The Polarimetric G Distribution for SAR Data

Analysis

(1)Corina C. Freitas (2)Alejandro C. Frery

(3)Antonio H. Correia

(1)Instituto Nacional de Pesquisas EspaciaisDivisao de Processamento de Imagens

Av. dos Astronautas, 175812227-010 Sao Jose dos Campos, SP – Brazil

(2)Departamento de Tecnologia da InformacaoUniversidade Federal de Alagoas

Campus A. C. SimoesBR 104 Norte km 14, Bloco 12

57072-970 Maceio, AL – Brazil

(3)Centro de Cartografia Automatizada do ExercitoEPCT km 4,5

70084-970 Sobradinho, DF – Brazil

November 17, 2003

Abstract

Remote sensing data, and radar data in particular, have become an essential tool forenviromental studies. Many airborne polarimetric sensors are currently operational, andmany more will be available in the near future including spaceborne platforms. Thesignal-to-noise ratio of this kind of imagery is lower than that of optical informationrequiring, thus, a careful statistical modelling. This modelling may lead to useful oruseless techniques for image processing and analysis, according to the agreement betweenthe data and their assumed properties. Several distributions have been used to describeSynthetic Aperture Radar (SAR) data. Many of these univaritate laws arise by assumingthe multiplicative model, such as Rayleigh, Square Root of Gamma, Exponential, Gamma,and the class of the KI distributions. The adequacy of these distributions depends on thedetection (amplitude, intensity, complex etc.), the number of looks, and the homogeneityof the data. In Frery, Muller, Yanasse and Sant’Anna (1997) another class of univariatedistributions, called G, was proposed to model extremely heterogeneous clutter, such asurban areas, as well as other types of clutter. This paper extends the univariate G familyto the multivariate multilook polarimetric situation: the GP law. The new family has theclassical polarimetric multilook KP distribution as a particular case, but another specialcase is shown more flexible and tractable, while having the same number of parametersand fully retaining their interpretability: the G0

P law. The main properties of this newmultivariate distribution are shown. Some results of modelling polarimetric data usingthe G0

P distribution are presented for two airborne polarimetric systems and a variety oftargets, showing its expresiveness beyond classical models.

Keywords: Speckle, Radar polarimetry, Synthetic aperture radar, Data models,Statistics, Covariance matrices, Radar clutter

1 Introduction

The last decade was marked by the affirmation of SAR images as a tool for Earth moni-

toring. Several studies were made confirming the relevance of these images, and specific

image processing techniques were developed. Some of the applications of SAR imagery to

environmental monitoring are deforestation and secondary forest regrowth for carbon cy-

cle assessment, biomass quantification, oil slick detection, crop growth monitoring, flood

prediction, stand-off day-and-night surveillance of military activity in crisis situations.

With the advent of new airborne and orbital sensors that will provide polarimetric

images, in a few years time there will be an extensive coverage of the Earth with this type

of images. Therefore, a better understanding of the polarimetric scattering mechanisms

of terrestrial targets is necessary in order to be prepared to fully extract information from

the polarimetric images.

Most of the SAR image processing techniques is based on the statistical properties of

data. These properties might be used for the development of tools for SAR image process-

ing and analysis, like filters for speckle noise reduction, segmentation and classification

algorithms, among others. There are plenty of statistical results regarding univariate

intensity and amplitude SAR imagery, but less is known about multivariate multichan-

nel polarimetric and interferometric multilook information (see, for instance, Frery et

al. 1997, Jakeman and Pusey 1976, Kuttikkad and Chellappa 2000, Lopes, Laur and

Nezry 1990, Lee, Du, Schuler and Grunes 1995, Lee, Grunes and Kwok 1994a, Lee, Hop-

pel, Mango and Miller 1994b, Lee, Schuler, Lang and Ranson 1994c, Yueh, Kong, Jao,

Shin and Novak 1989, Yueh, Kong, Jao, Shin, Zebker and Le Toan 1991).

Polarimetric sensors provide considerably more information about the target than

mere intensity and/or amplitude data. This information is embedded in the polarimetric

covariance matrix, and the aim of this paper is providing new good models for this matrix.

Desirable features of good statistical models for remote sensing are:

• flexibility, in the sense that they are suitable for a large variety of situations, in-

cluding different sensors, different polarization, different land use/land cover etc.;

• analytical tractability, in order to allow the biggest possible number of users to take

advantage of them;

• numerical stability, since every useful model eventually becomes software, and

• interpretability, since it is always desired to turn data into information.

In this sense, the G0 model, which is a special case of the G model, presented by Frery et

al. (1997), proved being an interesting univariate model for single complex, intensity or

amplitude SAR data.

1

In this article, the multivariate G distribution for multilook polarimetric images (co-

variance matrix) is developed and studied, and some results about the adequacy of this

new multivariate distribution for real data are presented.

