ThursdayACFreryEstimationDistanceMinimization

Parameter Estimation in SAR Imagery using

Stochastic Distances

Julia Cassetti, Juliana Gambini, Alejandro Frery

LaCCAN

Laboratório de Computação Científica e Análise Numérica

Universidade Federal de Alagoas, Brazil

APSAR 2013 September 26

Organization

1 Introduction

2 _{The GI}0 _Model

3 Maximum Likelihood Estimator

4 Stochastic Distances Minimization

5 Results

SAR Image Interpretation

In order to interpret SAR images, statistical modeling of the data is essential.

Speckled data have been described under the multiplicative model using the G family of distributions, since it is able to describe rough and extremely rough areas better than the K distribution.

Under the G model, regions with different degree of roughness can be characterized by the parameters.

SAR Image Interpretation

In order to interpret SAR images, statistical modeling of the data is essential.

Speckled data have been described under the multiplicative model using the G family of distributions, since it is able to describe rough and extremely rough areas better than the K distribution.

Under the G model, regions with different degree of roughness can be characterized by the parameters. Therefore, the accuracy of estimates is very important.

SAR Image Interpretation

In order to interpret SAR images, statistical modeling of the data is essential.

Speckled data have been described under the multiplicative model using the G family of distributions, since it is able to describe rough and extremely rough areas better than the K distribution.

Under the G model, regions with different degree of roughness can be characterized by the parameters.

SAR Image Interpretation

In order to interpret SAR images, statistical modeling of the data is essential.

Speckled data have been described under the multiplicative model using the G family of distributions, since it is able to describe rough and extremely rough areas better than the K distribution.

Under the G model, regions with different degree of roughness can be characterized by the parameters. Therefore, the accuracy of estimates is very important.

Information Theory

Information Theory (IT) is a branch of Probability and Statistics strongly influenced by Engineering

It is sometimes referred to as “statistical communication theory” and “communication theory”

Four big names in IT are Fisher, Shannon, Wiener and Kullback

Information Theory

Information Theory (IT) is a branch of Probability and Statistics strongly influenced by Engineering

It is sometimes referred to as “statistical communication theory” and “communication theory”

Four big names in IT are Fisher, Shannon, Wiener and Kullback

Information Theory

Information Theory (IT) is a branch of Probability and Statistics strongly influenced by Engineering

It is sometimes referred to as “statistical communication theory” and “communication theory”

Four big names in IT are Fisher, Shannon, Wiener and Kullback

Information Theory

Information Theory (IT) is a branch of Probability and Statistics strongly influenced by Engineering

It is sometimes referred to as “statistical communication theory” and “communication theory”

Four big names in IT are Fisher, Shannon, Wiener and Kullback

The proposal

Develop a method to find estimators with good properties of precision, accuracy, and robustness which employ small and medium size samples with relatively low computational cost.

We propose estimators based on the Hellinger, Bhattacharyya, R´enyi and Triangular distances, and we compare them with the Maximum Likelihood estimator.

Illustration

G0 I(−15, γ∗, 10) G0 I(−3, γ ∗_{, 10)} G0 I(−2, γ ∗_{, 10)} G0 I(−1.5, γ ∗_{, 10)} 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 Fitting Histogram Α=-1.5, L=10

Illustration

G0 I(−15, γ∗, 10) G0 I(−3, γ ∗_{, 10)} G0 I(−2, γ ∗_{, 10)} G0 I(−1.5, γ ∗_{, 10)} 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 Fitting Histogram Α=-1.5, L=10

Illustration

G0 I(−15, γ∗, 10) G0 I(−3, γ ∗_{, 10)} G0 I(−2, γ ∗_{, 10)} G0 I(−1.5, γ ∗_{, 10)} 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 Fitting Histogram Α=-1.5, L=10

Illustration

G0 I(−15, γ ∗_{, 10)} G0 I(−3, γ∗, 10) G0 I(−2, γ∗, 10) G0 I(−1.5, γ ∗_{, 10)} 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 Fitting Histogram Α=-1.5, L=10

Ingredients

The necessary ingredients are:

a model: our choice is the G0 distribution

distances: we choose from the family of h-φ distances an algorithm: sequential search

and a comparison is made with respect to maximum likelihood estimation.

Organization

1 Introduction

2 _{The GI}0 _Model

3 Maximum Likelihood Estimator

4 Stochastic Distances Minimization

5 Results

The Multiplicative Model

The return in monopolarized SAR images can be modeled as the product of two independent random variables.

