to every SAR image without the consideration of their inherent temporal relationships. In particular, most use local (spatial) approaches based on determining and averaging SAR backscattered signals related to neighboring SAR pixels. In this work, conversely, we explore the potential of a joint 3D space-time non-local mean filter (NLMF), which relies on the discrimination of similar features in a block of non-local SAR pixels extracted from the same or different SAR images. The theory behind non-local-mean filters is, first, shortly revised. Then, the developed space-time NLMF is applied to a real test case for the purposes of identifying flooded zones due to the massive inundations that hit the Kerala region, India, during the summer of 2018. To this aim, a set of 18 descending SAR images collected from the European (EU) Copernicus Sentinel-1 (S-1) sensor was exploited. The performance of the developed NLMF has also been assessed. It is worth remarking that the proposed method can be applied for the purposes of analyzing a heterogenous set of natural and/or artificial disastrous conditions. Further, it can also be helpful during the pre-processing stages of the sequences of SAR images for the purposes of CD applications.
... The NLM algorithm reconstructs image pixels by the weighted averaging of pixels in a predefined search window, where non-local weights are determined based on the similarity between different image patches rather than individual pixels. The NLM method, originally designed for Gaussian noise removal, has lately been applied to remove speckle noise from ultrasound images [29,30] and synthetic aperture radar images [31]. Furthermore, by designing a new weight calculation operator, some NLM-based filter filters are proposed for OCT image denoising. ...
Full-text available
Optical coherence tomography (OCT) is used in various fields such, as medical diagnosis and material inspection, as a non-invasive and high-resolution optical imaging modality. However, an OCT image is damaged by speckle noise during its generation, thus reducing the image quality. To address this problem, a non-local means (NLM) algorithm based on the fractional compact finite difference scheme (FCFDS) is proposed to remove the speckle noise in OCT images. FCFDS uses more local pixel information when compared to integer-order difference operators. The FCFDS operator is introduced into the NLM algorithm to construct a high-precision weight calculation so that the proposed algorithm can effectively reduce the speckle noise in the OCT images. Experiments on simulations and real OCT images show that the proposed method is comparable to other state-of-the-art despeckling methods and can substantially reduce noise and preserve image details such as edges and structures. Speckle noise removal can further promote the application of the proposed algorithm in medical diagnosis and industrial detection, as it has key research value.
... These issues preclude both the direct use of classical image processing techniques (which often assume additivity and Gaussianity) and the visual interpretation of textures and targets. As a consequence, defining adequate PolSAR models is a crucial step for several applications in polarimetry, such as classification [1,2], segmentation [3], speckle filtering [4,5], and boundary and change detection [6,7]. This paper is focused on distributions for multilook PolSAR data, which belong to the set of positive definite hermitian matrices, say Ω + , and are sometimes referred to as sample covariance matrices (SCM) [8]. ...
Full-text available
Polarimetric synthetic aperture radar (PolSAR) systems are an important remote sensing tool. Such systems can provide high spacial resolution images, but they are contaminated by an interference pattern called multidimensional speckle. This fact requires that PolSAR images receive specialised treatment; particularly, tailored models which are close to PolSAR physical formation are sought. In this paper, we propose two new matrix models which arise from applying the stochastic summation approach to PolSAR, called compound truncated Poisson complex Wishart (CTPCW) and compound geometric complex Wishart (CGCW) distributions. These models offer the unique ability to express multimodal data. Some of their mathematical properties are derived and discussed—characteristic function and Mellin-kind log-cumulants (MLCs). Moreover, maximum likelihood (ML) estimation procedures via expectation maximisation algorithm for CTPCW and CGCW parameters are furnished as well as MLC-based goodness-of-fit graphical tools. Monte Carlo experiment results indicate ML estimates perform at what is asymptotically expected (small bias and mean square error) even for small sample sizes. Finally, our proposals are employed to describe actual PolSAR images, presenting evidence that they can outperform other well-known distributions, such as WmC, Gm0, and Km.
... A robust approach to identify similar pixels consists of comparing image patches (Buades et al., 2005;Deledalle et al., 2011;Deledalle et al., 2012). So-called non-local methods perform a weighted averaging with weights derived from patch similarities (Deledalle et al., 2009;Parrilli et al., 2011;Cozzolino et al., 2013;Deledalle et al., 2010;Chen et al., 2010;Deledalle et al., 2014;Torres et al., 2014;Deledalle et al., 2015). All these pixel-selection methods form an estimateR i (or Σ i in PolSAR and InSAR) at pixel i based on the noisy observed values I j (or C j ) in the neighborhood N i centered on pixel i, using a MERLIN Figure 2.4 -Temporal evolution of some representative methods for speckle reduction in SAR images, from conventional methods which started emerging in the 1980s to the more recent deep learning methods. ...