Section 2 describes the multiplicative model, in which the G-model is based on, includ-

ing models for the backscatter, for the speckle and for the complex multilook polarimetric

covariance matrix return. It is shown that the Wishart, multivariate KP and multivariate

G0 distributions belong to the multivariate G-family. In Section 3 an application to real

data is presented, including aspects about parameter estimation. Conclusions and open

issues are given on Section 4.

2 Multiplicative Model for Polarimetric Data

There are several ways of representing SAR images, but all of them should consider the

phenomenon known as speckle, which is always present on images formed using coherent

illumination. Speckle, thus, is also present in sonar, laser and B-scan ultrasound imagery.

Speckle appears due to the interference phenomena among the coherent signals re-

turned by many individual scatterers, and affects our ability of interpreting SAR data.

Depending on the type of image (complex, amplitude, intensity, polarimetric etc.),

different models are used to represent SAR data. A very commom statistical framework

is the multiplicative model, which states that the observations are the outcome of the

product of two independent random variables: one (X) modelling the terrain backscatter,

and other (Y ) modelling the speckle noise. When dealing with a single image, this model

can be stated as Z = X · Y , where X is considered real and positive and Y has a unitary

mean and could be complex (if the image is in complex format) or positive real (intensity

and amplitude formats). This model was proposed by Goodman (1985) in the context of

optical statistics, and it allows the derivation of some of the most succesful distributions

for the return of SAR and other systems that employ coherent illumination, such as laser,

sonar and B-scan ultrasound.

The extension of this representation to multivariate images, as is the case of polarimet-

ric data, is not immediate. In order to retain the multiplicative representation, one has

to embed part of the terrain information within the speckle while making the backscatter

only responsible for the local fluctuation of the mean value. This will be formulated in

Section 2.1.

The following notation will be used henceforth: uppercase letters (X for instance)

will denote scalar random variables, while lowecase ones (x, for instance) their outcomes.

Bold letters (Z and z) will be used for vectors and matrices, uppercase if they are ran-

dom elements and lowercase otherwise. Subscripts I, C and P will denote, respectively,

intensity, complex and polarimetric formats, being the latter the covariance matrix form.

The expected value of random elements is denoted E.

2

2.1 Polarimetric Data

A polarimetric SAR measures the complex matrix S, the complex scattering matrix of the

ground. This matrix is formed, for each coordinate, as the sum of the return of individual

backscatterers in each polarization. This matrix, in complete form, can be written as

S =

(

Svv Svh

Shv Shh

)

, (1)

where

Spq = |Spq| exp (iφpq) =

k∑

j=1

|Spq,j| exp (iφpq,j) , (2)

being j the index of the individual scatterer with amplitude |Spq,j| and phase φpq,j, and

k the number of scatterers in the resolution cell (Sarabandi 1992); p and q denote either

horizontal (h) or vertical (v) polarization. Usually some assumptions are made on the

distributions of |Spq,j| and φpq,j, and on the number of scatterers k in order to derive

useful statistical properties for Spq.

Satellites usually employ the same antenna to both transmit and receive and, according

to Ulaby and Elachi (1990), one can consider that cross-polarizations are equal, i.e, that

Shv = Svh. In this manner, matrix (1) has redundant information and may be reduced to

ZC =

S1

S2

S3

, (3)

where the indexes 1, 2 and 3 denote the polarization hh, hv and vv in any order. This

matrix is called single look complex scattering matrix.

In order to employ the Multiplicative Model, assume that the following decomposition

for the single look complex scattering matrix holds:

ZC = X1/2YC, (4)

where YC is independent of X, the backscatter, a positive random variable such that

E (X) = 1. The physics of the imaging process leads to the result that YC follows a zero

mean Multivariate Complex Gaussian distribution with density

fYC(y) =

1

πm |C| exp(

−yC−1yt)

,

where m is the number of complex components of both ZC and YC , C = E (YCY∗tC ) and

“t” and “∗” denote the transpose and the conjugate, respectively (see Lee et al. 1994b).

The matrix C, unlike the univariate case derived by Frery et al. (1997), retains valuable

3

information about the terrain, while X only describes the fluctuation of the observations.

This departure from the single channel model is required in order to be able to model

channels with different intensity means, while using a single scalar (X) for the backscatter.

Under the aforementioned hypothesis holds that C = E (ZCZ∗tC ), a relation that will be

useful for estimating C. Different distributions for X will lead to different models for

the return, a subject that will be exploited in Section 2.2 in the context of multivariate

multilook polarimetric data.