Z = X · Y

where Z represents the return in each pixel, X corresponds to the backscatter, and Y to the speckle noise.

For intensity data, if we assume that Y ∼ Γ(L, L), where L is

the number of looks, and that X ∼ Γ−1(α, γ), then the return Z

The Multiplicative Model

The return in monopolarized SAR images can be modeled as the product of two independent random variables.

Z = X · Y

where Z represents the return in each pixel, X corresponds to the backscatter, and Y to the speckle noise.

For intensity data, if we assume that Y ∼ Γ(L, L), where L is the number of looks, and that X ∼ Γ−1(α, γ), then the return Z follows a G0 distribution.

Density and Moments

The model is specified by the density

fG0 I(z) = LLΓ(L − α) γα_{Γ(−α)Γ(L)} · zL−1 (γ + zL)L−α,

where −α, γ, z > 0 and L ≥ 1 are the texture, the scale and the number of looks.

Its r-order moments are given by

E(Zr) =γ L rΓ(−α − r) Γ(−α) · Γ(L + r) Γ(L) .

Density and Moments

The model is specified by the density

fG0 I(z) = LLΓ(L − α) γα_{Γ(−α)Γ(L)} · zL−1 (γ + zL)L−α,

where −α, γ, z > 0 and L ≥ 1 are the texture, the scale and the number of looks.

Its r-order moments are given by

E(Zr) =γ L rΓ(−α − r) Γ(−α) · Γ(L + r) Γ(L) .

Choice of scale

Double Purpose

Simplifying the calculations and making the results comparable, in the following we employ γ∗ such that E(Z) = 1.

Parameter Interpretation

One of the most important features of the G0 distribution is the interpretation of the α parameter, which is related to the roughness of the target.

α Value (−1, −3] (−3, −6] (−6, −∞)

Roughness Extremely Textured Textured Non Textured

This is one of the reasons why the accuracy in the estimation of the α parameter is important.

Parameter Interpretation

One of the most important features of the G0 distribution is the interpretation of the α parameter, which is related to the roughness of the target.

α Value (−1, −3] (−3, −6] (−6, −∞)

Roughness Extremely Textured Textured Non Textured

This is one of the reasons why the accuracy in the estimation of the α parameter is important.

Parameter Interpretation

One of the most important features of the G0 distribution is the interpretation of the α parameter, which is related to the roughness of the target.

α Value (−1, −3] (−3, −6] (−6, −∞)

Roughness Extremely Textured Textured Non Textured

This is one of the reasons why the accuracy in the estimation of the α parameter is important.

Organization

1 Introduction

2 _{The GI}0 _Model

3 Maximum Likelihood Estimator

4 Stochastic Distances Minimization

5 Results

Maximum Likelihood Estimator

Let z = (z1, . . . , zn) be a random sample of size n, the likelihood

function related to the G_I0(α, γ, L) distribution is given by L(α, γ, L, z) = L L_{Γ(L − α)} γα_{Γ(−α)Γ(L)} nYn i=1 z_iL−1(γ + Lzi)α−L

The ML estimator of α,α_bML, assuming γ = −α − 1 and L

known, based on z, is the solution of the following nonlinear equation ψ0(αbML) − ψ 0_{(L −} b αML) − log(1 −αbML) + b αML 1−αbML + 1 n Pn i=1log(1 −αbML+ Lzi) − b αML−L n Pn i=11−αbML1+Lzi = 0 where ψ0(·) is the digamma function.

Organization

1 Introduction

2 _{The GI}0 _Model

3 Maximum Likelihood Estimator

4 Stochastic Distances Minimization

5 Results

Why Stochastic Distances?

Stochastic Distances allow the comparison between probability distributions.

In this work we compare the G_I0 distribution with the empiric

Why Stochastic Distances?

Stochastic Distances allow the comparison between probability distributions.

In this work we compare the G_I0 distribution with the empiric distribution characterized by a histogram.

Stochastic Distances

Let V and W be two random variables defined over the same probability space whose density functions are fV(x; θ1) and

fW(x; θ2), respectively. Consider the following:

1 The Hellinger distance

dH(V, W ) = 1 −

Z ∞

−∞

p fVfW

2 The Bhattacharyya distance

dB(V, W ) = − log

Z ∞

−∞

p fVfW

Stochastic Distances

1 _{The Triangular distance}

dT(V, W ) =

Z ∞

−∞

(fV − fW)2

fV + fW

2 The R´enyi distance of order β

dβ_R(V, W ) = 1

2(β − 1)log Z ∞

−∞

But. . .