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Synthetic Aperture Radars (SARs) can collect data for Earth Observation purposes regardless of the daylight or cloud cover. Nowadays, thanks to the Copernicus program of the European Space Agency, a huge amount of SAR data is freely available. However, the exploitation of satellite SAR images is limited by the presence of strong fluctuations in the backscattered signal. Indeed, SAR images are corrupted by speckle, a phenomenon inherent to coherent imaging systems. In this Ph.D thesis, we aim to improve the interpretation of SAR images by resorting to speckle reduction techniques. Existing approaches are based on Goodman’s model, which describes the speckle component as a spatially uncorrelated multiplicative noise. In the computer vision field, denoising methods relying on Convolutional Neural Networks (deep learning approaches) have led to great improvements and provide nowadays state-of-the-art results. We propose to use deep learning-based denoising techniques to reduce speckle from SAR images (despeckling methods). At first, we study the adaptation of supervised techniques that minimize a certain distance between the estimation provided by the CNN and a reference image, also called “groundtruth”. We propose to create a dataset of reference images by averaging multi-temporal images acquired over the same area. Pairs of reference and corrupted images can be generated by synthetizing speckle following Goodman’s model. However, in real images the speckle component is spatially correlated which typically requires subsampling these images by a factor 2 to reduce the spatial correlations, which also degrades the spatial resolution. Given the limits of supervised approaches and inspired by noise2noise, a self-supervised denoising method, we propose to train our networks directly on actual SAR images. The principle of self-supervised denoising methods is the following: if a signal contains a deterministic component and a random component, then a network trained to predict a new signal realization from a first independent signal realization will only predict the deterministic component, i.e., the underlying scene, thereby suppressing the speckle. In the method we have developed, SAR2SAR, we leverage multi-temporal SAR series to obtain independent realizations of the same scene, under the hypothesis of temporally decorrelated speckle. Changes are compensated by devising an iterative training strategy. SAR2SAR is thus trained directly on images with spatially correlated speckle and can readily be applied on SAR images without subsampling, providing high-quality results. The training of SAR2SAR is quite heavy: it is articulated in several steps to compensate changes and a dataset comprising stacks of images must be built. With our approach “MERLIN”, we alleviate the training by proposing a single-image learning strategy. Indeed, in single-look-complex SAR images, real and imaginary parts are mutually independent and can benaturally exploited to train CNNs with self-supervision. We show the potential of this training framework for three imaging modalities, different in terms of spatial resolution, textures, and speckle spatial correlation. For the sake of open science, the code associated to each algorithm developed is made freely available.
Preserving image details is a challenging and significant task for synthetic aperture radar (SAR) image despeckling. In this paper, we proposed a high-frequency information extraction module based on block-matching and 3D filtering for SAR image despeckling (SAR-BM3D) and wavelet transformation to guide the denoisers based on a convolutional neural network (CNN). Firstly, SAR-BM3D is employed to conduct the initial speckle removal, and then Haar wavelet transformation takes the task of extracting high-frequency information of the initial despeckled image. The noisy image and the high-frequency information are merged together to be the input of the CNN, which makes it easier for the CNN-based denoisers to restore the textures and details of the SAR image. Extensive experiments on synthetic and real SAR images have been validated, which show that our method can effectively remove the speckle noise and improve the detail-preserving capacity of the CNN-based denoiser.
Interferometry and pixel offset tracking (POT) are two major technologies to measure ground deformation with repeat-pass synthetic aperture radar (SAR) observations. The SAR POT method can tolerate a much larger deformation gradient than interferometric SAR (InSAR), and thus it has been widely applied in monitoring earthquakes, volcanoes, landslides, glaciers, etc. However, there is an intrinsic trade-off between matching accuracy and detail preserving in terms of matching window size for the traditional SAR POT method using only single-polarization intensity. Taking advantage of polarimetric scattering properties, this study develops a novel POT method, named PCM-POT, towards mitigating the dependence of offset estimations on matching window size. It employs a generalized likelihood ratio test (GLRT) upon polarimetric covariance matrices (PCMs) to evaluate the similarity of two candidate pixels and estimate high-resolution surface displacements with high accuracy. The effectiveness and advantage of PCM-POT are demonstrated by an experiment of retrieving the displacements of the Slumgullion landslide from a pair of full-polarimetric UAVSAR images. Comparisons with the traditional POT indicate that using polarimetric scattering properties can reduce the uncertainty in offset estimations over heterogeneous sliding areas. Further experiments suggest that PCM-POT is also suitable for dual-polarimetric satellite SAR data.