Polarimetric SAR imagery is noisy, so it is frequently processed in order to diminish

the speckle (Lee, Grunes and Mango 1991, Touzi and Lopes 1994) and to compress the

data (Lee et al. 1994a). Multilook processing consists of using (ideally independent)

samples zC(1), . . . , zC(n) of the random vector ZC to form the n-looks complex covariance

matrix (Lee et al. 1995):

Z(n)C =

1

n

n∑

`=1

ZC(`) (Z∗

C(`))t , (5)

Using the definition given in equation (5) with the decomposition presented in equa-

tion (4), it is easy to see that Z(n)C can be written as

Z(n)C =

1

n

n∑

`=1

X(`)YC(`) (Y∗

C(`))t . (6)

Considering that X(`) does not vary from observation to observation in the resolution cell,

i.e., that X(`) = X for every `, if observations are made in a short timespan, equation (6)

can be written as

Z(n)C =

X

n

n∑

`=1

YC(`) (Y∗

C(`))t = XY(n)C , (7)

where the complex speckle Y(n)C obeys a scaled multivariate complex Wishart distribu-

tion (Goodman 1963, Lee et al. 1994b, Lee et al. 1994c, Srivastava 1965) with density

given by

fY

(n)C

(y) =nnm |y|n−m exp (−nTr (C−1y))

h(n, m) |C|n , (8)

where m is the dimension of the complex vector ZC , the scaling function h(n, m) is given

by h(n, m) = πm(m−1)/2Γ(n) · · ·Γ(n − m + 1) and, as before, C = E(ZCZ∗

C) = E(YCY∗

C);

Tr and |·| denote the trace and the determinant, respectively. This situation is denoted

Y(n)C ∼ W(C, n). The expected value of both Z

(n)C and Y

(n)C is also C.

It is, again, noteworthy that the random complex matrix Y(n)C (equation (7)) carries

the covariance structure C. Its diagonal elements describe the multilook intensity radar

cross sections (directly related to the desired and unobserved ground truth), while X

controls the fluctuation about the mean with E(X) = 1 (see Oliver and Quegan (1998,

Section 11.3.2) for a detailed explanation).

4

As previously said, several distributions have been proposed for the backscatter X, in

order to model homogeneous, heterogeneous and extremely heterogeneous clutter. The

simplest case is for homogeneous areas, where Pr(X = 1) = 1 and, therefore, the return

follows a complex Wishart law. Jakeman and Pusey (1976) used a Gamma distribution for

X to derive the polarimetric KP model for the return, suitable for heterogeneous targets.

Derivations of this multivariate multilook polarimetric KP model and its use in analysis

and segmentation are presented by Yueh et al. (1989), by Lee et al. (1994c) and by Novak,

Sechtin and Cardullo (1989), among other references.

Despite the usefulnes of the K model, it is often unable to explain extremely het-

erogeneous data such as urban clutter. Looking for a model, within the single channel

Multiplicative Model, able to describe this situation, Frery et al. (1997) proposed the

Generalized Inverse Gaussian distribution for the intensity backscatter X, which leads to

the so-called univariate G family of distributions for the single-polarization return Z. A

particular case of the Generalized Inverse Gaussian distribution, namely the Reciprocal

of Gamma, led to the G0 distribution for the return data and it was succesfully applied

to SAR imagery. The single-channel K model is another particular case of the univariate

G family, since the Gamma distribution is another special case of the Generalized Inverse

Gaussian law.

The univariate G0 distribution proved being a quite flexible model, capable of describ-

ing also heterogeneous and homogeneous data. Recent works (Correia, Freitas, Frery and

Sant’Anna 1998, Mejail, Jacobo-Berlles, Frery and Bustos 2000, Mejail, Frery, Jacobo-

Berlles and Bustos 2001, Mejail, Jacobo-Berlles, Frery and Bustos 2003) also validate its

use as the sole model for speckled imagery modelling and classification, so a multivariate

polarimetric version would be of valuable use. Some inference issues are treated by Bustos,

Lucini and Frery (2002) and by Cribari-Neto, Frery and Silva (2002).

This paper presents an extension of the G family for the multivariate multilook com-

plete polarimetric data. This model is obtained assuming that the backscatter obeys a

Generalized Inverse Gaussian distribution, with unitary mean, while the speckle noise

follows a Wishart distribution. The relationships between these laws for the backscat-

ter, the speckle noise and the return will be presented in Section 2.2 and illustrated in

Appendix A. Some particular cases of the new model will be analized in detail, the multi-

variate multilook KP and G0P models for polarimetric data among them. Other desirable

properties of the polarimetric G0P distribution, numerical and analytical tractability for

instance, will be addressed.