We are interested in minimizing distances. Due to the relation dB= − log(1 − dH),

and being the logarithm an increasing real function, it holds that

arg min

α dB(α) = arg minα dH(α).

Then, we use the Hellinger distance for its low computational cost.

Methodology

Let z = (z1, . . . , zn) be an independent sample of size n from

the G_I0(α, γ∗, L) distribution. The empirical density of z, estimated by the histogram and denoted f e, is computed using the Freedman-Diaconis method.

The estimator we will assess is given by

b

αD = arg min

−20≤α≤−1dD fGI0(α, γ

∗_{, L), f e(z)
,}

where dD ∈ {dH, dβ_R, dT} (Hellinger, R´enyi of order β or

Organization

1 Introduction

2 _{The GI}0 _Model

3 Maximum Likelihood Estimator

4 Stochastic Distances Minimization

5 Results

Experimental Results

We conducted a study with several parameter values; for each {α_b1, . . . ,αb1000} are obtained by simulation by each method.

In order to assess the proposed estimation method, the mean square error and the bias are estimated.

Estimators

The mean b E(α) =_b α =_b 1 1000 1000 X i=1 b αi The Bias b B(α) =b α − αb

The Mean Square Error

d mse(α) =_b 1 1000 1000 X (α_bi− α)2

Estimators

The mean b E(α) =_b α =_b 1 1000 1000 X i=1 b αi The Bias b B(α) =b α − αb The Mean Square Error

mse(α) = 1

1000 X

Estimators

The mean b E(α) =_b α =_b 1 1000 1000 X i=1 b αi The Bias b B(α) =b α − αb

The Mean Square Error

d mse(α) =_b 1 1000 1000 X (α_bi− α)2

Parameter space

Roughness

Textures ranging from extreme to absent: α = {−1.5, −3, −5, −8}

Number of Looks

Several processing levels L = {1, 3, 8}

Sample sizes

Small square windows, and large sample behavior n = {9, 25, 49, 81, 121, 1000}

Parameter space

Roughness

Textures ranging from extreme to absent: α = {−1.5, −3, −5, −8}

Number of Looks

Several processing levels L = {1, 3, 8} Sample sizes

Small square windows, and large sample behavior n = {9, 25, 49, 81, 121, 1000}

Parameter space

Roughness

Textures ranging from extreme to absent: α = {−1.5, −3, −5, −8}

Number of Looks

Several processing levels L = {1, 3, 8}

Sample sizes

Small square windows, and large sample behavior n = {9, 25, 49, 81, 121, 1000}

Mean square error L = 1

MSE mse 0 10 20 30 alpha=−1.5 L=1 10 100 1000 alpha=−3 L=1 alpha=−5 L=1 10 100 1000 alpha=−8 L=1 H MV R T

Mean square error L = 3 and L = 8

MSE mse 0 5 10 15 10 100 1000 alpha=−1.5 L=3 alpha=−3 L=3 10 100 1000 alpha=−5 L=3 alpha=−8 L=3 alpha=−1.5 L=8 10 100 1000 alpha=−3 L=8 alpha=−5 L=8 10 100 1000 0 5 10 15 alpha=−8 L=8 H MV R T

7 × 7 window

(c) R´enyi β = 0.8 (d) Triangular Distance (e) Maximum Likeli-hood

3 × 3 window

3 × 3 window

(h) R´enyi β = 0.8 (i) Triangular Distance (j) Maximum Likeli-hood

Organization

1 Introduction

2 _{The GI}0 _Model

3 Maximum Likelihood Estimator

4 Stochastic Distances Minimization

5 Results

Conclusions

This work is dedicated to estimate the texture parameter of the intensity G0 distribution using stochastic distances.

The procedures yield sensible values, being worst in less textured areas because the distance curve is too flat and minimization becomes unstable.

Conclusions

This work is dedicated to estimate the texture parameter of the intensity G0 distribution using stochastic distances. The procedures yield sensible values, being worst in less textured areas because the distance curve is too flat and minimization becomes unstable.

Conclusions

This work is dedicated to estimate the texture parameter of the intensity G0 distribution using stochastic distances. The procedures yield sensible values, being worst in less textured areas because the distance curve is too flat and minimization becomes unstable.

Future work

We will experiment with other distances, e.g. Kullback-Leibler, with contaminated data and with the estimation of γ and of L.