Speckle is a granular deformity that frequently appears in images acquired through coherent imaging sensors such as synthetic aperture radar (SAR). The existence of such noise in the image significantly restricts the perception of visual data resulting in low contrast and loss of singular structures. Among various speckle reduction methods, the nonlocal means (NLMs) and variational models (VMs) have shown promising performance. The NLMs restore the underlying image by selecting similar pixels through patches. For VMs, it is achieved by minimizing the objective function (combination of the forward and prior models) using a suitable optimization algorithm. A review of some major state-of-the-art speckle filtering algorithms for aforesaid methods available in the literature is presented in this manuscript. The different multiplicative noise models in SAR images along with their characteristics are explained. Brief descriptions regarding NLMs and VMs are presented. Different algorithms are listed by clearly stating their operating norms, strengths, and drawbacks.
Objective methods for assessing perceptual image quality have traditionally attempted to quantify the visibility of errors between a distorted image and a reference image using a variety of known properties of the human visual system. Under the assumption that human visual perception is highly adapted for extracting structural information from a scene, we introduce an alternative framework for quality assessment based on the degradation of structural information. As a specific example of this concept, we develop a Structural Similarity Index and demonstrate its promise through a set of intuitive examples, as well as comparison to both subjective ratings and state-of-the-art objective methods on a database of images compressed with JPEG and JPEG2000. A MatLab implementation of the proposed algorithm is available online at
PURPOSE: To evaluate the clinical results of the pilon fractures treated with arthroscopically assisted limited open reduction and Ilizarov external fixation. MATERIALS AND METHODS: This is a retrospective study of the clinical result, bone union, complication and postoperative ankle function of 22 pilon fractures treated with arthroscopically assisted limited open reduction and Ilizarov external fixation between January 1999 to March 2004. RESULTS: Clinical follow up averaged 16 months, with an average age of 39.2. All patients with type 1 and 2 fracture had excellent or good score by Ovadia and Beals criteria. Closed fractures healed within 13 weeks and open fractures within 16 weeks after surgery in average. Average range of motion of the ankle was 12o dorsiflexion (0~20 degree) and 25o plantar flexion (15~35 degree). CONCLUSION: Minimal soft tissue dissection and anatomical reduction are very important factor for minimizing complication and satisfactory ankle function. So, arthroscopically assisted limited open reduction and Ilizarov external fixation is an effective treatment option for tibial pilon fractures.
Conference Paper
We propose an algorithm for minimizing the total variation of an image, and provide a proof of convergence. We show applications to image denoising, zooming, and the computation of the mean curvature motion of interfaces.
The recent launches of three fully polarimetric synthetic aperture radar (PolSAR) satellites have shown that polarimetric radar imaging can provide abundant data on the Earth’s environment, such as biomass and forest height estimation, snow cover mapping, glacier monitoring, and damage assessment. Written by two of the most recognized leaders in this field, Polarimetric Radar Imaging: From Basics to Applications presents polarimetric radar imaging and processing techniques and shows how to develop remote sensing applications using PolSAR imaging radar. The book provides a substantial and balanced introduction to the basic theory and advanced concepts of polarimetric scattering mechanisms, speckle statistics and speckle filtering, polarimetric information analysis and extraction techniques, and applications typical to radar polarimetric remote sensing. It explains the importance of wave polarization theory and the speckle phenomenon in the information retrieval problem of microwave imaging and inverse scattering. The authors demonstrate how to devise intelligent information extraction algorithms for remote sensing applications. They also describe more advanced polarimetric analysis techniques for polarimetric target decompositions, polarization orientation effects, polarimetric scattering modeling, speckle filtering, terrain and forest classification, manmade target analysis, and PolSAR interferometry. With sample PolSAR data sets and software available for download, this self-contained, hands-on book encourages you to analyze space-borne and airborne PolSAR and polarimetric interferometric SAR (Pol-InSAR) data and then develop applications using this data.
We propose an algorithm for minimizing the total variation of an image, and provide a proof of convergence. We show applications to image denoising, zooming, and the computation of the mean curvature motion of interfaces.