5

2.2 Models for the Backscatter

The most general model to be considered in this paper for the intensity backscatter is the

Generalized Inverse Gaussian distribution, characterized by the density

fX(x) =(λ/γ)α/2

2Kα

(√λγ)xα−1 exp

(

−1

2

(γ

x+ λx

)

)

, x > 0, (9)

where Kν denotes the modified Bessel function of the third kind and order ν, with the

domain of variation of the parameters given by

γ > 0, λ ≥ 0 if α < 0

γ > 0, λ > 0 if α = 0

γ ≥ 0, λ > 0 if α > 0.

(10)

The distribution defined above is denoted here as XI ∼ N−1(α, γ, λ). For detailed

properties and applications of the Generalized Inverse Gaussian distribution the reader is

referred to the works by Barndorff-Nielsen and Blæsild (1981) and Jørgensen (1982).

Its r-th order moments are given by

E(Xr) =(γ

λ

)r/2 Kα+r

(√γλ)

Kα

(√γλ) . (11)

This distribution can be reduced to several particular cases, but the following two are of

special interest in our study:

1. the Gamma distribution, when γ = 0, denoted here as Γ(α, λ);

2. the distribution of the reciprocal of a Gamma distributed random variable, when

λ = 0, denoted here as Γ−1(α, γ).

A third particular model, namely the Inverse Gaussian law, is studied in detail by Seshadri

(1993).

Another important parametrization of the Generalized Inverse Gaussian distribution,

convenient for dealing with polarimetric models, is obtained making ω =√

γλ and η =√

γ/λ. The density (9) can now be written as

fX(x) =1

2ηαKα (ω)xα−1 exp

(

−ω

2

(

η

x+

x

η

))

, x > 0,

with the parameters space obtained applying the transformation on the set given in rela-

tions (10). This distribution will be denoted here as N−1(α, ω, η).

A random variable obeying a N−1(α, ω, η) law is scale invariant, in the sense that if

X ∼ N−1(α, ω, η) then cX ∼ N−1(α, ω, cη) for any positive constant c. This property,

6

along with the moments presented in equation (11), allows us to easily derive the dis-

tribution of a unitary mean random variable obeying the Generalized Inverse Gaussian

distribution, required to model the polarimetric backscatter. Consider X ′ ∼ N−1(α, ω, η),

then

E(X ′) =

ηrα,ω if ω > 0,2αλ

if α > 0, γ = 0,

− γ2(α+1)

if α < −1, λ = 0,

∞ if −1 ≤ α < 0, λ = 0,

(12)

where rα,ω = Kα+1(ω)/Kα(ω). Therefore, the scaled random variable X = X ′/rα,ω has

unitary mean whenever α /∈ [−1, 0) (incidentally, this guarantees that all polarimetric

distributions considered in this paper have finite mean), and its density is given by

fX(x) =rαα,ω

2Kα (ω)xα−1 exp

(

−ω

2

(

1

rα,ωx+ rα,ωx

))

, x > 0. (13)

The shape of this density is shown in Figures 8 and 9 (Appendix A).

Two particular cases of the Generalized Inverse Gaussian distribution, whichever the

chosen parametrization, are the Gamma and Reciprocal of Gamma laws (see Frery et

al. (1997) for detailed properties). In order to derive these laws with unitary mean, one

can start using equation (13) and the fact that, for small values of the argument µ and

positive order ν, the function Kv(µ) can be approximated by 2ν−1Γ(ν)µ−ν; also the fact

that K−v(ν) = Kv(ν) is useful. The density that characterizes the unitary mean Gamma

distributions is

fX(x) =ααxα−1

Γ(α)exp (−αx) , α, x > 0, (14)

while the one corresponding to the unitary mean Reciprocal of Gamma law is given by

fX(x) =xα−1

(−α − 1)α Γ(−α)exp

(

α + 1

x

)

,−α, x > 0. (15)

Equation (14) was derived assuming γ → 0 and α > 0, while density (15) stems

from the hypothesis λ → 0 and α < 0 in equation (13) making the proper change of

parametrization. Figure 10 (Appendix A) illustrates these two densities.

A graphical representation of the relationships among distributions for the backscatter

is shown in the scheme (21), Appendix A. The next section presents the derivation of the

distribution for the complex multilook polarimetric covariance matrix Z(n)

C assuming the

law characterized by density (13) for the backscatter. This new multivariate distribution

for polarimetric data will allow a very general and convenient modelling of SAR data.

7

2.3 Models for the complex multilook polarimetric covariance

matrix return

Assuming the Multiplicative Model, the return of multilook polarimetric data in the

form of the complex covariance matrix can be described by equation (7). In this section

we will derive the distribution that arises assuming that X obeys the law induced by

density (13) that, as depicted in the scheme (21) Appendix A, is a very general model for

the backscatter. The random variable Y(n)C will always follow the Wishart distribution

characterized by density (8), and will be independent of X.

