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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

S˜ao Jos´e dos Campos – SP, Brazil

2Universidade Federal de Alagoas – UFAL

Laborat´orio de Computa¸c˜ao Cient´ıﬁca e An´alise Num´erica – LaCCAN

Av. Lourival Melo Mota, s/n, Tabuleiro dos Martins, 57072-900

Macei´o – AL, Brazil

{ljmtorres,sidnei,corina}@dpi.inpe.br,acfrery@gmail.com

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

distribution.

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

is

f(Y;Σ) = 1

π3|Σ|exp−Y∗tΣ−1Y,

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=L−1PL

ι=1 YιY∗

ι,where

Lis the number of looks.

Goodman [16] proved that Zfollows a scaled multilook complex Wishart

distribution, denoted by Z∼W(Σ, L), and characterized by the following prob-

ability density function:

fZ(Z0;Σ, L) = L3L|Z0|L−3

|Σ|LΓ3(L)exp−LtrΣ−1Z0,(1)

where, for L≥3, Γ3(L) = π3Q2

i=0 Γ(L−i), Γ(·) is the gamma function, tr(·)

is the trace operator, and the covariance matrix Zis given by

Σ=E{Y Y ∗t}=

E{SHH S∗

HH }E{SHH S∗

VH }E{SHHS∗

VV }

E{SVH S∗

HH }E{SVH S∗

VH }E{SVH S∗

VV }

E{SVV S∗

HH }E{SVV S∗

VH }E{SVV S∗

VV }

,

where E{·} denote expectation. Anﬁnsen et al. [2] removed the restriction L≥3.

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

image.

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+ (L−3) log |Zr| − Llog |Σ| − 3 log π−

2

X

q=0

log Γ(L−q)−Ltr(Σ−1Zr),

resulting in the following score function:

∇`r=Lvec(Σ−1ZrΣ−1−Σ−1)

3(log L+ 1) + log |Zr| − log |Σ| − P2

j=0 ψ0(L−j)−tr(Σ−1Zr),

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

Σ,b

L) of its param-

eters is b

Σ=Z=N−1ΣN

r=1Zrand the solution of

3 log b

L+1

N

N

X

r=1

log |Zr| − log |Z| −

2

X

q=0

ψ0(b

L−q)=0.(2)

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,v∈Wf(x+u, y +v)w(u, v)

Pu,v∈Ww(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) = expn−1

hX

k∈P

|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 h→0 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

ﬁlter.

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

data.

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,

2

ηp(1, i)−1 if η

2< p(1, i)< η,

0 otherwise,

(4)

Speckle Reduction in Polarimetric SAR Imagery with SD and NL-means 7

Z1

Z2

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.

0

1

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

arbitrarily.

–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

cost.

–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 ≤i≤25. This will be performed using

tests derived from stochastic distances between samples.

Denote b

θ1the estimated parameter in the central region Z1, and b

θ2,...,b

θ25

the estimated parameters in the remaining areas. To account for possible de-

partures from the homogeneous model, we estimate b

θ= ( b

Σ,b

L) by maximum

likelihood.

The proposal is based on the use of stochastic distances between the patches.

Consider that Z1and Zi, 2 ≤i≤25, 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

by

Dh

φ(Z1,Zi) = hZA

φfZ1(Z0;θ1)

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

dh

φ(Z1,Zi) = Dh

φ(Z1,Zi) + Dh

φ(Zi,Z1)

2.

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]:

Sh

φ(b

θ1,b

θi) = 2mn

(m+n)h0(0)φ00(1) dh

φ(b

θ1,b

θi),(5)

where b

θ1and b

θiare maximum likelihood estimators based on samples size m

and n, respectively. When θ1=θi, under mild conditions Sh

φ(b

θ1,b

θi) is asymp-

totically χ2

Mdistributed, being Mthe dimension of θ1. Observing Sh

φ(b

θ1,b

θi),

the (asymptotic) p-value of the test is p= Pr(χ2

M>Sh

φ(b

θ1,b

θi)), and the null

hypothesis θ1=θican be rejected at signiﬁcance level ηif p≤η; 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

φ(b

θ1,b

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:

SH(b

θ1,b

θi) = 8mn

m+n"1−b

L1

b

Σ−1

1+

b

Li

b

Σ−1

i

2−1

(

b

L1+

b

Li)/2

|b

Σ1|

b

L1/2|b

Σi|

b

Li/2qb

L3

b

L1

1b

L3

b

Li

i

2

Y

q=0

Γ(

b

L1+

b

Li

2−q)

qΓ(b

L1−q)Γ(b

Li−q)#,(6)

provided L≥3. If L < 3, the following expression can be used

SH(b

θ1,b

θi) = 8mn

m+n"1−b

Σ−1

1+

b

Σ−1

i

2−1

q|b

Σ1| | b

Σi|(

b

L1+

b

Li)/2#;

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

used.

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

by

SSIM(f, g) = Cov(f, g ) + C1

bσfbσg+C1

2fg +C2

f2+g2+C2

2bσfbσg+C3

bσ2

f+bσ2

g+C3

,

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

SDNLM).

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

variations.

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

HH HV VV HH HV VV

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, |SHH−SV 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.

band

Filter ENL SSIM Index

HH HV VV HH HV VV

L-

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

C-

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

X-

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 http://earth.eo.esa.int/polsarpro/

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 http://goo.gl/maps/HJkPf).

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

HH HV VV HH HV VV

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

indexes.

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

HH HV VV HH HV VV

Real data 3.867 4.227 4.494 58.258 70.845 61.593

1-iteration

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

3-iterations

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

5-iterations

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].

Acknowledgements

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.

Speckle Reduction in Polarimetric SAR Imagery with SD and NL-means 19

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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

b

Σ1=

7.60830 −0.74901 −2.29165i1.38157 + 8.39200i

24.8580 −5.90346 −0.45011i

32.2771

·10−4

b

Σ2=

128.592 12.1941 −7.12246i39.1107 + 18.7954i

336.959 −8.49716 −11.8210i

154.343

·10−4

b

Σ3=

29.6303 4.86985 + 1.55848i3.41851 + 1.43502i

86.8985 −2.03628 −8.24319i

43.3504

·10−4

b

Σ4=

14.0576 −0.25731 −1.48967i4.36926 + 9.41493i

60.5614 −4.92951 −2.16850i

42.3767

·10−4

b

Σ5=

4.89301 −0.52225 −0.62765i1.38866 + 5.29889i

12.1149 −3.30897 −0.85846i

25.6761

·10−4

b

Σ6=

18.7013 0.81235 −1.72513i1.26677 + 6.08878i

32.8094 −3.01618 −1.67916i

25.8651

·10−4.

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-

eration.

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

L-band

(b) Boxcar ﬁl-

ter

(c) Reﬁned Lee

ﬁlter

(d) IDAN ﬁlter (e) SDNLM ﬁl-

ter

(f) Simulated

C-band

(g) Boxcar ﬁl-

ter

(h) Reﬁned Lee

ﬁlter

(i) IDAN ﬁlter (j) SDNLM ﬁl-

ter

(k) Simulated

X-band

(l) Boxcar ﬁlter (m) Reﬁned

Lee ﬁlter

(n) IDAN ﬁlter (o) SDNLM ﬁl-

ter

Fig. 5. Original physical-based single-look images and their ﬁltered versions after one

iteration.

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

technique.

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

1-iteration

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

3-iterations

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

5-iterations

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