In this manner, the distribution of the random variable Z(n)C can be derived computing

the distribution of XY(n)C , which can be done by

fZ

(n)C

(z) =

∫

R+

fxY

(n)C

(z)fX(x)dx,

where fY

(n)C

and fX are given in equations (8) and (13). The density of the scale trans-

formation xY(n)C is f

xY(n)C

(z) = x−m2fY

(n)C

(x−1z), leading to

fxY

(n)C

(z) =nmnx−mn |z|n−m exp (−nTr (C−1z)x−1)

h(n, m) |C|n .

Therefore,

fZ

(n)C

(z) =nmn |z|n−m rα

α,ω

h(n, m) |C|n 2Kα (ω)·

·∫

R+

xα−mn−1 exp

(

−1

x

(

nTr(

C−1z)

+ω

2rα,ω

)

− ωrα,ω

2x

)

dx.

Using the following integral definition of modified Bessel functions:

Kν(2√

ab) =(a/b)ν/2

2

∫

R+

xν−1 exp (−ax − b/x) dx,

one obtains that

fZ

(n)C

(z) =nmn |z|n−m rα

α,ω

h(n, m) |C|n Kα (ω)

(

2nTr (C−1z) + ωrα,ω

ωrα,ω

)α−mn

2

·

· Kα−mn

(√

ωrα,ω

(

2nTr (C−1z) +ω

rα,ω

)

)

. (16)

This distribution, denoted by GP (α, ω, C, n), is a multivariate extension of the uni-

variate GI law presented in Frery et al. (1997), using a different parametrization, being

the latter the special case m = 1. The intensity of each polarization Ij = Z(n)C (j, j),

8

j ∈ {hh, hv, vv}, has a GI(α, ωj, C(j, j), n) law, with C(j, j) = E(Ij) = (γj/λj)1/2rα,ωj

=

ηjrα,ωj, where it is clear that the roughness parameter α does not vary among polariza-

tions. This marginal density for the intensity, derived by Frery et al. (1997), is given

by

fIj(z) =

nnzn−1

Γ (n) ηnKα (ω)

(

2nz + ωη

ηω

)(α−n)/2

Kα−n

(

√

ω (2nz/η + ω))

.

Two important special cases of the multivariate GP (α, ω, C, n) distribution are ob-

tained making ω → 0 with α > 0 and α < −1, respectively. The former is equivalent to

making γ → 0 with λ > 0, while the latter to λ → 0 with γ > 0. In order to make these

derivations the relations for Bessel K functions presented in Section 2.2 are needed.

A model for multivariate multilook polarimetric heterogeneous clutter is obtained

assuming ω → 0 with α > 0, leading to the density

fZ

(n)C

(z) =2 |z|n−m (nα)

α+mn2

h(n, m) |C|n Γ(α)

(

Tr(

C−1z))

α−mn2 Kα−mn

(

2√

nαTr (C−1z))

. (17)

This is the multivariate KP distribution for the multilook polarimetric covariance ma-

trix, presented by Lee et al. (1994c), and denoted here KP (α, C, n). When m = 1 this

multivariate law reduces to the univariate multilook intensity KI distribution (Oliver and

Quegan 1998). The multivariate multilook polarimetric distribution can also be derived

as the product of two independent random variables X and Y(n)C , where the latter obeys

a Wishart distribution with parameters C and n (equation (8)), and the former obeys a

Γ law with unitary mean (equation (14)) .

A model for extremely heterogeneous multivariate multilook polarimetric clutter is

obtained assuming ω → 0 with α < −1, leading to the density

fZ

(n)C

(z) =nmn |z|n−m Γ(mn − α)

h(n, m) |C|n Γ(−α)(−α − 1)α

(

nTr(

C−1z)

+ (−α − 1))α−mn

. (18)

This multivariate distribution, denoted here G0P (α, C, n), is an extension of the univariate

G0I law (Frery et al. 1997), being the latter the special case when m = 1. The multivariate

G0P distribution can also be derived as the product of two independent random variables

X and Y(n)C , where the latter obeys a Wishart distribution with parameters C and n

(equation (8)), and the former obeys a Γ−1 law with unitary mean (equation (15)). It is

noteworthy that, in fully accordance with what is presented by Frery et al. (1997) and

by Mejail et al. (2001) for the univariate case, the G0P can be also used to describe het-

erogeneous and homogeneous data, beyond extremely heterogeneous polarimetric clutter;

examples are shown in Section 3.2.

Comparing equations (17) and (18) it is immediate that the latter does not depend

on the modified Bessel function of the third kind Kν. This is a desirable feature of

the G0P distribution, since this function is only available through numerical evaluation of

9

integrals (Cody 1993, Gordon and Ritcey 1995) and, as presented by Yanasse, Frery and

Sant’Anna (1995), it is subjected to severe instabilities.

The expected value of random matrices obeying multivariate GP , KP and G0P distri-

butions is the complex matrix C.

A graphical representation of the relationships among distributions for the return is

shown in the scheme (22), appendix A.

3 Application to real data

Estimating all the parameters of a GP (α, ω, C, n) distribution is a hard computational task,

whichever estimation procedure is chosen. Besides this, as noted by Frery et al. (1997),

in most of the analyzed areas the parameter λ was very small, so it might be possible

to assume that the parameter ω tends to zero. It is shown (Mejail et al. 2000, Mejail

et al. 2001) that the G0I distribution can describe well homogeneous and heterogeneous

areas, besides extremely heterogeneous clutter. These three reasons lead to the use of the

G0P (α, C, n) distribution for polarimetric SAR data analysis.

Since the density given in equation (18) is a function of the form fZ

(n)C

: C9 → R+, the

goodness-of-fit of the data to the distribution is hard to validate using multidimentional

histograms. Following the work by Lee et al. (1994c), the validation of the model will be

performed using the marginal intensity distributions for each channel j ∈ {hh, hv, vv}.These univariate marginal distributions are characterized by the density

fIj(z) =

2nnnΓ (n − α)

γαj Γ (n) Γ (−α)

zn−1

(2nz + γj)n−α , α < −1, γ > 0, z > 0 (19)

which is the same density for the intensity G0I distribution (Frery et al. 1997) using a slight

different parametrization. In order to obtain this marginal distribution, it is enough to

set m = 1 in equation (18).

The first and second moment of the distribution given by density (19) are useful

for parameter estimation (Section 3.1) using the analogy method (Manski 1988). These

quantities are given by

E(Ij) =γ

2(−α − 1),

E(I2j ) =

γ2(n + 1)

4nα(α + 1). (20)

Another useful feature of the G0 model is that the computation of the cumulative

distribution function of the intensities is immediate using the relationship proved by Mejail

et al. (2001) Pr(Ij ≤ t) = Υ2n,−2α(−αt/γj), where Υη,κ is the cumulative distribution

function of an F -distributed random variable with η and κ degrees of freedom. Since the

10

F distribution is a commonplace in statistical software, there are plenty of dependable

routines (and tables, also) that provide these values.

3.1 Parameter estimation

In order to estimate the parameters of the G0P (α, C, n) distribution one starts calculating

the equivalent number of looks n for the whole image using homogeneous areas (see

Yanasse, Frery, Sant’Anna, Hernandez Filho and Dutra 1993). The matrix C is then

estimated with

C(i, j) =

{

mi if i = j

mij if i 6= j,

where mi is the sample mean of the intensity of channel i, i.e., of Z(n)C (i, i), and mij is the

sample mean of the random variable Z(n)C (i, j).

The roughness parameter α is estimated as α = (αhh + αhv + αvv)/3, and each αj is

estimated using the first and second moments set of equations, given in (20), in marginal

intensity data. Numerical maximization of the likelihood or log-likelihood is also possible.

3.2 Data analysis

Data from two airborne sensors were used to assess the proposed model, namely from

DLR’s experimental E-SAR system and AeroSensing’s RadarSysteme GmbH, the former

in L-band and the latter in P-band. Images from the first were taken over small Bavarian

towns, while the ones from the latter correspond to Tapajos National Forest, Brazil.

Figure 1 shows a color composite (R: hh, G: vv, B: hv) from part of the AeroSensing

P-band image locating some of samples taken from primary forest, old regeneration and

bare soil.

Table 1 describes the type of area employed in the analysis, as well as the estimated

parameters of the marginal intensity data: Γ(n, λ), KI(α, λ, n) and G0I (α, γ, n). The

equivalent number of looks n was estimated beforehand for the whole datasets using

homogeneous targets, resulting in n = 2.50 and n = 4.04 for the first and second sensor,

respectively. The Urban area data were obtained by E-SAR, while all the other datasets

were generated by AeroSensing’s sensor.

Figure 2 shows the fitted distributions to the histogram of urban data. The G0I law

gives the best fit for the three polarizations and, as can be seen in Table 1, the roughness

parameters α of this distribution are of the same order of magnitude for every polarization,

in accordance with the proposed model.

Figure 3 shows the fitted distributions to the histogram of bare soil data. These dataset

exhibited again extreme heterogeneity, as can be seen in the estimated α parameters

reported in Table 1, which are close to zero. This is another example where neither the

11

Type Sample Size λ · 105 α, λ · 106 α, γ · 10−4 Polarization

Urban 138293.38.83.7

0.38, 5.060.80, 0.280.25, 3.65

−1.84, 6.65−1.72, 2.26−1.62, 4.50

hhvvhv

Bare Soil 18399991.2910.58781.5

0.48, 1188.92.03, 4566.01.78, 38651.3

−2.29, 0.05−5.12, 0.18−4.26, 0.01

hhvvhv

Old Regeneration 11860296.6234.81830.8

15.15, 11117.234.76, 20189.876.17, 344956.0

−21.25, 2.76−29.25, 4.89−29.14, 0.63

hhvvhv

Primary Forest 13829198.9168.01355.9

3.94, 1937.74.45, 1847.48.49, 28491.2

−6.41, 1.10−6.32, 1.28−11.51, 0.31

hhvvhv

Table 1: Estimated parameters for the three intensity distributions, four types of area and three polarization datasets.

12

KI nor the Γ laws are able to adequately describe the data. The roughness parameters of

the G0I distribution are, again, similar.

Figure 4 shows the fitted distributions to the histogram of old regeneration data. The

KI and Γ densities overlap in the two last plots, i.e., for the vv and hv polarizations the

roughness parameter of the KI law is big enough to ensure its equivalence to the Γ law.

The estimated parameters shown in Table 1 confirm that the three distributions will fit

well the data since, as presented in the scheme (22), the roughness parameter α is, in

absolute value, large in homogeneous targets.

Figure 5 shows the fitted distributions to the histogram of primary forest data. As

can be seen in the estimated roughness parameters α, this type of area corresponds to

heterogeneous clutter, so the fact that the Γ distribution fails to describe the values is

in accordance with the theory. Both KI and G0I laws are able to explain the data, being

the latter slightly better than the former. It is noticeable that the roughness parameter

of these two laws is significatively different in the hv polarization, being this a departure

from the hypothesized conditions for both polarimetric distributions KP and G0P .

4 Conclusions and future work

This paper presented a multivariate extension of the polarimetric distributions available

in the literature under the Multiplicative Model framework: the multilook polarimetric

family of GP laws. This extension was obtained using the Generalized Inverse Gaussian

distribution as the model for the backscatter. This law generalizes the Gamma distribu-

tion used to derive the multivariate polarimetric KP model. Another particular situation

of the Generalized Inverse Gaussian distribution is the Inverse Gamma distribution, that

leads to the G0P polarimetric distribution.

This new law for multilook polarimetric data offers a number of desirable features

such as analytical and numerical tractability. Furthermore, the the G0P distribution is

capable of explaining well the data the KP explains, and is capable of describing ex-

tremely heterogeneous clutter the latter fails to describe. Since the G0P law was derived

within the multiplicative model, its parameters have fully interpretability and they can

be conveniently estimated from the data.

Data from two airborne polarimetric systems were analyzed for a variety of types

of targets ranging from homogeneous (old regeneration) to heterogeneous (forest) to ex-

tremely heterogeneous (bare soil and urban spots). Observations in the three intensity

channels were well described by the proposed distribution.

The presented results can be used for designing techniques for filtering, segmenting

and classifying polarimetric data. Preliminary classification results can be seen in Freitas,

Sant’Anna, Soler, Santos, Dutra, de Araujo, Mura and Hernandez Filho (2001), where

classification of polarimetric SAR data is performed with, among others, the distributions

13

presented in this work. In that work it is shown that six out of seven classes were better

explained by the G0P distribution.

A limitation of the models presented in this paper is that the roughness parameter α

is the same for the nine components of the complex multilook polarimetric matrix while,

in practice, in some areas different textures are observed (see Figure 5). An extension in

this direction is being sought.

5 Acknowledgements

The authors received partial support from CNPq and Vitae. E-SAR data were kindly

provided by DLR (through Hans-J. Muller).

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A Distributions and their relationships

Figure 6 shows densities of the Generalized Inverse Gaussian (equation (13)) for −α ∈{1.1, 3, 10, 20} with ω = 1. It is noticeable that the smaller the parameter α the more

symmetric the density is, the less the dispersion is and, also, the closer the mode is to the

expected value 1; also, the mode increases when α decreases. Figure 7 presents this density

for ω ∈ {1, 2, 10, 30} with α = −2. The bigger the parameter ω the more symmetric the

density is and, also, the closer the mode is to the expected value 1.

The shape of the density of the Generalized Inverse Gaussian distribution correspond-

ing to the positive branch of the shape parameter α is illustrated in Figures 8 and 9.

The behaviour is analogous to that observed in Figures 6 and 7 though, for the same ω,

the densities are quite different for small values of |α|. The negative branch of the N−1

distribution is particularly effective for modelling extremely heterogeneous backscatter

(Section 3.2).

Figure 10 shows four cases of the densities given in equations (14) and (15), namely

those corresponding to |α| ∈ {1.2, 1.5, 3, 10, 20} (continuous line, dots, dashes, dash-dot,

dot-dot-dot-dash respectively): positive values of α (unitary Gamma distribution) to the

left and negative values of α (unitary Reciprocal of Gamma distribution) to the right.

In both cases the closer α is to 0, the bigger the difference between densities is, and the

bigger |α| the closer the mode to the expected value 1 is. The negative branch is more

flexible than the positive counterpart, in the sense that richer shapes are attainable when

α < 0 than when α > 0, specially for small values of |α|.The relationships among the univariate distributions that describe the backscatter in

the Multiplicative model are depicted below, where “D−→” and “

Pr−→” denote the conver-

gences in distribution and probability, respectively, of the associated random variables.

These properties show that either homogeneous (the degenerate model for which Pr(X =

1) = 1), heterogeneous (X ∼ Γ(α, λ)) or extremely heterogeneous (X ∼ Γ−1(α, γ))

17

backscatters can be treated as the outcome of a X ∼ N −1(α, γ, λ) random variable.

D ↗ α, λ > 0

γ → 0 Γ(α, λ)

Heterogeneous

Pr

−→α, λ → ∞α/λ → 1 1

Homogeneous

N−1 (α, γ, λ)

General

Situation

D ↘ −α, γ > 0

λ → 0Γ−1(α, γ)

Extremely

Heterogeneous

Pr

−→−α, γ → ∞−α/γ → 1 1

Homogeneous

(21)

The relationships among the univariate distributions that describe the backscatter in

the Multiplicative model are depicted below. These properties show that either homoge-

neous, heterogeneous or extremely heterogeneous return can be treated as the outcome

of a Z(n)C ∼ GP (α, ω, C, n) random variable.

D ↗ α, λ > 0

ω → 0 KP (α, C, n)

Heterogeneous

D−→

α, λ → ∞α/λ → 1 W(C, n)

GP (α, ω, C, n),

ω =√

γλ

General

Situation

Homogeneous

D ↘ −α, γ > 0

ω → 0G0

P (α, C, n)

Extremely

heterogeneous

D−→

−α, γ → ∞−α/γ → 1 W(C, n)

(22)

The simulation of outcomes from the G0P (α, C, n) distribution is straightforward using

its multiplicative representation. It is enough to return the product xy where x is the

outcome of a random variable obeying the law given in equation (15) and y is the outcome

18

of a Complex Wishart distribution. Techniques for sampling from the Wishart law can

be seen in Johnson (1987), while the former can be obtained using the property that if

X ∼ Γ−1(α, 2(−α − 1)) then V = X−1 ∼ Γ(−α, 2(−α − 1)), so v can be sampled from

a Γ(−α, 2(−α − 1)) random variable, and v−1y returned as the sample from the G0P -

distributed random matrix. Algorithms for obtaining Γ samples can be found in Devroye

(1986).

19

Figure 1: Part of the image over Tapajos, taken in P-Band by the AeroSensing system:samples of primary forest in yellow, of old regeneration in purple and of bare soil in red.

20

Figure 2: Fitted distributions to urban data from L-band E-SAR system: G0I (continuous

line), KI (dashes) and Γ (dash-dot) in three polarizations: hh (left), vv (middle) and hv(right).

Figure 3: Fitted distributions to bare soil data from P-band AeroSensing system: G0I

(continuous line), KI (dashes) and Γ (dash-dot) in three polarizations: hh (left), vv(middle) and hv (right).

21

Figure 4: Fitted distributions to old regeneration data from P-band AeroSensing system:G0

I (continuous line), KI (dashes) and Γ (dash-dot) in three polarizations: hh (left), vv(middle) and hv (right).

Figure 5: Fitted distributions to primary forest data from P-band AeroSensing system:G0

I (continuous line), KI (dashes) and Γ (dash-dot) in three polarizations: hh (left), vv(middle) and hv (right).

22

Figure 6: Densities of the Generalized Inverse Gaussian Distribution with unitary mean,ω = 1 and −α ∈ {1.1, 3, 10, 20} (solid, dots, dashes and dot-dash respectively).

23

Figure 7: Densities of the Generalized Inverse Gaussian Distribution with unitary mean,α = −2 and ω ∈ (1, 2, 10, 30) (solid, dots, dashes and dot-dash respectively).

24

Figure 8: Densities of the Generalized Inverse Gaussian Distribution with unitary mean,ω = 1 and α ∈ (1.1, 3, 10, 20) (solid, dots, dashes and dot-dash respectively).

25

Figure 9: Densities of the Generalized Inverse Gaussian Distribution with unitary mean,α = 2 and ω ∈ (1, 2, 10, 30) (solid, dots, dashes and dot-dash respectively).

26

Figure 10: Densities of the unitary Gamma (α > 0) and Reciprocal of Gamma (α < 0)distributions with |α| ∈ {1.2, 1.5, 3, 10, 20} (continuous line, dots, dashes, dash-dot, dot-dot-dot-dash).

27